1624:
1484:
1616:
2161:
2025:
1343:
76:
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.
2256:
2031:
1892:
470:
2503:
2392:
2976:
1634:
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
2600:
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."
1874:
1655:
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.
1771:
1479:{\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}}\,,}
1329:
2582:
680:
42:, 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.
1220:
1650:
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
374:
2172:
518:
2156:{\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}}}}
108:(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 This amalgamation, which Maxwell himself had actually originally made at equation (112) in "On Physical Lines of Force", is the one that modifies Ampère's Circuital Law to include Maxwell's
262:
569:
312:
2020:{\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}}}}
1664:
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
394:
1147:
614:
1808:
1702:
196:
2400:
2289:
785:
150:
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.
2745:
219:
1816:
2281:
907:
875:
813:
744:
712:
2842:
1005:
971:
1710:
941:
843:
68:
72:
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,
2636:
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.
85:
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
1274:
2514:
2886:
1233:) 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 "
630:
2654:
1603:
1178:
2961:
2251:{\displaystyle \nabla \times \left(\nabla \times \mathbf {V} \right)=\nabla \left(\nabla \cdot \mathbf {V} \right)-\nabla ^{2}\mathbf {V} }
89:, until this term became applied instead to a vectorized set of four equations selected in 1884, which had all appeared in his 1861 paper "
328:
1245:
483:
1588:. However, if we trace the signs through the previous two triplets of equations, we see that what seem to be the components of
2866:
2628:
the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good
224:
534:
278:
2966:
1646:
He obtained a wave equation with a speed in close agreement to experimental determinations of the speed of light. He commented,
1043:
465:{\displaystyle \mathbf {f} =\mu (\mathbf {v} \times \mathbf {H} )-{\frac {\partial \mathbf {A} }{\partial t}}-\nabla \phi }
1640:
1500:
1489:
1106:
385:
2608:
1056:
1091:
319:
105:
1116:
816:
127:
585:
104:). Another of Heaviside's four equations is an amalgamation of Maxwell's law of total currents (equation "A") with
90:
2951:
2946:
2498:{\displaystyle {\partial ^{2}\mathbf {H} \over \partial t^{2}}\ -\ c^{2}\cdot \nabla ^{2}\mathbf {H} \ \ =\ \ 0}
2387:{\displaystyle {\partial ^{2}\mathbf {E} \over \partial t^{2}}\ -\ c^{2}\cdot \nabla ^{2}\mathbf {E} \ \ =\ \ 0}
1779:
1673:
1159:
878:
269:
2894:
161:
96:
Heaviside's versions of
Maxwell's equations are distinct by virtue of the fact that they are written in modern
17:
2641:
753:
1496:
may be described, from left to right, as the motional term, the transformer term, and the conservative term.
2971:
2956:
974:
2621:
2617:
1234:
86:
2818:
1519:
63:
2918:
2765:
1869:{\displaystyle \nabla \times \mathbf {H} =\varepsilon _{o}{\frac {\partial \mathbf {E} }{\partial t}}}
66:, the Society's physical sciences secretary, on 23 March 1865. It was approved for publication in the
1636:
788:
621:
109:
201:
2786:
2684:
1507:
of the medium, and accordingly drops the cross-product term. But he still works from equation
1225:
This is simply the
Lorentz force law on a per-unit-charge basis â although Maxwell's equation
1032:
914:
35:
2264:
890:
858:
796:
727:
695:
2716:
1883:
1628:
1076:
944:
1623:
1230:
54:
on 8 December 1864, having been sent by Maxwell to the society on 27 October. It then underwent
1766:{\displaystyle \nabla \times \mathbf {E} =-\mu _{o}{\frac {\partial \mathbf {H} }{\partial t}}}
2862:
990:
956:
2700:
747:
198:
139:
121:
39:
926:
828:
2708:
2604:
2597:
97:
1324:{\displaystyle \mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}-\nabla \phi \,}
1150:
715:
576:
101:
2688:
2577:{\displaystyle c={1 \over {\sqrt {\mu _{o}\varepsilon _{o}}}}=2.99792458\times 10^{8}}
2940:
2720:
1239:
1102:
381:
51:
1023:
Maxwell did not consider completely general materials; his initial formulation used
2660:
1615:
1036:
1237:". The cross-product term in the Lorentz force law is the source of the so-called
525:
675:{\displaystyle \nabla \cdot \mathbf {J} =-{\frac {\partial \rho }{\partial t}}\,}
1250:
1050:
982:
846:
59:
55:
1574:, viewed in isolation as printed in the 1864 paper, at first seems to say that
1504:
1072:
2977:
Works originally published in Philosophical Transactions of the Royal Society
1511:, in contrast to modern textbooks which tend to work from Faraday's law (see
1067:) is not included in the above list, but follows directly from equation
913:" by Maxwell, not to be confused with the scalar quantity that is now called
2712:
1253:â we can drop the cross-product term, and the force per unit charge (called
1028:
2704:
1215:{\displaystyle \mathbf {f} =\mathbf {E} +\mathbf {v} \times \mathbf {B} \,}
50:
Following standard procedure for the time, the paper was first read to the
1335:
138:
below, each of which represents a group of three original equations in
1024:
100:. They actually only contain one of the original eightâequation "G" (
369:{\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{\rm {tot}}}
2726:(Paper read at a meeting of the Royal Society on 8 December 1864).
1622:
1614:
1249:). Where there is no motion through the magnetic field â e.g., in
513:{\displaystyle \mathbf {f} ={\frac {1}{\varepsilon }}\mathbf {D} }
2283:
is any vector function of space, we recover the wave equations
142:. The 19th and 20th of Maxwell's component equations appear as
1009:
257:{\displaystyle +\,{\frac {\partial \mathbf {D} }{\partial t}}}
564:{\displaystyle \mathbf {f} ={\frac {1}{\sigma }}\mathbf {J} }
2596:
Of this paper and Maxwell's related works, fellow physicist
1529:
are now usually written in the rest frame of the medium as
307:{\displaystyle \mu \mathbf {H} =\nabla \times \mathbf {A} }
2642:
extending general relativity into five physical dimensions
1607:
is different, and avoids the misleading first impression.
1492:. Thus the three terms on the right side of equation
2919:"'Look Ma, No Wires': Marconi and the Invention of Radio"
2693:
Philosophical Transactions of the Royal Society of London
2843:
A Dynamical Theory of the Electromagnetic Field/Part VI
2607:
used Maxwell's equations as the starting point for his
126:
Eighteen of Maxwell's twenty original equations can be
2857:
Maxwell, James C.; Torrance, Thomas F. (March 1996).
2517:
2403:
2292:
2267:
2175:
2034:
1895:
1819:
1782:
1713:
1676:
1346:
1277:
1181:
1119:
993:
959:
929:
893:
861:
831:
799:
756:
730:
698:
633:
588:
537:
486:
397:
331:
281:
227:
204:
164:
2576:
2497:
2386:
2275:
2250:
2155:
2019:
1868:
1802:
1765:
1696:
1478:
1323:
1214:
1141:
999:
965:
935:
901:
869:
837:
807:
779:
738:
706:
674:
608:
563:
512:
464:
368:
306:
256:
213:
190:
2689:"A dynamical theory of the electromagnetic field"
977:(Maxwell called the inverse of conductivity the "
1503:, Maxwell considers the situation only from the
1142:{\displaystyle \partial \mathbf {A} /\partial t}
1049:, although he also discussed the possibility of
2859:A Dynamical Theory of the Electromagnetic Field
1648:
609:{\displaystyle \nabla \cdot \mathbf {D} =\rho }
69:Philosophical Transactions of the Royal Society
32:A Dynamical Theory of the Electromagnetic Field
18:A dynamical theory of the electromagnetic field
1437:
1419:
1090:yields the familiar differential form of the
8:
2913:Oxford University Press. ISBN 978-0198505945
2878:The Scientific Papers of James Clerk Maxwell
1512:
2640:Maxwell's equations can also be derived by
1803:{\displaystyle \nabla \cdot \mathbf {H} =0}
1697:{\displaystyle \nabla \cdot \mathbf {E} =0}
62:) on 24 December 1864. It was then sent to
2736:Royal Society archives; register of papers
909:is the force per unit charge (called the "
787:being the total current density including
2923:History of Communications Infrastructures
2911:Electromagnetism from Ampère to Einstein.
2787:"On the presentation of Maxwell's theory"
2568:
2545:
2535:
2529:
2524:
2516:
2472:
2466:
2453:
2431:
2417:
2411:
2404:
2402:
2361:
2355:
2342:
2320:
2306:
2300:
2293:
2291:
2268:
2266:
2243:
2237:
2220:
2193:
2174:
2144:
2130:
2124:
2117:
2111:
2101:
2086:
2065:
2059:
2047:
2033:
2008:
1994:
1988:
1981:
1975:
1965:
1950:
1929:
1923:
1908:
1894:
1850:
1844:
1838:
1826:
1818:
1789:
1781:
1747:
1741:
1735:
1720:
1712:
1683:
1675:
1472:
1456:
1450:
1446:
1442:
1436:
1435:
1430:
1418:
1417:
1402:
1398:
1394:
1378:
1372:
1362:
1358:
1353:
1345:
1320:
1295:
1289:
1278:
1276:
1211:
1206:
1198:
1190:
1182:
1180:
1128:
1123:
1118:
992:
958:
928:
894:
892:
862:
860:
830:
800:
798:
764:
763:
758:
755:
731:
729:
699:
697:
671:
651:
640:
632:
595:
587:
556:
546:
538:
536:
505:
495:
487:
485:
437:
431:
420:
412:
398:
396:
353:
352:
347:
338:
330:
299:
285:
280:
238:
232:
231:
226:
206:
205:
203:
191:{\displaystyle \mathbf {J} _{\rm {tot}}=}
172:
171:
166:
163:
1334:Taking curls, noting that the curl of a
780:{\displaystyle \mathbf {J} _{\rm {tot}}}
2891:James Clerk Maxwell â The Great Unknown
2808:A Treatise on Electricity and Magnetism
2679:
2677:
2675:
2671:
2655:A Treatise on Electricity and Magnetism
1601:. The notation used in Maxwell's later
58:, being sent to William Thomson (later
2759:
2757:
2755:
2753:
2588:is the speed of light in free space.
1604:Treatise on Electricity and Magnetism
476:(E) The electric elasticity equation
7:
2917:Katz, Randy H. (February 22, 1997).
2613:The Electrodynamics of Moving Bodies
1611:Maxwell â electromagnetic light wave
1594:are in fact the components of
1246:Moving magnet and conductor problem
27:1865 physics paper by James Maxwell
2463:
2424:
2408:
2352:
2313:
2297:
2234:
2214:
2206:
2187:
2176:
2137:
2121:
2080:
2071:
2067:
2041:
2035:
2001:
1985:
1944:
1935:
1931:
1902:
1896:
1857:
1847:
1820:
1783:
1754:
1744:
1714:
1677:
1490:differential form of Faraday's law
1463:
1453:
1424:
1408:
1404:
1385:
1375:
1366:
1347:
1314:
1302:
1292:
1265:, so that Maxwell's equation
1133:
1120:
994:
771:
768:
765:
662:
654:
634:
589:
456:
444:
434:
360:
357:
354:
332:
293:
245:
235:
179:
176:
173:
80:
25:
1243:in electric generators (see also
2473:
2418:
2362:
2307:
2269:
2244:
2221:
2194:
2131:
2087:
2048:
1995:
1951:
1909:
1886:of the curl equations we obtain
1851:
1827:
1790:
1748:
1721:
1684:
1619:Father of Electromagnetic Theory
1457:
1431:
1379:
1354:
1296:
1279:
1259:) reduces to the electric field
1207:
1199:
1191:
1183:
1124:
895:
863:
801:
759:
732:
700:
641:
596:
557:
539:
506:
488:
438:
421:
413:
399:
348:
339:
300:
286:
239:
207:
167:
2880:. Vol. 1. New York: Dover.
2810:. Oxford: Clarendon Press. Vol.
2785:Cf. Tai, Chen-To (1972),
2166:If we note the vector identity
1075:(because the divergence of the
2861:. Eugene, OR: Wipf and Stock.
425:
409:
214:{\displaystyle \,\mathbf {J} }
154:(A) The law of total currents
1:
2806:Maxwell, James Clerk (1873).
2764:Maxwell, James Clerk (1861).
1641:electromagnetic wave equation
1501:electromagnetic wave equation
1105:and the differential form of
2962:Works by James Clerk Maxwell
2766:"On physical lines of force"
2609:special theory of relativity
2276:{\displaystyle \mathbf {V} }
1229:first appeared at equation (
947:(which Maxwell also called "
918:
902:{\displaystyle \mathbf {f} }
870:{\displaystyle \mathbf {A} }
851:quantity of free electricity
808:{\displaystyle \mathbf {D} }
739:{\displaystyle \mathbf {J} }
718:, which Maxwell called the "
707:{\displaystyle \mathbf {H} }
130:into six equations, labeled
81:Maxwell's original equations
2887:"The electromagnetic field"
2885:Johnson, Kevin (May 2002).
1627:A postcard from Maxwell to
2993:
2909:Darrigol, Olivier (2000).
1107:Faraday's law of induction
981:", what is now called the
386:Faraday's law of induction
119:
91:On Physical Lines of Force
74:Philosophical Transactions
2615:, one of Einstein's 1905
1057:Gauss's law for magnetism
115:
2797: (8): 936â45.
1101:implicitly contains the
2791:Proceedings of the IEEE
1660:Modern equation methods
1007:is the vector operator
1000:{\displaystyle \nabla }
975:electrical conductivity
966:{\displaystyle \sigma }
268:(B) Definition of the
2773:Philosophical Magazine
2705:10.1098/rstl.1865.0008
2578:
2499:
2388:
2277:
2252:
2157:
2021:
1870:
1804:
1767:
1698:
1653:
1631:
1620:
1571:
1526:
1522:
1520:constitutive equations
1508:
1493:
1480:
1325:
1266:
1226:
1216:
1170:
1143:
1098:
1087:
1083:
1068:
1001:
967:
937:
903:
871:
839:
809:
781:
740:
708:
676:
610:
565:
514:
466:
370:
320:Ampère's circuital law
308:
258:
215:
192:
147:
143:
135:
131:
106:Ampère's circuital law
2897:on September 15, 2008
2876:Niven, W. D. (1952).
2579:
2500:
2389:
2278:
2253:
2158:
2022:
1871:
1805:
1768:
1699:
1626:
1618:
1481:
1326:
1217:
1144:
1002:
968:
938:
936:{\displaystyle \phi }
904:
872:
849:density (called the "
840:
838:{\displaystyle \rho }
821:electric displacement
810:
782:
741:
709:
677:
611:
566:
515:
467:
371:
309:
259:
216:
193:
116:Heaviside's equations
64:George Gabriel Stokes
2685:Maxwell, James Clerk
2515:
2401:
2290:
2265:
2173:
2032:
1893:
1817:
1780:
1711:
1674:
1637:displacement current
1344:
1275:
1179:
1117:
991:
957:
927:
891:
859:
829:
797:
789:displacement current
754:
728:
696:
631:
622:continuity of charge
586:
535:
484:
395:
329:
279:
225:
202:
162:
110:displacement current
2967:Maxwell's equations
2624:. In it is stated:
1570:Maxwell's equation
1338:is zero, we obtain
1235:Maxwell's equations
979:specific resistance
915:electromotive force
911:electromotive force
87:Maxwell's equations
36:James Clerk Maxwell
2584:meters per second
2574:
2495:
2384:
2273:
2248:
2153:
2017:
1866:
1800:
1763:
1694:
1632:
1621:
1476:
1321:
1212:
1149:vanishes, and the
1139:
1092:Maxwell-Ampère law
997:
963:
949:electric potential
945:electric potential
933:
899:
879:magnetic potential
867:
835:
817:displacement field
805:
777:
736:
720:magnetic intensity
704:
672:
606:
561:
510:
462:
366:
304:
270:magnetic potential
254:
211:
188:
2789:(Invited Paper),
2592:Legacy and impact
2553:
2551:
2491:
2488:
2482:
2479:
2448:
2442:
2438:
2380:
2377:
2371:
2368:
2337:
2331:
2327:
2151:
2078:
2015:
1942:
1879:
1878:
1864:
1761:
1639:, to derive the
1470:
1415:
1392:
1309:
1103:Lorentz force law
669:
620:(H) Equation of
554:
503:
451:
252:
16:(Redirected from
2984:
2952:Electromagnetism
2947:1860s in science
2933:
2931:
2929:
2906:
2904:
2902:
2893:. Archived from
2881:
2872:
2845:
2840:
2834:
2832:
2828:
2822:
2816:
2813:
2804:
2798:
2783:
2777:
2776:
2770:
2761:
2748:
2746:royalsociety.org
2743:
2737:
2734:
2728:
2724:
2681:
2583:
2581:
2580:
2575:
2573:
2572:
2554:
2552:
2550:
2549:
2540:
2539:
2530:
2525:
2504:
2502:
2501:
2496:
2489:
2486:
2480:
2477:
2476:
2471:
2470:
2458:
2457:
2446:
2440:
2439:
2437:
2436:
2435:
2422:
2421:
2416:
2415:
2405:
2393:
2391:
2390:
2385:
2378:
2375:
2369:
2366:
2365:
2360:
2359:
2347:
2346:
2335:
2329:
2328:
2326:
2325:
2324:
2311:
2310:
2305:
2304:
2294:
2282:
2280:
2279:
2274:
2272:
2257:
2255:
2254:
2249:
2247:
2242:
2241:
2229:
2225:
2224:
2202:
2198:
2197:
2162:
2160:
2159:
2154:
2152:
2150:
2149:
2148:
2135:
2134:
2129:
2128:
2118:
2116:
2115:
2106:
2105:
2090:
2079:
2077:
2066:
2064:
2063:
2051:
2026:
2024:
2023:
2018:
2016:
2014:
2013:
2012:
1999:
1998:
1993:
1992:
1982:
1980:
1979:
1970:
1969:
1954:
1943:
1941:
1930:
1928:
1927:
1912:
1875:
1873:
1872:
1867:
1865:
1863:
1855:
1854:
1845:
1843:
1842:
1830:
1809:
1807:
1806:
1801:
1793:
1772:
1770:
1769:
1764:
1762:
1760:
1752:
1751:
1742:
1740:
1739:
1724:
1703:
1701:
1700:
1695:
1687:
1668:
1667:
1600:
1593:
1587:
1577:
1566:
1559:
1555:
1547:
1540:
1536:
1499:In deriving the
1485:
1483:
1482:
1477:
1471:
1469:
1461:
1460:
1451:
1441:
1440:
1434:
1423:
1422:
1416:
1414:
1403:
1393:
1391:
1383:
1382:
1373:
1357:
1330:
1328:
1327:
1322:
1310:
1308:
1300:
1299:
1290:
1282:
1264:
1258:
1221:
1219:
1218:
1213:
1210:
1202:
1194:
1186:
1168:
1162:and is given by
1157:
1148:
1146:
1145:
1140:
1132:
1127:
1113:magnetic field,
1066:
1006:
1004:
1003:
998:
972:
970:
969:
964:
942:
940:
939:
934:
908:
906:
905:
900:
898:
876:
874:
873:
868:
866:
844:
842:
841:
836:
814:
812:
811:
806:
804:
786:
784:
783:
778:
776:
775:
774:
762:
748:electric current
745:
743:
742:
737:
735:
713:
711:
710:
705:
703:
681:
679:
678:
673:
670:
668:
660:
652:
644:
615:
613:
612:
607:
599:
570:
568:
567:
562:
560:
555:
547:
542:
519:
517:
516:
511:
509:
504:
496:
491:
471:
469:
468:
463:
452:
450:
442:
441:
432:
424:
416:
402:
375:
373:
372:
367:
365:
364:
363:
351:
342:
313:
311:
310:
305:
303:
289:
263:
261:
260:
255:
253:
251:
243:
242:
233:
220:
218:
217:
212:
210:
197:
195:
194:
189:
184:
183:
182:
170:
122:Oliver Heaviside
40:electromagnetism
34:" is a paper by
21:
2992:
2991:
2987:
2986:
2985:
2983:
2982:
2981:
2937:
2936:
2927:
2925:
2916:
2900:
2898:
2884:
2875:
2869:
2856:
2853:
2851:Further reading
2848:
2841:
2837:
2830:
2826:
2820:
2814:
2811:
2805:
2801:
2784:
2780:
2768:
2763:
2762:
2751:
2744:
2740:
2735:
2731:
2683:
2682:
2673:
2669:
2650:
2618:Annus Mirabilis
2611:, presented in
2605:Albert Einstein
2598:Richard Feynman
2594:
2586:
2564:
2541:
2531:
2513:
2512:
2506:
2462:
2449:
2427:
2423:
2407:
2406:
2399:
2398:
2395:
2351:
2338:
2316:
2312:
2296:
2295:
2288:
2287:
2263:
2262:
2259:
2233:
2213:
2209:
2186:
2182:
2171:
2170:
2164:
2140:
2136:
2120:
2119:
2107:
2097:
2070:
2055:
2030:
2029:
2004:
2000:
1984:
1983:
1971:
1961:
1934:
1919:
1891:
1890:
1882:If we take the
1880:
1856:
1846:
1834:
1815:
1814:
1778:
1777:
1753:
1743:
1731:
1709:
1708:
1672:
1671:
1662:
1613:
1595:
1589:
1578:
1575:
1557:
1553:
1549:
1538:
1534:
1530:
1486:
1462:
1452:
1407:
1384:
1374:
1342:
1341:
1332:
1301:
1291:
1273:
1272:
1260:
1254:
1223:
1177:
1176:
1163:
1153:
1115:
1114:
1060:
1021:
1016:
989:
988:
955:
954:
925:
924:
889:
888:
883:angular impulse
857:
856:
827:
826:
795:
794:
757:
752:
751:
726:
725:
694:
693:
684:
661:
653:
629:
628:
617:
584:
583:
572:
533:
532:
521:
482:
481:
473:
443:
433:
393:
392:
377:
346:
327:
326:
315:
277:
276:
265:
244:
234:
223:
222:
200:
199:
165:
160:
159:
124:
118:
98:vector notation
83:
48:
28:
23:
22:
15:
12:
11:
5:
2990:
2988:
2980:
2979:
2974:
2972:1865 documents
2969:
2964:
2959:
2957:Physics papers
2954:
2949:
2939:
2938:
2935:
2934:
2914:
2907:
2882:
2873:
2867:
2852:
2849:
2847:
2846:
2835:
2799:
2778:
2749:
2738:
2729:
2670:
2668:
2665:
2664:
2663:
2658:
2649:
2646:
2638:
2637:
2630:
2629:
2593:
2590:
2571:
2567:
2563:
2560:
2557:
2548:
2544:
2538:
2534:
2528:
2523:
2520:
2510:
2494:
2485:
2475:
2469:
2465:
2461:
2456:
2452:
2445:
2434:
2430:
2426:
2420:
2414:
2410:
2396:
2383:
2374:
2364:
2358:
2354:
2350:
2345:
2341:
2334:
2323:
2319:
2315:
2309:
2303:
2299:
2285:
2271:
2246:
2240:
2236:
2232:
2228:
2223:
2219:
2216:
2212:
2208:
2205:
2201:
2196:
2192:
2189:
2185:
2181:
2178:
2168:
2147:
2143:
2139:
2133:
2127:
2123:
2114:
2110:
2104:
2100:
2096:
2093:
2089:
2085:
2082:
2076:
2073:
2069:
2062:
2058:
2054:
2050:
2046:
2043:
2040:
2037:
2011:
2007:
2003:
1997:
1991:
1987:
1978:
1974:
1968:
1964:
1960:
1957:
1953:
1949:
1946:
1940:
1937:
1933:
1926:
1922:
1918:
1915:
1911:
1907:
1904:
1901:
1898:
1888:
1877:
1876:
1862:
1859:
1853:
1849:
1841:
1837:
1833:
1829:
1825:
1822:
1811:
1810:
1799:
1796:
1792:
1788:
1785:
1774:
1773:
1759:
1756:
1750:
1746:
1738:
1734:
1730:
1727:
1723:
1719:
1716:
1705:
1704:
1693:
1690:
1686:
1682:
1679:
1666:
1661:
1658:
1612:
1609:
1475:
1468:
1465:
1459:
1455:
1449:
1445:
1439:
1433:
1429:
1426:
1421:
1413:
1410:
1406:
1401:
1397:
1390:
1387:
1381:
1377:
1371:
1368:
1365:
1361:
1356:
1352:
1349:
1340:
1319:
1316:
1313:
1307:
1304:
1298:
1294:
1288:
1285:
1281:
1271:
1209:
1205:
1201:
1197:
1193:
1189:
1185:
1175:
1151:electric field
1138:
1135:
1131:
1126:
1122:
1019:Clarifications
1017:
1015:
1014:
996:
986:
962:
952:
932:
922:
897:
886:
885:" by Maxwell).
865:
854:
853:" by Maxwell).
834:
824:
823:" by Maxwell).
803:
792:
773:
770:
767:
761:
750:density (with
734:
723:
716:magnetic field
702:
690:
689:
688:
667:
664:
659:
656:
650:
647:
643:
639:
636:
626:
625:
624:
605:
602:
598:
594:
591:
581:
580:
579:
559:
553:
550:
545:
541:
530:
529:
528:
508:
502:
499:
494:
490:
479:
478:
477:
461:
458:
455:
449:
446:
440:
436:
430:
427:
423:
419:
415:
411:
408:
405:
401:
390:
389:
388:
362:
359:
356:
350:
345:
341:
337:
334:
324:
323:
322:
302:
298:
295:
292:
288:
284:
274:
273:
272:
250:
247:
241:
237:
230:
209:
187:
181:
178:
175:
169:
157:
156:
155:
140:component form
117:
114:
82:
79:
47:
44:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2989:
2978:
2975:
2973:
2970:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2944:
2942:
2924:
2920:
2915:
2912:
2908:
2896:
2892:
2888:
2883:
2879:
2874:
2870:
2868:1-57910-015-5
2864:
2860:
2855:
2854:
2850:
2844:
2839:
2836:
2824:
2809:
2803:
2800:
2796:
2792:
2788:
2782:
2779:
2774:
2767:
2760:
2758:
2756:
2754:
2750:
2747:
2742:
2739:
2733:
2730:
2727:
2722:
2718:
2714:
2710:
2706:
2702:
2698:
2694:
2690:
2686:
2680:
2678:
2676:
2672:
2666:
2662:
2659:
2657:
2656:
2652:
2651:
2647:
2645:
2643:
2635:
2634:
2633:
2627:
2626:
2625:
2623:
2620:
2619:
2614:
2610:
2606:
2602:
2599:
2591:
2589:
2585:
2569:
2565:
2561:
2558:
2555:
2546:
2542:
2536:
2532:
2526:
2521:
2518:
2509:
2505:
2492:
2483:
2467:
2459:
2454:
2450:
2443:
2432:
2428:
2412:
2394:
2381:
2372:
2356:
2348:
2343:
2339:
2332:
2321:
2317:
2301:
2284:
2258:
2238:
2230:
2226:
2217:
2210:
2203:
2199:
2190:
2183:
2179:
2167:
2163:
2145:
2141:
2125:
2112:
2108:
2102:
2098:
2094:
2091:
2083:
2074:
2060:
2056:
2052:
2044:
2038:
2027:
2009:
2005:
1989:
1976:
1972:
1966:
1962:
1958:
1955:
1947:
1938:
1924:
1920:
1916:
1913:
1905:
1899:
1887:
1885:
1860:
1839:
1835:
1831:
1823:
1813:
1812:
1797:
1794:
1786:
1776:
1775:
1757:
1736:
1732:
1728:
1725:
1717:
1707:
1706:
1691:
1688:
1680:
1670:
1669:
1665:
1659:
1657:
1652:
1647:
1644:
1642:
1638:
1630:
1625:
1617:
1610:
1608:
1606:
1605:
1599:
1592:
1585:
1581:
1573:
1568:
1565:
1562:
1552:
1546:
1543:
1533:
1528:
1524:
1521:
1516:
1514:
1510:
1506:
1502:
1497:
1495:
1491:
1488:which is the
1473:
1466:
1447:
1443:
1427:
1411:
1399:
1395:
1388:
1369:
1363:
1359:
1350:
1339:
1337:
1317:
1311:
1305:
1286:
1283:
1270:
1268:
1263:
1257:
1252:
1248:
1247:
1242:
1241:
1236:
1232:
1228:
1203:
1195:
1187:
1174:
1172:
1167:
1161:
1156:
1152:
1136:
1129:
1112:
1108:
1104:
1100:
1095:
1093:
1089:
1085:
1082:Substituting
1080:
1078:
1074:
1070:
1064:
1058:
1054:
1052:
1048:
1045:
1041:
1038:
1034:
1033:nondispersive
1030:
1026:
1020:
1012:
1011:
987:
984:
980:
976:
960:
953:
950:
946:
930:
923:
920:
916:
912:
887:
884:
881:(called the "
880:
855:
852:
848:
832:
825:
822:
819:(called the "
818:
793:
790:
749:
724:
721:
717:
692:
691:
686:
685:
683:
665:
657:
648:
645:
637:
623:
619:
618:
616:
603:
600:
592:
578:
574:
573:
571:
551:
548:
543:
527:
523:
522:
520:
500:
497:
492:
475:
474:
472:
459:
453:
447:
428:
417:
406:
403:
387:
383:
382:Lorentz force
379:
378:
376:
343:
335:
321:
317:
316:
314:
296:
290:
282:
271:
267:
266:
264:
248:
228:
221:
185:
153:
152:
151:
149:
145:
141:
137:
133:
129:
123:
113:
111:
107:
103:
99:
94:
92:
88:
78:
75:
71:
70:
65:
61:
57:
53:
52:Royal Society
45:
43:
41:
37:
33:
19:
2926:. Retrieved
2922:
2910:
2901:September 7,
2899:. Retrieved
2895:the original
2890:
2877:
2858:
2838:
2807:
2802:
2794:
2790:
2781:
2772:
2741:
2732:
2725:
2696:
2692:
2661:Gauge theory
2653:
2639:
2631:
2616:
2612:
2603:
2595:
2587:
2511:
2507:
2397:
2286:
2260:
2169:
2165:
2028:
1889:
1881:
1663:
1654:
1649:
1645:
1633:
1602:
1597:
1590:
1583:
1579:
1569:
1563:
1560:
1550:
1544:
1541:
1531:
1517:
1498:
1487:
1333:
1261:
1255:
1251:transformers
1244:
1240:motional emf
1238:
1224:
1165:
1160:conservative
1154:
1110:
1096:
1081:
1062:
1055:
1046:
1044:permeability
1039:
1037:permittivity
1022:
1018:
1008:
978:
948:
910:
882:
850:
820:
719:
627:
582:
531:
480:
391:
325:
275:
158:
125:
95:
84:
73:
67:
49:
31:
29:
2699:: 459â512.
1582:+ ââ
1269:reduces to
1173:reduces to
1073:divergences
1053:materials.
1051:anisotropic
1035:media with
983:resistivity
847:free charge
577:Gauss's law
102:Gauss's Law
60:Lord Kelvin
56:peer review
46:Publication
2941:Categories
2667:References
2559:2.99792458
1629:Peter Tait
1505:rest frame
1169:, so that
1079:is zero).
1071:by taking
128:vectorized
120:See also:
2721:186207827
2713:25533062M
2562:×
2543:ε
2533:μ
2464:∇
2460:⋅
2444:−
2425:∂
2409:∂
2353:∇
2349:⋅
2333:−
2314:∂
2298:∂
2235:∇
2231:−
2218:⋅
2215:∇
2207:∇
2191:×
2188:∇
2180:×
2177:∇
2138:∂
2122:∂
2109:ε
2099:μ
2095:−
2084:×
2081:∇
2072:∂
2068:∂
2057:ε
2045:×
2042:∇
2039:×
2036:∇
2002:∂
1986:∂
1973:ε
1963:μ
1959:−
1948:×
1945:∇
1936:∂
1932:∂
1921:μ
1917:−
1906:×
1903:∇
1900:×
1897:∇
1858:∂
1848:∂
1836:ε
1824:×
1821:∇
1787:⋅
1784:∇
1755:∂
1745:∂
1733:μ
1729:−
1718:×
1715:∇
1681:⋅
1678:∇
1464:∂
1454:∂
1448:−
1428:×
1425:∇
1409:∂
1405:∂
1400:−
1386:∂
1376:∂
1370:×
1367:∇
1364:−
1351:×
1348:∇
1318:ϕ
1315:∇
1312:−
1303:∂
1293:∂
1287:−
1204:×
1134:∂
1121:∂
1097:Equation
1061:ââ
1029:isotropic
995:∇
961:σ
931:ϕ
833:ρ
663:∂
658:ρ
655:∂
649:−
638:⋅
635:∇
604:ρ
593:⋅
590:∇
552:σ
526:Ohm's law
501:ε
460:ϕ
457:∇
454:−
445:∂
435:∂
429:−
418:×
407:μ
380:(D) The
336:×
333:∇
297:×
294:∇
283:μ
246:∂
236:∂
2687:(1865).
2648:See also
1336:gradient
1158:becomes
1109:. For a
687:Notation
2827:
2821:
2812:
2793:
1576:
1558:
1554:
1539:
1535:
973:is the
943:is the
877:is the
845:is the
815:is the
746:is the
714:is the
2928:Sep 7,
2865:
2719:
2711:
2622:papers
2508:where
2490:
2487:
2481:
2478:
2447:
2441:
2379:
2376:
2370:
2367:
2336:
2330:
2261:where
1111:static
1025:linear
917:; see
2825:, eq.
2769:(PDF)
2717:S2CID
1651:laws.
1513:below
1086:into
919:below
575:(G)
524:(F)
318:(C)
2930:2009
2903:2009
2863:ISBN
2632:and
1884:curl
1548:and
1525:and
1518:The
1077:curl
1042:and
384:and
146:and
2823:233
2701:doi
2697:155
1586:= 0
1572:(G)
1527:(F)
1523:(E)
1515:).
1509:(D)
1494:(D)
1267:(D)
1227:(D)
1171:(D)
1099:(D)
1088:(C)
1084:(A)
1069:(B)
1065:= 0
1010:del
951:").
148:(H)
144:(G)
136:(F)
134:to
132:(A)
93:".
38:on
2943::
2921:.
2889:.
2833:).
2819:p.
2817:,
2815:II
2795:60
2771:.
2752:^
2715:.
2709:OL
2707:.
2695:.
2691:.
2674:^
2644:.
2566:10
1643:.
1567:.
1231:77
1164:ââ
1094:.
1031:,
1027:,
985:).
921:).
791:).
722:".
682:.
112:.
2932:.
2905:.
2871:.
2831:J
2829:(
2775:.
2723:.
2703::
2570:8
2556:=
2547:o
2537:o
2527:1
2522:=
2519:c
2493:0
2484:=
2474:H
2468:2
2455:2
2451:c
2433:2
2429:t
2419:H
2413:2
2382:0
2373:=
2363:E
2357:2
2344:2
2340:c
2322:2
2318:t
2308:E
2302:2
2270:V
2245:V
2239:2
2227:)
2222:V
2211:(
2204:=
2200:)
2195:V
2184:(
2146:2
2142:t
2132:H
2126:2
2113:o
2103:o
2092:=
2088:E
2075:t
2061:o
2053:=
2049:H
2010:2
2006:t
1996:E
1990:2
1977:o
1967:o
1956:=
1952:H
1939:t
1925:o
1914:=
1910:E
1861:t
1852:E
1840:o
1832:=
1828:H
1798:0
1795:=
1791:H
1758:t
1749:H
1737:o
1726:=
1722:E
1692:0
1689:=
1685:E
1598:D
1596:â
1591:D
1584:D
1580:Ď
1564:E
1561:Ď
1556:=
1551:J
1545:E
1542:Ďľ
1537:=
1532:D
1474:,
1467:t
1458:B
1444:=
1438:)
1432:A
1420:(
1412:t
1396:=
1389:t
1380:A
1360:=
1355:E
1331:.
1306:t
1297:A
1284:=
1280:E
1262:E
1256:f
1222:.
1208:B
1200:v
1196:+
1192:E
1188:=
1184:f
1166:Ď
1155:E
1137:t
1130:/
1125:A
1063:B
1059:(
1047:Îź
1040:Ďľ
1013:.
896:f
864:A
802:D
772:t
769:o
766:t
760:J
733:J
701:H
666:t
646:=
642:J
601:=
597:D
558:J
549:1
544:=
540:f
507:D
498:1
493:=
489:f
448:t
439:A
426:)
422:H
414:v
410:(
404:=
400:f
361:t
358:o
355:t
349:J
344:=
340:H
301:A
291:=
287:H
249:t
240:D
229:+
208:J
186:=
180:t
177:o
174:t
168:J
30:"
20:)
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