158:
1159:
45:
476:, each of which is neutral. In the example of an insulating dielectric between metal capacitor plates, the only free charges are on the metal plates and dielectric contains only dipoles. If the dielectric is replaced by a doped semiconductor or an ionised gas, etc, then electrons move relative to the ions, and if the system is finite they both contribute to
779:, and in nonhomogeneous media it is a function of position inside the medium. It may also depend upon the electric field (nonlinear materials) and have a time dependent response. Explicit time dependence can arise if the materials are physically moving or changing in time (e.g. reflections off a moving interface give rise to
1614:
and either the voltage difference between the plates will be smaller by this factor, or the charge must be higher. The partial cancellation of fields in the dielectric allows a larger amount of free charge to dwell on the two plates of the capacitor per unit of potential drop than would be possible
1724:
1045:
1574:
219:. If an electric field is applied to an insulator, then (for instance) the negative charges can move slightly towards the positive side of the field, and the positive charges in the other direction. This leads to an induced dipole which is described as a
1106:
who reformulated the complicated
Maxwell's equations to the modern form. It wasn't until 1884 that Heaviside, concurrently with Willard Gibbs and Heinrich Hertz, grouped the equations together into a distinct set. This group of four equations was
1463:
722:
884:
406:
300:
1236:
553:
1633:
1139:
end on free charges, and there are the same number of uniformly distributed charges of opposite sign on both plates, then the flux lines must all simply traverse the capacitor from one side to the other. In
632:
1404:
1468:
1124:
1361:
1309:
932:
918:. In fact, all physical materials have some material dispersion because they cannot respond instantaneously to applied fields, but for many problems (those concerned with a narrow enough
1135:
where the space between the plates is empty or contains a neutral, insulating medium. In both cases, the free charges are only on the metal capacitor plates. Since the flux lines
1081:
661:
1188:
1612:
1363:
is the free surface charge density on the positive plate. If the space between the capacitor plates is filled with a linear homogeneous isotropic dielectric with permittivity
501:
470:
435:
327:
1111:
as the Hertz–Heaviside equations and the
Maxwell–Hertz equations, and is sometimes still known as the Maxwell–Heaviside equations; hence, it was probably Heaviside who lent
561:, the electric analogue to a bar magnet. There is no free charge in such a material, but the inherent polarization gives rise to an electric field, demonstrating that the
826:
351:
904:
1409:
652:
254:
213:
190:
1127:
A parallel plate capacitor. Using an imaginary box, it is possible to use Gauss's law to explain the relationship between electric displacement and free charge.
565:
field is not determined entirely by the free charge. The electric field is determined by using the above relation along with other boundary conditions on the
516:
437:
is the number of free charges per unit volume. These charges are the ones that have made the volume non-neutral, and they are sometimes referred to as the
1734:
1108:
223:. There can be slightly different movements of the negative electrons and positive nuclei in molecules, or different displacements of the atoms in an
591:
66:
1807:
1193:
787:
medium, as there can be a time delay between the imposition of the electric field and the resulting polarization of the material. In this case,
127:
1246:
is perpendicular to the field, so the integral over this section is zero, as is the integral on the face that is outside the capacitor where
914:, which place limitations upon the form of the frequency dependence. The phenomenon of a frequency-dependent permittivity is an example of
1253:
1366:
231:
and always have a polarization; in others spatially varying strains can break the inversion symmetry and lead to polarization, the
1626:
parallel plate capacitor is much smaller than its lateral dimensions we can approximate it using the infinite case and obtain its
1250:
is zero. The only surface that contributes to the integral is therefore the surface of the box inside the capacitor, and hence
17:
1719:{\displaystyle C={\frac {Q_{\text{free}}}{V}}\approx {\frac {Q_{\text{free}}}{|\mathbf {E} |d}}={\frac {A}{d}}\varepsilon ,}
1040:{\displaystyle (\mathbf {D_{1}} -\mathbf {D_{2}} )\cdot {\hat {\mathbf {n} }}=D_{1,\perp }-D_{2,\perp }=\sigma _{\text{f}}}
1318:
1569:{\displaystyle V=|\mathbf {E} |d={\frac {|\mathbf {D} |d}{\varepsilon }}={\frac {|Q_{\text{free}}|d}{\varepsilon A}}}
337:
is the (macroscopic) density of the permanent and induced electric dipole moments in the material, called the
919:
557:
The effect of this equation can be seen in the case of an object with a "frozen in" polarization like a bar
235:. Other stimuli such as magnetic fields can lead to polarization in some materials, this being called the
1057:
911:
1165:
1099:, specific capacity of electric induction, in a form different from the modern and familiar notations.
1590:
479:
448:
413:
305:
1744:
655:
70:
1754:
1458:{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} =\varepsilon \mathbf {E} }
236:
111:
889:
820:
761:
1739:
717:{\displaystyle \mathbf {D} =\varepsilon _{0}(1+\chi )\mathbf {E} =\varepsilon \mathbf {E} }
637:
566:
338:
220:
119:
60:
33:
8:
1091:
The earliest known use of the term is from the year 1864, in James Clerk
Maxwell's paper
915:
816:
330:
151:
195:
172:
1152:, by integrating over a small rectangular box straddling one plate of the capacitor:
812:
577:
1103:
808:
796:
232:
228:
166:
143:
139:
879:{\displaystyle \mathbf {D} (\omega )=\varepsilon (\omega )\mathbf {E} (\omega ),}
401:{\displaystyle \nabla \cdot \mathbf {D} =\rho -\rho _{\text{b}}=\rho _{\text{f}}}
115:
54:
1749:
1149:
1123:
784:
345:
224:
157:
123:
1801:
780:
295:{\displaystyle \mathbf {D} \equiv \varepsilon _{0}\mathbf {E} +\mathbf {P} ,}
1144:
units, the charge density on the plates is proportional to the value of the
744:
569:
to yield the bound charges, which will, in turn, yield the electric field.
438:
107:
1231:{\displaystyle \mathbf {D} \cdot \mathrm {d} \mathbf {A} =Q_{\text{free}}}
138:
to an electric field, and how shapes can change due to electric fields in
1627:
792:
772:
584:
dielectric with instantaneous response to changes in the electric field,
131:
135:
1132:
907:
581:
548:{\displaystyle \nabla \times \mathbf {D} =\nabla \times \mathbf {P} }
558:
1735:
History of
Maxwell's equations § The term Maxwell's equations
91:
513:
has a curl of zero in electrostatic situations, it follows that
776:
627:{\displaystyle \mathbf {P} =\varepsilon _{0}\chi \mathbf {E} ,}
573:
473:
216:
1399:{\displaystyle \varepsilon =\varepsilon _{0}\varepsilon _{r}}
1118:
1083:
points in the direction from medium 2 to medium 1.
29:
Vector field related to displacement current and flux density
147:
227:. Materials which do not have an inversion center display
906:
is the frequency of the applied field. The constraint of
441:. This equation says, in effect, that the flux lines of
1141:
472:
is the density of all those charges that are part of a
1406:, then there is a polarization induced in the medium,
1169:
783:). A different form of time dependence can arise in a
1636:
1593:
1471:
1412:
1369:
1321:
1256:
1196:
1168:
1148:
field between the plates. This follows directly from
1060:
935:
892:
829:
664:
640:
594:
519:
509:
is not determined exclusively by the free charge. As
482:
451:
416:
354:
308:
257:
198:
175:
161:
Illustration of polarization due to a negative charge
1465:
and so the voltage difference between the plates is
807:. Such a convolution takes on a simpler form in the
445:
must begin and end on the free charges. In contrast
1315:is the surface area of the top face of the box and
1718:
1606:
1568:
1457:
1398:
1356:{\displaystyle Q_{\text{free}}/A=\rho _{\text{f}}}
1355:
1304:{\displaystyle |\mathbf {D} |A=|Q_{\text{free}}|,}
1303:
1230:
1182:
1075:
1039:
898:
878:
716:
646:
626:
547:
495:
464:
429:
400:
321:
294:
207:
184:
1067:
1799:
1789:A Dynamical Theory of the Electromagnetic Field
1093:A Dynamical Theory of the Electromagnetic Field
1054:is the free charge density and the unit normal
333:(also called permittivity of free space), and
130:. It plays a major role in topics such as the
1773:
819:, one obtains the following relation for a
1119:Example: Displacement field in a capacitor
215:are the same. This means that there is no
134:of a material, as well as the response of
767:In linear, homogeneous, isotropic media,
1791:PART V. — THEORY OF CONDENSERS, page 494
1767:
1615:if the plates were separated by vacuum.
1122:
588:depends linearly on the electric field,
156:
32:For broader coverage of this topic, see
14:
1808:Electric and magnetic fields in matter
1800:
634:where the constant of proportionality
1583:Introducing the dielectric increases
1115:the present significance it now has.
1131:Consider an infinite parallel plate
38:
1076:{\displaystyle \mathbf {\hat {n}} }
24:
1206:
815:the relationship and applying the
771:is a constant. However, in linear
534:
520:
355:
169:then the charge at, for instance,
25:
1819:
1183:{\displaystyle \scriptstyle _{A}}
344:The displacement field satisfies
247:The electric displacement field "
1682:
1607:{\displaystyle \varepsilon _{r}}
1508:
1484:
1451:
1440:
1432:
1414:
1263:
1211:
1198:
1157:
1064:
976:
960:
956:
945:
941:
860:
831:
710:
699:
666:
617:
596:
541:
527:
496:{\displaystyle \rho _{\text{f}}}
465:{\displaystyle \rho _{\text{b}}}
430:{\displaystyle \rho _{\text{f}}}
362:
322:{\displaystyle \varepsilon _{0}}
285:
277:
259:
165:In any material, if there is an
43:
1776:Introduction to Electrodynamics
1782:
1687:
1677:
1548:
1533:
1513:
1503:
1489:
1479:
1294:
1279:
1268:
1258:
1095:. Maxwell introduced the term
980:
966:
936:
922:) the frequency-dependence of
870:
864:
856:
850:
841:
835:
695:
683:
13:
1:
1760:
242:
7:
1728:
150:and charge transfer due to
146:as well as the creation of
96:electric displacement field
57:the scope of other articles
10:
1824:
1242:On the sides of the box, d
1086:
126:, combining the two in an
31:
1622:between the plates of a
912:Kramers–Kronig relations
1745:Electric susceptibility
899:{\displaystyle \omega }
803:and the electric field
656:electric susceptibility
1755:Electric dipole moment
1720:
1608:
1570:
1459:
1400:
1357:
1305:
1232:
1184:
1128:
1077:
1041:
900:
880:
718:
658:of the material. Thus
648:
628:
549:
497:
466:
431:
402:
323:
296:
237:magnetoelectric effect
209:
186:
162:
114:. It accounts for the
1721:
1609:
1580:is their separation.
1571:
1460:
1401:
1358:
1306:
1233:
1185:
1126:
1078:
1042:
901:
881:
821:linear time-invariant
762:relative permittivity
719:
649:
647:{\displaystyle \chi }
629:
550:
498:
467:
432:
403:
324:
297:
210:
187:
160:
69:and help introduce a
18:Electric displacement
1778:(3rd 1999 ed.).
1740:Polarization density
1634:
1591:
1469:
1410:
1367:
1319:
1254:
1194:
1166:
1058:
933:
890:
827:
813:Fourier transforming
662:
638:
592:
567:polarization density
517:
480:
449:
414:
352:
339:polarization density
306:
255:
233:flexoelectric effect
196:
173:
61:Displacement current
34:Displacement current
916:material dispersion
817:convolution theorem
331:vacuum permittivity
112:Maxwell's equations
1716:
1604:
1566:
1455:
1396:
1353:
1301:
1228:
1180:
1179:
1129:
1073:
1037:
926:can be neglected.
896:
876:
714:
644:
624:
545:
493:
462:
427:
410:In this equation,
398:
319:
292:
208:{\displaystyle -x}
205:
185:{\displaystyle +x}
182:
163:
104:electric induction
67:discuss this issue
1774:David Griffiths.
1708:
1695:
1672:
1658:
1652:
1564:
1544:
1524:
1350:
1329:
1290:
1225:
1070:
1034:
983:
764:of the material.
490:
459:
424:
395:
382:
348:in a dielectric:
88:
87:
16:(Redirected from
1815:
1792:
1786:
1780:
1779:
1771:
1725:
1723:
1722:
1717:
1709:
1701:
1696:
1694:
1690:
1685:
1680:
1674:
1673:
1670:
1664:
1659:
1654:
1653:
1650:
1644:
1618:If the distance
1613:
1611:
1610:
1605:
1603:
1602:
1575:
1573:
1572:
1567:
1565:
1563:
1555:
1551:
1546:
1545:
1542:
1536:
1530:
1525:
1520:
1516:
1511:
1506:
1500:
1492:
1487:
1482:
1464:
1462:
1461:
1456:
1454:
1443:
1435:
1430:
1429:
1417:
1405:
1403:
1402:
1397:
1395:
1394:
1385:
1384:
1362:
1360:
1359:
1354:
1352:
1351:
1348:
1336:
1331:
1330:
1327:
1310:
1308:
1307:
1302:
1297:
1292:
1291:
1288:
1282:
1271:
1266:
1261:
1238:
1237:
1235:
1234:
1229:
1227:
1226:
1223:
1214:
1209:
1201:
1190:
1189:
1187:
1186:
1181:
1178:
1177:
1161:
1160:
1104:Oliver Heaviside
1082:
1080:
1079:
1074:
1072:
1071:
1063:
1046:
1044:
1043:
1038:
1036:
1035:
1032:
1023:
1022:
1004:
1003:
985:
984:
979:
974:
965:
964:
963:
950:
949:
948:
905:
903:
902:
897:
885:
883:
882:
877:
863:
834:
809:frequency domain
797:impulse response
759:
742:
723:
721:
720:
715:
713:
702:
682:
681:
669:
653:
651:
650:
645:
633:
631:
630:
625:
620:
612:
611:
599:
554:
552:
551:
546:
544:
530:
502:
500:
499:
494:
492:
491:
488:
471:
469:
468:
463:
461:
460:
457:
436:
434:
433:
428:
426:
425:
422:
407:
405:
404:
399:
397:
396:
393:
384:
383:
380:
365:
328:
326:
325:
320:
318:
317:
301:
299:
298:
293:
288:
280:
275:
274:
262:
229:piezoelectricity
214:
212:
211:
206:
191:
189:
188:
183:
167:inversion center
144:flexoelectricity
140:piezoelectricity
110:that appears in
83:
80:
74:
47:
46:
39:
21:
1823:
1822:
1818:
1817:
1816:
1814:
1813:
1812:
1798:
1797:
1796:
1795:
1787:
1783:
1772:
1768:
1763:
1731:
1700:
1686:
1681:
1676:
1675:
1669:
1665:
1663:
1649:
1645:
1643:
1635:
1632:
1631:
1598:
1594:
1592:
1589:
1588:
1556:
1547:
1541:
1537:
1532:
1531:
1529:
1512:
1507:
1502:
1501:
1499:
1488:
1483:
1478:
1470:
1467:
1466:
1450:
1439:
1431:
1425:
1421:
1413:
1411:
1408:
1407:
1390:
1386:
1380:
1376:
1368:
1365:
1364:
1347:
1343:
1332:
1326:
1322:
1320:
1317:
1316:
1293:
1287:
1283:
1278:
1267:
1262:
1257:
1255:
1252:
1251:
1222:
1218:
1210:
1205:
1197:
1195:
1192:
1191:
1173:
1170:
1167:
1164:
1163:
1162:
1158:
1156:
1121:
1109:known variously
1089:
1062:
1061:
1059:
1056:
1055:
1053:
1031:
1027:
1012:
1008:
993:
989:
975:
973:
972:
959:
955:
954:
944:
940:
939:
934:
931:
930:
929:At a boundary,
891:
888:
887:
859:
830:
828:
825:
824:
799:susceptibility
754:
748:
741:
735:
725:
709:
698:
677:
673:
665:
663:
660:
659:
639:
636:
635:
616:
607:
603:
595:
593:
590:
589:
540:
526:
518:
515:
514:
487:
483:
481:
478:
477:
456:
452:
450:
447:
446:
421:
417:
415:
412:
411:
392:
388:
379:
375:
361:
353:
350:
349:
313:
309:
307:
304:
303:
284:
276:
270:
266:
258:
256:
253:
252:
251:" is defined as
245:
197:
194:
193:
174:
171:
170:
128:auxiliary field
122:and that of an
116:electromagnetic
84:
78:
75:
73:to the article.
64:
59:, specifically
48:
44:
37:
30:
23:
22:
15:
12:
11:
5:
1821:
1811:
1810:
1794:
1793:
1781:
1765:
1764:
1762:
1759:
1758:
1757:
1752:
1750:Magnetic field
1747:
1742:
1737:
1730:
1727:
1715:
1712:
1707:
1704:
1699:
1693:
1689:
1684:
1679:
1668:
1662:
1657:
1648:
1642:
1639:
1601:
1597:
1562:
1559:
1554:
1550:
1540:
1535:
1528:
1523:
1519:
1515:
1510:
1505:
1498:
1495:
1491:
1486:
1481:
1477:
1474:
1453:
1449:
1446:
1442:
1438:
1434:
1428:
1424:
1420:
1416:
1393:
1389:
1383:
1379:
1375:
1372:
1346:
1342:
1339:
1335:
1325:
1300:
1296:
1286:
1281:
1277:
1274:
1270:
1265:
1260:
1240:
1239:
1221:
1217:
1213:
1208:
1204:
1200:
1176:
1172:
1120:
1117:
1088:
1085:
1069:
1066:
1051:
1030:
1026:
1021:
1018:
1015:
1011:
1007:
1002:
999:
996:
992:
988:
982:
978:
971:
968:
962:
958:
953:
947:
943:
938:
895:
875:
872:
869:
866:
862:
858:
855:
852:
849:
846:
843:
840:
837:
833:
785:time-invariant
781:Doppler shifts
775:media it is a
752:
739:
733:
712:
708:
705:
701:
697:
694:
691:
688:
685:
680:
676:
672:
668:
654:is called the
643:
623:
619:
615:
610:
606:
602:
598:
543:
539:
536:
533:
529:
525:
522:
503:at the edges.
486:
455:
420:
391:
387:
378:
374:
371:
368:
364:
360:
357:
316:
312:
291:
287:
283:
279:
273:
269:
265:
261:
244:
241:
225:ionic compound
204:
201:
181:
178:
124:electric field
86:
85:
51:
49:
42:
28:
9:
6:
4:
3:
2:
1820:
1809:
1806:
1805:
1803:
1790:
1785:
1777:
1770:
1766:
1756:
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1736:
1733:
1732:
1726:
1713:
1710:
1705:
1702:
1697:
1691:
1666:
1660:
1655:
1646:
1640:
1637:
1629:
1625:
1621:
1616:
1599:
1595:
1586:
1581:
1579:
1560:
1557:
1552:
1538:
1526:
1521:
1517:
1496:
1493:
1475:
1472:
1447:
1444:
1436:
1426:
1422:
1418:
1391:
1387:
1381:
1377:
1373:
1370:
1344:
1340:
1337:
1333:
1323:
1314:
1298:
1284:
1275:
1272:
1249:
1245:
1219:
1215:
1202:
1174:
1171:
1155:
1154:
1153:
1151:
1147:
1143:
1138:
1134:
1125:
1116:
1114:
1110:
1105:
1100:
1098:
1094:
1084:
1050:
1028:
1024:
1019:
1016:
1013:
1009:
1005:
1000:
997:
994:
990:
986:
969:
951:
927:
925:
921:
917:
913:
910:leads to the
909:
893:
873:
867:
853:
847:
844:
838:
822:
818:
814:
810:
806:
802:
798:
794:
790:
786:
782:
778:
774:
770:
765:
763:
758:
751:
746:
738:
732:
728:
706:
703:
692:
689:
686:
678:
674:
670:
657:
641:
621:
613:
608:
604:
600:
587:
583:
579:
575:
570:
568:
564:
560:
555:
537:
531:
523:
512:
508:
504:
484:
475:
453:
444:
440:
418:
408:
389:
385:
376:
372:
369:
366:
358:
347:
342:
340:
336:
332:
314:
310:
289:
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250:
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234:
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137:
133:
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125:
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117:
113:
109:
105:
101:
97:
93:
82:
72:
71:summary style
68:
62:
58:
56:
52:This article
50:
41:
40:
35:
27:
19:
1788:
1784:
1775:
1769:
1623:
1619:
1617:
1587:by a factor
1584:
1582:
1577:
1312:
1247:
1243:
1241:
1145:
1136:
1130:
1112:
1101:
1096:
1092:
1090:
1048:
928:
923:
804:
800:
788:
768:
766:
756:
749:
745:permittivity
736:
730:
726:
585:
571:
562:
556:
510:
506:
505:
442:
439:space charge
409:
343:
334:
248:
246:
221:polarization
164:
120:polarization
108:vector field
103:
99:
98:(denoted by
95:
89:
76:
53:
26:
1628:capacitance
1150:Gauss's law
793:convolution
773:anisotropic
578:homogeneous
346:Gauss's law
136:dielectrics
132:capacitance
118:effects of
1761:References
243:Definition
55:duplicates
1711:ε
1661:≈
1596:ε
1558:ε
1522:ε
1448:ε
1423:ε
1388:ε
1378:ε
1371:ε
1345:ρ
1203:⋅
1133:capacitor
1068:^
1029:σ
1020:⊥
1006:−
1001:⊥
981:^
970:⋅
952:−
920:bandwidth
908:causality
894:ω
868:ω
854:ω
848:ε
839:ω
707:ε
693:χ
675:ε
642:χ
614:χ
605:ε
582:isotropic
538:×
535:∇
524:×
521:∇
485:ρ
454:ρ
419:ρ
390:ρ
377:ρ
373:−
370:ρ
359:⋅
356:∇
311:ε
268:ε
264:≡
200:−
154:strains.
79:July 2023
1802:Category
1729:See also
1047:, where
823:medium:
559:electret
148:voltages
1102:It was
1087:History
795:of the
743:is the
329:is the
152:elastic
92:physics
65:Please
1624:finite
1576:where
1311:where
886:where
777:tensor
755:= 1 +
747:, and
724:where
574:linear
474:dipole
302:where
217:dipole
94:, the
811:: by
791:is a
572:In a
106:is a
102:) or
1671:free
1651:free
1543:free
1328:free
1289:free
1224:free
760:the
192:and
1630:as
142:or
90:In
1804::
1142:SI
729:=
580:,
576:,
341:.
239:.
1714:,
1706:d
1703:A
1698:=
1692:d
1688:|
1683:E
1678:|
1667:Q
1656:V
1647:Q
1641:=
1638:C
1620:d
1600:r
1585:ε
1578:d
1561:A
1553:d
1549:|
1539:Q
1534:|
1527:=
1518:d
1514:|
1509:D
1504:|
1497:=
1494:d
1490:|
1485:E
1480:|
1476:=
1473:V
1452:E
1445:=
1441:P
1437:+
1433:E
1427:0
1419:=
1415:D
1392:r
1382:0
1374:=
1349:f
1341:=
1338:A
1334:/
1324:Q
1313:A
1299:,
1295:|
1285:Q
1280:|
1276:=
1273:A
1269:|
1264:D
1259:|
1248:D
1244:A
1220:Q
1216:=
1212:A
1207:d
1199:D
1175:A
1146:D
1137:D
1113:D
1097:D
1065:n
1052:f
1049:σ
1033:f
1025:=
1017:,
1014:2
1010:D
998:,
995:1
991:D
987:=
977:n
967:)
961:2
957:D
946:1
942:D
937:(
924:ε
874:,
871:)
865:(
861:E
857:)
851:(
845:=
842:)
836:(
832:D
805:E
801:χ
789:P
769:ε
757:χ
753:r
750:ε
740:r
737:ε
734:0
731:ε
727:ε
711:E
704:=
700:E
696:)
690:+
687:1
684:(
679:0
671:=
667:D
622:,
618:E
609:0
601:=
597:P
586:P
563:D
542:P
532:=
528:D
511:E
507:D
489:f
458:b
443:D
423:f
394:f
386:=
381:b
367:=
363:D
335:P
315:0
290:,
286:P
282:+
278:E
272:0
260:D
249:D
203:x
180:x
177:+
100:D
81:)
77:(
63:.
36:.
20:)
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