38:
1572:
111:
2626:
1841:
2109:
1666:
1971:
1242:
1836:{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int {{\frac {\mathbf {J} (\mathbf {r} ')\times \left(\mathbf {r} -\mathbf {r} '\right)}{|\mathbf {r} -\mathbf {r} '|^{3}}}\mathrm {d} ^{3}\mathbf {r} '}}
1129:
1175:
2203:
2370:
2479:
1616:
1861:. One advantage of this technique is that, if a coil has a complex geometry, it can be divided into sections and the integral evaluated for each section. Since this equation is primarily used to solve
1966:
1077:
1526:
1460:
2285:
2239:
2403:
1653:
1395:
944:
events that occur on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static – as long as the currents do not
2104:{\displaystyle \mathbf {A} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int {{\frac {\mathbf {J(\mathbf {r} ')} }{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}\mathbf {r} '}.}
1304:
2508:
Hiebert, W; Ballentine, G; Freeman, M (2002). "Comparison of experimental and numerical micromagnetic dynamics in coherent precessional switching and modal oscillations".
1920:
1552:
1482:
1424:
1368:
997:
2430:
2312:
1346:
1279:
668:
1181:
1885:
1011:. The fields are independent of time and each other. The magnetostatic equations, in both differential and integral forms, are shown in the table below.
641:
2151:
2321:
653:
2438:
3003:
1492:
A common technique is to solve a series of magnetostatic problems at incremental time steps and then use these solutions to approximate the term
2888:
1932:
2658:
1083:
2977:
1142:
904:
673:
2246:
2210:
683:
2856:
508:
1578:
523:
518:
145:
17:
1046:
533:
1404:
and considering the importance of the terms that have been removed. Of particular significance is the comparison of the
2405:
has a role analogous to the electric charge in electrostatics and is often referred to as an effective charge density
1495:
1429:
2651:
2606:
135:
81:
59:
403:
52:
940:
are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast
2630:
1850:
318:
2377:
1625:
1575:
Summary of magnetostatic relations between magnetic vector potential, magnetic field and current density. Here,
2804:
2572:
897:
663:
140:
1529:
1025:
678:
383:
2644:
1400:
The quality of this approximation may be guessed by comparing the above equations with the full version of
1037:
543:
283:
150:
273:
2941:
1484:
term is substantially larger, then the smaller term may be ignored without significant loss of accuracy.
836:
711:
608:
583:
503:
2772:
2745:
336:
1926:
1897:
890:
851:
378:
368:
308:
303:
243:
1373:
2878:
1325:
1284:
388:
46:
1903:
1535:
1465:
1407:
1351:
980:
821:
323:
2998:
701:
228:
218:
213:
2207:
Except in the case of conductors, electric currents can be ignored. Then Ampère's law is simply
826:
796:
2962:
2824:
2408:
2290:
1555:
1401:
974:
941:
648:
418:
193:
63:
1622:
If all currents in a system are known (i.e., if a complete description of the current density
2967:
2936:
2799:
2720:
1866:
1258:
746:
433:
423:
373:
363:
1896:
calculation uses a modified form of the magnetostatic equations above in order to calculate
2926:
2750:
2710:
2519:
1858:
871:
771:
736:
488:
353:
253:
238:
173:
110:
8:
3008:
2901:
2760:
2755:
2740:
2715:
2692:
2668:
2119:
1660:
945:
831:
811:
806:
613:
598:
483:
453:
348:
278:
2523:
2957:
2834:
2829:
2782:
2562:
1331:
1307:
1264:
1237:{\displaystyle \oint _{C}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =I_{\mathrm {enc} }}
706:
446:
248:
208:
1135:
528:
2972:
2921:
2866:
2846:
2732:
2602:
2592:
2568:
2510:
2489:
2148:. In such materials the magnetization must be explicitly included using the relation
1854:
925:
766:
2911:
2851:
2777:
2527:
2435:
The vector potential method can also be employed with an effective current density
1881:
1877:
1571:
953:
866:
781:
741:
731:
618:
573:
556:
473:
408:
178:
102:
2931:
2841:
2787:
2687:
2598:
2544:
1315:
957:
937:
801:
726:
721:
588:
463:
428:
288:
188:
841:
2861:
2816:
2794:
2682:
2588:
2531:
2145:
2133:
2129:
2123:
1893:
1889:
1008:
1004:
1000:
949:
933:
929:
921:
761:
756:
578:
468:
393:
343:
293:
223:
198:
168:
161:
2992:
2871:
2141:
2137:
1873:
876:
861:
846:
786:
498:
413:
398:
313:
298:
203:
2636:
2896:
1554:(which had previously been ignored). This method is not a true solution of
856:
751:
716:
658:
593:
513:
478:
358:
233:
968:
27:
Branch of physics about magnetism in systems with steady electric currents
1865:
problems, the contributions can be added. For a very difficult geometry,
776:
628:
458:
120:
1872:
For problems where the dominant magnetic material is a highly permeable
2916:
2906:
1655:
is available) then the magnetic field can be determined, at a position
1250:
977:
and assuming that charges are either fixed or move as a steady current
493:
1124:{\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0}
2767:
2315:
1880:
approach is useful. When the air gaps are large in comparison to the
816:
791:
603:
125:
1929:. Since the divergence of the magnetic flux density is always zero,
1030:
568:
563:
183:
1170:{\displaystyle \mathbf {\nabla } \times \mathbf {H} =\mathbf {J} }
2198:{\displaystyle \mathbf {B} =\mu _{0}(\mathbf {M} +\mathbf {H} ).}
1558:
but can provide a good approximation for slowly changing fields.
538:
2625:
2365:{\displaystyle \nabla ^{2}\Phi _{M}=\nabla \cdot \mathbf {M} .}
1862:
1846:
623:
130:
2545:
The
Feynman Lectures on Physics Vol. II Ch. 13: Magnetostatics
1328:, the second integral is a line integral around a closed loop
1845:
This technique works well for problems where the medium is a
2474:{\displaystyle \mathbf {J_{M}} =\nabla \times \mathbf {M} .}
948:
rapidly. Magnetostatics is widely used in applications of
932:(not changing with time). It is the magnetic analogue of
2507:
1968:
and the relation of the vector potential to current is:
1961:{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} ,}
1611:{\displaystyle \mathbf {r} =\mathbf {x} -\mathbf {x'} }
969:
Magnetostatics as a special case of
Maxwell's equations
1072:{\displaystyle \mathbf {\nabla } \cdot \mathbf {B} =0}
2441:
2411:
2380:
2324:
2293:
2249:
2213:
2154:
1974:
1935:
1906:
1669:
1628:
1581:
1538:
1498:
1468:
1432:
1410:
1376:
1354:
1334:
1287:
1267:
1184:
1145:
1086:
1049:
983:
999:, the equations separate into two equations for the
2473:
2424:
2397:
2364:
2306:
2279:
2233:
2197:
2103:
1960:
1914:
1835:
1647:
1610:
1546:
1520:
1476:
1454:
1418:
1389:
1362:
1340:
1298:
1273:
1236:
1169:
1123:
1071:
991:
2039:
2026:
1561:
2990:
1521:{\displaystyle \partial \mathbf {B} /\partial t}
1455:{\displaystyle \partial \mathbf {D} /\partial t}
2280:{\displaystyle \mathbf {H} =-\nabla \Phi _{M},}
1487:
2666:
2652:
2234:{\displaystyle \nabla \times \mathbf {H} =0.}
898:
2594:Introduction to the Theory of Ferromagnetism
2556:
2554:
2552:
2374:Thus, the divergence of the magnetization,
1925:The magnetic field can be derived from the
1888:becomes significant and usually requires a
2659:
2645:
2398:{\displaystyle \nabla \cdot \mathbf {M} ,}
1922:can be found from the magnetic potential.
1648:{\displaystyle \mathbf {J} (\mathbf {r} )}
905:
891:
109:
2318:. Substituting this in Gauss's law gives
82:Learn how and when to remove this message
2549:
1570:
1370:. The current going through the loop is
45:This article includes a list of general
2587:
2560:
1849:or air or some similar material with a
1261:, the first integral is over a surface
654:Electromagnetism and special relativity
14:
3004:Electric and magnetic fields in matter
2991:
2857:Atomic, molecular, and optical physics
2501:
2640:
674:Maxwell equations in curved spacetime
31:
2128:Strongly magnetic materials (i.e.,
24:
2457:
2381:
2348:
2336:
2326:
2295:
2265:
2261:
2214:
2077:
1944:
1876:with relatively small air gaps, a
1812:
1566:
1512:
1499:
1446:
1433:
1228:
1225:
1222:
1204:
1147:
1106:
1051:
51:it lacks sufficient corresponding
25:
3020:
2618:
2567:(2nd ed.). New York: Wiley.
1306:. Where â with the cross denotes
2624:
2464:
2448:
2444:
2388:
2355:
2251:
2221:
2185:
2177:
2156:
2113:
2089:
2059:
2050:
2031:
2023:
1984:
1976:
1951:
1937:
1908:
1824:
1787:
1778:
1757:
1748:
1728:
1719:
1679:
1671:
1638:
1630:
1600:
1591:
1583:
1540:
1503:
1470:
1437:
1412:
1356:
1292:
1209:
1196:
1163:
1155:
1111:
1098:
1059:
985:
36:
2978:Timeline of physics discoveries
963:
2581:
2538:
2243:This has the general solution
2189:
2173:
2068:
2045:
1988:
1980:
1797:
1773:
1736:
1723:
1683:
1675:
1642:
1634:
1562:Solving for the magnetic field
1390:{\displaystyle I_{\text{enc}}}
1281:with oriented surface element
13:
1:
2495:
1299:{\displaystyle d\mathbf {S} }
1249:Where â with the dot denotes
679:Relativistic electromagnetism
2561:Jackson, John David (1975).
1915:{\displaystyle \mathbf {B} }
1547:{\displaystyle \mathbf {E} }
1528:. Plugging this result into
1488:Re-introducing Faraday's law
1477:{\displaystyle \mathbf {J} }
1419:{\displaystyle \mathbf {J} }
1363:{\displaystyle \mathbf {l} }
992:{\displaystyle \mathbf {J} }
7:
2942:Quantum information science
2483:
1659:, from the currents by the
10:
3025:
2773:Classical electromagnetism
2532:10.1103/PhysRevB.65.140404
2117:
404:LiĂŠnardâWiechert potential
2950:
2887:
2815:
2731:
2703:
2675:
2564:Classical electrodynamics
2425:{\displaystyle \rho _{M}}
2307:{\displaystyle \Phi _{M}}
2144:that is primarily due to
1019:
1016:
669:Mathematical descriptions
379:Electromagnetic radiation
369:Electromagnetic induction
309:Magnetic vector potential
304:Magnetic scalar potential
2879:Condensed matter physics
1326:magnetic field intensity
219:Electrostatic induction
214:Electrostatic discharge
66:more precise citations.
2963:Nobel Prize in Physics
2825:Relativistic mechanics
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1125:
1073:
993:
649:Electromagnetic tensor
2968:Philosophy of physics
2476:
2427:
2400:
2367:
2309:
2282:
2236:
2200:
2118:Further information:
2106:
1963:
1917:
1867:numerical integration
1859:air-core transformers
1851:relative permeability
1838:
1650:
1613:
1574:
1549:
1523:
1479:
1457:
1421:
1392:
1365:
1343:
1301:
1276:
1259:magnetic flux density
1239:
1172:
1126:
1074:
994:
924:in systems where the
642:Covariant formulation
434:Synchrotron radiation
374:Electromagnetic pulse
364:Electromagnetic field
18:Static magnetic field
2927:Mathematical physics
2633:at Wikimedia Commons
2439:
2409:
2378:
2322:
2291:
2247:
2211:
2152:
1972:
1933:
1904:
1853:of 1. This includes
1667:
1661:BiotâSavart equation
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1430:
1408:
1374:
1352:
1332:
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1265:
1182:
1143:
1084:
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684:Stressâenergy tensor
609:Reluctance (complex)
354:Displacement current
2902:Atmospheric physics
2741:Classical mechanics
2669:branches of physics
2524:2002PhRvB..65n0404H
2120:Demagnetizing field
1556:Maxwell's equations
1402:Maxwell's equations
975:Maxwell's equations
599:Magnetomotive force
484:Electromotive force
454:Alternating current
389:Jefimenko equations
349:Cyclotron radiation
2958:History of physics
2471:
2422:
2395:
2362:
2304:
2277:
2231:
2195:
2101:
1958:
1912:
1898:magnetic potential
1855:air-core inductors
1833:
1645:
1620:
1608:
1544:
1532:finds a value for
1518:
1474:
1452:
1416:
1387:
1360:
1348:with line element
1338:
1296:
1271:
1234:
1167:
1121:
1069:
1007:) and two for the
989:
952:such as models of
942:magnetic switching
447:Electrical network
284:Gauss magnetic law
249:Static electricity
209:Electric potential
2986:
2985:
2973:Physics education
2922:Materials science
2889:Interdisciplinary
2847:Quantum mechanics
2629:Media related to
2511:Physical Review B
2490:Darwin Lagrangian
2073:
2014:
1892:calculation. The
1808:
1709:
1426:term against the
1384:
1341:{\displaystyle C}
1274:{\displaystyle S}
1247:
1246:
915:
914:
614:Reluctance (real)
584:Gyratorâcapacitor
529:Resonant cavities
419:Maxwell equations
92:
91:
84:
16:(Redirected from
3016:
2912:Chemical physics
2852:Particle physics
2778:Classical optics
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2201:
2196:
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2102:
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2015:
2013:
2005:
2004:
1995:
1987:
1979:
1967:
1965:
1964:
1959:
1954:
1940:
1927:vector potential
1921:
1919:
1918:
1913:
1911:
1882:magnetic circuit
1878:magnetic circuit
1842:
1840:
1839:
1834:
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1807:
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1302:
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1277:
1272:
1243:
1241:
1240:
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1231:
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1207:
1199:
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1054:
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998:
996:
995:
990:
988:
954:magnetic storage
920:is the study of
907:
900:
893:
574:Electric machine
557:Magnetic circuit
519:Parallel circuit
509:Network analysis
474:Electric current
409:London equations
254:Triboelectricity
244:Potential energy
113:
103:Electromagnetism
94:
93:
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80:
76:
73:
67:
62:this article by
53:inline citations
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2842:Nuclear physics
2811:
2805:Non-equilibrium
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2599:Clarendon Press
2589:Aharoni, Amikam
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1901:
1900:. The value of
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1801:
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1786:
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1777:
1772:
1771:
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1567:Current sources
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1371:
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1329:
1319:
1316:current density
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1179:
1162:
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1136:Ampère's law
1038:Gauss's law
1026:Differential
972:
964:Applications
936:, where the
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659:Four-current
594:Linear motor
479:Electrolysis
359:Eddy current
319:Permeability
266:
239:Polarization
234:Permittivity
78:
69:
50:
29:
2800:Statistical
2716:Theoretical
2693:Engineering
629:Transformer
459:Capacitance
384:Faraday law
179:Coulomb law
121:Electricity
64:introducing
3009:Potentials
2993:Categories
2917:Geophysics
2907:Biophysics
2751:Analytical
2704:Approaches
2574:047143132X
2496:References
1251:divergence
696:Scientists
544:Waveguides
524:Resistance
494:Inductance
274:Ampère law
47:references
2867:Molecular
2768:Acoustics
2761:Continuum
2756:Celestial
2746:Newtonian
2733:Classical
2676:Divisions
2461:×
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2414:ρ
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2382:∇
2352:⋅
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2337:Φ
2327:∇
2316:potential
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2266:Φ
2262:∇
2259:−
2218:×
2215:∇
2165:μ
2140:) have a
2055:−
2017:∫
2011:π
1998:μ
1948:×
1945:∇
1783:−
1753:−
1740:×
1712:∫
1706:π
1693:μ
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1500:∂
1447:∂
1434:∂
1201:⋅
1187:∮
1152:×
1148:∇
1103:⋅
1089:∮
1056:⋅
1052:∇
946:alternate
852:Steinmetz
782:Kirchhoff
767:Jefimenko
762:Hopkinson
747:Helmholtz
742:Heaviside
604:Permeance
489:Impedance
229:Insulator
224:Gauss law
174:Conductor
151:Phenomena
146:Textbooks
126:Magnetism
2591:(1996).
2484:See also
2094:′
2064:′
2036:′
1886:fringing
1884:length,
1829:′
1792:′
1762:′
1733:′
1604:′
1031:Integral
926:currents
877:Wiechert
832:Poynting
722:Einstein
569:DC motor
564:AC motor
399:Lenz law
184:Electret
2951:Related
2835:General
2830:Special
2688:Applied
2520:Bibcode
1324:is the
1314:is the
1257:is the
938:charges
862:Thomson
837:Ritchie
827:Poisson
812:Neumann
807:Maxwell
802:Lorentz
797:LiĂŠnard
727:Faraday
712:Coulomb
539:Voltage
514:Ohm law
136:History
60:improve
2862:Atomic
2817:Modern
2667:Major
2605:
2571:
2287:where
1863:linear
1847:vacuum
1253:, and
930:steady
847:Singer
842:Savart
822:Ărsted
787:Larmor
777:Kelvin
732:Fizeau
702:Ampère
624:Stator
131:Optics
49:, but
1020:Form
1017:Name
1003:(see
872:Weber
867:Volta
857:Tesla
772:Joule
757:Hertz
752:Henry
737:Gauss
619:Rotor
2788:Wave
2683:Pure
2603:ISBN
2569:ISBN
2122:and
1857:and
1318:and
1308:curl
928:are
792:Lenz
717:Davy
707:Biot
2783:Ray
2528:doi
2136:or
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817:Ohm
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1312:J
1293:S
1289:d
1269:S
1255:B
1229:c
1226:n
1223:e
1218:I
1214:=
1210:l
1205:d
1197:H
1191:C
1164:J
1160:=
1156:H
1119:0
1116:=
1112:S
1107:d
1099:B
1093:S
1067:0
1064:=
1060:B
986:J
906:e
899:t
892:v
85:)
79:(
74:)
70:(
56:.
20:)
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