2872:
2306:
900:
2127:
36:
3137:). However, Jefimenko's equations show an alternative point of view. Jefimenko says, "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents."
1826:
3172:
Maxwell's equations where both sides take place simultaneously. In the typical expressions for
Maxwell's equations there is no doubt that both sides are equal to each other, but as Jefimenko notes, "... since each of these equations connects quantities simultaneous in time, none of these equations can represent a causal relation."
2122:{\displaystyle {\begin{aligned}&\varphi (\mathbf {r} ,t)={\dfrac {1}{4\pi \varepsilon _{0}}}\int {\dfrac {\rho (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}dV',\\&\mathbf {A} (\mathbf {r} ,t)={\dfrac {\mu _{0}}{4\pi }}\int {\dfrac {\mathbf {J} (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}dV',\end{aligned}}}
3171:
Essential features of these equations are easily observed which is that the right hand sides involve "retarded" time which reflects the "causality" of the expressions. In other words, the left side of each equation is actually "caused" by the right side, unlike the normal differential expressions for
2220:
2933:
which is the propagation time of the electric field. Heuristically, this can be regarded as nature "attempting" to forecast what the present field would be by linear extrapolation to the present time. The last term, proportional to the second derivative of the
1361:
1680:
3156:, however, clarifies that "the earliest explicit statement of which I am aware was by Oleg Jefimenko, in 1966" and characterizes equations in Panofsky and Phillips's textbook as only "closely related expressions". According to
2645:
2140:
3132:
There is a widespread interpretation of
Maxwell's equations indicating that spatially varying electric and magnetic fields can cause each other to change in time, thus giving rise to a propagating electromagnetic wave
1767:
2586:
2279:
1831:
2798:
2931:
948:
3419:, Addison-Wesley (2nd. ed - 1962), Section 14.3. The electric field is written in a slightly different - but completely equivalent - form. Reprint: Dover Publications (2005),
2736:
3107:
3007:
2899:
2689:
2667:
1365:
2967:
2866:
3067:
2823:
2590:
3034:
2760:
2709:
3141:
593:
2347:
2875:
Illustration of the retarded charge position for a particle moving in one spatial dimension: the observer sees the particle where it was, not where it is.
566:
1699:
2969:, is sensitive to charge motion perpendicular to the line of sight. It can be shown that the electric field generated by this term is proportional to
3157:
578:
2225:
2901:
represents the
Coulomb's law for the static electric field. The second term is the time derivative of the first Coulombic term multiplied by
2215:{\displaystyle \mathbf {E} =-\nabla \varphi -{\dfrac {\partial \mathbf {A} }{\partial t}}\,,\quad \mathbf {B} =\nabla \times \mathbf {A} }
829:
598:
608:
3486:
433:
3424:
3212:
3109:
Coulombic behavior, this term is responsible for the long-range electromagnetic radiation caused by the accelerating charge.
448:
443:
70:
458:
3400:
The relation between expressions for time-dependent electromagnetic fields given by
Jefimenko and by Panofsky and Phillips
3276:
The
Feynman Lectures on Physics - 21.5 The potentials of a moving charge; the general solution of Liénard and Wiechert
3457:
3386:
3362:
3246:
60:
3181:
328:
2331:
243:
822:
588:
65:
2326:
2321:
electric charge version of
Jefimenko's equations. Actually, it can be (non trivially) deduced from them using
2130:
3241:
Introduction to
Electrodynamics (3rd Edition), D. J. Griffiths, Pearson Education, Dorling Kindersley, 2007,
603:
308:
468:
208:
75:
2871:
198:
3436:
Andrew
Zangwill, Modern Electrodynamics, Cambridge University Press, 1st edition (2013), pp. 726â727, 765
761:
636:
533:
508:
428:
2305:
2769:
261:
2714:
3481:
2904:
2337:
2134:
815:
776:
303:
293:
233:
228:
168:
3476:
2882:
2672:
2650:
746:
248:
3145:
626:
153:
143:
138:
751:
721:
3165:
3113:
883:
573:
343:
118:
3207:, Appleton-Century-Crofts (New-York - 1966). 2nd ed.: Electret Scientific (Star City - 1989),
3072:
2972:
2826:
1356:{\displaystyle \mathbf {E} (\mathbf {r} ,t)={\frac {1}{4\pi \varepsilon _{0}}}\int \leftdV',}
671:
358:
348:
298:
288:
3289:
2940:
2832:
3320:
3012:
2829:
propagates at the speed of light, both these quantities are evaluated at the retarded time
796:
696:
661:
413:
278:
178:
163:
98:
3039:
35:
8:
3275:
2739:
1789:
1689:
756:
736:
731:
538:
523:
408:
378:
273:
203:
3324:
3205:
Electricity and
Magnetism: An Introduction to the Theory of Electric and Magnetic Fields
2803:
3336:
3310:
3216:
3153:
3117:
2745:
2694:
1813:
631:
371:
173:
133:
453:
3453:
3445:
3420:
3382:
3374:
3358:
3332:
3242:
3208:
3200:
1675:{\displaystyle \mathbf {B} (\mathbf {r} ,t)=-{\frac {\mu _{0}}{4\pi }}\int \leftdV',}
851:
691:
3340:
3260:
Solutions of
Maxwell's equations for electric and magnetic fields in arbitrary media
3328:
3161:
3134:
2341:
1785:
867:
843:
791:
706:
666:
656:
543:
498:
481:
398:
333:
103:
27:
1793:
937:
863:
726:
651:
646:
513:
388:
353:
213:
113:
899:
766:
3381:, 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, Sec. 1-5, page 16
3357:, 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, Sec. 1-4, page 16
3168:
in his treatise Electromagnetic Radiation (University Press, Cambridge, 1912).
3149:
3121:
2763:
2640:{\displaystyle \mathbf {B} =-\mathbf {e} _{r'}\times {\frac {\mathbf {E} }{c}}}
2322:
2318:
930:
923:
916:
875:
859:
855:
686:
681:
503:
393:
318:
268:
218:
191:
148:
123:
93:
86:
3452:, 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, Sec. 1-1, page 6
3144:, Jefimenko's equations seem to appear first in 1962 in the second edition of
3470:
3036:
is the transverse acceleration in the retarded time. As it decreases only as
1801:
1797:
1770:
871:
801:
786:
771:
711:
423:
338:
323:
238:
223:
128:
781:
676:
641:
583:
518:
438:
403:
283:
158:
2935:
701:
553:
383:
45:
3290:
The Feynman Lectures on Physics Vol. I Ch. 28: Electromagnetic Radiation
2309:
Explanation of the variables relevant for the HeavisideâFeynman formula.
418:
3301:
Kinsler, P. (2011). "How to be causal: time, spacetime, and spectra".
2581:{\displaystyle \mathbf {E} ={\frac {-q}{4\pi \varepsilon _{0}}}\left}
741:
716:
528:
50:
493:
488:
108:
3315:
3221:
Time-dependent generalizations of the BiotâSavart and Coulomb laws
2317:, also known as the JefimenkoâFeynman formula, can be seen as the
1762:{\displaystyle t_{r}=t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}
463:
548:
55:
2800:
is a unit vector pointing from the observer to the charge and
3160:, the equations analogous to Jefimenko's but in the Fourier
929:
produced by an arbitrary charge or current distribution, of
882:
charges and currents. They are the particular solutions to
2274:{\displaystyle c^{2}={\frac {1}{\varepsilon _{0}\mu _{0}}}}
878:
and relativistic effects. Therefore, they can be used for
2336:
where it was used to introduce and describe the origin of
1784:
These equations are the time-dependent generalization of
870:
in space, that takes into account the propagation delay (
3124:
for overall radiation power of the accelerating charge.
886:
for any arbitrary distribution of charges and currents.
2825:
is the distance between observer and charge. Since the
2340:. The formula provides a natural generalization of the
3075:
3042:
3015:
2975:
2943:
2907:
2885:
2835:
2806:
2772:
2748:
2717:
2697:
2675:
2653:
2593:
2350:
2228:
2165:
2143:
2030:
2003:
1889:
1859:
1829:
1702:
1368:
951:
3450:
Causality Electromagnetic Induction and Gravitation
3379:
Causality Electromagnetic Induction and Gravitation
3355:
Causality Electromagnetic Induction and Gravitation
2691:are the electric and magnetic fields respectively,
3112:The HeavisideâFeynman formula can be derived from
3101:
3061:
3028:
3001:
2961:
2925:
2893:
2860:
2817:
2792:
2754:
2730:
2703:
2683:
2661:
2639:
2580:
2273:
2214:
2121:
1807:
1761:
1674:
1355:
3285:
3283:
3468:
3120:. It allows, for example, the derivation of the
2131:Maxwell's equations in the potential formulation
894:
3280:
2133:, then substituting in the definitions of the
2344:for cases where the source charge is moving:
823:
2300:
1812:Jefimenko's equations can be found from the
3300:
830:
816:
34:
3415:Wolfgang K. H. Panofsky, Melba Phillips,
3314:
2188:
2870:
2304:
898:
3069:with distance compared to the standard
579:Electromagnetism and special relativity
3469:
1796:, which were originally true only for
599:Maxwell equations in curved spacetime
3237:
3235:
3233:
1773:. There are similar expressions for
3417:Classical Electricity And Magnetism
13:
2879:The first term in the formula for
2201:
2178:
2168:
2155:
1644:
1605:
1325:
1286:
1217:
1180:
874:) of the fields due to the finite
14:
3498:
3230:
2793:{\displaystyle \mathbf {e} _{r'}}
2887:
2775:
2731:{\displaystyle \varepsilon _{0}}
2677:
2655:
2628:
2607:
2595:
2558:
2471:
2398:
2352:
2208:
2194:
2172:
2145:
2084:
2075:
2043:
2034:
1986:
1978:
1941:
1932:
1900:
1842:
1740:
1731:
1618:
1609:
1566:
1557:
1541:
1532:
1501:
1492:
1465:
1456:
1440:
1431:
1378:
1370:
1299:
1290:
1250:
1241:
1191:
1144:
1135:
1119:
1110:
1079:
1048:
1039:
1023:
1014:
961:
953:
3439:
3430:
3409:
2926:{\displaystyle {\frac {r'}{c}}}
2332:The Feynman Lectures on Physics
2192:
1808:Origin from retarded potentials
915:Jefimenko's equations give the
3487:Eponymous equations of physics
3402:, American Journal of Physics
3392:
3368:
3347:
3294:
3269:
3262:, American Journal of Physics
3252:
3223:, American Journal of Physics
3194:
2742:(electric field constant) and
2093:
2070:
2064:
2038:
1996:
1982:
1950:
1927:
1921:
1895:
1852:
1838:
1749:
1726:
1639:
1613:
1576:
1552:
1522:
1496:
1475:
1451:
1388:
1374:
1320:
1294:
1259:
1236:
1212:
1186:
1154:
1130:
1100:
1074:
1058:
1034:
971:
957:
1:
3127:
1804:fields, and steady currents.
604:Relativistic electromagnetism
16:Equations of electromagnetism
2894:{\displaystyle \mathbf {E} }
2684:{\displaystyle \mathbf {B} }
2662:{\displaystyle \mathbf {E} }
895:Electric and magnetic fields
889:
7:
3175:
3116:using the technique of the
2327:Liénard-Wiechert potentials
2129:which are the solutions to
10:
3503:
3333:10.1088/0143-0807/32/6/022
3182:LiĂ©nardâWiechert potential
2329:. It is mostly known from
2135:electromagnetic potentials
329:LiĂ©nardâWiechert potential
2338:electromagnetic radiation
2315:HeavisideâFeynman formula
2301:HeavisideâFeynman formula
1696:is a point in space, and
911:âČ used in the calculation
862:due to a distribution of
594:Mathematical descriptions
304:Electromagnetic radiation
294:Electromagnetic induction
234:Magnetic vector potential
229:Magnetic scalar potential
3187:
3102:{\displaystyle 1/r'^{2}}
3002:{\displaystyle a_{t}/r'}
2711:is the electric charge,
2281:replaces the potentials
2222:and using the relation
144:Electrostatic induction
139:Electrostatic discharge
3166:George Adolphus Schott
3164:were first derived by
3103:
3063:
3030:
3003:
2963:
2962:{\displaystyle e_{r'}}
2927:
2895:
2876:
2862:
2861:{\displaystyle t-r'/c}
2819:
2794:
2756:
2732:
2705:
2685:
2663:
2641:
2582:
2310:
2275:
2216:
2123:
1763:
1676:
1357:
912:
574:Electromagnetic tensor
3152:'s classic textbook.
3104:
3064:
3031:
3029:{\displaystyle a_{t}}
3004:
2964:
2936:unit direction vector
2928:
2896:
2874:
2863:
2827:electromagnetic field
2820:
2795:
2757:
2733:
2706:
2686:
2664:
2642:
2583:
2308:
2276:
2217:
2124:
1764:
1677:
1358:
902:
848:Jefimenko's equations
567:Covariant formulation
359:Synchrotron radiation
299:Electromagnetic pulse
289:Electromagnetic field
3073:
3062:{\displaystyle 1/r'}
3040:
3013:
2973:
2941:
2905:
2883:
2833:
2804:
2770:
2746:
2715:
2695:
2673:
2651:
2591:
2348:
2226:
2141:
1827:
1700:
1688:âČ is a point in the
1366:
949:
609:Stressâenergy tensor
534:Reluctance (complex)
279:Displacement current
3353:Oleg D. Jefimenko,
3325:2011EJPh...32.1687K
3258:Oleg D. Jefimenko,
3114:Maxwell's equations
2740:vacuum permittivity
1814:retarded potentials
1690:charge distribution
884:Maxwell's equations
524:Magnetomotive force
409:Electromotive force
379:Alternating current
314:Jefimenko equations
274:Cyclotron radiation
3406:(1997), 1074-1076.
3398:Kirk T. McDonald,
3217:David J. Griffiths
3140:As pointed out by
3118:retarded potential
3099:
3059:
3026:
2999:
2959:
2923:
2891:
2877:
2858:
2818:{\displaystyle r'}
2815:
2790:
2752:
2728:
2701:
2681:
2659:
2637:
2578:
2311:
2271:
2212:
2186:
2119:
2117:
2099:
2024:
1956:
1883:
1759:
1672:
1353:
913:
372:Electrical network
209:Gauss magnetic law
174:Static electricity
134:Electric potential
3446:Oleg D. Jefimenko
3425:978-0-486-43924-2
3375:Oleg D. Jefimenko
3219:, Mark A. Heald,
3213:978-0-917406-08-9
3201:Oleg D. Jefimenko
2921:
2755:{\displaystyle c}
2704:{\displaystyle q}
2635:
2554:
2525:
2501:
2461:
2446:
2428:
2387:
2269:
2185:
2098:
2023:
1955:
1882:
1757:
1651:
1600:
1587:
1486:
1417:
1332:
1281:
1264:
1224:
1175:
1165:
1069:
1000:
903:Position vectors
852:Oleg D. Jefimenko
840:
839:
539:Reluctance (real)
509:Gyratorâcapacitor
454:Resonant cavities
344:Maxwell equations
3494:
3482:Electromagnetism
3461:
3443:
3437:
3434:
3428:
3413:
3407:
3396:
3390:
3372:
3366:
3351:
3345:
3344:
3318:
3298:
3292:
3287:
3278:
3273:
3267:
3266:(1992), 899â902.
3256:
3250:
3239:
3228:
3227:(1991), 111-117.
3198:
3162:frequency domain
3135:electromagnetism
3108:
3106:
3105:
3100:
3098:
3097:
3096:
3083:
3068:
3066:
3065:
3060:
3058:
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3035:
3033:
3032:
3027:
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3024:
3008:
3006:
3005:
3000:
2998:
2990:
2985:
2984:
2968:
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2958:
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2956:
2932:
2930:
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2924:
2922:
2917:
2909:
2900:
2898:
2897:
2892:
2890:
2867:
2865:
2864:
2859:
2854:
2849:
2824:
2822:
2821:
2816:
2814:
2799:
2797:
2796:
2791:
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2787:
2778:
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2759:
2758:
2753:
2737:
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2734:
2729:
2727:
2726:
2710:
2708:
2707:
2702:
2690:
2688:
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2682:
2680:
2668:
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2643:
2638:
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2619:
2610:
2598:
2587:
2585:
2584:
2579:
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2573:
2572:
2571:
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2551:
2538:
2537:
2528:
2526:
2524:
2523:
2511:
2506:
2502:
2500:
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2498:
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2474:
2468:
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2412:
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2395:
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2384:
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2360:
2355:
2280:
2278:
2277:
2272:
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2268:
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2256:
2243:
2238:
2237:
2221:
2219:
2218:
2213:
2211:
2197:
2187:
2184:
2176:
2175:
2166:
2148:
2128:
2126:
2125:
2120:
2118:
2111:
2100:
2097:
2096:
2091:
2087:
2078:
2073:
2067:
2063:
2062:
2050:
2046:
2037:
2031:
2025:
2022:
2014:
2013:
2004:
1989:
1981:
1975:
1968:
1957:
1954:
1953:
1948:
1944:
1935:
1930:
1924:
1920:
1919:
1907:
1903:
1890:
1884:
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1879:
1860:
1845:
1833:
1768:
1766:
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1760:
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1747:
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1711:
1681:
1679:
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1668:
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1642:
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1593:
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1579:
1573:
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1373:
1362:
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1331:
1323:
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1302:
1293:
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1267:
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1257:
1253:
1244:
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1215:
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1198:
1194:
1178:
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1127:
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1067:
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1051:
1042:
1037:
1031:
1030:
1026:
1017:
1011:
1001:
999:
998:
997:
978:
964:
956:
868:electric current
864:electric charges
844:electromagnetism
832:
825:
818:
499:Electric machine
482:Magnetic circuit
444:Parallel circuit
434:Network analysis
399:Electric current
334:London equations
179:Triboelectricity
169:Potential energy
38:
28:Electromagnetism
19:
18:
3502:
3501:
3497:
3496:
3495:
3493:
3492:
3491:
3477:Electrodynamics
3467:
3466:
3465:
3464:
3444:
3440:
3435:
3431:
3414:
3410:
3397:
3393:
3373:
3369:
3352:
3348:
3299:
3295:
3288:
3281:
3274:
3270:
3257:
3253:
3240:
3231:
3199:
3195:
3190:
3178:
3158:Andrew Zangwill
3154:David Griffiths
3130:
3092:
3088:
3084:
3079:
3074:
3071:
3070:
3051:
3046:
3041:
3038:
3037:
3020:
3016:
3014:
3011:
3010:
2991:
2986:
2980:
2976:
2974:
2971:
2970:
2949:
2948:
2944:
2942:
2939:
2938:
2910:
2908:
2906:
2903:
2902:
2886:
2884:
2881:
2880:
2850:
2842:
2834:
2831:
2830:
2807:
2805:
2802:
2801:
2780:
2779:
2774:
2773:
2771:
2768:
2767:
2747:
2744:
2743:
2722:
2718:
2716:
2713:
2712:
2696:
2693:
2692:
2676:
2674:
2671:
2670:
2654:
2652:
2649:
2648:
2627:
2625:
2612:
2611:
2606:
2605:
2594:
2592:
2589:
2588:
2563:
2562:
2557:
2556:
2547:
2543:
2539:
2533:
2529:
2527:
2519:
2515:
2510:
2494:
2490:
2486:
2476:
2475:
2470:
2469:
2467:
2463:
2453:
2448:
2435:
2433:
2421:
2417:
2413:
2403:
2402:
2397:
2396:
2394:
2393:
2389:
2380:
2376:
2369:
2361:
2359:
2351:
2349:
2346:
2345:
2325:, or using the
2323:Dirac functions
2303:
2262:
2258:
2252:
2248:
2247:
2242:
2233:
2229:
2227:
2224:
2223:
2207:
2193:
2177:
2171:
2167:
2164:
2144:
2142:
2139:
2138:
2116:
2115:
2104:
2092:
2083:
2082:
2074:
2069:
2068:
2058:
2054:
2042:
2041:
2033:
2032:
2029:
2015:
2009:
2005:
2002:
1985:
1977:
1973:
1972:
1961:
1949:
1940:
1939:
1931:
1926:
1925:
1915:
1911:
1899:
1898:
1891:
1888:
1875:
1871:
1864:
1858:
1841:
1830:
1828:
1825:
1824:
1810:
1794:electrodynamics
1790:BiotâSavart law
1748:
1739:
1738:
1730:
1725:
1724:
1722:
1707:
1703:
1701:
1698:
1697:
1661:
1643:
1633:
1629:
1617:
1616:
1608:
1604:
1602:
1592:
1580:
1575:
1574:
1565:
1564:
1556:
1551:
1550:
1540:
1539:
1531:
1530:
1528:
1516:
1512:
1500:
1499:
1491:
1479:
1474:
1473:
1464:
1463:
1455:
1450:
1449:
1439:
1438:
1430:
1429:
1427:
1426:
1422:
1409:
1403:
1399:
1397:
1377:
1369:
1367:
1364:
1363:
1342:
1324:
1314:
1310:
1298:
1297:
1289:
1285:
1283:
1275:
1271:
1266:
1258:
1249:
1248:
1240:
1235:
1234:
1229:
1216:
1206:
1202:
1190:
1189:
1179:
1177:
1167:
1158:
1153:
1152:
1143:
1142:
1134:
1129:
1128:
1118:
1117:
1109:
1108:
1106:
1094:
1090:
1078:
1077:
1062:
1057:
1056:
1047:
1046:
1038:
1033:
1032:
1022:
1021:
1013:
1012:
1010:
1009:
1005:
993:
989:
982:
977:
960:
952:
950:
947:
946:
938:current density
897:
892:
836:
807:
806:
622:
614:
613:
569:
559:
558:
514:Induction motor
484:
474:
473:
389:Current density
374:
364:
363:
354:Poynting vector
264:
262:Electrodynamics
254:
253:
249:Right-hand rule
214:Magnetic dipole
204:BiotâSavart law
194:
184:
183:
119:Electric dipole
114:Electric charge
89:
17:
12:
11:
5:
3500:
3490:
3489:
3484:
3479:
3463:
3462:
3438:
3429:
3408:
3391:
3367:
3346:
3293:
3279:
3268:
3251:
3229:
3192:
3191:
3189:
3186:
3185:
3184:
3177:
3174:
3129:
3126:
3122:Larmor formula
3095:
3091:
3087:
3082:
3078:
3057:
3054:
3049:
3045:
3023:
3019:
2997:
2994:
2989:
2983:
2979:
2955:
2952:
2947:
2920:
2916:
2913:
2889:
2857:
2853:
2848:
2845:
2841:
2838:
2813:
2810:
2786:
2783:
2777:
2764:speed of light
2751:
2725:
2721:
2700:
2679:
2657:
2634:
2630:
2624:
2618:
2615:
2609:
2604:
2601:
2597:
2576:
2569:
2566:
2560:
2550:
2546:
2542:
2536:
2532:
2522:
2518:
2514:
2509:
2505:
2497:
2493:
2489:
2482:
2479:
2473:
2466:
2459:
2456:
2452:
2445:
2441:
2438:
2432:
2424:
2420:
2416:
2409:
2406:
2400:
2392:
2383:
2379:
2375:
2372:
2367:
2364:
2358:
2354:
2302:
2299:
2289:by the fields
2265:
2261:
2255:
2251:
2246:
2241:
2236:
2232:
2210:
2206:
2203:
2200:
2196:
2191:
2183:
2180:
2174:
2170:
2163:
2160:
2157:
2154:
2151:
2147:
2114:
2110:
2107:
2103:
2095:
2090:
2086:
2081:
2077:
2072:
2066:
2061:
2057:
2053:
2049:
2045:
2040:
2036:
2028:
2021:
2018:
2012:
2008:
2001:
1998:
1995:
1992:
1988:
1984:
1980:
1976:
1974:
1971:
1967:
1964:
1960:
1952:
1947:
1943:
1938:
1934:
1929:
1923:
1918:
1914:
1910:
1906:
1902:
1897:
1894:
1887:
1878:
1874:
1870:
1867:
1863:
1857:
1854:
1851:
1848:
1844:
1840:
1837:
1834:
1832:
1809:
1806:
1756:
1751:
1746:
1742:
1737:
1733:
1728:
1721:
1718:
1715:
1710:
1706:
1671:
1667:
1664:
1660:
1656:
1649:
1646:
1641:
1636:
1632:
1628:
1624:
1620:
1615:
1611:
1607:
1599:
1596:
1591:
1583:
1578:
1572:
1568:
1563:
1559:
1554:
1547:
1543:
1538:
1534:
1527:
1524:
1519:
1515:
1511:
1507:
1503:
1498:
1494:
1490:
1482:
1477:
1471:
1467:
1462:
1458:
1453:
1446:
1442:
1437:
1433:
1425:
1421:
1415:
1412:
1406:
1402:
1396:
1393:
1390:
1387:
1384:
1380:
1376:
1372:
1352:
1348:
1345:
1341:
1337:
1330:
1327:
1322:
1317:
1313:
1309:
1305:
1301:
1296:
1292:
1288:
1278:
1274:
1270:
1261:
1256:
1252:
1247:
1243:
1238:
1233:
1228:
1222:
1219:
1214:
1209:
1205:
1201:
1197:
1193:
1188:
1185:
1182:
1174:
1171:
1161:
1156:
1150:
1146:
1141:
1137:
1132:
1125:
1121:
1116:
1112:
1105:
1102:
1097:
1093:
1089:
1085:
1081:
1076:
1073:
1065:
1060:
1054:
1050:
1045:
1041:
1036:
1029:
1025:
1020:
1016:
1008:
1004:
996:
992:
988:
985:
981:
976:
973:
970:
967:
963:
959:
955:
931:charge density
924:magnetic field
917:electric field
896:
893:
891:
888:
876:speed of light
860:magnetic field
856:electric field
838:
837:
835:
834:
827:
820:
812:
809:
808:
805:
804:
799:
794:
789:
784:
779:
774:
769:
764:
759:
754:
749:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
623:
620:
619:
616:
615:
612:
611:
606:
601:
596:
591:
589:Four-potential
586:
581:
576:
570:
565:
564:
561:
560:
557:
556:
551:
546:
541:
536:
531:
526:
521:
516:
511:
506:
504:Electric motor
501:
496:
491:
485:
480:
479:
476:
475:
472:
471:
466:
461:
459:Series circuit
456:
451:
446:
441:
436:
431:
429:Kirchhoff laws
426:
421:
416:
411:
406:
401:
396:
394:Direct current
391:
386:
381:
375:
370:
369:
366:
365:
362:
361:
356:
351:
349:Maxwell tensor
346:
341:
336:
331:
326:
321:
319:Larmor formula
316:
311:
306:
301:
296:
291:
286:
281:
276:
271:
269:Bremsstrahlung
265:
260:
259:
256:
255:
252:
251:
246:
241:
236:
231:
226:
221:
219:Magnetic field
216:
211:
206:
201:
195:
192:Magnetostatics
190:
189:
186:
185:
182:
181:
176:
171:
166:
161:
156:
151:
146:
141:
136:
131:
126:
124:Electric field
121:
116:
111:
106:
101:
96:
94:Charge density
90:
87:Electrostatics
85:
84:
81:
80:
79:
78:
73:
68:
63:
58:
53:
48:
40:
39:
31:
30:
24:
23:
22:Articles about
15:
9:
6:
4:
3:
2:
3499:
3488:
3485:
3483:
3480:
3478:
3475:
3474:
3472:
3459:
3458:0-917406-23-0
3455:
3451:
3447:
3442:
3433:
3426:
3422:
3418:
3412:
3405:
3401:
3395:
3388:
3387:0-917406-23-0
3384:
3380:
3376:
3371:
3364:
3363:0-917406-23-0
3360:
3356:
3350:
3342:
3338:
3334:
3330:
3326:
3322:
3317:
3312:
3308:
3304:
3297:
3291:
3286:
3284:
3277:
3272:
3265:
3261:
3255:
3248:
3247:81-7758-293-3
3244:
3238:
3236:
3234:
3226:
3222:
3218:
3214:
3210:
3206:
3202:
3197:
3193:
3183:
3180:
3179:
3173:
3169:
3167:
3163:
3159:
3155:
3151:
3147:
3143:
3138:
3136:
3125:
3123:
3119:
3115:
3110:
3093:
3089:
3085:
3080:
3076:
3055:
3052:
3047:
3043:
3021:
3017:
2995:
2992:
2987:
2981:
2977:
2953:
2950:
2945:
2937:
2918:
2914:
2911:
2873:
2869:
2855:
2851:
2846:
2843:
2839:
2836:
2828:
2811:
2808:
2784:
2781:
2766:. The vector
2765:
2749:
2741:
2723:
2719:
2698:
2632:
2622:
2616:
2613:
2602:
2599:
2574:
2567:
2564:
2548:
2544:
2540:
2534:
2530:
2520:
2516:
2512:
2507:
2503:
2495:
2491:
2487:
2480:
2477:
2464:
2457:
2454:
2450:
2443:
2439:
2436:
2430:
2422:
2418:
2414:
2407:
2404:
2390:
2381:
2377:
2373:
2370:
2365:
2362:
2356:
2343:
2342:Coulomb's law
2339:
2335:
2333:
2328:
2324:
2320:
2316:
2307:
2298:
2296:
2292:
2288:
2284:
2263:
2259:
2253:
2249:
2244:
2239:
2234:
2230:
2204:
2198:
2189:
2181:
2161:
2158:
2152:
2149:
2136:
2132:
2112:
2108:
2105:
2101:
2088:
2079:
2059:
2055:
2051:
2047:
2026:
2019:
2016:
2010:
2006:
1999:
1993:
1990:
1969:
1965:
1962:
1958:
1945:
1936:
1916:
1912:
1908:
1904:
1892:
1885:
1876:
1872:
1868:
1865:
1861:
1855:
1849:
1846:
1835:
1822:
1818:
1815:
1805:
1803:
1802:magnetostatic
1799:
1798:electrostatic
1795:
1791:
1787:
1786:Coulomb's law
1782:
1780:
1776:
1772:
1771:retarded time
1754:
1744:
1735:
1719:
1716:
1713:
1708:
1704:
1695:
1691:
1687:
1682:
1669:
1665:
1662:
1658:
1654:
1647:
1634:
1630:
1626:
1622:
1597:
1594:
1589:
1581:
1570:
1561:
1545:
1536:
1525:
1517:
1513:
1509:
1505:
1488:
1480:
1469:
1460:
1444:
1435:
1423:
1419:
1413:
1410:
1404:
1400:
1394:
1391:
1385:
1382:
1350:
1346:
1343:
1339:
1335:
1328:
1315:
1311:
1307:
1303:
1276:
1272:
1268:
1254:
1245:
1231:
1226:
1220:
1207:
1203:
1199:
1195:
1183:
1172:
1169:
1159:
1148:
1139:
1123:
1114:
1103:
1095:
1091:
1087:
1083:
1071:
1063:
1052:
1043:
1027:
1018:
1006:
1002:
994:
990:
986:
983:
979:
974:
968:
965:
944:
942:
939:
935:
932:
928:
925:
921:
918:
910:
906:
901:
887:
885:
881:
877:
873:
872:retarded time
869:
865:
861:
857:
853:
850:(named after
849:
845:
833:
828:
826:
821:
819:
814:
813:
811:
810:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
624:
618:
617:
610:
607:
605:
602:
600:
597:
595:
592:
590:
587:
585:
582:
580:
577:
575:
572:
571:
568:
563:
562:
555:
552:
550:
547:
545:
542:
540:
537:
535:
532:
530:
527:
525:
522:
520:
517:
515:
512:
510:
507:
505:
502:
500:
497:
495:
492:
490:
487:
486:
483:
478:
477:
470:
467:
465:
462:
460:
457:
455:
452:
450:
447:
445:
442:
440:
437:
435:
432:
430:
427:
425:
424:Joule heating
422:
420:
417:
415:
412:
410:
407:
405:
402:
400:
397:
395:
392:
390:
387:
385:
382:
380:
377:
376:
373:
368:
367:
360:
357:
355:
352:
350:
347:
345:
342:
340:
339:Lorentz force
337:
335:
332:
330:
327:
325:
322:
320:
317:
315:
312:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
285:
282:
280:
277:
275:
272:
270:
267:
266:
263:
258:
257:
250:
247:
245:
242:
240:
239:Magnetization
237:
235:
232:
230:
227:
225:
224:Magnetic flux
222:
220:
217:
215:
212:
210:
207:
205:
202:
200:
197:
196:
193:
188:
187:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
155:
152:
150:
147:
145:
142:
140:
137:
135:
132:
130:
129:Electric flux
127:
125:
122:
120:
117:
115:
112:
110:
107:
105:
102:
100:
97:
95:
92:
91:
88:
83:
82:
77:
74:
72:
69:
67:
66:Computational
64:
62:
59:
57:
54:
52:
49:
47:
44:
43:
42:
41:
37:
33:
32:
29:
26:
25:
21:
20:
3449:
3441:
3432:
3416:
3411:
3403:
3399:
3394:
3378:
3370:
3354:
3349:
3306:
3303:Eur. J. Phys
3302:
3296:
3271:
3263:
3259:
3254:
3224:
3220:
3215:. See also:
3204:
3196:
3170:
3139:
3131:
3111:
2878:
2330:
2314:
2312:
2294:
2290:
2286:
2282:
2137:themselves:
1820:
1816:
1811:
1783:
1778:
1774:
1693:
1685:
1683:
945:
940:
933:
926:
919:
914:
908:
904:
879:
847:
841:
584:Four-current
519:Linear motor
404:Electrolysis
313:
284:Eddy current
244:Permeability
164:Polarization
159:Permittivity
3309:(6): 1687.
854:) give the
554:Transformer
384:Capacitance
309:Faraday law
104:Coulomb law
46:Electricity
3471:Categories
3128:Discussion
2319:point-like
621:Scientists
469:Waveguides
449:Resistance
419:Inductance
199:AmpĂšre law
3316:1106.1792
2840:−
2720:ε
2623:×
2603:−
2378:ε
2374:π
2363:−
2260:μ
2250:ε
2205:×
2202:∇
2179:∂
2169:∂
2162:−
2159:φ
2156:∇
2153:−
2080:−
2027:∫
2020:π
2007:μ
1937:−
1893:ρ
1886:∫
1873:ε
1869:π
1836:φ
1736:−
1720:−
1645:∂
1606:∂
1590:×
1562:−
1537:−
1489:×
1461:−
1436:−
1420:∫
1414:π
1401:μ
1395:−
1326:∂
1287:∂
1246:−
1227:−
1218:∂
1184:ρ
1181:∂
1140:−
1115:−
1072:ρ
1044:−
1019:−
1003:∫
991:ε
987:π
890:Equations
777:Steinmetz
707:Kirchhoff
692:Jefimenko
687:Hopkinson
672:Helmholtz
667:Heaviside
529:Permeance
414:Impedance
154:Insulator
149:Gauss law
99:Conductor
76:Phenomena
71:Textbooks
51:Magnetism
3341:56034806
3176:See also
3150:Phillips
3146:Panofsky
3142:McDonald
3090:′
3056:′
3009:, where
2996:′
2954:′
2915:′
2847:′
2812:′
2785:′
2617:′
2568:′
2492:′
2481:′
2440:′
2419:′
2408:′
2109:′
2089:′
2048:′
1966:′
1946:′
1905:′
1788:and the
1745:′
1666:′
1623:′
1571:′
1546:′
1506:′
1470:′
1445:′
1347:′
1304:′
1255:′
1196:′
1149:′
1124:′
1084:′
1053:′
1028:′
802:Wiechert
757:Poynting
647:Einstein
494:DC motor
489:AC motor
324:Lenz law
109:Electret
3404:65 (11)
3321:Bibcode
3264:60 (10)
2762:is the
2738:is the
1769:is the
787:Thomson
762:Ritchie
752:Poisson
737:Neumann
732:Maxwell
727:Lorentz
722:Liénard
652:Faraday
637:Coulomb
464:Voltage
439:Ohm law
61:History
3456:
3423:
3385:
3361:
3339:
3245:
3225:59 (2)
3211:
2647:Here,
1684:where
880:moving
772:Singer
767:Savart
747:Ărsted
712:Larmor
702:Kelvin
657:Fizeau
627:AmpĂšre
549:Stator
56:Optics
3337:S2CID
3311:arXiv
3188:Notes
797:Weber
792:Volta
782:Tesla
697:Joule
682:Hertz
677:Henry
662:Gauss
544:Rotor
3454:ISBN
3421:ISBN
3383:ISBN
3359:ISBN
3243:ISBN
3209:ISBN
3148:and
2669:and
2313:The
2293:and
2285:and
1819:and
1800:and
1777:and
936:and
922:and
907:and
866:and
858:and
717:Lenz
642:Davy
632:Biot
3329:doi
1792:to
842:In
742:Ohm
3473::
3448:,
3377:,
3335:.
3327:.
3319:.
3307:32
3305:.
3282:^
3232:^
3203:,
2868:.
2297:.
1823::
1781:.
1692:,
943::
846:,
3460:.
3427:.
3389:.
3365:.
3343:.
3331::
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3313::
3249:.
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2011:0
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1979:A
1970:,
1963:V
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817:v
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