<!DOCTYPE html><html lang="en"><head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="shortcut icon" href="/img/favicon.ico">
<meta name="keywords" content="Finite Element Library,Finite Element Code,Finite Element Method Library,Finite Element Method Code,Finite Element C++,Finite Element Method C++,Finite Element Library C++,Finite Element Code C++,FEM Code,FEM Library,FEM C++">
<meta name="google-site-verification" content="45K56TawSdzbgy-uDtfJdYwp7zaVdL3fIAGYt869wwU">
<!-- <title></title> -->
<title>MFEM - Finite Element Discretization Library</title>
<link href="../../doc/web/bootstrap-mfem.min.css" rel="stylesheet">
<link href="../../doc/web/font-awesome-4.0.3.css" rel="stylesheet">
<link href="../../doc/web/highlight.css" rel="stylesheet">
<link href="../../doc/web/base.css" rel="stylesheet">
<!-- HTML5 shim and Respond.js IE8 support of HTML5 elements and media queries -->
<!--[if lt IE 9]>
<script src="https://oss.maxcdn.com/libs/html5shiv/3.7.0/html5shiv.js"></script>
<script src="https://oss.maxcdn.com/libs/respond.js/1.3.0/respond.min.js"></script>
<![endif]-->
<style type="text/css">.MathJax_Hover_Frame {border-radius: .25em; -webkit-border-radius: .25em; -moz-border-radius: .25em; -khtml-border-radius: .25em; box-shadow: 0px 0px 15px #83A; -webkit-box-shadow: 0px 0px 15px #83A; -moz-box-shadow: 0px 0px 15px #83A; -khtml-box-shadow: 0px 0px 15px #83A; border: 1px solid #A6D ! important; display: inline-block; position: absolute}
.MathJax_Menu_Button .MathJax_Hover_Arrow {position: absolute; cursor: pointer; display: inline-block; border: 2px solid #AAA; border-radius: 4px; -webkit-border-radius: 4px; -moz-border-radius: 4px; -khtml-border-radius: 4px; font-family: 'Courier New',Courier; font-size: 9px; color: #F0F0F0}
.MathJax_Menu_Button .MathJax_Hover_Arrow span {display: block; background-color: #AAA; border: 1px solid; border-radius: 3px; line-height: 0; padding: 4px}
.MathJax_Hover_Arrow:hover {color: white!important; border: 2px solid #CCC!important}
.MathJax_Hover_Arrow:hover span {background-color: #CCC!important}
</style><style type="text/css">#MathJax_About {position: fixed; left: 50%; width: auto; text-align: center; border: 3px outset; padding: 1em 2em; background-color: #DDDDDD; color: black; cursor: default; font-family: message-box; font-size: 120%; font-style: normal; text-indent: 0; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; z-index: 201; border-radius: 15px; -webkit-border-radius: 15px; -moz-border-radius: 15px; -khtml-border-radius: 15px; box-shadow: 0px 10px 20px #808080; -webkit-box-shadow: 0px 10px 20px #808080; -moz-box-shadow: 0px 10px 20px #808080; -khtml-box-shadow: 0px 10px 20px #808080; filter: progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color='gray', Positive='true')}
#MathJax_About.MathJax_MousePost {outline: none}
.MathJax_Menu {position: absolute; background-color: white; color: black; width: auto; padding: 5px 0px; border: 1px solid #CCCCCC; margin: 0; cursor: default; font: menu; text-align: left; text-indent: 0; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; z-index: 201; border-radius: 5px; -webkit-border-radius: 5px; -moz-border-radius: 5px; -khtml-border-radius: 5px; box-shadow: 0px 10px 20px #808080; -webkit-box-shadow: 0px 10px 20px #808080; -moz-box-shadow: 0px 10px 20px #808080; -khtml-box-shadow: 0px 10px 20px #808080; filter: progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color='gray', Positive='true')}
.MathJax_MenuItem {padding: 1px 2em; background: transparent}
.MathJax_MenuArrow {position: absolute; right: .5em; padding-top: .25em; color: #666666; font-size: .75em}
.MathJax_MenuActive .MathJax_MenuArrow {color: white}
.MathJax_MenuArrow.RTL {left: .5em; right: auto}
.MathJax_MenuCheck {position: absolute; left: .7em}
.MathJax_MenuCheck.RTL {right: .7em; left: auto}
.MathJax_MenuRadioCheck {position: absolute; left: .7em}
.MathJax_MenuRadioCheck.RTL {right: .7em; left: auto}
.MathJax_MenuLabel {padding: 1px 2em 3px 1.33em; font-style: italic}
.MathJax_MenuRule {border-top: 1px solid #DDDDDD; margin: 4px 3px}
.MathJax_MenuDisabled {color: GrayText}
.MathJax_MenuActive {background-color: #606872; color: white}
.MathJax_MenuDisabled:focus, .MathJax_MenuLabel:focus {background-color: #E8E8E8}
.MathJax_ContextMenu:focus {outline: none}
.MathJax_ContextMenu .MathJax_MenuItem:focus {outline: none}
#MathJax_AboutClose {top: .2em; right: .2em}
.MathJax_Menu .MathJax_MenuClose {top: -10px; left: -10px}
.MathJax_MenuClose {position: absolute; cursor: pointer; display: inline-block; border: 2px solid #AAA; border-radius: 18px; -webkit-border-radius: 18px; -moz-border-radius: 18px; -khtml-border-radius: 18px; font-family: 'Courier New',Courier; font-size: 24px; color: #F0F0F0}
.MathJax_MenuClose span {display: block; background-color: #AAA; border: 1.5px solid; border-radius: 18px; -webkit-border-radius: 18px; -moz-border-radius: 18px; -khtml-border-radius: 18px; line-height: 0; padding: 8px 0 6px}
.MathJax_MenuClose:hover {color: white!important; border: 2px solid #CCC!important}
.MathJax_MenuClose:hover span {background-color: #CCC!important}
.MathJax_MenuClose:hover:focus {outline: none}
</style><style type="text/css">.MathJax_Preview .MJXf-math {color: inherit!important}
</style><style type="text/css">.MJX_Assistive_MathML {position: absolute!important; top: 0; left: 0; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0 0 0!important; border: 0!important; height: 1px!important; width: 1px!important; overflow: hidden!important; display: block!important; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none}
.MJX_Assistive_MathML.MJX_Assistive_MathML_Block {width: 100%!important}
</style><style type="text/css">#MathJax_Zoom {position: absolute; background-color: #F0F0F0; overflow: auto; display: block; z-index: 301; padding: .5em; border: 1px solid black; margin: 0; font-weight: normal; font-style: normal; text-align: left; text-indent: 0; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; -webkit-box-sizing: content-box; -moz-box-sizing: content-box; box-sizing: content-box; box-shadow: 5px 5px 15px #AAAAAA; -webkit-box-shadow: 5px 5px 15px #AAAAAA; -moz-box-shadow: 5px 5px 15px #AAAAAA; -khtml-box-shadow: 5px 5px 15px #AAAAAA; filter: progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color='gray', Positive='true')}
#MathJax_ZoomOverlay {position: absolute; left: 0; top: 0; z-index: 300; display: inline-block; width: 100%; height: 100%; border: 0; padding: 0; margin: 0; background-color: white; opacity: 0; filter: alpha(opacity=0)}
#MathJax_ZoomFrame {position: relative; display: inline-block; height: 0; width: 0}
#MathJax_ZoomEventTrap {position: absolute; left: 0; top: 0; z-index: 302; display: inline-block; border: 0; padding: 0; margin: 0; background-color: white; opacity: 0; filter: alpha(opacity=0)}
</style><style type="text/css">.MathJax_Preview {color: #888}
#MathJax_Message {position: fixed; left: 1px; bottom: 2px; background-color: #E6E6E6; border: 1px solid #959595; margin: 0px; padding: 2px 8px; z-index: 102; color: black; font-size: 80%; width: auto; white-space: nowrap}
#MathJax_MSIE_Frame {position: absolute; top: 0; left: 0; width: 0px; z-index: 101; border: 0px; margin: 0px; padding: 0px}
.MathJax_Error {color: #CC0000; font-style: italic}
</style><style type="text/css">.MJXp-script {font-size: .8em}
.MJXp-right {-webkit-transform-origin: right; -moz-transform-origin: right; -ms-transform-origin: right; -o-transform-origin: right; transform-origin: right}
.MJXp-bold {font-weight: bold}
.MJXp-italic {font-style: italic}
.MJXp-scr {font-family: MathJax_Script,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-frak {font-family: MathJax_Fraktur,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-sf {font-family: MathJax_SansSerif,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-cal {font-family: MathJax_Caligraphic,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-mono {font-family: MathJax_Typewriter,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-largeop {font-size: 150%}
.MJXp-largeop.MJXp-int {vertical-align: -.2em}
.MJXp-math {display: inline-block; line-height: 1.2; text-indent: 0; font-family: 'Times New Roman',Times,STIXGeneral,serif; white-space: nowrap; border-collapse: collapse}
.MJXp-display {display: block; text-align: center; margin: 1em 0}
.MJXp-math span {display: inline-block}
.MJXp-box {display: block!important; text-align: center}
.MJXp-box:after {content: " "}
.MJXp-rule {display: block!important; margin-top: .1em}
.MJXp-char {display: block!important}
.MJXp-mo {margin: 0 .15em}
.MJXp-mfrac {margin: 0 .125em; vertical-align: .25em}
.MJXp-denom {display: inline-table!important; width: 100%}
.MJXp-denom > * {display: table-row!important}
.MJXp-surd {vertical-align: top}
.MJXp-surd > * {display: block!important}
.MJXp-script-box > * {display: table!important; height: 50%}
.MJXp-script-box > * > * {display: table-cell!important; vertical-align: top}
.MJXp-script-box > *:last-child > * {vertical-align: bottom}
.MJXp-script-box > * > * > * {display: block!important}
.MJXp-mphantom {visibility: hidden}
.MJXp-munderover {display: inline-table!important}
.MJXp-over {display: inline-block!important; text-align: center}
.MJXp-over > * {display: block!important}
.MJXp-munderover > * {display: table-row!important}
.MJXp-mtable {vertical-align: .25em; margin: 0 .125em}
.MJXp-mtable > * {display: inline-table!important; vertical-align: middle}
.MJXp-mtr {display: table-row!important}
.MJXp-mtd {display: table-cell!important; text-align: center; padding: .5em 0 0 .5em}
.MJXp-mtr > .MJXp-mtd:first-child {padding-left: 0}
.MJXp-mtr:first-child > .MJXp-mtd {padding-top: 0}
.MJXp-mlabeledtr {display: table-row!important}
.MJXp-mlabeledtr > .MJXp-mtd:first-child {padding-left: 0}
.MJXp-mlabeledtr:first-child > .MJXp-mtd {padding-top: 0}
.MJXp-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 1px 3px; font-style: normal; font-size: 90%}
.MJXp-scale0 {-webkit-transform: scaleX(.0); -moz-transform: scaleX(.0); -ms-transform: scaleX(.0); -o-transform: scaleX(.0); transform: scaleX(.0)}
.MJXp-scale1 {-webkit-transform: scaleX(.1); -moz-transform: scaleX(.1); -ms-transform: scaleX(.1); -o-transform: scaleX(.1); transform: scaleX(.1)}
.MJXp-scale2 {-webkit-transform: scaleX(.2); -moz-transform: scaleX(.2); -ms-transform: scaleX(.2); -o-transform: scaleX(.2); transform: scaleX(.2)}
.MJXp-scale3 {-webkit-transform: scaleX(.3); -moz-transform: scaleX(.3); -ms-transform: scaleX(.3); -o-transform: scaleX(.3); transform: scaleX(.3)}
.MJXp-scale4 {-webkit-transform: scaleX(.4); -moz-transform: scaleX(.4); -ms-transform: scaleX(.4); -o-transform: scaleX(.4); transform: scaleX(.4)}
.MJXp-scale5 {-webkit-transform: scaleX(.5); -moz-transform: scaleX(.5); -ms-transform: scaleX(.5); -o-transform: scaleX(.5); transform: scaleX(.5)}
.MJXp-scale6 {-webkit-transform: scaleX(.6); -moz-transform: scaleX(.6); -ms-transform: scaleX(.6); -o-transform: scaleX(.6); transform: scaleX(.6)}
.MJXp-scale7 {-webkit-transform: scaleX(.7); -moz-transform: scaleX(.7); -ms-transform: scaleX(.7); -o-transform: scaleX(.7); transform: scaleX(.7)}
.MJXp-scale8 {-webkit-transform: scaleX(.8); -moz-transform: scaleX(.8); -ms-transform: scaleX(.8); -o-transform: scaleX(.8); transform: scaleX(.8)}
.MJXp-scale9 {-webkit-transform: scaleX(.9); -moz-transform: scaleX(.9); -ms-transform: scaleX(.9); -o-transform: scaleX(.9); transform: scaleX(.9)}
.MathJax_PHTML .noError {vertical-align: ; font-size: 90%; text-align: left; color: black; padding: 1px 3px; border: 1px solid}
</style><style type="text/css">.MathJax_SVG_Display {text-align: center; margin: 1em 0em; position: relative; display: block!important; text-indent: 0; max-width: none; max-height: none; min-width: 0; min-height: 0; width: 100%}
.MathJax_SVG .MJX-monospace {font-family: monospace}
.MathJax_SVG .MJX-sans-serif {font-family: sans-serif}
#MathJax_SVG_Tooltip {background-color: InfoBackground; color: InfoText; border: 1px solid black; box-shadow: 2px 2px 5px #AAAAAA; -webkit-box-shadow: 2px 2px 5px #AAAAAA; -moz-box-shadow: 2px 2px 5px #AAAAAA; -khtml-box-shadow: 2px 2px 5px #AAAAAA; padding: 3px 4px; z-index: 401; position: absolute; left: 0; top: 0; width: auto; height: auto; display: none}
.MathJax_SVG {display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 100%; font-size-adjust: none; text-indent: 0; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; padding: 0; margin: 0}
.MathJax_SVG * {transition: none; -webkit-transition: none; -moz-transition: none; -ms-transition: none; -o-transition: none}
.mjx-svg-href {fill: blue; stroke: blue}
.MathJax_SVG_Processing {visibility: hidden; position: absolute; top: 0; left: 0; width: 0; height: 0; overflow: hidden; display: block!important}
.MathJax_SVG_Processed {display: none!important}
.MathJax_SVG_ExBox {display: block!important; overflow: hidden; width: 1px; height: 60ex; min-height: 0; max-height: none; padding: 0; border: 0; margin: 0}
.MathJax_SVG_LineBox {display: table!important}
.MathJax_SVG_LineBox span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MathJax_SVG .noError {vertical-align: ; font-size: 90%; text-align: left; color: black; padding: 1px 3px; border: 1px solid}
</style></head>
<body style=""><div style="visibility: hidden; overflow: hidden; position: absolute; top: 0px; height: 1px; width: auto; padding: 0px; border: 0px; margin: 0px; text-align: left; text-indent: 0px; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal;"><div id="MathJax_SVG_Hidden"></div><svg><defs id="MathJax_SVG_glyphs"><path stroke-width="1" id="MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="MJMAIN-2207" d="M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z"></path><path stroke-width="1" id="MJMAIN-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path><path stroke-width="1" id="MJMATHI-48" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path><path stroke-width="1" id="MJMAIN-20D7" d="M-123 694Q-123 702 -118 708T-103 714Q-93 714 -88 706T-80 687T-67 660T-40 633Q-29 626 -29 615Q-29 606 -36 600T-53 590T-83 571T-121 531Q-135 516 -143 516T-157 522T-163 536T-152 559T-129 584T-116 595H-287L-458 596Q-459 597 -461 599T-466 602T-469 607T-471 615Q-471 622 -458 635H-99Q-123 673 -123 694Z"></path><path stroke-width="1" id="MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="MJMAIN-2202" d="M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z"></path><path stroke-width="1" id="MJMATHI-44" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></path><path stroke-width="1" id="MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="1" id="MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="MJMATHI-45" d="M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z"></path><path stroke-width="1" id="MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="1" id="MJMATHI-42" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path stroke-width="1" id="MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="1" id="MJMAIN-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path stroke-width="1" id="MJMATHI-3C1" d="M58 -216Q25 -216 23 -186Q23 -176 73 26T127 234Q143 289 182 341Q252 427 341 441Q343 441 349 441T359 442Q432 442 471 394T510 276Q510 219 486 165T425 74T345 13T266 -10H255H248Q197 -10 165 35L160 41L133 -71Q108 -168 104 -181T92 -202Q76 -216 58 -216ZM424 322Q424 359 407 382T357 405Q322 405 287 376T231 300Q217 269 193 170L176 102Q193 26 260 26Q298 26 334 62Q367 92 389 158T418 266T424 322Z"></path><path stroke-width="1" id="MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path stroke-width="1" id="MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path stroke-width="1" id="MJMATHI-3F5" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path><path stroke-width="1" id="MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path stroke-width="1" id="MJMAIN-36" d="M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z"></path><path stroke-width="1" id="MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path><path stroke-width="1" id="MJMATHI-4D" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path><path stroke-width="1" id="MJMATHI-3C7" d="M576 -125Q576 -147 547 -175T487 -204H476Q394 -204 363 -157Q334 -114 293 26L284 59Q283 58 248 19T170 -66T92 -151T53 -191Q49 -194 43 -194Q36 -194 31 -189T25 -177T38 -154T151 -30L272 102L265 131Q189 405 135 405Q104 405 87 358Q86 351 68 351Q48 351 48 361Q48 369 56 386T89 423T148 442Q224 442 258 400Q276 375 297 320T330 222L341 180Q344 180 455 303T573 429Q579 431 582 431Q600 431 600 414Q600 407 587 392T477 270Q356 138 353 134L362 102Q392 -10 428 -89T490 -168Q504 -168 517 -156T536 -126Q539 -116 543 -115T557 -114T571 -115Q576 -118 576 -125Z"></path><path stroke-width="1" id="MJMATHI-3C6" d="M92 210Q92 176 106 149T142 108T185 85T220 72L235 70L237 71L250 112Q268 170 283 211T322 299T370 375T429 423T502 442Q547 442 582 410T618 302Q618 224 575 152T457 35T299 -10Q273 -10 273 -12L266 -48Q260 -83 252 -125T241 -179Q236 -203 215 -212Q204 -218 190 -218Q159 -215 159 -185Q159 -175 214 -2L209 0Q204 2 195 5T173 14T147 28T120 46T94 71T71 103T56 142T50 190Q50 238 76 311T149 431H162Q183 431 183 423Q183 417 175 409Q134 361 114 300T92 210ZM574 278Q574 320 550 344T486 369Q437 369 394 329T323 218Q309 184 295 109L286 64Q304 62 306 62Q423 62 498 131T574 278Z"></path><path stroke-width="1" id="MJMAIN-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path stroke-width="1" id="MJMAIN-38" d="M70 417T70 494T124 618T248 666Q319 666 374 624T429 515Q429 485 418 459T392 417T361 389T335 371T324 363L338 354Q352 344 366 334T382 323Q457 264 457 174Q457 95 399 37T249 -22Q159 -22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368Q70 417 70 494ZM286 386L292 390Q298 394 301 396T311 403T323 413T334 425T345 438T355 454T364 471T369 491T371 513Q371 556 342 586T275 624Q268 625 242 625Q201 625 165 599T128 534Q128 511 141 492T167 463T217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194T353 216T330 234T301 253T274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239T107 157Q107 98 150 60T250 21Z"></path><path stroke-width="1" id="MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="MJMAIN-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path><path stroke-width="1" id="MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="1" id="MJMAIN-39" d="M352 287Q304 211 232 211Q154 211 104 270T44 396Q42 412 42 436V444Q42 537 111 606Q171 666 243 666Q245 666 249 666T257 665H261Q273 665 286 663T323 651T370 619T413 560Q456 472 456 334Q456 194 396 97Q361 41 312 10T208 -22Q147 -22 108 7T68 93T121 149Q143 149 158 135T173 96Q173 78 164 65T148 49T135 44L131 43Q131 41 138 37T164 27T206 22H212Q272 22 313 86Q352 142 352 280V287ZM244 248Q292 248 321 297T351 430Q351 508 343 542Q341 552 337 562T323 588T293 615T246 625Q208 625 181 598Q160 576 154 546T147 441Q147 358 152 329T172 282Q197 248 244 248Z"></path><path stroke-width="1" id="MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="1" id="MJMATHI-4B" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path><path stroke-width="1" id="MJMATHI-3C3" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path><path stroke-width="1" id="MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path><path stroke-width="1" id="MJMATHI-3B2" d="M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z"></path><path stroke-width="1" id="MJMATHI-3B3" d="M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z"></path><path stroke-width="1" id="MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="1" id="MJMATHI-65" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"></path><path stroke-width="1" id="MJMAIN-2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z"></path><path stroke-width="1" id="MJMATHI-3B7" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="MJSZ1-221A" d="M263 249Q264 249 315 130T417 -108T470 -228L725 302Q981 837 982 839Q989 850 1001 850Q1008 850 1013 844T1020 832V826L741 243Q645 43 540 -176Q479 -303 469 -324T453 -348Q449 -350 436 -350L424 -349L315 -96Q206 156 205 156L171 130Q138 104 137 104L111 130L263 249Z"></path><path stroke-width="1" id="MJMAIN-64" d="M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 688Q393 689 413 690T443 693T454 694H457V390Q457 84 458 81Q461 61 472 55T517 46H535V0Q533 0 459 -5T380 -11H373V44L365 37Q307 -11 235 -11Q158 -11 96 50T34 215Q34 315 97 378T244 442Q319 442 376 393V495ZM373 342Q328 405 260 405Q211 405 173 369Q146 341 139 305T131 211Q131 155 138 120T173 59Q203 26 251 26Q322 26 373 103V342Z"></path><path stroke-width="1" id="MJMAIN-69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z"></path><path stroke-width="1" id="MJMAIN-76" d="M338 431Q344 429 422 429Q479 429 503 431H508V385H497Q439 381 423 345Q421 341 356 172T288 -2Q283 -11 263 -11Q244 -11 239 -2Q99 359 98 364Q93 378 82 381T43 385H19V431H25L33 430Q41 430 53 430T79 430T104 429T122 428Q217 428 232 431H240V385H226Q187 384 184 370Q184 366 235 234L286 102L377 341V349Q377 363 367 372T349 383T335 385H331V431H338Z"></path><path stroke-width="1" id="MJMATHI-46" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z"></path><path stroke-width="1" id="MJMAIN-3A6" d="M312 622Q310 623 307 625T303 629T297 631T286 634T270 635T246 636T211 637H184V683H196Q220 680 361 680T526 683H538V637H511Q468 637 447 635T422 631T411 622V533L425 531Q525 519 595 466T665 342Q665 301 642 267T583 209T506 172T425 152L411 150V61Q417 55 421 53T447 48T511 46H538V0H526Q502 3 361 3T196 0H184V46H211Q231 46 245 46T270 47T286 48T297 51T303 54T307 57T312 61V150H310Q309 151 289 153T232 166T160 195Q149 201 136 210T103 238T69 284T56 342Q56 414 128 467T294 530Q309 532 310 533H312V622ZM170 342Q170 207 307 188H312V495H309Q301 495 282 491T231 469T186 423Q170 389 170 342ZM415 188Q487 199 519 236T551 342Q551 384 539 414T507 459T470 481T434 491T415 495H410V188H415Z"></path><path stroke-width="1" id="MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="1" id="MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="1" id="MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="MJMATHI-63" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path></defs></svg></div><div id="MathJax_Message" style="display: none;"></div>
<div class="navbar navbar-default navbar-fixed-top" role="navigation">
<div class="container">
<!-- Collapsed navigation -->
<div class="navbar-header">
<!-- Expander button -->
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-collapse">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<!-- Main title -->
<a class="navbar-brand" href="http://mfem.org">MFEM</a>
</div>
<!-- Expanded navigation -->
<div class="navbar-collapse collapse">
<!-- Main navigation -->
<ul class="nav navbar-nav">
<li>
<!-- Replace "nav_item.url|url" with "nav_item.url" for older mkdocs (before 1.0) -->
<a href="http://mfem.org/features/">Features</a>
</li>
<li>
<!-- Replace "nav_item.url|url" with "nav_item.url" for older mkdocs (before 1.0) -->
<a href="http://mfem.org/examples/">Examples</a>
</li>
<li class="dropdown active">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Documentation <b class="caret"></b></a>
<ul class="dropdown-menu">
<li>
<a href="http://mfem.org/building/">Building MFEM</a>
</li>
<li>
<a href="http://mfem.org/serial-tutorial/">Serial Tutorial</a>
</li>
<li>
<a href="http://mfem.org/parallel-tutorial/">Parallel Tutorial</a>
</li>
<li>
<a href="http://mfem.org/fem/">Finite Elements</a>
</li>
<li class="active">
<a href="./">Electromagnetics</a>
</li>
<li>
<a href="http://mfem.org/meshing/">Meshing</a>
</li>
<li>
<a href="http://mfem.org/performance/">Performance</a>
</li>
<li>
<a href="http://mfem.org/tools/">Tools</a>
</li>
<li>
<a href="http://mfem.org/toys/">Toys</a>
</li>
<li>
<a href="http://mfem.org/publications/">Publications</a>
</li>
<li>
<a href="http://mfem.org/about/">About</a>
</li>
</ul>
</li>
<li>
<!-- Replace "nav_item.url|url" with "nav_item.url" for older mkdocs (before 1.0) -->
<a href="http://mfem.org/gallery/">Gallery</a>
</li>
<li>
<!-- Replace "nav_item.url|url" with "nav_item.url" for older mkdocs (before 1.0) -->
<a href="http://mfem.org/download/">Download</a>
</li>
</ul>
<!-- Search, Navigation and Repo links -->
<ul class="nav navbar-nav navbar-right">
<li>
<a href="https://github.com/mfem/mfem/">
<i class="fa fa-github-square"></i>
GitHub
</a>
</li>
</ul>
</div>
</div>
</div>
<div class="container">
<div class="col-md-3 visible-xs visible-sm"><div class="bs-sidebar hidden-print affix well" role="complementary">
<ul class="nav bs-sidenav">
<li class="main active"><a href="#electromagnetics-mini-applications">Electromagnetics Mini Applications</a></li>
<li><a href="#electromagnetics">Electromagnetics</a></li>
<li><a href="#electrostatics">Electrostatics</a></li>
<li><a href="#magnetostatics">Magnetostatics</a></li>
<li><a href="#transient-full-wave-electromagnetics">Transient Full-Wave Electromagnetics</a></li>
<li><a href="#transient-magnetics-and-joule-heating">Transient Magnetics and Joule Heating</a></li>
</ul>
</div></div> <!-- toc on top on mobile -->
<div class="col-md-9" role="main">
<h1 id="electromagnetics-mini-applications">Electromagnetics Mini Applications</h1>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" />" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0" height="0.264ex" viewBox="0 -56.8 0 113.7" role="img" focusable="false" style="vertical-align: -0.132ex;" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"></math></span></span><script type="math/tex" id="MathJax-Element-1">\newcommand{\A}{\vec{A}}\newcommand{\B}{\vec{B}}
\newcommand{\D}{\vec{D}}\newcommand{\E}{\vec{E}}
\newcommand{\H}{\vec{H}}\newcommand{\J}{\vec{J}}
\newcommand{\M}{\vec{M}}\newcommand{\P}{\vec{P}}
\newcommand{\F}{\vec{F}}
\newcommand{\dd}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\cross}{\times}\newcommand{\inner}{\cdot}
\newcommand{\div}{\nabla\cdot}\newcommand{\curl}{\nabla\times}
\newcommand{\grad}{\nabla}</script></p>
<p>The <code>miniapps/electromagnetics</code> directory contains a collection of
electromagnetic miniapps based on MFEM.</p>
<p>Compared to the <a href="http://mfem.org/examples/">example codes</a>, the miniapps are more complex,
demonstrating more advanced usage of the library. They are intended to be more
representative of MFEM-based application codes. We recommend that new users
start with the example codes before moving to the miniapps.</p>
<p>The current electromagnetic miniapps are described below.</p>
<h2 id="electromagnetics">Electromagnetics</h2>
<p>The equations describing electromagnetic phenomena are known collectively as
the Maxwell Equations. They are usually given as:</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-ampere"><mtext>(1)</mtext></mtd><mtd><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>&#x2212;</mo><mfrac><mrow><mi mathvariant="normal">&#x2202;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mrow><mrow><mi mathvariant="normal">&#x2202;</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-faraday"><mtext>(2)</mtext></mtd><mtd><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>+</mo><mfrac><mrow><mi mathvariant="normal">&#x2202;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mrow><mrow><mi mathvariant="normal">&#x2202;</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-gauss"><mtext>(3)</mtext></mtd><mtd><mi mathvariant="normal">&#x2207;</mi><mo>&#x22C5;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mi>&#x03C1;</mi></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-divb"><mtext>(4)</mtext></mtd><mtd><mi mathvariant="normal">&#x2207;</mi><mo>&#x22C5;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn></mtd></mlabeledtr></mtable></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="43.057ex" height="18.99ex" viewBox="0 -4335 18538.5 8176.3" role="img" focusable="false" style="vertical-align: -8.922ex; max-width: 338px;" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(17259,0)"><g id="mjx-eqn-ampere" transform="translate(0,2633)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g><g id="mjx-eqn-faraday" transform="translate(0,-26)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-32" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g><g id="mjx-eqn-gauss" transform="translate(0,-2008)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-33" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g><g id="mjx-eqn-divb" transform="translate(0,-3526)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-34" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></g><g transform="translate(5429,0)"><g transform="translate(-14,0)"><g transform="translate(0,2633)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-D7" x="1055" y="0"></use><g transform="translate(2056,0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g><use xlink:href="#MJMAIN-2212" x="3167" y="0"></use><g transform="translate(3945,0)"><g transform="translate(342,0)"><rect stroke="none" width="1516" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#MJMAIN-2202" x="0" y="0"></use><g transform="translate(567,0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="704" y="251"></use></g></g><g transform="translate(293,-694)"><use xlink:href="#MJMAIN-2202" x="0" y="0"></use><use xlink:href="#MJMATHI-74" x="567" y="0"></use></g></g></g></g><g transform="translate(193,-26)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-D7" x="1055" y="0"></use><g transform="translate(2056,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g><use xlink:href="#MJMAIN-2B" x="3043" y="0"></use><g transform="translate(3821,0)"><g transform="translate(342,0)"><rect stroke="none" width="1447" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#MJMAIN-2202" x="0" y="0"></use><g transform="translate(567,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></g><g transform="translate(259,-694)"><use xlink:href="#MJMAIN-2202" x="0" y="0"></use><use xlink:href="#MJMATHI-74" x="567" y="0"></use></g></g></g></g><g transform="translate(3538,-2008)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-22C5" x="1055" y="0"></use><g transform="translate(1556,0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="704" y="251"></use></g></g><g transform="translate(3607,-3526)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-22C5" x="1055" y="0"></use><g transform="translate(1556,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></g></g><g transform="translate(5910,0)"><g transform="translate(0,2633)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><g transform="translate(1334,0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use></g></g><g transform="translate(0,-26)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><use xlink:href="#MJMAIN-30" x="1334" y="0"></use></g><g transform="translate(0,-2008)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><use xlink:href="#MJMATHI-3C1" x="1334" y="0"></use></g><g transform="translate(0,-3526)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><use xlink:href="#MJMAIN-30" x="1334" y="0"></use></g></g></g></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-ampere"><mtext>(1)</mtext></mtd><mtd><mi mathvariant="normal">∇</mi><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">→</mo></mover></mrow></mrow><mrow><mi mathvariant="normal">∂</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-faraday"><mtext>(2)</mtext></mtd><mtd><mi mathvariant="normal">∇</mi><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></mrow><mrow><mi mathvariant="normal">∂</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-gauss"><mtext>(3)</mtext></mtd><mtd><mi mathvariant="normal">∇</mi><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mi>ρ</mi></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-divb"><mtext>(4)</mtext></mtd><mtd><mi mathvariant="normal">∇</mi><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn></mtd></mlabeledtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-2">\begin{align}
\curl\H - \dd{\D}{t} & = \J \label{ampere} \\
\curl\E + \dd{\B}{t} & = 0 \label{faraday} \\
\div\D & = \rho \label{gauss} \\
\div\B & = 0 \label{divb}
\end{align}</script></p>
<p>Where equation <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-ampere"><mtext>(1)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-ampere"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-ampere"><mtext>(1)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-3">\eqref{ampere}</script> can be referred to as <em>Ampére's Law</em>, equation
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-faraday"><mtext>(2)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-faraday"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-32" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-faraday"><mtext>(2)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-4">\eqref{faraday}</script> is called <em>Faraday's Law</em>, equation <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-gauss"><mtext>(3)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-gauss"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-33" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-gauss"><mtext>(3)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-5">\eqref{gauss}</script> is <em>Gauss's
Law</em>, and equation <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-6-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-divb"><mtext>(4)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-divb"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-34" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-divb"><mtext>(4)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-6">\eqref{divb}</script> doesn't generally have a name but is related to
the nonexistence of magnetic monopoles. The various fields in these equations
are:</p>
<table class="table table-striped table-hover">
<thead>
<tr>
<th>Symbol</th>
<th>Name</th>
<th>SI Units</th>
</tr>
</thead>
<tbody>
<tr>
<td><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.064ex" height="2.684ex" viewBox="0 -1044.1 888.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-7">\H</script></td>
<td>magnetic field</td>
<td>Ampere/meter</td>
</tr>
<tr>
<td><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.684ex" viewBox="0 -1044.1 759.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-8">\B</script></td>
<td>magnetic flux density</td>
<td>Tesla</td>
</tr>
<tr>
<td><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.776ex" height="2.684ex" viewBox="0 -1044.1 764.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-9">\E</script></td>
<td>electric field</td>
<td>Volts/meter</td>
</tr>
<tr>
<td><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.684ex" viewBox="0 -1044.1 828.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="704" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-10">\D</script></td>
<td>electric displacement</td>
<td>Coulomb/meter<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi></mi><mn>2</mn></msup></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.054ex" height="2.302ex" viewBox="0 -934.4 453.9 991.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.132ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-32" x="0" y="513"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi></mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-11">^2</script></td>
</tr>
<tr>
<td><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.788ex" height="2.684ex" viewBox="0 -1044.1 770 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-12">\J</script></td>
<td>current density</td>
<td>Ampere/meter<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi></mi><mn>2</mn></msup></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.054ex" height="2.302ex" viewBox="0 -934.4 453.9 991.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.132ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-32" x="0" y="513"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi></mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-13">^2</script></td>
</tr>
<tr>
<td><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C1;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.202ex" height="1.92ex" viewBox="0 -550.5 517.5 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C1" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi></math></span></span><script type="math/tex" id="MathJax-Element-14">\rho</script></td>
<td>charge density</td>
<td>Coulomb/meter<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi></mi><mn>3</mn></msup></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.054ex" height="2.302ex" viewBox="0 -934.4 453.9 991.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.132ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33" x="0" y="513"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi></mi><mn>3</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-15">^3</script></td>
</tr>
</tbody>
</table>
<p>In the literature these names do vary, particularly those for <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-16-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.064ex" height="2.684ex" viewBox="0 -1044.1 888.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-16">\H</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-17-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.684ex" viewBox="0 -1044.1 759.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-17">\B</script>,
but in this document we will try to adhere to the convention laid out above.</p>
<p>Generally we also need constitutive relations between <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.776ex" height="2.684ex" viewBox="0 -1044.1 764.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-18">\E</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.684ex" viewBox="0 -1044.1 828.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="704" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-19">\D</script> and/or
between <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.064ex" height="2.684ex" viewBox="0 -1044.1 888.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-20">\H</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.684ex" viewBox="0 -1044.1 759.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-21">\B</script>. These relations start with the definitions:</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-22-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-const_d"><mtext>(5)</mtext></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><msub><mi>&#x03F5;</mi><mn>0</mn></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>P</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-const_b"><mtext>(6)</mtext></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><msub><mi>&#x03BC;</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo stretchy="false">)</mo></mtd></mlabeledtr></mtable></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="43.057ex" height="6.761ex" viewBox="0 -1702.3 18538.5 2910.9" role="img" focusable="false" aria-hidden="true" style="vertical-align: -2.807ex; max-width: 338px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(17259,0)"><g id="mjx-eqn-const_d" transform="translate(0,651)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-35" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g><g id="mjx-eqn-const_b" transform="translate(0,-867)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-36" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></g><g transform="translate(5862,0)"><g transform="translate(-14,0)"><g transform="translate(0,651)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="704" y="251"></use></g><g transform="translate(69,-867)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></g><g transform="translate(814,0)"><g transform="translate(0,651)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><g transform="translate(1334,0)"><use xlink:href="#MJMATHI-3F5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="574" y="-213"></use></g><g transform="translate(2194,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g><use xlink:href="#MJMAIN-2B" x="3181" y="0"></use><g transform="translate(4181,0)"><use xlink:href="#MJMATHI-50" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="764" y="251"></use></g></g><g transform="translate(0,-867)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><g transform="translate(1334,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="853" y="-213"></use></g><use xlink:href="#MJMAIN-28" x="2391" y="0"></use><g transform="translate(2780,0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g><use xlink:href="#MJMAIN-2B" x="3891" y="0"></use><g transform="translate(4892,0)"><use xlink:href="#MJMATHI-4D" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="896" y="252"></use></g><use xlink:href="#MJMAIN-29" x="5943" y="0"></use></g></g></g></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-const_d"><mtext>(5)</mtext></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><msub><mi>ϵ</mi><mn>0</mn></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>P</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-const_b"><mtext>(6)</mtext></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><msub><mi>μ</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mtd></mlabeledtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-22">\begin{align}
\D & = \epsilon_0\E + \P \label{const_d} \\
\B & = \mu_0(\H + \M) \label{const_b}
\end{align}</script></p>
<p>Where <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-23-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>P</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.778ex" height="2.684ex" viewBox="0 -1044.1 765.4 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-50" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="764" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>P</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-23">\P</script> is the <em>polarization density</em>, and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-24-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.442ex" height="2.684ex" viewBox="0 -1044.1 1051.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4D" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="896" y="252"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-24">\M</script> is the <em>magnetization</em>.
Also, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-25-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03F5;</mi><mn>0</mn></msub></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.998ex" height="1.793ex" viewBox="0 -495.6 860.4 771.9" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3F5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="574" y="-213"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϵ</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-25">\epsilon_0</script> is the <em>permittivity of free space</em> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-26-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03BC;</mi><mn>0</mn></msub></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.456ex" height="1.92ex" viewBox="0 -550.5 1057.4 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="853" y="-213"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-26">\mu_0</script> is the
<em>permeability of free space</em> which are both constants of nature. In many common
materials the polarization density can be approximated as a scalar multiple of
the electric field, i.e., <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-27-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>P</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><msub><mi>&#x03F5;</mi><mn>0</mn></msub><mi>&#x03C7;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.105ex" height="3.067ex" viewBox="0 -1044.1 4350.9 1320.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-50" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="764" y="251"></use><use xlink:href="#MJMAIN-3D" x="1043" y="0"></use><g transform="translate(2099,0)"><use xlink:href="#MJMATHI-3F5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="574" y="-213"></use></g><use xlink:href="#MJMATHI-3C7" x="2959" y="0"></use><g transform="translate(3586,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>P</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><msub><mi>ϵ</mi><mn>0</mn></msub><mi>χ</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-27">\P = \epsilon_0\chi\E</script>, where <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-28-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C7;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.455ex" height="1.92ex" viewBox="0 -550.5 626.5 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C7" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>χ</mi></math></span></span><script type="math/tex" id="MathJax-Element-28">\chi</script> is called the
<em>electric susceptibility</em>. In such cases we usually use the relation <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-29-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mi>&#x03F5;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.742ex" height="2.684ex" viewBox="0 -1044.1 3333.6 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="704" y="251"></use><use xlink:href="#MJMAIN-3D" x="1106" y="0"></use><use xlink:href="#MJMATHI-3F5" x="2162" y="0"></use><g transform="translate(2569,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mi>ϵ</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-29">\D =
\epsilon\E</script> with <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-30-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03F5;</mi><mo>=</mo><msub><mi>&#x03F5;</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>&#x03C7;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.308ex" height="2.684ex" viewBox="0 -824.7 5729.9 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3F5" x="0" y="0"></use><use xlink:href="#MJMAIN-3D" x="684" y="0"></use><g transform="translate(1740,0)"><use xlink:href="#MJMATHI-3F5" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="574" y="-213"></use></g><use xlink:href="#MJMAIN-28" x="2600" y="0"></use><use xlink:href="#MJMAIN-31" x="2990" y="0"></use><use xlink:href="#MJMAIN-2B" x="3713" y="0"></use><use xlink:href="#MJMATHI-3C7" x="4713" y="0"></use><use xlink:href="#MJMAIN-29" x="5340" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi><mo>=</mo><msub><mi>ϵ</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>χ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-30">\epsilon = \epsilon_0(1 + \chi)</script> and call <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-31-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03F5;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.944ex" height="1.411ex" viewBox="0 -495.6 406.5 607.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3F5" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math></span></span><script type="math/tex" id="MathJax-Element-31">\epsilon</script> the
<em>permittivity</em> of the material.</p>
<p>The nature of magnetization is more complicated but we will take a very
simplified view which is valid in many situations. Specifically, we will assume
that either <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-32-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.442ex" height="2.684ex" viewBox="0 -1044.1 1051.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4D" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="896" y="252"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-32">\M</script> is proportional to <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-33-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.064ex" height="2.684ex" viewBox="0 -1044.1 888.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-33">\H</script> yielding the relation <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-34-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mi>&#x03BC;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.328ex" height="3.067ex" viewBox="0 -1044.1 3585.6 1320.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use><use xlink:href="#MJMAIN-3D" x="1037" y="0"></use><use xlink:href="#MJMATHI-3BC" x="2093" y="0"></use><g transform="translate(2697,0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mi>μ</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-34">\B = \mu\H</script>
where <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-35-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi><mo>=</mo><msub><mi>&#x03BC;</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>&#x03C7;</mi><mi>M</mi></msub><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.182ex" height="2.684ex" viewBox="0 -824.7 6967.4 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><use xlink:href="#MJMAIN-3D" x="881" y="0"></use><g transform="translate(1937,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="853" y="-213"></use></g><use xlink:href="#MJMAIN-28" x="2994" y="0"></use><use xlink:href="#MJMAIN-31" x="3384" y="0"></use><use xlink:href="#MJMAIN-2B" x="4107" y="0"></use><g transform="translate(5107,0)"><use xlink:href="#MJMATHI-3C7" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-4D" x="886" y="-213"></use></g><use xlink:href="#MJMAIN-29" x="6577" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><msub><mi>μ</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>χ</mi><mi>M</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-35">\mu = \mu_0(1 + \chi_M)</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-36-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C7;</mi><mi>M</mi></msub></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.414ex" height="1.92ex" viewBox="0 -550.5 1470 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C7" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-4D" x="886" y="-213"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>χ</mi><mi>M</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-36">\chi_M</script> is the <em>magnetic susceptibility</em>
or that <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-37-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.442ex" height="2.684ex" viewBox="0 -1044.1 1051.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4D" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="896" y="252"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-37">\M</script> is independent of the applied field. The former case pertains to
both diamagnetic and paramagnetic materials and the latter to ferromagnetic
materials.</p>
<p>Finally we should note that equations <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-38-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-ampere"><mtext>(1)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-ampere"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-ampere"><mtext>(1)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-38">\eqref{ampere}</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-39-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-gauss"><mtext>(3)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-gauss"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-33" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-gauss"><mtext>(3)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-39">\eqref{gauss}</script> can be
combined to yield the equation of charge continuity <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-40-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">&#x2202;</mi><mi>&#x03C1;</mi></mrow><mrow><mi mathvariant="normal">&#x2202;</mi><mi>t</mi></mrow></mfrac><mo>+</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x22C5;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mn>0</mn></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.123ex" height="3.831ex" viewBox="0 -1153.8 6511.1 1649.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -1.151ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="887" height="60" x="0" y="220"></rect><g transform="translate(60,566)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2202" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-3C1" x="567" y="0"></use></g><g transform="translate(115,-420)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2202" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="567" y="0"></use></g></g><use xlink:href="#MJMAIN-2B" x="1349" y="0"></use><use xlink:href="#MJMAIN-2207" x="2350" y="0"></use><use xlink:href="#MJMAIN-22C5" x="3405" y="0"></use><g transform="translate(3906,0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use></g><use xlink:href="#MJMAIN-3D" x="4954" y="0"></use><use xlink:href="#MJMAIN-30" x="6010" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>t</mi></mrow></mfrac><mo>+</mo><mi mathvariant="normal">∇</mi><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-40">\dd{\rho}{t} + \div\J = 0</script>
which can be important in plasma physics and magnetohydrodynamics (MHD).</p>
<h2 id="electrostatics">Electrostatics</h2>
<p>Electrostatic problems come in a variety of subtypes but they all derive from
Gauss's Law and Faraday's Law (equations <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-41-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-gauss"><mtext>(3)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-gauss"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-33" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-gauss"><mtext>(3)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-41">\eqref{gauss}</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-42-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-faraday"><mtext>(2)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-faraday"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-32" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-faraday"><mtext>(2)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-42">\eqref{faraday}</script>).
When we assume no time variation, Faraday's Law becomes simply <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-43-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mn>0</mn></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.813ex" height="2.684ex" viewBox="0 -1044.1 4655.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-D7" x="1055" y="0"></use><g transform="translate(2056,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g><use xlink:href="#MJMAIN-3D" x="3098" y="0"></use><use xlink:href="#MJMAIN-30" x="4155" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∇</mi><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-43">\curl\E = 0</script>.
This suggests that the electric field can be expressed as the gradient of a
scalar field which is traditionally taken to be <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-44-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2212;</mo><mi>&#x03C6;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.328ex" height="2.175ex" viewBox="0 -660.2 1433 936.4" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><use xlink:href="#MJMATHI-3C6" x="778" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mi>φ</mi></math></span></span><script type="math/tex" id="MathJax-Element-44">-\varphi</script>, i.e.</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-45-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-gradphi"><mtext>(7)</mtext></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mo>&#x2212;</mo><mi mathvariant="normal">&#x2207;</mi><mi>&#x03C6;</mi></mtd></mlabeledtr></mtable></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="43.057ex" height="3.194ex" viewBox="0 -934.4 18538.5 1375.2" role="img" focusable="false" aria-hidden="true" style="vertical-align: -1.024ex; max-width: 338px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(17259,0)"><g id="mjx-eqn-gradphi" transform="translate(0,-107)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-37" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></g><g transform="translate(7260,0)"><g transform="translate(-14,0)"><g transform="translate(0,-107)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use><use xlink:href="#MJMAIN-3D" x="1042" y="0"></use><use xlink:href="#MJMAIN-2212" x="2098" y="0"></use><use xlink:href="#MJMAIN-2207" x="2877" y="0"></use><use xlink:href="#MJMATHI-3C6" x="3710" y="0"></use></g></g></g></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-gradphi"><mtext>(7)</mtext></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo>−</mo><mi mathvariant="normal">∇</mi><mi>φ</mi></mtd></mlabeledtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-45">\E = -\grad\varphi \label{gradphi}</script></p>
<p>where <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-46-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C6;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.52ex" height="1.92ex" viewBox="0 -550.5 654.5 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C6" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi></math></span></span><script type="math/tex" id="MathJax-Element-46">\varphi</script> is called the <em>electric potential</em> and has units of Volts in
the SI system. Inserting this definition into equation <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-47-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-gauss"><mtext>(3)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-gauss"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-33" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-gauss"><mtext>(3)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-47">\eqref{gauss}</script> gives:</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-48-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-poisson"><mtext>(8)</mtext></mtd><mtd><mo>&#x2212;</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x22C5;</mo><mi>&#x03F5;</mi><mi mathvariant="normal">&#x2207;</mi><mi>&#x03C6;</mi><mo>=</mo><mi>&#x03C1;</mi><mo>&#x2212;</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x22C5;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>P</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd></mlabeledtr></mtable></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="43.057ex" height="3.194ex" viewBox="0 -934.4 18538.5 1375.2" role="img" focusable="false" aria-hidden="true" style="vertical-align: -1.024ex; max-width: 338px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(17259,0)"><g id="mjx-eqn-poisson" transform="translate(0,-108)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-38" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></g><g transform="translate(4630,0)"><g transform="translate(-14,0)"><g transform="translate(0,-108)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><use xlink:href="#MJMAIN-2207" x="778" y="0"></use><use xlink:href="#MJMAIN-22C5" x="1834" y="0"></use><use xlink:href="#MJMATHI-3F5" x="2334" y="0"></use><use xlink:href="#MJMAIN-2207" x="2741" y="0"></use><use xlink:href="#MJMATHI-3C6" x="3574" y="0"></use><use xlink:href="#MJMAIN-3D" x="4507" y="0"></use><use xlink:href="#MJMATHI-3C1" x="5563" y="0"></use><use xlink:href="#MJMAIN-2212" x="6303" y="0"></use><use xlink:href="#MJMAIN-2207" x="7303" y="0"></use><use xlink:href="#MJMAIN-22C5" x="8359" y="0"></use><g transform="translate(8860,0)"><use xlink:href="#MJMATHI-50" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="764" y="251"></use></g></g></g></g></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-poisson"><mtext>(8)</mtext></mtd><mtd><mo>−</mo><mi mathvariant="normal">∇</mi><mo>⋅</mo><mi>ϵ</mi><mi mathvariant="normal">∇</mi><mi>φ</mi><mo>=</mo><mi>ρ</mi><mo>−</mo><mi mathvariant="normal">∇</mi><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>P</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mlabeledtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-48">-\div\epsilon\grad\varphi = \rho - \div\P \label{poisson}</script></p>
<p>which is <em>Poisson's equation</em> for the electric potential, where we have assumed
a linear constitutive relation between <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-49-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.684ex" viewBox="0 -1044.1 828.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="704" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-49">\D</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-50-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.776ex" height="2.684ex" viewBox="0 -1044.1 764.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-50">\E</script> of the form <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-51-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mi>&#x03F5;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>P</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.361ex" height="2.812ex" viewBox="0 -1044.1 5321.9 1210.6" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.387ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="704" y="251"></use><use xlink:href="#MJMAIN-3D" x="1106" y="0"></use><use xlink:href="#MJMATHI-3F5" x="2162" y="0"></use><g transform="translate(2569,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g><use xlink:href="#MJMAIN-2B" x="3555" y="0"></use><g transform="translate(4556,0)"><use xlink:href="#MJMATHI-50" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="764" y="251"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mi>ϵ</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>P</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-51">\D =
\epsilon\E + \P</script>. This allows a polarization which is proportional to <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-52-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.776ex" height="2.684ex" viewBox="0 -1044.1 764.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-52">\E</script> as
well a polarization independent of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-53-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.776ex" height="2.684ex" viewBox="0 -1044.1 764.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-53">\E</script>. If this relation happens to be
nonlinear then Poisson's equation would need to be replaced with a more
complicated nonlinear expression.</p>
<p>The solutions to equation <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-54-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-poisson"><mtext>(8)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-poisson"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-38" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-poisson"><mtext>(8)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-54">\eqref{poisson}</script> are non unique because they can be
shifted by any additive constant. This means that we must apply a Dirichlet
boundary condition at least at one point in the problem domain in order to
obtain a solution. Typically this point will be on the boundary but it need not
be so. Such a Dirichlet value is equivalent to fixing the voltage (a.k.a.
potential) at one or more locations. Additionally, this equation admits a normal
derivative boundary condition. This corresponds to setting <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-55-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>n</mi><mo stretchy="false">&#x005E;</mo></mover></mrow><mo>&#x22C5;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.998ex" height="2.684ex" viewBox="0 -1044.1 2151.9 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-6E" x="0" y="0"></use><use xlink:href="#MJMAIN-5E" x="50" y="-4"></use><use xlink:href="#MJMAIN-22C5" x="822" y="0"></use><g transform="translate(1323,0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="704" y="251"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>n</mi><mo stretchy="false">^</mo></mover></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-55">\hat{n}\cdot\D</script> to a
prescribed value on some portion of the boundary. This is equivalent to defining
a surface charge density on that portion of the boundary.</p>
<h3 id="volta-mini-application">Volta Mini Application</h3>
<p>The electrostatics mini application, named <code>volta</code> after the inventor of the
<a href="http://en.wikipedia.org/wiki/Voltaic_pile">voltaic pile</a>, is intended to
demonstrate how to solve standard electrostatics problems in MFEM. Its source
terms and boundary conditions are simple but they should indicate how more
specialized sources or boundary conditions could be implemented.</p>
<p><img alt="" src="../../doc/web/examples/volta-screenshot-small.png"></p>
<p>Note that this application assumes the mesh coordinates are given in meters.</p>
<h4 id="mini-application-features">Mini Application Features</h4>
<p><strong>Permittivity:</strong> The permittivity, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-56-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03F5;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.944ex" height="1.411ex" viewBox="0 -495.6 406.5 607.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3F5" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math></span></span><script type="math/tex" id="MathJax-Element-56">\epsilon</script>, is assumed to be that of free
space except for an optional sphere of dielectric material which can be
defined by the user. The command line option <code>-ds</code> can be used to set the
parameters for this dielectric sphere. For example, to produce a sphere at the
origin with a radius of 0.5 and a relative permittivity of 3 the user would
specify: <code>-ds '0 0 0 0.5 3'</code>.</p>
<p><strong>Charge Density:</strong> The charge density, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-57-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C1;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.202ex" height="1.92ex" viewBox="0 -550.5 517.5 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C1" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi></math></span></span><script type="math/tex" id="MathJax-Element-57">\rho</script>, is assumed to be zero except for
an optional sphere of uniform charge density which can be defined by the user.
The command line option for this is <code>-cs</code> which follows the same pattern as
the dielectric sphere. Note that the last entry is the total charge of the
sphere and not its charge density.</p>
<p><strong>Polarization:</strong> A polarization vector function, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-58-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>P</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.778ex" height="2.684ex" viewBox="0 -1044.1 765.4 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-50" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="764" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>P</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-58">\P</script>, can be imposed as a
source of the electric field. The command line option <code>-vp</code> creates a
polarization due to a simple voltaic pile, i.e., a cylinder which is
electrically polarized along its axis. The user should specify the two end
points of the cylinder axis, its radius and the magnitude of the polarization
vector.</p>
<p><strong>Dirichlet BC:</strong> Dirichlet boundary conditions can either specify piecewise
constant voltages on a collection of surfaces or they can specify a gradient
field which approximates a uniform applied electric field. In either case the
user specifies the surfaces where the Dirichlet boundary condition should be
applied using the <code>-dbcs</code> option followed by a list of boundary attributes.
For example to select surfaces 2, 3, and 4 the user would use the following:
<code>-dbcs '2 3 4'</code>.</p>
<p>To apply a gradient field on these surfaces the user would also use the
<code>-dbcg</code> option. This defaults to the uniform field <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-59-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.239ex" height="3.194ex" viewBox="0 -1044.1 5269.4 1375.2" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use><use xlink:href="#MJMAIN-3D" x="1042" y="0"></use><use xlink:href="#MJMAIN-28" x="2098" y="0"></use><use xlink:href="#MJMAIN-30" x="2488" y="0"></use><use xlink:href="#MJMAIN-2C" x="2988" y="0"></use><use xlink:href="#MJMAIN-30" x="3433" y="0"></use><use xlink:href="#MJMAIN-2C" x="3934" y="0"></use><use xlink:href="#MJMAIN-31" x="4379" y="0"></use><use xlink:href="#MJMAIN-29" x="4879" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-59">\E = (0,0,1)</script> in 3D or
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-60-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.042ex" height="3.194ex" viewBox="0 -1044.1 4323.7 1375.2" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use><use xlink:href="#MJMAIN-3D" x="1042" y="0"></use><use xlink:href="#MJMAIN-28" x="2098" y="0"></use><use xlink:href="#MJMAIN-30" x="2488" y="0"></use><use xlink:href="#MJMAIN-2C" x="2988" y="0"></use><use xlink:href="#MJMAIN-31" x="3433" y="0"></use><use xlink:href="#MJMAIN-29" x="3934" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-60">\E = (0,1)</script> in 2D. An arbitrary vector can be specified with <code>-uebc</code>
followed by the desired vector, e.g., to apply <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-61-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.239ex" height="3.194ex" viewBox="0 -1044.1 5269.4 1375.2" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use><use xlink:href="#MJMAIN-3D" x="1042" y="0"></use><use xlink:href="#MJMAIN-28" x="2098" y="0"></use><use xlink:href="#MJMAIN-31" x="2488" y="0"></use><use xlink:href="#MJMAIN-2C" x="2988" y="0"></use><use xlink:href="#MJMAIN-32" x="3433" y="0"></use><use xlink:href="#MJMAIN-2C" x="3934" y="0"></use><use xlink:href="#MJMAIN-33" x="4379" y="0"></use><use xlink:href="#MJMAIN-29" x="4879" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-61">\E = (1,2,3)</script> the user would
supply: <code>-uebc '1 2 3'</code>.</p>
<p>To specify piecewise constant potential values the user would list the
desired values after <code>-dbcv</code> as follows: <code>-dbcv '0.0 1.0 -1.0'</code>.</p>
<p><strong>Neumann BC:</strong> Neumann boundary conditions set the normal component of the
electric displacement on portions of the boundary. This normal component is
equivalent to the surface charge density on the surface. This is rarely used
because surface charge densities are rarely known unless they are know to be
zero. However, if the surface charge density is zero then the Neumann BCs are
not needed because this is the natural boundary condition. Only piecewise
constant Neumann boundary conditions are supported. They can be set
analogously to piecewise Dirichlet boundary conditions but using options
<code>-nbcs</code> and <code>-nbcv</code>.</p>
<h2 id="magnetostatics">Magnetostatics</h2>
<p>Magnetostatic problems arise when we assume no time variation in Ampére's Law
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-62-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-ampere"><mtext>(1)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-ampere"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-ampere"><mtext>(1)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-62">\eqref{ampere}</script> which leads to:</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-63-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.727ex" height="2.684ex" viewBox="0 -1044.1 5049 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-D7" x="1055" y="0"></use><g transform="translate(2056,0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g><use xlink:href="#MJMAIN-3D" x="3222" y="0"></use><g transform="translate(4279,0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use></g></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="normal">∇</mi><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-63">\curl\H = \J \nonumber</script></p>
<p>We will again assume a somewhat more general constitutive relation between <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-64-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.064ex" height="2.684ex" viewBox="0 -1044.1 888.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-64">\H</script>
and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-65-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.684ex" viewBox="0 -1044.1 759.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-65">\vec{B}</script> than is normally seen:</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-66-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mi>&#x03BC;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>+</mo><msub><mi>&#x03BC;</mi><mn>0</mn></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><msub><mi>&#x03BC;</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>&#x03C7;</mi><mi>M</mi></msub><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>+</mo><msub><mi>&#x03BC;</mi><mn>0</mn></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="40.649ex" height="3.194ex" viewBox="0 -1044.1 17501.7 1375.2" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use><use xlink:href="#MJMAIN-3D" x="1037" y="0"></use><use xlink:href="#MJMATHI-3BC" x="2093" y="0"></use><g transform="translate(2697,0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g><use xlink:href="#MJMAIN-2B" x="3807" y="0"></use><g transform="translate(4808,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="853" y="-213"></use></g><g transform="translate(5865,0)"><use xlink:href="#MJMATHI-4D" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="896" y="252"></use></g><use xlink:href="#MJMAIN-3D" x="7195" y="0"></use><g transform="translate(8251,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="853" y="-213"></use></g><use xlink:href="#MJMAIN-28" x="9308" y="0"></use><use xlink:href="#MJMAIN-31" x="9698" y="0"></use><use xlink:href="#MJMAIN-2B" x="10421" y="0"></use><g transform="translate(11421,0)"><use xlink:href="#MJMATHI-3C7" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-4D" x="886" y="-213"></use></g><use xlink:href="#MJMAIN-29" x="12891" y="0"></use><g transform="translate(13281,0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g><use xlink:href="#MJMAIN-2B" x="14392" y="0"></use><g transform="translate(15392,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="853" y="-213"></use></g><g transform="translate(16450,0)"><use xlink:href="#MJMATHI-4D" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="896" y="252"></use></g></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mi>μ</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><msub><mi>μ</mi><mn>0</mn></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><msub><mi>μ</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>χ</mi><mi>M</mi></msub><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><msub><mi>μ</mi><mn>0</mn></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-66">\B = \mu\H + \mu_0\M = \mu_0(1 + \chi_M)\H + \mu_0\M \nonumber</script></p>
<p>Where the magnetization is split into two portions; one which is proportional
to <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-67-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.064ex" height="2.684ex" viewBox="0 -1044.1 888.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-67">\H</script> and given by <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-68-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C7;</mi><mi>M</mi></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.478ex" height="3.067ex" viewBox="0 -1044.1 2358.5 1320.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C7" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-4D" x="886" y="-213"></use><g transform="translate(1470,0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>χ</mi><mi>M</mi></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-68">\chi_M\H</script>, and another which is independent of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-69-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.064ex" height="2.684ex" viewBox="0 -1044.1 888.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-69">\H</script> and
is given by <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-70-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.442ex" height="2.684ex" viewBox="0 -1044.1 1051.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4D" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="896" y="252"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-70">\M</script>. This allows for paramagnetic and/or diamagnetic materials
defined through <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-71-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.92ex" viewBox="0 -550.5 603.5 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></span></span><script type="math/tex" id="MathJax-Element-71">\mu</script> as well as ferromagnetic materials represented by <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-72-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.442ex" height="2.684ex" viewBox="0 -1044.1 1051.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4D" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="896" y="252"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-72">\M</script>.
This choice yields:</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-73-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><msup><mi>&#x03BC;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>+</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><msup><mi>&#x03BC;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><msub><mi>&#x03BC;</mi><mn>0</mn></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="31.411ex" height="3.067ex" viewBox="0 -1044.1 13524 1320.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-D7" x="1055" y="0"></use><g transform="translate(2056,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><g transform="translate(603,412)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="778" y="0"></use></g></g><g transform="translate(3664,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g><use xlink:href="#MJMAIN-3D" x="4701" y="0"></use><g transform="translate(5757,0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use></g><use xlink:href="#MJMAIN-2B" x="6750" y="0"></use><use xlink:href="#MJMAIN-2207" x="7750" y="0"></use><use xlink:href="#MJMAIN-D7" x="8806" y="0"></use><g transform="translate(9807,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><g transform="translate(603,412)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="778" y="0"></use></g></g><g transform="translate(11415,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="853" y="-213"></use></g><g transform="translate(12472,0)"><use xlink:href="#MJMATHI-4D" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="896" y="252"></use></g></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="normal">∇</mi><mo>×</mo><msup><mi>μ</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mi mathvariant="normal">∇</mi><mo>×</mo><msup><mi>μ</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><msub><mi>μ</mi><mn>0</mn></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-73">\curl\mu^{-1}\B = \J + \curl\mu^{-1}\mu_0\M \nonumber</script></p>
<p>Which, when combined with equation <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-74-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-divb"><mtext>(4)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-divb"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-34" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-divb"><mtext>(4)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-74">\eqref{divb}</script>, becomes:</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-75-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-9"><mtext>(9)</mtext></mtd><mtd><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><msup><mi>&#x03BC;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>+</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><msup><mi>&#x03BC;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><msub><mi>&#x03BC;</mi><mn>0</mn></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd></mlabeledtr></mtable></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="43.057ex" height="3.194ex" viewBox="0 -934.4 18538.5 1375.2" role="img" focusable="false" aria-hidden="true" style="vertical-align: -1.024ex; max-width: 338px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(17259,0)"><g id="mjx-eqn-9" transform="translate(0,-125)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-39" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></g><g transform="translate(1068,0)"><g transform="translate(-14,0)"><g transform="translate(0,-125)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-D7" x="1055" y="0"></use><g transform="translate(2056,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><g transform="translate(603,412)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="778" y="0"></use></g></g><use xlink:href="#MJMAIN-2207" x="3664" y="0"></use><use xlink:href="#MJMAIN-D7" x="4720" y="0"></use><g transform="translate(5720,0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="749" y="284"></use></g><use xlink:href="#MJMAIN-3D" x="6749" y="0"></use><g transform="translate(7805,0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use></g><use xlink:href="#MJMAIN-2B" x="8797" y="0"></use><use xlink:href="#MJMAIN-2207" x="9798" y="0"></use><use xlink:href="#MJMAIN-D7" x="10853" y="0"></use><g transform="translate(11854,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><g transform="translate(603,412)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="778" y="0"></use></g></g><g transform="translate(13462,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="853" y="-213"></use></g><g transform="translate(14519,0)"><use xlink:href="#MJMATHI-4D" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="896" y="252"></use></g></g></g></g></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-9"><mtext>(9)</mtext></mtd><mtd><mi mathvariant="normal">∇</mi><mo>×</mo><msup><mi>μ</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mi mathvariant="normal">∇</mi><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mi mathvariant="normal">∇</mi><mo>×</mo><msup><mi>μ</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><msub><mi>μ</mi><mn>0</mn></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mlabeledtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-75">\curl\mu^{-1}\curl\A = \J + \curl\mu^{-1}\mu_0\M </script></p>
<p>If <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-76-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.788ex" height="2.684ex" viewBox="0 -1044.1 770 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-76">\J</script> happens to be zero we have another option because we can assume that
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-77-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>=</mo><mo>&#x2212;</mo><mi mathvariant="normal">&#x2207;</mi><msub><mi>&#x03C6;</mi><mi>M</mi></msub></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.385ex" height="3.067ex" viewBox="0 -1044.1 5332.6 1320.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use><use xlink:href="#MJMAIN-3D" x="1166" y="0"></use><use xlink:href="#MJMAIN-2212" x="2222" y="0"></use><use xlink:href="#MJMAIN-2207" x="3001" y="0"></use><g transform="translate(3834,0)"><use xlink:href="#MJMATHI-3C6" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-4D" x="925" y="-213"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo>−</mo><mi mathvariant="normal">∇</mi><msub><mi>φ</mi><mi>M</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-77">\H = -\grad\varphi_M</script> for some scalar potential <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-78-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C6;</mi><mi>M</mi></msub></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.479ex" height="1.92ex" viewBox="0 -550.5 1498 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C6" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-4D" x="925" y="-213"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>φ</mi><mi>M</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-78">\varphi_M</script>. When combined
with equation <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-79-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-divb"><mtext>(4)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-divb"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-34" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-divb"><mtext>(4)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-79">\eqref{divb}</script> this leads to:</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-80-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-10"><mtext>(10)</mtext></mtd><mtd><mi mathvariant="normal">&#x2207;</mi><mo>&#x22C5;</mo><mi>&#x03BC;</mi><mi mathvariant="normal">&#x2207;</mi><msub><mi>&#x03C6;</mi><mi>M</mi></msub><mo>=</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x22C5;</mo><msub><mi>&#x03BC;</mi><mn>0</mn></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd></mlabeledtr></mtable></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="43.057ex" height="3.194ex" viewBox="0 -934.4 18538.5 1375.2" role="img" focusable="false" aria-hidden="true" style="vertical-align: -1.024ex; max-width: 338px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(16758,0)"><g id="mjx-eqn-10" transform="translate(0,-109)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-30" x="890" y="0"></use><use xlink:href="#MJMAIN-29" x="1390" y="0"></use></g></g><g transform="translate(4697,0)"><g transform="translate(-14,0)"><g transform="translate(0,-109)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-22C5" x="1055" y="0"></use><use xlink:href="#MJMATHI-3BC" x="1556" y="0"></use><use xlink:href="#MJMAIN-2207" x="2159" y="0"></use><g transform="translate(2993,0)"><use xlink:href="#MJMATHI-3C6" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-4D" x="925" y="-213"></use></g><use xlink:href="#MJMAIN-3D" x="4769" y="0"></use><use xlink:href="#MJMAIN-2207" x="5825" y="0"></use><use xlink:href="#MJMAIN-22C5" x="6881" y="0"></use><g transform="translate(7381,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="853" y="-213"></use></g><g transform="translate(8439,0)"><use xlink:href="#MJMATHI-4D" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="896" y="252"></use></g></g></g></g></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-10"><mtext>(10)</mtext></mtd><mtd><mi mathvariant="normal">∇</mi><mo>⋅</mo><mi>μ</mi><mi mathvariant="normal">∇</mi><msub><mi>φ</mi><mi>M</mi></msub><mo>=</mo><mi mathvariant="normal">∇</mi><mo>⋅</mo><msub><mi>μ</mi><mn>0</mn></msub><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mlabeledtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-80">\div\mu\grad\varphi_M = \div\mu_0\M </script></p>
<p>Currently only the vector potential equation is used so we will focus on that
for the remainder of this document.</p>
<p>The vector potential is again non unique so we must apply additional
constraints in order to arrive at a solution for <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-81-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.743ex" height="2.812ex" viewBox="0 -1099 750.5 1210.6" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="749" y="284"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-81">\A</script>. When working
analytically it is common to constrain the solution by restricting the
divergence of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-82-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.743ex" height="2.812ex" viewBox="0 -1099 750.5 1210.6" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="749" y="284"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-82">\A</script> but numerically this leads to other complications. For our
problems of interest it will be necessary to require Dirichlet boundary
conditions on the entire outer surface in order to sufficiently constrain the
solution.</p>
<p>Dirichlet boundary conditions for the vector potential on a surface provide a
means to specify the component of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-83-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.684ex" viewBox="0 -1044.1 759.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-83">\B</script> normal to that surface. For example,
setting the tangential components of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-84-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.743ex" height="2.812ex" viewBox="0 -1099 750.5 1210.6" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="749" y="284"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-84">\A</script> to be zero on a particular surface
results in a magnetic flux density which must be tangent to that surface.</p>
<h3 id="tesla-mini-application">Tesla Mini Application</h3>
<p>The magnetostatics mini application, named <code>tesla</code> after the unit of magnetic
field strength (and of course the man Nikola Tesla), is intended to demonstrate
how to solve standard magnetostatics problems in MFEM. Its source terms and
boundary conditions are simple but they should indicate how more specialized
sources of boundary conditions could be implemented.</p>
<p><img alt="" src="../../doc/web/examples/tesla-screenshot-small.png"></p>
<p>Note that this application assumes the mesh coordinates are given in meters.</p>
<h4 id="mini-application-features_1">Mini Application Features</h4>
<p><strong>Permeability:</strong> The permeability, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-85-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.92ex" viewBox="0 -550.5 603.5 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></span></span><script type="math/tex" id="MathJax-Element-85">\mu</script>, is assumed to be that of free space
except for an optional spherical shell of diamagnetic or paramagnetic material
which can be defined by the user. The command line option <code>-ms</code> can be used to
set the parameters for this shell.</p>
<p>For example, to produce a shell at the origin with inner and outer radii of
0.4 and 0.5 respectively and a relative permeability of 3 the user would
specify: <code>-ms '0 0 0 0.4 0.5 3'</code>.</p>
<p><strong>Current Density:</strong> The current density, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-86-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.788ex" height="2.684ex" viewBox="0 -1044.1 770 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-86">\J</script>, is assumed to be zero except for
an optional ring of constant current which can be defined by the user. The
command line option for this is <code>-cr</code> which requires two points giving the end
points of the ring's axis, inner and outer radii, and a constant total
current.</p>
<p>For example, to specify a ring centered at the origin and laying in the XY
plane with a thickness of 0.2 and radii 0.4 and 0.5, and a current of 2 amps
the user would give: <code>-cr 0 0 -0.1 0 0 0.1 0.4 0.5 2</code>.</p>
<p><strong>Magnetization:</strong> A permanent magnetization, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-87-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.442ex" height="2.684ex" viewBox="0 -1044.1 1051.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4D" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="896" y="252"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-87">\M</script>, can be applied in the form
of a cylindrical magnet with poles at its circular ends. The command line
option is <code>-bm</code> which indicates a 'bar magnet'. The option requires the two
end points of the cylinder's axis, its radius, and the magnitude of the
magnetization.</p>
<p><strong>Surface Current Density:</strong> A surface current can be imposed indirectly by
specifying separate surface patches with different voltages as well as a
collection of surface patches connecting the voltages through which the
current will flow. The voltage surfaces and their voltages can be specified
using <code>-vbcs</code> followed by the indices of the surfaces and <code>-vbcv</code> followed by
their voltages. The path for the surface current (<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-88-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>K</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.066ex" height="2.684ex" viewBox="0 -1044.1 889.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4B" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="761" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>K</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-88">\vec{K}</script>) is specified by
using <code>-kbcs</code> followed by a set of surface indices.</p>
<p>For example, applying voltages 1 and -1 to surfaces 2 and 3 with a current
path along surfaces 4 and 6 would be specified as:
<code>-vbcs '2 3' -vbcv '1 -1' -kbcs '4 6'</code>.</p>
<p>Any surfaces not listed as voltage or current surfaces will be assigned as
homogeneous Dirichlet boundaries. Note that when this option is selected an
auxiliary electrostatic problem will be solved on the surface of the
geometry to compute the surface current.</p>
<p><strong>Dirichlet BC:</strong> Dirichlet boundary conditions are required if a surface
current density is not defined. For this reason the user need not specify
boundary surfaces by number since the boundary condition must be applied on
all of them. The default boundary condition is a homogeneous Dirichlet
boundary condition on all outer surfaces. This means that the normal
component of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-89-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.684ex" viewBox="0 -1044.1 759.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-89">\B</script> will be zero at the outer boundary. An alternative is to
specify a desired uniform magnetic flux density on the entire outer surface.
This is accomplished with the <code>-ubbc</code> command line option followed by the
desired <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-90-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.684ex" viewBox="0 -1044.1 759.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-90">\B</script> vector.</p>
<h2 id="transient-full-wave-electromagnetics">Transient Full-Wave Electromagnetics</h2>
<p>Transient electromagnetics problems are governed by the time-dependent
Maxwell equations <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-91-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-ampere"><mtext>(1)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-ampere"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-ampere"><mtext>(1)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-91">\eqref{ampere}</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-92-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-faraday"><mtext>(2)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-faraday"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-32" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-faraday"><mtext>(2)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-92">\eqref{faraday}</script> when combined
using the constitutive relations <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-93-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-const_d"><mtext>(5)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-const_d"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-35" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-const_d"><mtext>(5)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-93">\eqref{const_d}</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-94-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-const_b"><mtext>(6)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-const_b"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-36" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-const_b"><mtext>(6)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-94">\eqref{const_b}</script>.
When combined these equations can describe the evolution and
propagation of electromagnetic waves.</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-95-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-11"><mtext>(11)</mtext></mtd><mtd><mfrac><mrow><mi mathvariant="normal">&#x2202;</mi><mo stretchy="false">(</mo><mi>&#x03F5;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">&#x2202;</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><mo stretchy="false">(</mo><msup><mi>&#x03BC;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>&#x2212;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-12"><mtext>(12)</mtext></mtd><mtd><mfrac><mrow><mi mathvariant="normal">&#x2202;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mrow><mrow><mi mathvariant="normal">&#x2202;</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mo>&#x2212;</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd></mlabeledtr></mtable></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="43.057ex" height="12.111ex" viewBox="0 -2854.1 18538.5 5214.5" role="img" focusable="false" aria-hidden="true" style="vertical-align: -5.482ex; max-width: 338px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(16758,0)"><g id="mjx-eqn-11" transform="translate(0,1091)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-31" x="890" y="0"></use><use xlink:href="#MJMAIN-29" x="1390" y="0"></use></g><g id="mjx-eqn-12" transform="translate(0,-1568)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-32" x="890" y="0"></use><use xlink:href="#MJMAIN-29" x="1390" y="0"></use></g></g><g transform="translate(2171,0)"><g transform="translate(-14,0)"><g transform="translate(0,1091)"><g transform="translate(120,0)"><rect stroke="none" width="2637" height="60" x="0" y="220"></rect><g transform="translate(60,728)"><use xlink:href="#MJMAIN-2202" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="567" y="0"></use><use xlink:href="#MJMATHI-3F5" x="957" y="0"></use><g transform="translate(1363,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g><use xlink:href="#MJMAIN-29" x="2128" y="0"></use></g><g transform="translate(854,-694)"><use xlink:href="#MJMAIN-2202" x="0" y="0"></use><use xlink:href="#MJMATHI-74" x="567" y="0"></use></g></g></g><g transform="translate(1190,-1568)"><g transform="translate(120,0)"><rect stroke="none" width="1447" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#MJMAIN-2202" x="0" y="0"></use><g transform="translate(567,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></g><g transform="translate(259,-694)"><use xlink:href="#MJMAIN-2202" x="0" y="0"></use><use xlink:href="#MJMATHI-74" x="567" y="0"></use></g></g></g></g><g transform="translate(2863,0)"><g transform="translate(0,1091)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><use xlink:href="#MJMAIN-2207" x="1334" y="0"></use><use xlink:href="#MJMAIN-D7" x="2389" y="0"></use><use xlink:href="#MJMAIN-28" x="3390" y="0"></use><g transform="translate(3780,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><g transform="translate(603,412)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="778" y="0"></use></g></g><g transform="translate(5387,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g><use xlink:href="#MJMAIN-29" x="6147" y="0"></use><use xlink:href="#MJMAIN-2212" x="6759" y="0"></use><use xlink:href="#MJMATHI-3C3" x="7759" y="0"></use><g transform="translate(8332,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g><use xlink:href="#MJMAIN-2212" x="9319" y="0"></use><g transform="translate(10319,0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use></g></g><g transform="translate(0,-1568)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><use xlink:href="#MJMAIN-2212" x="1334" y="0"></use><use xlink:href="#MJMAIN-2207" x="2112" y="0"></use><use xlink:href="#MJMAIN-D7" x="3168" y="0"></use><g transform="translate(4169,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></g></g></g></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-11"><mtext>(11)</mtext></mtd><mtd><mfrac><mrow><mi mathvariant="normal">∂</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">∂</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mi mathvariant="normal">∇</mi><mo>×</mo><mo stretchy="false">(</mo><msup><mi>μ</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>−</mo><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-12"><mtext>(12)</mtext></mtd><mtd><mfrac><mrow><mi mathvariant="normal">∂</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></mrow><mrow><mi mathvariant="normal">∂</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mo>−</mo><mi mathvariant="normal">∇</mi><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mlabeledtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-95">\begin{align}
\dd{(\epsilon\E)}{t} & = \curl(\mu^{-1}\B) - \sigma \E - \J \\
\dd{\B}{t} & = - \curl\E
\end{align}</script></p>
<p>The term <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-96-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.105ex" height="2.684ex" viewBox="0 -1044.1 1337 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C3" x="0" y="0"></use><g transform="translate(572,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-96">\sigma\E</script> arises in the presence of electrically conductive
materials where the electric field induces a current which can be
separated from <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-97-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.788ex" height="2.684ex" viewBox="0 -1044.1 770 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-97">\J</script>. In such cases the total current appearing in
Ampére's Law <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-98-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-ampere"><mtext>(1)</mtext></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.684ex" viewBox="0 -824.7 1279.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><a class="mjx-svg-href" xlink:href="#mjx-eqn-ampere"><rect width="1279.5" height="1002" y="-251" fill="none" stroke="none" pointer-events="all"></rect><g class="MathJax_ref"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-29" x="890" y="0"></use></g></a></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MathJax_ref" href="#mjx-eqn-ampere"><mtext>(1)</mtext></mrow></math></span></span><script type="math/tex" id="MathJax-Element-98">\eqref{ampere}</script> can be expressed as the sum of an applied
current (also labeled as <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-99-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.788ex" height="2.684ex" viewBox="0 -1044.1 770 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-99">\J</script>) and an induced current <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-100-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.105ex" height="2.684ex" viewBox="0 -1044.1 1337 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C3" x="0" y="0"></use><g transform="translate(572,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-100">\sigma\E</script>.</p>
<p>Solving these equations requires initial conditions for both the
electric and magnetic fields <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-101-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.776ex" height="2.684ex" viewBox="0 -1044.1 764.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-101">\E</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-102-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.684ex" viewBox="0 -1044.1 759.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-102">\B</script> as well as boundary
conditions related to the tangential components of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-103-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.776ex" height="2.684ex" viewBox="0 -1044.1 764.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-103">\E</script> or <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-104-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.064ex" height="2.684ex" viewBox="0 -1044.1 888.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-104">\H</script>.
Other formulations are possible such as evolving <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-105-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.064ex" height="2.684ex" viewBox="0 -1044.1 888.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-105">\H</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-106-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.924ex" height="2.684ex" viewBox="0 -1044.1 828.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-44" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="704" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-106">\D</script> or the
potentials <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-107-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C6;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.52ex" height="1.92ex" viewBox="0 -550.5 654.5 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C6" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi></math></span></span><script type="math/tex" id="MathJax-Element-107">\varphi</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-108-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.743ex" height="2.812ex" viewBox="0 -1099 750.5 1210.6" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="749" y="284"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-108">\A</script>. This system of equations can also be
written as a single second order equation involving only <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-109-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.776ex" height="2.684ex" viewBox="0 -1044.1 764.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-109">\E</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-110-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.064ex" height="2.684ex" viewBox="0 -1044.1 888.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="771" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-110">\H</script>,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-111-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C6;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.52ex" height="1.92ex" viewBox="0 -550.5 654.5 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C6" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi></math></span></span><script type="math/tex" id="MathJax-Element-111">\varphi</script>, or <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-112-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.743ex" height="2.812ex" viewBox="0 -1099 750.5 1210.6" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-41" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="749" y="284"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-112">\A</script>. Each of these formulations has a different set of
sources, initial and boundary conditions for which it is well-suited.
The choice we make here is perhaps the most common but it may not be
the most convenient choice for a given application.</p>
<p>These equations can be used to evolve their initial conditions or they
can be driven by either a current source or through time-varying
boundary conditions. It is also possible to combine all three of
these sources in a single simulation.</p>
<h3 id="maxwell-mini-application">Maxwell Mini Application</h3>
<p>The electrodynamics mini application, named <code>maxwell</code> after James Clerk Maxwell
who first formulated the classical theory of electromagnetic radiation, is
intended to demonstrate how to solve transient wave problems in MFEM. Its source
terms and boundary conditions are simple but they should indicate how more
specialized sources or boundary conditions could be implemented.</p>
<p>An example simulation is depicted below (click to animate the wave propagation).</p>
<p><a href="http://mfem.org/img/examples/maxwell.gif"><img alt="" src="../../doc/web/examples/maxwell.png"></a></p>
<p>Time integration is handled by a variable order symplectic time integration
algorithm. This algorithm is designed for systems of equations which are
derived from a Hamiltonian and it helps to ensure energy conservation within
some tolerance. The time step used during integration is automatically chosen
based on the largest stable time step as computed from the largest eigenvalue of
the update equations. This determination involves a user-adjustable factor
which creates a safety margin. By default the actual time step is less than 95%
of the estimate for the largest stable time step.</p>
<p>Note that this application assumes the mesh coordinates are given in meters.
Internally the code assumes time is in seconds but the command line options use
nanoseconds for convenience.</p>
<h4 id="mini-application-features_2">Mini Application Features</h4>
<p><strong>Time Evolution:</strong> The initial and final times for the simulation can be
specified, in nanoseconds, with the <code>-ti</code> and <code>-tf</code> options. Visualization
snapshots of data will be written out after time intervals specified by <code>-ts</code>
which again given in nanoseconds. The order of the time integration can be
specified, from 1 to 4, using the <code>-to</code> option.</p>
<p><strong>Permittivity:</strong> The permittivity, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-113-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03F5;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.944ex" height="1.411ex" viewBox="0 -495.6 406.5 607.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3F5" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math></span></span><script type="math/tex" id="MathJax-Element-113">\epsilon</script>, is assumed to be that of free
space except for an optional sphere of dielectric material which can be
defined by the user. The command line option <code>-ds</code> can be used to set the
parameters for this dielectric sphere. For example, to produce a sphere at the
origin with a radius of 0.5 and a relative permittivity of 3 the user would
specify: <code>-ds '0 0 0 0.5 3'</code>.</p>
<p><strong>Permeability:</strong> The permeability, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-114-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.92ex" viewBox="0 -550.5 603.5 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></span></span><script type="math/tex" id="MathJax-Element-114">\mu</script>, is assumed to be that of free space
except for an optional spherical shell of diamagnetic or paramagnetic material
which can be defined by the user. The command line option <code>-ms</code> can be used to
set the parameters for this shell.</p>
<p>For example, to produce a shell at the origin with inner and outer radii of
0.4 and 0.5 respectively and a relative permeability of 3 the user would
specify: <code>-ms '0 0 0 0.4 0.5 3'</code>.</p>
<p><strong>Conductivity:</strong> The conductivity, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-115-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C3;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.33ex" height="1.411ex" viewBox="0 -495.6 572.5 607.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C3" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi></math></span></span><script type="math/tex" id="MathJax-Element-115">\sigma</script>, is assumed to be zero except for
an optional sphere of conductive material which can be defined by the user.
The command line option <code>-cs</code> can be used to set the parameters for this
conductive sphere. For example, to produce a sphere at the origin with a
radius of 0.5 and a conductivity of 3,000,000 S/m the user would specify: <code>-cs
'0 0 0 0.5 3e6'</code>.</p>
<p><strong>Current Density:</strong> The current density, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-116-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.788ex" height="2.684ex" viewBox="0 -1044.1 770 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-116">\J</script>, is assumed to be zero except for
an optional cylinder of pulsed current which can be defined by the user. The
command line option for this is <code>-dp</code>, short for 'dipole pulse', which
requires two points giving the end points of the cylinder's axis, radius,
amplitude (<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-117-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B1;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.538ex" viewBox="0 -550.5 640.5 662.2" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3B1" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math></span></span><script type="math/tex" id="MathJax-Element-117">\alpha</script>), pulse center (<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-118-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B2;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.332ex" height="2.43ex" viewBox="0 -769.9 573.5 1046.1" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3B2" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math></span></span><script type="math/tex" id="MathJax-Element-118">\beta</script>), and a pulse width (<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-119-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B3;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.262ex" height="1.92ex" viewBox="0 -550.5 543.5 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3B3" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math></span></span><script type="math/tex" id="MathJax-Element-119">\gamma</script>).
The time dependence of this pulse is given by: <span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-120-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-13"><mtext>(13)</mtext></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>a</mi><mo stretchy="false">&#x005E;</mo></mover></mrow><mi>&#x03B1;</mi><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mo stretchy="false">(</mo><mi>t</mi><mo>&#x2212;</mo><mi>&#x03B2;</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mn>2</mn><msup><mi>&#x03B3;</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></msup></mtd></mlabeledtr></mtable></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="43.057ex" height="3.449ex" viewBox="0 -989.3 18538.5 1484.9" role="img" focusable="false" aria-hidden="true" style="vertical-align: -1.151ex; max-width: 338px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(16758,0)"><g id="mjx-eqn-13" transform="translate(0,-167)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-33" x="890" y="0"></use><use xlink:href="#MJMAIN-29" x="1390" y="0"></use></g></g><g transform="translate(4650,0)"><g transform="translate(-14,0)"><g transform="translate(0,-167)"><use xlink:href="#MJMATHI-4A" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="769" y="251"></use><use xlink:href="#MJMAIN-28" x="769" y="0"></use><use xlink:href="#MJMATHI-74" x="1159" y="0"></use><use xlink:href="#MJMAIN-29" x="1520" y="0"></use><use xlink:href="#MJMAIN-3D" x="2188" y="0"></use><g transform="translate(3244,0)"><use xlink:href="#MJMATHI-61" x="0" y="0"></use><use xlink:href="#MJMAIN-5E" x="14" y="-6"></use></g><use xlink:href="#MJMATHI-3B1" x="3774" y="0"></use><g transform="translate(4414,0)"><use xlink:href="#MJMATHI-65" x="0" y="0"></use><g transform="translate(466,412)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-28" x="778" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="1168" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="1529" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-3B2" x="2308" y="0"></use><g transform="translate(2037,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-29" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#MJMAIN-32" x="550" y="675"></use></g><use transform="scale(0.707)" xlink:href="#MJMAIN-2F" x="3724" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-28" x="4225" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-32" x="4614" y="0"></use><g transform="translate(3617,0)"><use transform="scale(0.707)" xlink:href="#MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#MJMAIN-32" x="779" y="513"></use></g><use transform="scale(0.707)" xlink:href="#MJMAIN-29" x="6120" y="0"></use></g></g></g></g></g></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-13"><mtext>(13)</mtext></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>J</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>a</mi><mo stretchy="false">^</mo></mover></mrow><mi>α</mi><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><mi>β</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mn>2</mn><msup><mi>γ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></msup></mtd></mlabeledtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-120">\J(t) = \hat{a} \alpha
e^{-(t-\beta)^2/(2\gamma^2)}</script> Where <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-121-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>a</mi><mo stretchy="false">&#x005E;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.23ex" height="2.048ex" viewBox="0 -769.9 529.5 881.6" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61" x="0" y="0"></use><use xlink:href="#MJMAIN-5E" x="14" y="-6"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>a</mi><mo stretchy="false">^</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-121">\hat{a}</script> is the unit vector along the
cylinder's axis and both <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-122-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B2;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.332ex" height="2.43ex" viewBox="0 -769.9 573.5 1046.1" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3B2" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math></span></span><script type="math/tex" id="MathJax-Element-122">\beta</script> and <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-123-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B3;</mi></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.262ex" height="1.92ex" viewBox="0 -550.5 543.5 826.7" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3B3" x="0" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math></span></span><script type="math/tex" id="MathJax-Element-123">\gamma</script> are specified in nanoseconds.</p>
<p><strong>Dirichlet BC:</strong> Homogeneous Dirichlet boundary conditions, which constrain the
tangential components of <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-124-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">&#x2202;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mrow><mrow><mi mathvariant="normal">&#x2202;</mi><mi>t</mi></mrow></mfrac></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.024ex" height="3.958ex" viewBox="0 -1208.6 1301.9 1704.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -1.151ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="1061" height="60" x="0" y="220"></rect><g transform="translate(59,429)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2202" x="0" y="0"></use><g transform="translate(401,0)"><use transform="scale(0.707)" xlink:href="#MJMATHI-45" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-20D7" x="751" y="282"></use></g></g><g transform="translate(202,-420)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2202" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="567" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">∂</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></mrow><mrow><mi mathvariant="normal">∂</mi><mi>t</mi></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-124">\frac{\partial\E}{\partial t}</script> to be zero, can be
activated on a portion of the boundary by specifying a list of boundary
attributes such as <code>-dbcs '4 8'</code>. For convenience a boundary attribute of
'-1' can be used to specify all boundary surfaces. Non-Homogeneous,
time-dependent Dirichlet boundary conditions are supported by the Maxwell
solver so a user can edit <code>maxwell.cpp</code> and supply their own function if
desired.</p>
<p><strong>Absorbing BC:</strong> A first order Sommerfeld absorbing boundary condition can be
applied to a portion of the boundary using the <code>-abcs</code> option along with a
list of boundary attributes such as <code>-abcs '4 18'</code>. Again, the special
purpose boundary attribute '-1' can be used to specify all boundary surfaces.
This boundary condition depends on a coefficient,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-125-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&#x03B7;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mo>=</mo><msqrt><mi>&#x03F5;</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>&#x03BC;</mi></msqrt></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.437ex" height="3.321ex" viewBox="0 -989.3 5354.8 1430" role="img" focusable="false" aria-hidden="true" style="vertical-align: -1.024ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3B7" x="0" y="0"></use><g transform="translate(505,362)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="778" y="0"></use></g><use xlink:href="#MJMAIN-3D" x="1787" y="0"></use><g transform="translate(2843,0)"><use xlink:href="#MJSZ1-221A" x="0" y="18"></use><rect stroke="none" width="1510" height="60" x="1000" y="809"></rect><g transform="translate(1000,0)"><use xlink:href="#MJMATHI-3F5" x="0" y="0"></use><use xlink:href="#MJMAIN-2F" x="406" y="0"></use><use xlink:href="#MJMATHI-3BC" x="907" y="0"></use></g></g></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>η</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msqrt><mi>ϵ</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>μ</mi></msqrt></math></span></span><script type="math/tex" id="MathJax-Element-125">\eta^{-1}=\sqrt{\epsilon/\mu}</script>, which must be matched to the materials just
inside the boundary. The code assumes that the permittivity and permeability
are those of the vacuum near the surface but, if this is not the case, an
ambitious user can replace <code>etaInvCoef_</code> with a more appropriate function.</p>
<h2 id="transient-magnetics-and-joule-heating">Transient Magnetics and Joule Heating</h2>
<h3 id="joule-mini-application">Joule Mini Application</h3>
<p>The transient magnetics mini application, named <code>joule</code> after the SI unit of energy (and the
scientist James Prescott Joule, who was also a brewer), is intended to demonstrate how to solve
transient implicit diffusion problems. The equations of low-frequency electromagnetics are coupled
with the equations of heat transfer. The coupling is one way, electromagnetics generates Joule
heating, but the heating does not affect the electromagnetics. The thermal problem
is solved using an <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-126-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi></mrow><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.04ex" height="2.684ex" viewBox="0 -824.7 3031 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="888" y="0"></use><g transform="translate(1278,0)"><use xlink:href="#MJMAIN-64" x="0" y="0"></use><use xlink:href="#MJMAIN-69" x="556" y="0"></use><use xlink:href="#MJMAIN-76" x="835" y="0"></use></g><use xlink:href="#MJMAIN-29" x="2641" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi></mrow><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-126">H(\mathrm{div})</script> method, i.e. temperature is discontinuous and the
thermal flux <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-127-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>F</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.771ex" height="2.684ex" viewBox="0 -1044.1 762.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-46" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="761" y="248"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>F</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-127">\F</script> is in <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-128-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi></mrow><mo stretchy="false">)</mo></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.04ex" height="2.684ex" viewBox="0 -824.7 3031 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="888" y="0"></use><g transform="translate(1278,0)"><use xlink:href="#MJMAIN-64" x="0" y="0"></use><use xlink:href="#MJMAIN-69" x="556" y="0"></use><use xlink:href="#MJMAIN-76" x="835" y="0"></use></g><use xlink:href="#MJMAIN-29" x="2641" y="0"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi></mrow><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-128">H(\mathrm{div})</script>.
There are three linear solves per time step:</p>
<ol>
<li>Poisson's equation for the scalar electric potential is solved using the AMG
preconditioner,</li>
<li>the electric diffusion equation is solved using the AMS preconditioner, and</li>
<li>the thermal diffusion equation is solved using the ADS preconditioner.</li>
</ol>
<p>Two example meshes are provided, one is a straight circular metal rod in vacuum, the other is a helical
coil in vacuum (the latter is 21MB and can be downloaded from <a href="https://github.com/mfem/data/blob/master/coil.gen">here</a>).
The idea is that a voltage is applied to the ends of the rod/coil, the electric field diffuses
into the metal, the metal is heated by Joule heating, the heat diffuses out.</p>
<p>The equations are:</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-129-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-14"><mtext>(14)</mtext></mtd><mtd><mi mathvariant="normal">&#x2207;</mi><mo>&#x22C5;</mo><mi>&#x03C3;</mi><mi mathvariant="normal">&#x2207;</mi><mi mathvariant="normal">&#x03A6;</mi></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-15"><mtext>(15)</mtext></mtd><mtd><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><msup><mi>&#x03BC;</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>&#x2212;</mo><mi>&#x03C3;</mi><mi mathvariant="normal">&#x2207;</mi><mi mathvariant="normal">&#x03A6;</mi></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-16"><mtext>(16)</mtext></mtd><mtd><mfrac><mrow><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mo>&#x2212;</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x00D7;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-17"><mtext>(17)</mtext></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>F</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mo>&#x2212;</mo><mi>k</mi><mi mathvariant="normal">&#x2207;</mi><mi>T</mi></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-18"><mtext>(18)</mtext></mtd><mtd><mi>c</mi><mfrac><mrow><mi>d</mi><mi>T</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mo>&#x2212;</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x22C5;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>F</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>+</mo><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow><mo>&#x22C5;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></mtd></mlabeledtr></mtable></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="43.057ex" height="21.538ex" viewBox="0 -4883.4 18538.5 9273.3" role="img" focusable="false" aria-hidden="true" style="vertical-align: -10.196ex; max-width: 338px;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(16758,0)"><g id="mjx-eqn-14" transform="translate(0,3996)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-34" x="890" y="0"></use><use xlink:href="#MJMAIN-29" x="1390" y="0"></use></g><g id="mjx-eqn-15" transform="translate(0,2479)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-35" x="890" y="0"></use><use xlink:href="#MJMAIN-29" x="1390" y="0"></use></g><g id="mjx-eqn-16" transform="translate(0,285)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-36" x="890" y="0"></use><use xlink:href="#MJMAIN-29" x="1390" y="0"></use></g><g id="mjx-eqn-17" transform="translate(0,-1676)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-37" x="890" y="0"></use><use xlink:href="#MJMAIN-29" x="1390" y="0"></use></g><g id="mjx-eqn-18" transform="translate(0,-3599)"><use xlink:href="#MJMAIN-28"></use><use xlink:href="#MJMAIN-31" x="389" y="0"></use><use xlink:href="#MJMAIN-38" x="890" y="0"></use><use xlink:href="#MJMAIN-29" x="1390" y="0"></use></g></g><g transform="translate(3045,0)"><g transform="translate(-14,0)"><g transform="translate(0,3996)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-22C5" x="1055" y="0"></use><use xlink:href="#MJMATHI-3C3" x="1556" y="0"></use><use xlink:href="#MJMAIN-2207" x="2128" y="0"></use><use xlink:href="#MJMAIN-3A6" x="2962" y="0"></use></g><g transform="translate(2347,2479)"><use xlink:href="#MJMATHI-3C3" x="0" y="0"></use><g transform="translate(572,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></g><g transform="translate(2041,285)"><g transform="translate(120,0)"><rect stroke="none" width="1403" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use><g transform="translate(523,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></g><g transform="translate(259,-686)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use><use xlink:href="#MJMATHI-74" x="523" y="0"></use></g></g></g><g transform="translate(2922,-1676)"><use xlink:href="#MJMATHI-46" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="761" y="248"></use></g><g transform="translate(1663,-3599)"><use xlink:href="#MJMATHI-63" x="0" y="0"></use><g transform="translate(433,0)"><g transform="translate(120,0)"><rect stroke="none" width="1348" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use><use xlink:href="#MJMATHI-54" x="523" y="0"></use></g><g transform="translate(231,-686)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use><use xlink:href="#MJMATHI-74" x="523" y="0"></use></g></g></g></g></g><g transform="translate(3671,0)"><g transform="translate(0,3996)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><use xlink:href="#MJMAIN-30" x="1334" y="0"></use></g><g transform="translate(0,2479)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><use xlink:href="#MJMAIN-2207" x="1334" y="0"></use><use xlink:href="#MJMAIN-D7" x="2389" y="0"></use><g transform="translate(3390,0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use><g transform="translate(603,412)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="778" y="0"></use></g></g><g transform="translate(4998,0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g><use xlink:href="#MJMAIN-2212" x="5980" y="0"></use><use xlink:href="#MJMATHI-3C3" x="6980" y="0"></use><use xlink:href="#MJMAIN-2207" x="7553" y="0"></use><use xlink:href="#MJMAIN-3A6" x="8386" y="0"></use></g><g transform="translate(0,285)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><use xlink:href="#MJMAIN-2212" x="1334" y="0"></use><use xlink:href="#MJMAIN-2207" x="2112" y="0"></use><use xlink:href="#MJMAIN-D7" x="3168" y="0"></use><g transform="translate(4169,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></g><g transform="translate(0,-1676)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><use xlink:href="#MJMAIN-2212" x="1334" y="0"></use><use xlink:href="#MJMATHI-6B" x="2112" y="0"></use><use xlink:href="#MJMAIN-2207" x="2634" y="0"></use><use xlink:href="#MJMATHI-54" x="3467" y="0"></use></g><g transform="translate(0,-3599)"><use xlink:href="#MJMAIN-3D" x="277" y="0"></use><use xlink:href="#MJMAIN-2212" x="1334" y="0"></use><use xlink:href="#MJMAIN-2207" x="2112" y="0"></use><use xlink:href="#MJMAIN-22C5" x="3168" y="0"></use><g transform="translate(3669,0)"><use xlink:href="#MJMATHI-46" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="761" y="248"></use></g><use xlink:href="#MJMAIN-2B" x="4653" y="0"></use><use xlink:href="#MJMATHI-3C3" x="5654" y="0"></use><g transform="translate(6226,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g><use xlink:href="#MJMAIN-22C5" x="7213" y="0"></use><g transform="translate(7714,0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="717" y="248"></use></g></g></g></g></g></svg><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mlabeledtr><mtd id="mjx-eqn-14"><mtext>(14)</mtext></mtd><mtd><mi mathvariant="normal">∇</mi><mo>⋅</mo><mi>σ</mi><mi mathvariant="normal">∇</mi><mi mathvariant="normal">Φ</mi></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-15"><mtext>(15)</mtext></mtd><mtd><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mi mathvariant="normal">∇</mi><mo>×</mo><msup><mi>μ</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mi>σ</mi><mi mathvariant="normal">∇</mi><mi mathvariant="normal">Φ</mi></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-16"><mtext>(16)</mtext></mtd><mtd><mfrac><mrow><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mo>−</mo><mi mathvariant="normal">∇</mi><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-17"><mtext>(17)</mtext></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>F</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mo>−</mo><mi>k</mi><mi mathvariant="normal">∇</mi><mi>T</mi></mtd></mlabeledtr><mlabeledtr><mtd id="mjx-eqn-18"><mtext>(18)</mtext></mtd><mtd><mi>c</mi><mfrac><mrow><mi>d</mi><mi>T</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mo>−</mo><mi mathvariant="normal">∇</mi><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>F</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mlabeledtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-129">\begin{align}
\div\sigma\grad\Phi &= 0 \\
\sigma \E &= \curl\mu^{-1} \B - \sigma \grad \Phi \\
\frac{d \B}{d t} &= - \curl \E \\
\F &= -k \grad T \\
c \frac{d T}{d t} &= - \div \F + \sigma \E \cdot \E
\end{align}</script></p>
<p>The equations are integrated in time using implicit time integration, either midpoint or
higher order SDIRK.</p>
<p>Since there are three solves, three sets of boundary conditions must be specified. The
essential BC's are the scalar potential, the electric field, and the thermal flux. These are not
set via command line arguments, you have to edit the code to change these. To change these,
search the code for <code>ess_bdr</code></p>
<p>There are conducting and non-conducting material regions, and the mesh must have integer attributes
to specify these regions. To change these, search the code for <code>std::map<int, double></code> this maps the
integer attribute to the floating-point material value.</p>
<p>Note that this application assumes the mesh coordinates are given in meters.</p>
<p><a href="../../doc/web/examples/joule-screenshot.png"><img alt="" src="../../doc/web/examples/joule.png"></a></p>
<p>The above picture shows Joule heating of a cylinder using the mesh <code>cylinder-hex.mesh</code>. The cylinder is
surrounded by vacuum. The black arrows show the magnetic field <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-130-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="2.684ex" viewBox="0 -1044.1 759.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-42" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="698" y="251"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-130">\B</script>, the magenta arrows show the heat
flux <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax_SVG" id="MathJax-Element-131-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>F</mi><mo stretchy="false">&#x2192;</mo></mover></mrow></math>" role="presentation" style="font-size: 100%; display: inline-block; position: relative;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.771ex" height="2.684ex" viewBox="0 -1044.1 762.5 1155.8" role="img" focusable="false" aria-hidden="true" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-46" x="0" y="0"></use><use xlink:href="#MJMAIN-20D7" x="761" y="248"></use></g></svg><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>F</mi><mo stretchy="false">→</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-131">\F</script>, and the pseudocolor in the center of the cylinder shows the temperature.</p>
<h4 id="mini-application-features_3">Mini Application Features</h4>
<p><strong>Boundary Conditions:</strong> Since there are three solves, three sets of boundary conditions must be specified. The
essential BC's are the voltage for the scalar potential, the tangential electric field, and the normal thermal flux.
These are not
set via command line arguments, you have to edit the code to change these. To change these,
search the code for <code>ess_bdr</code>. Note that the essential BC's can be time varying.</p>
<p><strong>Material Properties:</strong> There are conducting and non-conducting material regions, and the mesh must have integer attributes
to specify these regions. To change these, search the code for <code>std::map<int, double></code> this maps the
integer attribute to the floating-point material value.</p>
<script type="text/x-mathjax-config;executed=true">MathJax.Hub.Config({TeX: {equationNumbers: {autoNumber: "all"}}, tex2jax: {inlineMath: [['$','$']]}});</script>
<script type="text/javascript" src=""></script></div>
<div class="col-md-3 hidden-xs hidden-sm"><div class="bs-sidebar hidden-print affix well" role="complementary">
<ul class="nav bs-sidenav">
<li class="main active"><a href="#electromagnetics-mini-applications">Electromagnetics Mini Applications</a></li>
<li><a href="#electromagnetics">Electromagnetics</a></li>
<li><a href="#electrostatics">Electrostatics</a></li>
<li><a href="#magnetostatics">Magnetostatics</a></li>
<li><a href="#transient-full-wave-electromagnetics">Transient Full-Wave Electromagnetics</a></li>
<li><a href="#transient-magnetics-and-joule-heating">Transient Magnetics and Joule Heating</a></li>
</ul>
</div></div>
</div>
<footer class="footer">
<div class="container">
<p class="text-muted alignright">LLNL-WEB-676715</p>
<p class="text-muted alignleft">
Developed by the <a href="/about/">MFEM team</a> at
<a href="http://computation.llnl.gov/casc/">CASC</a>,
<a href="https://www.llnl.gov/">LLNL</a></p>
</div>
</footer>
<script src="../../doc/web/jquery-1.10.2.min.js"></script>
<script src="../../doc/web/bootstrap-3.0.3.min.js"></script>
<script src="../../doc/web/highlight.pack.js"></script>
<script src="../../doc/web/base.js"></script>
<script src="../../doc/web/retina.min.js"></script>
<script src="../search/main.js"></script>
</body></html>
