1613:
1473:
1605:
2150:
2014:
1332:
65:
was only published as a bound volume once a year, and would have been prepared for the society's anniversary day on 30 November (the exact date is not recorded). However, the printer would have prepared and delivered to
Maxwell offprints, for the author to distribute as he wished, soon after 16 June.
2245:
2020:
1881:
459:
2492:
2381:
2965:
1623:
In part VI of "A Dynamical Theory of the
Electromagnetic Field", subtitled "Electromagnetic theory of light", Maxwell uses the correction to Ampère's Circuital Law made in part III of his 1862 paper, "On Physical Lines of Force", which is defined as
2589:
said: "From the long view of this history of mankind â seen from, say, 10,000 years from now â there can be little doubt that the most significant event of the 19th century will be judged as
Maxwell's discovery of the laws of electromagnetism."
1863:
1644:
Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics by a much less cumbersome method which combines the corrected version of Ampère's
Circuital Law with Faraday's law of electromagnetic induction.
1760:
1468:{\displaystyle \nabla \times \mathbf {E} \,=\,-\nabla \times {\frac {\partial \mathbf {A} }{\partial t}}\,=\,-{\frac {\partial }{\partial t}}{\big (}\nabla \times \mathbf {A} {\big )}\,=\,-{\frac {\partial \mathbf {B} }{\partial t}}\,,}
1318:
2571:
669:
31:, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave.
1209:
1639:
The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic
363:
2161:
507:
2145:{\displaystyle \nabla \times \nabla \times \mathbf {H} =\varepsilon _{o}{\frac {\partial }{\partial t}}\nabla \times \mathbf {E} =-\mu _{o}\varepsilon _{o}{\frac {\partial ^{2}\mathbf {H} }{\partial t^{2}}}}
97:(equation "C"). This amalgamation, which Maxwell himself had actually originally made at equation (112) in "On Physical Lines of Force", is the one that modifies Ampère's Circuital Law to include Maxwell's
251:
558:
301:
2009:{\displaystyle \nabla \times \nabla \times \mathbf {E} =-\mu _{o}{\frac {\partial }{\partial t}}\nabla \times \mathbf {H} =-\mu _{o}\varepsilon _{o}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}
1653:
To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of
Maxwell's equations. Using (SI units) in a vacuum, these equations are
383:
1136:
603:
1797:
1691:
185:
2389:
2278:
774:
139:
below, making a total of eight vector equations. These are listed below in
Maxwell's original order, designated by the letters that Maxwell assigned to them in his 1864 paper.
2734:
208:
1805:
2270:
896:
864:
802:
733:
701:
2831:
994:
960:
1699:
930:
832:
57:
61:
on 15 June 1865, by the
Committee of Papers (essentially the society's governing council) and sent to the printer the following day (16 June). During this period,
2625:
Any ray of light moves in the "stationary" system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body.
74:
In part III of the paper, which is entitled "General
Equations of the Electromagnetic Field", Maxwell formulated twenty equations which were to become known as
1263:
2503:
2875:
1222:) in "On Physical Lines of Force" in 1861, 34 years before Lorentz derived his force law, which is now usually presented as a supplement to the four "
619:
2643:
1592:
1167:
2950:
2240:{\displaystyle \nabla \times \left(\nabla \times \mathbf {V} \right)=\nabla \left(\nabla \cdot \mathbf {V} \right)-\nabla ^{2}\mathbf {V} }
78:, until this term became applied instead to a vectorized set of four equations selected in 1884, which had all appeared in his 1861 paper "
317:
1234:
472:
1577:. However, if we trace the signs through the previous two triplets of equations, we see that what seem to be the components of
2855:
2617:
the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good
213:
523:
267:
2955:
1635:
He obtained a wave equation with a speed in close agreement to experimental determinations of the speed of light. He commented,
1032:
454:{\displaystyle \mathbf {f} =\mu (\mathbf {v} \times \mathbf {H} )-{\frac {\partial \mathbf {A} }{\partial t}}-\nabla \phi }
1629:
1489:
1478:
1095:
374:
2597:
1045:
1080:
308:
94:
1105:
805:
116:
574:
93:). Another of Heaviside's four equations is an amalgamation of Maxwell's law of total currents (equation "A") with
79:
2940:
2935:
2487:{\displaystyle {\partial ^{2}\mathbf {H} \over \partial t^{2}}\ -\ c^{2}\cdot \nabla ^{2}\mathbf {H} \ \ =\ \ 0}
2376:{\displaystyle {\partial ^{2}\mathbf {E} \over \partial t^{2}}\ -\ c^{2}\cdot \nabla ^{2}\mathbf {E} \ \ =\ \ 0}
1768:
1662:
1148:
867:
258:
2883:
150:
85:
Heaviside's versions of
Maxwell's equations are distinct by virtue of the fact that they are written in modern
2630:
742:
1485:
may be described, from left to right, as the motional term, the transformer term, and the conservative term.
2960:
2945:
963:
2610:
2606:
1223:
75:
2807:
1508:
52:
2907:
2754:
1858:{\displaystyle \nabla \times \mathbf {H} =\varepsilon _{o}{\frac {\partial \mathbf {E} }{\partial t}}}
55:, the Society's physical sciences secretary, on 23 March 1865. It was approved for publication in the
1625:
777:
610:
98:
190:
2775:
2673:
1496:
of the medium, and accordingly drops the cross-product term. But he still works from equation
1214:
This is simply the
Lorentz force law on a per-unit-charge basis â although Maxwell's equation
1021:
903:
24:
2253:
879:
847:
785:
716:
684:
2705:
1872:
1617:
1065:
933:
1612:
1219:
43:
on 8 December 1864, having been sent by Maxwell to the society on 27 October. It then underwent
1755:{\displaystyle \nabla \times \mathbf {E} =-\mu _{o}{\frac {\partial \mathbf {H} }{\partial t}}}
2851:
979:
945:
2689:
736:
187:
128:
110:
28:
915:
817:
2697:
2593:
2586:
86:
1313:{\displaystyle \mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}-\nabla \phi \,}
1139:
704:
565:
90:
2677:
2566:{\displaystyle c={1 \over {\sqrt {\mu _{o}\varepsilon _{o}}}}=2.99792458\times 10^{8}}
2929:
2709:
1228:
1091:
370:
40:
1012:
Maxwell did not consider completely general materials; his initial formulation used
2649:
1604:
1025:
1226:". The cross-product term in the Lorentz force law is the source of the so-called
514:
664:{\displaystyle \nabla \cdot \mathbf {J} =-{\frac {\partial \rho }{\partial t}}\,}
1239:
1039:
971:
835:
48:
44:
1563:, viewed in isolation as printed in the 1864 paper, at first seems to say that
1493:
1061:
2966:
Works originally published in Philosophical Transactions of the Royal Society
1500:, in contrast to modern textbooks which tend to work from Faraday's law (see
1056:) is not included in the above list, but follows directly from equation
902:" by Maxwell, not to be confused with the scalar quantity that is now called
2701:
1242:â we can drop the cross-product term, and the force per unit charge (called
1017:
2693:
1204:{\displaystyle \mathbf {f} =\mathbf {E} +\mathbf {v} \times \mathbf {B} \,}
39:
Following standard procedure for the time, the paper was first read to the
1324:
127:
below, each of which represents a group of three original equations in
1013:
89:. They actually only contain one of the original eightâequation "G" (
358:{\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{\rm {tot}}}
2715:(Paper read at a meeting of the Royal Society on 8 December 1864).
1611:
1603:
1238:). Where there is no motion through the magnetic field â e.g., in
502:{\displaystyle \mathbf {f} ={\frac {1}{\varepsilon }}\mathbf {D} }
2272:
is any vector function of space, we recover the wave equations
131:. The 19th and 20th of Maxwell's component equations appear as
998:
246:{\displaystyle +\,{\frac {\partial \mathbf {D} }{\partial t}}}
553:{\displaystyle \mathbf {f} ={\frac {1}{\sigma }}\mathbf {J} }
2585:
Of this paper and Maxwell's related works, fellow physicist
1518:
are now usually written in the rest frame of the medium as
296:{\displaystyle \mu \mathbf {H} =\nabla \times \mathbf {A} }
2631:
extending general relativity into five physical dimensions
1596:
is different, and avoids the misleading first impression.
1481:. Thus the three terms on the right side of equation
2908:"'Look Ma, No Wires': Marconi and the Invention of Radio"
2682:
Philosophical Transactions of the Royal Society of London
2832:
A Dynamical Theory of the Electromagnetic Field/Part VI
2596:
used Maxwell's equations as the starting point for his
115:
Eighteen of Maxwell's twenty original equations can be
2846:
Maxwell, James C.; Torrance, Thomas F. (March 1996).
2506:
2392:
2281:
2256:
2164:
2023:
1884:
1808:
1771:
1702:
1665:
1335:
1266:
1170:
1108:
982:
948:
918:
882:
850:
820:
788:
745:
719:
687:
622:
577:
526:
475:
386:
320:
270:
216:
193:
153:
2565:
2486:
2375:
2264:
2239:
2144:
2008:
1857:
1791:
1754:
1685:
1467:
1312:
1203:
1130:
988:
954:
924:
890:
858:
826:
796:
768:
727:
695:
663:
597:
552:
501:
453:
357:
295:
245:
202:
179:
2678:"A dynamical theory of the electromagnetic field"
966:(Maxwell called the inverse of conductivity the "
1492:, Maxwell considers the situation only from the
1131:{\displaystyle \partial \mathbf {A} /\partial t}
1038:, although he also discussed the possibility of
2848:A Dynamical Theory of the Electromagnetic Field
1637:
598:{\displaystyle \nabla \cdot \mathbf {D} =\rho }
58:Philosophical Transactions of the Royal Society
21:A Dynamical Theory of the Electromagnetic Field
1426:
1408:
1079:yields the familiar differential form of the
8:
2902:Oxford University Press. ISBN 978-0198505945
2867:The Scientific Papers of James Clerk Maxwell
1501:
2629:Maxwell's equations can also be derived by
1792:{\displaystyle \nabla \cdot \mathbf {H} =0}
1686:{\displaystyle \nabla \cdot \mathbf {E} =0}
51:) on 24 December 1864. It was then sent to
2725:Royal Society archives; register of papers
898:is the force per unit charge (called the "
776:being the total current density including
2912:History of Communications Infrastructures
2900:Electromagnetism from Ampère to Einstein.
2776:"On the presentation of Maxwell's theory"
2557:
2534:
2524:
2518:
2513:
2505:
2461:
2455:
2442:
2420:
2406:
2400:
2393:
2391:
2350:
2344:
2331:
2309:
2295:
2289:
2282:
2280:
2257:
2255:
2232:
2226:
2209:
2182:
2163:
2133:
2119:
2113:
2106:
2100:
2090:
2075:
2054:
2048:
2036:
2022:
1997:
1983:
1977:
1970:
1964:
1954:
1939:
1918:
1912:
1897:
1883:
1839:
1833:
1827:
1815:
1807:
1778:
1770:
1736:
1730:
1724:
1709:
1701:
1672:
1664:
1461:
1445:
1439:
1435:
1431:
1425:
1424:
1419:
1407:
1406:
1391:
1387:
1383:
1367:
1361:
1351:
1347:
1342:
1334:
1309:
1284:
1278:
1267:
1265:
1200:
1195:
1187:
1179:
1171:
1169:
1117:
1112:
1107:
981:
947:
917:
883:
881:
851:
849:
819:
789:
787:
753:
752:
747:
744:
720:
718:
688:
686:
660:
640:
629:
621:
584:
576:
545:
535:
527:
525:
494:
484:
476:
474:
426:
420:
409:
401:
387:
385:
342:
341:
336:
327:
319:
288:
274:
269:
227:
221:
220:
215:
195:
194:
192:
180:{\displaystyle \mathbf {J} _{\rm {tot}}=}
161:
160:
155:
152:
1323:Taking curls, noting that the curl of a
769:{\displaystyle \mathbf {J} _{\rm {tot}}}
2880:James Clerk Maxwell â The Great Unknown
2797:A Treatise on Electricity and Magnetism
2668:
2666:
2664:
2660:
2644:A Treatise on Electricity and Magnetism
1590:. The notation used in Maxwell's later
47:, being sent to William Thomson (later
2748:
2746:
2744:
2742:
2577:is the speed of light in free space.
1593:Treatise on Electricity and Magnetism
465:(E) The electric elasticity equation
7:
2906:Katz, Randy H. (February 22, 1997).
2602:The Electrodynamics of Moving Bodies
1600:Maxwell â electromagnetic light wave
1583:are in fact the components of
1235:Moving magnet and conductor problem
16:1865 physics paper by James Maxwell
2452:
2413:
2397:
2341:
2302:
2286:
2223:
2203:
2195:
2176:
2165:
2126:
2110:
2069:
2060:
2056:
2030:
2024:
1990:
1974:
1933:
1924:
1920:
1891:
1885:
1846:
1836:
1809:
1772:
1743:
1733:
1703:
1666:
1479:differential form of Faraday's law
1452:
1442:
1413:
1397:
1393:
1374:
1364:
1355:
1336:
1303:
1291:
1281:
1254:, so that Maxwell's equation
1122:
1109:
983:
760:
757:
754:
651:
643:
623:
578:
445:
433:
423:
349:
346:
343:
321:
282:
234:
224:
168:
165:
162:
69:
14:
1232:in electric generators (see also
2462:
2407:
2351:
2296:
2258:
2233:
2210:
2183:
2120:
2076:
2037:
1984:
1940:
1898:
1875:of the curl equations we obtain
1840:
1816:
1779:
1737:
1710:
1673:
1608:Father of Electromagnetic Theory
1446:
1420:
1368:
1343:
1285:
1268:
1248:) reduces to the electric field
1196:
1188:
1180:
1172:
1113:
884:
852:
790:
748:
721:
689:
630:
585:
546:
528:
495:
477:
427:
410:
402:
388:
337:
328:
289:
275:
228:
196:
156:
2869:. Vol. 1. New York: Dover.
2799:. Oxford: Clarendon Press. Vol.
2774:Cf. Tai, Chen-To (1972),
2155:If we note the vector identity
1064:(because the divergence of the
2850:. Eugene, OR: Wipf and Stock.
414:
398:
203:{\displaystyle \,\mathbf {J} }
143:(A) The law of total currents
1:
2795:Maxwell, James Clerk (1873).
2753:Maxwell, James Clerk (1861).
1630:electromagnetic wave equation
1490:electromagnetic wave equation
1094:and the differential form of
2951:Works by James Clerk Maxwell
2755:"On physical lines of force"
2598:special theory of relativity
2265:{\displaystyle \mathbf {V} }
1218:first appeared at equation (
936:(which Maxwell also called "
907:
891:{\displaystyle \mathbf {f} }
859:{\displaystyle \mathbf {A} }
840:quantity of free electricity
797:{\displaystyle \mathbf {D} }
728:{\displaystyle \mathbf {J} }
707:, which Maxwell called the "
696:{\displaystyle \mathbf {H} }
119:into six equations, labeled
70:Maxwell's original equations
2876:"The electromagnetic field"
2874:Johnson, Kevin (May 2002).
1616:A postcard from Maxwell to
2982:
2898:Darrigol, Olivier (2000).
1096:Faraday's law of induction
970:", what is now called the
375:Faraday's law of induction
108:
80:On Physical Lines of Force
63:Philosophical Transactions
2604:, one of Einstein's 1905
1046:Gauss's law for magnetism
104:
2786: (8): 936â45.
1090:implicitly contains the
2780:Proceedings of the IEEE
1649:Modern equation methods
996:is the vector operator
989:{\displaystyle \nabla }
964:electrical conductivity
955:{\displaystyle \sigma }
257:(B) Definition of the
2762:Philosophical Magazine
2694:10.1098/rstl.1865.0008
2567:
2488:
2377:
2266:
2241:
2146:
2010:
1859:
1793:
1756:
1687:
1642:
1620:
1609:
1560:
1515:
1511:
1509:constitutive equations
1497:
1482:
1469:
1314:
1255:
1215:
1205:
1159:
1132:
1087:
1076:
1072:
1057:
990:
956:
926:
892:
860:
828:
798:
770:
729:
697:
665:
599:
554:
503:
455:
359:
309:Ampère's circuital law
297:
247:
204:
181:
136:
132:
124:
120:
95:Ampère's circuital law
2886:on September 15, 2008
2865:Niven, W. D. (1952).
2568:
2489:
2378:
2267:
2242:
2147:
2011:
1860:
1794:
1757:
1688:
1615:
1607:
1470:
1315:
1206:
1133:
991:
957:
927:
925:{\displaystyle \phi }
893:
861:
838:density (called the "
829:
827:{\displaystyle \rho }
810:electric displacement
799:
771:
730:
698:
666:
600:
555:
504:
456:
360:
298:
248:
205:
182:
105:Heaviside's equations
53:George Gabriel Stokes
2674:Maxwell, James Clerk
2504:
2390:
2279:
2254:
2162:
2021:
1882:
1806:
1769:
1700:
1663:
1626:displacement current
1333:
1264:
1168:
1106:
980:
946:
916:
880:
848:
818:
786:
778:displacement current
743:
717:
685:
620:
611:continuity of charge
575:
524:
473:
384:
318:
268:
214:
191:
151:
99:displacement current
2956:Maxwell's equations
2613:. In it is stated:
1559:Maxwell's equation
1327:is zero, we obtain
1224:Maxwell's equations
968:specific resistance
904:electromotive force
900:electromotive force
76:Maxwell's equations
25:James Clerk Maxwell
2573:meters per second
2563:
2484:
2373:
2262:
2237:
2142:
2006:
1855:
1789:
1752:
1683:
1621:
1610:
1465:
1310:
1201:
1138:vanishes, and the
1128:
1081:Maxwell-Ampère law
986:
952:
938:electric potential
934:electric potential
922:
888:
868:magnetic potential
856:
824:
806:displacement field
794:
766:
725:
709:magnetic intensity
693:
661:
595:
550:
499:
451:
355:
293:
259:magnetic potential
243:
200:
177:
2778:(Invited Paper),
2581:Legacy and impact
2542:
2540:
2480:
2477:
2471:
2468:
2437:
2431:
2427:
2369:
2366:
2360:
2357:
2326:
2320:
2316:
2140:
2067:
2004:
1931:
1868:
1867:
1853:
1750:
1628:, to derive the
1459:
1404:
1381:
1298:
1092:Lorentz force law
658:
609:(H) Equation of
543:
492:
440:
241:
2973:
2941:Electromagnetism
2936:1860s in science
2922:
2920:
2918:
2895:
2893:
2891:
2882:. Archived from
2870:
2861:
2834:
2829:
2823:
2821:
2817:
2811:
2805:
2802:
2793:
2787:
2772:
2766:
2765:
2759:
2750:
2737:
2735:royalsociety.org
2732:
2726:
2723:
2717:
2713:
2670:
2572:
2570:
2569:
2564:
2562:
2561:
2543:
2541:
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2538:
2529:
2528:
2519:
2514:
2493:
2491:
2490:
2485:
2478:
2475:
2469:
2466:
2465:
2460:
2459:
2447:
2446:
2435:
2429:
2428:
2426:
2425:
2424:
2411:
2410:
2405:
2404:
2394:
2382:
2380:
2379:
2374:
2367:
2364:
2358:
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2349:
2348:
2336:
2335:
2324:
2318:
2317:
2315:
2314:
2313:
2300:
2299:
2294:
2293:
2283:
2271:
2269:
2268:
2263:
2261:
2246:
2244:
2243:
2238:
2236:
2231:
2230:
2218:
2214:
2213:
2191:
2187:
2186:
2151:
2149:
2148:
2143:
2141:
2139:
2138:
2137:
2124:
2123:
2118:
2117:
2107:
2105:
2104:
2095:
2094:
2079:
2068:
2066:
2055:
2053:
2052:
2040:
2015:
2013:
2012:
2007:
2005:
2003:
2002:
2001:
1988:
1987:
1982:
1981:
1971:
1969:
1968:
1959:
1958:
1943:
1932:
1930:
1919:
1917:
1916:
1901:
1864:
1862:
1861:
1856:
1854:
1852:
1844:
1843:
1834:
1832:
1831:
1819:
1798:
1796:
1795:
1790:
1782:
1761:
1759:
1758:
1753:
1751:
1749:
1741:
1740:
1731:
1729:
1728:
1713:
1692:
1690:
1689:
1684:
1676:
1657:
1656:
1589:
1582:
1576:
1566:
1555:
1548:
1544:
1536:
1529:
1525:
1488:In deriving the
1474:
1472:
1471:
1466:
1460:
1458:
1450:
1449:
1440:
1430:
1429:
1423:
1412:
1411:
1405:
1403:
1392:
1382:
1380:
1372:
1371:
1362:
1346:
1319:
1317:
1316:
1311:
1299:
1297:
1289:
1288:
1279:
1271:
1253:
1247:
1210:
1208:
1207:
1202:
1199:
1191:
1183:
1175:
1157:
1151:and is given by
1146:
1137:
1135:
1134:
1129:
1121:
1116:
1102:magnetic field,
1055:
995:
993:
992:
987:
961:
959:
958:
953:
931:
929:
928:
923:
897:
895:
894:
889:
887:
865:
863:
862:
857:
855:
833:
831:
830:
825:
803:
801:
800:
795:
793:
775:
773:
772:
767:
765:
764:
763:
751:
737:electric current
734:
732:
731:
726:
724:
702:
700:
699:
694:
692:
670:
668:
667:
662:
659:
657:
649:
641:
633:
604:
602:
601:
596:
588:
559:
557:
556:
551:
549:
544:
536:
531:
508:
506:
505:
500:
498:
493:
485:
480:
460:
458:
457:
452:
441:
439:
431:
430:
421:
413:
405:
391:
364:
362:
361:
356:
354:
353:
352:
340:
331:
302:
300:
299:
294:
292:
278:
252:
250:
249:
244:
242:
240:
232:
231:
222:
209:
207:
206:
201:
199:
186:
184:
183:
178:
173:
172:
171:
159:
111:Oliver Heaviside
29:electromagnetism
23:" is a paper by
2981:
2980:
2976:
2975:
2974:
2972:
2971:
2970:
2926:
2925:
2916:
2914:
2905:
2889:
2887:
2873:
2864:
2858:
2845:
2842:
2840:Further reading
2837:
2830:
2826:
2819:
2815:
2809:
2803:
2800:
2794:
2790:
2773:
2769:
2757:
2752:
2751:
2740:
2733:
2729:
2724:
2720:
2672:
2671:
2662:
2658:
2639:
2607:Annus Mirabilis
2600:, presented in
2594:Albert Einstein
2587:Richard Feynman
2583:
2575:
2553:
2530:
2520:
2502:
2501:
2495:
2451:
2438:
2416:
2412:
2396:
2395:
2388:
2387:
2384:
2340:
2327:
2305:
2301:
2285:
2284:
2277:
2276:
2252:
2251:
2248:
2222:
2202:
2198:
2175:
2171:
2160:
2159:
2153:
2129:
2125:
2109:
2108:
2096:
2086:
2059:
2044:
2019:
2018:
1993:
1989:
1973:
1972:
1960:
1950:
1923:
1908:
1880:
1879:
1871:If we take the
1869:
1845:
1835:
1823:
1804:
1803:
1767:
1766:
1742:
1732:
1720:
1698:
1697:
1661:
1660:
1651:
1602:
1584:
1578:
1567:
1564:
1546:
1542:
1538:
1527:
1523:
1519:
1475:
1451:
1441:
1396:
1373:
1363:
1331:
1330:
1321:
1290:
1280:
1262:
1261:
1249:
1243:
1212:
1166:
1165:
1152:
1142:
1104:
1103:
1049:
1010:
1005:
978:
977:
944:
943:
914:
913:
878:
877:
872:angular impulse
846:
845:
816:
815:
784:
783:
746:
741:
740:
715:
714:
683:
682:
673:
650:
642:
618:
617:
606:
573:
572:
561:
522:
521:
510:
471:
470:
462:
432:
422:
382:
381:
366:
335:
316:
315:
304:
266:
265:
254:
233:
223:
212:
211:
189:
188:
154:
149:
148:
113:
107:
87:vector notation
72:
37:
17:
12:
11:
5:
2979:
2977:
2969:
2968:
2963:
2961:1865 documents
2958:
2953:
2948:
2946:Physics papers
2943:
2938:
2928:
2927:
2924:
2923:
2903:
2896:
2871:
2862:
2856:
2841:
2838:
2836:
2835:
2824:
2788:
2767:
2738:
2727:
2718:
2659:
2657:
2654:
2653:
2652:
2647:
2638:
2635:
2627:
2626:
2619:
2618:
2582:
2579:
2560:
2556:
2552:
2549:
2546:
2537:
2533:
2527:
2523:
2517:
2512:
2509:
2499:
2483:
2474:
2464:
2458:
2454:
2450:
2445:
2441:
2434:
2423:
2419:
2415:
2409:
2403:
2399:
2385:
2372:
2363:
2353:
2347:
2343:
2339:
2334:
2330:
2323:
2312:
2308:
2304:
2298:
2292:
2288:
2274:
2260:
2235:
2229:
2225:
2221:
2217:
2212:
2208:
2205:
2201:
2197:
2194:
2190:
2185:
2181:
2178:
2174:
2170:
2167:
2157:
2136:
2132:
2128:
2122:
2116:
2112:
2103:
2099:
2093:
2089:
2085:
2082:
2078:
2074:
2071:
2065:
2062:
2058:
2051:
2047:
2043:
2039:
2035:
2032:
2029:
2026:
2000:
1996:
1992:
1986:
1980:
1976:
1967:
1963:
1957:
1953:
1949:
1946:
1942:
1938:
1935:
1929:
1926:
1922:
1915:
1911:
1907:
1904:
1900:
1896:
1893:
1890:
1887:
1877:
1866:
1865:
1851:
1848:
1842:
1838:
1830:
1826:
1822:
1818:
1814:
1811:
1800:
1799:
1788:
1785:
1781:
1777:
1774:
1763:
1762:
1748:
1745:
1739:
1735:
1727:
1723:
1719:
1716:
1712:
1708:
1705:
1694:
1693:
1682:
1679:
1675:
1671:
1668:
1655:
1650:
1647:
1601:
1598:
1464:
1457:
1454:
1448:
1444:
1438:
1434:
1428:
1422:
1418:
1415:
1410:
1402:
1399:
1395:
1390:
1386:
1379:
1376:
1370:
1366:
1360:
1357:
1354:
1350:
1345:
1341:
1338:
1329:
1308:
1305:
1302:
1296:
1293:
1287:
1283:
1277:
1274:
1270:
1260:
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1164:
1140:electric field
1127:
1124:
1120:
1115:
1111:
1008:Clarifications
1006:
1004:
1003:
985:
975:
951:
941:
921:
911:
886:
875:
874:" by Maxwell).
854:
843:
842:" by Maxwell).
823:
813:
812:" by Maxwell).
792:
781:
762:
759:
756:
750:
739:density (with
723:
712:
705:magnetic field
691:
679:
678:
677:
656:
653:
648:
645:
639:
636:
632:
628:
625:
615:
614:
613:
594:
591:
587:
583:
580:
570:
569:
568:
548:
542:
539:
534:
530:
519:
518:
517:
497:
491:
488:
483:
479:
468:
467:
466:
450:
447:
444:
438:
435:
429:
425:
419:
416:
412:
408:
404:
400:
397:
394:
390:
379:
378:
377:
351:
348:
345:
339:
334:
330:
326:
323:
313:
312:
311:
291:
287:
284:
281:
277:
273:
263:
262:
261:
239:
236:
230:
226:
219:
198:
176:
170:
167:
164:
158:
146:
145:
144:
129:component form
106:
103:
71:
68:
36:
33:
15:
13:
10:
9:
6:
4:
3:
2:
2978:
2967:
2964:
2962:
2959:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2933:
2931:
2913:
2909:
2904:
2901:
2897:
2885:
2881:
2877:
2872:
2868:
2863:
2859:
2857:1-57910-015-5
2853:
2849:
2844:
2843:
2839:
2833:
2828:
2825:
2813:
2798:
2792:
2789:
2785:
2781:
2777:
2771:
2768:
2763:
2756:
2749:
2747:
2745:
2743:
2739:
2736:
2731:
2728:
2722:
2719:
2716:
2711:
2707:
2703:
2699:
2695:
2691:
2687:
2683:
2679:
2675:
2669:
2667:
2665:
2661:
2655:
2651:
2648:
2646:
2645:
2641:
2640:
2636:
2634:
2632:
2624:
2623:
2622:
2616:
2615:
2614:
2612:
2609:
2608:
2603:
2599:
2595:
2591:
2588:
2580:
2578:
2574:
2558:
2554:
2550:
2547:
2544:
2535:
2531:
2525:
2521:
2515:
2510:
2507:
2498:
2494:
2481:
2472:
2456:
2448:
2443:
2439:
2432:
2421:
2417:
2401:
2383:
2370:
2361:
2345:
2337:
2332:
2328:
2321:
2310:
2306:
2290:
2273:
2247:
2227:
2219:
2215:
2206:
2199:
2192:
2188:
2179:
2172:
2168:
2156:
2152:
2134:
2130:
2114:
2101:
2097:
2091:
2087:
2083:
2080:
2072:
2063:
2049:
2045:
2041:
2033:
2027:
2016:
1998:
1994:
1978:
1965:
1961:
1955:
1951:
1947:
1944:
1936:
1927:
1913:
1909:
1905:
1902:
1894:
1888:
1876:
1874:
1849:
1828:
1824:
1820:
1812:
1802:
1801:
1786:
1783:
1775:
1765:
1764:
1746:
1725:
1721:
1717:
1714:
1706:
1696:
1695:
1680:
1677:
1669:
1659:
1658:
1654:
1648:
1646:
1641:
1636:
1633:
1631:
1627:
1619:
1614:
1606:
1599:
1597:
1595:
1594:
1588:
1581:
1574:
1570:
1562:
1557:
1554:
1551:
1541:
1535:
1532:
1522:
1517:
1513:
1510:
1505:
1503:
1499:
1495:
1491:
1486:
1484:
1480:
1477:which is the
1462:
1455:
1436:
1432:
1416:
1400:
1388:
1384:
1377:
1358:
1352:
1348:
1339:
1328:
1326:
1306:
1300:
1294:
1275:
1272:
1259:
1257:
1252:
1246:
1241:
1237:
1236:
1231:
1230:
1225:
1221:
1217:
1192:
1184:
1176:
1163:
1161:
1156:
1150:
1145:
1141:
1125:
1118:
1101:
1097:
1093:
1089:
1084:
1082:
1078:
1074:
1071:Substituting
1069:
1067:
1063:
1059:
1053:
1047:
1043:
1041:
1037:
1034:
1030:
1027:
1023:
1022:nondispersive
1019:
1015:
1009:
1001:
1000:
976:
973:
969:
965:
949:
942:
939:
935:
919:
912:
909:
905:
901:
876:
873:
870:(called the "
869:
844:
841:
837:
821:
814:
811:
808:(called the "
807:
782:
779:
738:
713:
710:
706:
681:
680:
675:
674:
672:
654:
646:
637:
634:
626:
612:
608:
607:
605:
592:
589:
581:
567:
563:
562:
560:
540:
537:
532:
516:
512:
511:
509:
489:
486:
481:
464:
463:
461:
448:
442:
436:
417:
406:
395:
392:
376:
372:
371:Lorentz force
368:
367:
365:
332:
324:
310:
306:
305:
303:
285:
279:
271:
260:
256:
255:
253:
237:
217:
210:
174:
142:
141:
140:
138:
134:
130:
126:
122:
118:
112:
102:
100:
96:
92:
88:
83:
81:
77:
67:
64:
60:
59:
54:
50:
46:
42:
41:Royal Society
34:
32:
30:
26:
22:
2915:. Retrieved
2911:
2899:
2890:September 7,
2888:. Retrieved
2884:the original
2879:
2866:
2847:
2827:
2796:
2791:
2783:
2779:
2770:
2761:
2730:
2721:
2714:
2685:
2681:
2650:Gauge theory
2642:
2628:
2620:
2605:
2601:
2592:
2584:
2576:
2500:
2496:
2386:
2275:
2249:
2158:
2154:
2017:
1878:
1870:
1652:
1643:
1638:
1634:
1622:
1591:
1586:
1579:
1572:
1568:
1558:
1552:
1549:
1539:
1533:
1530:
1520:
1506:
1487:
1476:
1322:
1250:
1244:
1240:transformers
1233:
1229:motional emf
1227:
1213:
1154:
1149:conservative
1143:
1099:
1085:
1070:
1051:
1044:
1035:
1033:permeability
1028:
1026:permittivity
1011:
1007:
997:
967:
937:
899:
871:
839:
809:
708:
616:
571:
520:
469:
380:
314:
264:
147:
114:
84:
73:
62:
56:
38:
20:
18:
2688:: 459â512.
1571:+ ââ
1258:reduces to
1162:reduces to
1062:divergences
1042:materials.
1040:anisotropic
1024:media with
972:resistivity
836:free charge
566:Gauss's law
91:Gauss's Law
49:Lord Kelvin
45:peer review
35:Publication
2930:Categories
2656:References
2548:2.99792458
1618:Peter Tait
1494:rest frame
1158:, so that
1068:is zero).
1060:by taking
117:vectorized
109:See also:
2710:186207827
2702:25533062M
2551:×
2532:ε
2522:μ
2453:∇
2449:⋅
2433:−
2414:∂
2398:∂
2342:∇
2338:⋅
2322:−
2303:∂
2287:∂
2224:∇
2220:−
2207:⋅
2204:∇
2196:∇
2180:×
2177:∇
2169:×
2166:∇
2127:∂
2111:∂
2098:ε
2088:μ
2084:−
2073:×
2070:∇
2061:∂
2057:∂
2046:ε
2034:×
2031:∇
2028:×
2025:∇
1991:∂
1975:∂
1962:ε
1952:μ
1948:−
1937:×
1934:∇
1925:∂
1921:∂
1910:μ
1906:−
1895:×
1892:∇
1889:×
1886:∇
1847:∂
1837:∂
1825:ε
1813:×
1810:∇
1776:⋅
1773:∇
1744:∂
1734:∂
1722:μ
1718:−
1707:×
1704:∇
1670:⋅
1667:∇
1453:∂
1443:∂
1437:−
1417:×
1414:∇
1398:∂
1394:∂
1389:−
1375:∂
1365:∂
1359:×
1356:∇
1353:−
1340:×
1337:∇
1307:ϕ
1304:∇
1301:−
1292:∂
1282:∂
1276:−
1193:×
1123:∂
1110:∂
1086:Equation
1050:ââ
1018:isotropic
984:∇
950:σ
920:ϕ
822:ρ
652:∂
647:ρ
644:∂
638:−
627:⋅
624:∇
593:ρ
582:⋅
579:∇
541:σ
515:Ohm's law
490:ε
449:ϕ
446:∇
443:−
434:∂
424:∂
418:−
407:×
396:μ
369:(D) The
325:×
322:∇
286:×
283:∇
272:μ
235:∂
225:∂
2676:(1865).
2637:See also
1325:gradient
1147:becomes
1098:. For a
676:Notation
2816:
2810:
2801:
2782:
1565:
1547:
1543:
1528:
1524:
962:is the
932:is the
866:is the
834:is the
804:is the
735:is the
703:is the
2917:Sep 7,
2854:
2708:
2700:
2611:papers
2497:where
2479:
2476:
2470:
2467:
2436:
2430:
2368:
2365:
2359:
2356:
2325:
2319:
2250:where
1100:static
1014:linear
906:; see
2814:, eq.
2758:(PDF)
2706:S2CID
1640:laws.
1502:below
1075:into
908:below
564:(G)
513:(F)
307:(C)
2919:2009
2892:2009
2852:ISBN
2621:and
1873:curl
1537:and
1514:and
1507:The
1066:curl
1031:and
373:and
135:and
2812:233
2690:doi
2686:155
1575:= 0
1561:(G)
1516:(F)
1512:(E)
1504:).
1498:(D)
1483:(D)
1256:(D)
1216:(D)
1160:(D)
1088:(D)
1077:(C)
1073:(A)
1058:(B)
1054:= 0
999:del
940:").
137:(H)
133:(G)
125:(F)
123:to
121:(A)
82:".
27:on
2932::
2910:.
2878:.
2822:).
2808:p.
2806:,
2804:II
2784:60
2760:.
2741:^
2704:.
2698:OL
2696:.
2684:.
2680:.
2663:^
2633:.
2555:10
1632:.
1556:.
1220:77
1153:ââ
1083:.
1020:,
1016:,
974:).
910:).
780:).
711:".
671:.
101:.
2921:.
2894:.
2860:.
2820:J
2818:(
2764:.
2712:.
2692::
2559:8
2545:=
2536:o
2526:o
2516:1
2511:=
2508:c
2482:0
2473:=
2463:H
2457:2
2444:2
2440:c
2422:2
2418:t
2408:H
2402:2
2371:0
2362:=
2352:E
2346:2
2333:2
2329:c
2311:2
2307:t
2297:E
2291:2
2259:V
2234:V
2228:2
2216:)
2211:V
2200:(
2193:=
2189:)
2184:V
2173:(
2135:2
2131:t
2121:H
2115:2
2102:o
2092:o
2081:=
2077:E
2064:t
2050:o
2042:=
2038:H
1999:2
1995:t
1985:E
1979:2
1966:o
1956:o
1945:=
1941:H
1928:t
1914:o
1903:=
1899:E
1850:t
1841:E
1829:o
1821:=
1817:H
1787:0
1784:=
1780:H
1747:t
1738:H
1726:o
1715:=
1711:E
1681:0
1678:=
1674:E
1587:D
1585:â
1580:D
1573:D
1569:Ď
1553:E
1550:Ď
1545:=
1540:J
1534:E
1531:Ďľ
1526:=
1521:D
1463:,
1456:t
1447:B
1433:=
1427:)
1421:A
1409:(
1401:t
1385:=
1378:t
1369:A
1349:=
1344:E
1320:.
1295:t
1286:A
1273:=
1269:E
1251:E
1245:f
1211:.
1197:B
1189:v
1185:+
1181:E
1177:=
1173:f
1155:Ď
1144:E
1126:t
1119:/
1114:A
1052:B
1048:(
1036:Îź
1029:Ďľ
1002:.
885:f
853:A
791:D
761:t
758:o
755:t
749:J
722:J
690:H
655:t
635:=
631:J
590:=
586:D
547:J
538:1
533:=
529:f
496:D
487:1
482:=
478:f
437:t
428:A
415:)
411:H
403:v
399:(
393:=
389:f
350:t
347:o
344:t
338:J
333:=
329:H
290:A
280:=
276:H
238:t
229:D
218:+
197:J
175:=
169:t
166:o
163:t
157:J
19:"
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