27:
1561:
100:
2615:
1830:
2098:
1655:
1960:
1231:
1825:{\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} '}}
1118:
1164:
2192:
2359:
2468:
1605:
1850:. 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
1955:
1066:
1515:
1449:
2274:
2228:
2392:
1642:
1384:
933:
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
2093:{\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} '}.}
1293:
2497:
Hiebert, W; Ballentine, G; Freeman, M (2002). "Comparison of experimental and numerical micromagnetic dynamics in coherent precessional switching and modal oscillations".
1909:
1541:
1471:
1413:
1357:
986:
2419:
2301:
1335:
1268:
657:
1170:
1874:
1000:. The fields are independent of time and each other. The magnetostatic equations, in both differential and integral forms, are shown in the table below.
630:
2140:
2310:
642:
2427:
2992:
1481:
A common technique is to solve a series of magnetostatic problems at incremental time steps and then use these solutions to approximate the term
2877:
1921:
2647:
1072:
2966:
1131:
893:
662:
2235:
2199:
672:
2845:
497:
1567:
512:
507:
134:
1035:
522:
1393:
and considering the importance of the terms that have been removed. Of particular significance is the comparison of the
2394:
has a role analogous to the electric charge in electrostatics and is often referred to as an effective charge density
1484:
1418:
2640:
2595:
124:
70:
48:
392:
41:
929:
are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast
2619:
1839:
307:
2366:
1614:
1564:
Summary of magnetostatic relations between magnetic vector potential, magnetic field and current density. Here,
2793:
2561:
886:
652:
129:
1518:
1014:
667:
372:
2633:
1389:
The quality of this approximation may be guessed by comparing the above equations with the full version of
1026:
532:
272:
139:
262:
2930:
1473:
term is substantially larger, then the smaller term may be ignored without significant loss of accuracy.
825:
700:
597:
572:
492:
2761:
2734:
325:
1915:
1886:
879:
840:
367:
357:
297:
292:
232:
1362:
2867:
1314:
1273:
377:
35:
1892:
1524:
1454:
1396:
1340:
969:
810:
312:
2987:
690:
217:
207:
202:
2196:
Except in the case of conductors, electric currents can be ignored. Then Ampère's law is simply
815:
785:
2951:
2813:
2397:
2279:
1544:
1390:
963:
930:
637:
407:
182:
52:
1611:
If all currents in a system are known (i.e., if a complete description of the current density
2956:
2925:
2788:
2709:
1855:
1247:
735:
422:
412:
362:
352:
1885:
calculation uses a modified form of the magnetostatic equations above in order to calculate
2915:
2739:
2699:
2508:
1847:
860:
760:
725:
477:
342:
242:
227:
162:
99:
8:
2997:
2890:
2749:
2744:
2729:
2704:
2681:
2657:
2108:
1649:
934:
820:
800:
795:
602:
587:
472:
442:
337:
267:
2512:
2946:
2823:
2818:
2771:
2551:
1320:
1296:
1253:
1226:{\displaystyle \oint _{C}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =I_{\mathrm {enc} }}
695:
435:
237:
197:
1124:
517:
2961:
2910:
2855:
2835:
2721:
2591:
2581:
2557:
2499:
2478:
2137:. In such materials the magnetization must be explicitly included using the relation
1843:
914:
755:
2900:
2840:
2766:
2516:
2424:
The vector potential method can also be employed with an effective current density
1870:
1866:
1560:
942:
855:
770:
730:
720:
607:
562:
545:
462:
397:
167:
91:
2920:
2830:
2776:
2676:
2587:
2533:
1304:
946:
926:
790:
715:
710:
577:
452:
417:
277:
177:
830:
2850:
2805:
2783:
2671:
2577:
2520:
2134:
2122:
2118:
2112:
1882:
1878:
997:
993:
989:
938:
922:
918:
910:
750:
745:
567:
457:
382:
332:
282:
212:
187:
157:
150:
2981:
2860:
2130:
2126:
1862:
865:
850:
835:
775:
487:
402:
387:
302:
287:
192:
2625:
2885:
1543:(which had previously been ignored). This method is not a true solution of
845:
740:
705:
647:
582:
502:
467:
347:
222:
957:
16:
Branch of physics about magnetism in systems with steady electric currents
1854:
problems, the contributions can be added. For a very difficult geometry,
765:
617:
447:
109:
1861:
For problems where the dominant magnetic material is a highly permeable
2905:
2895:
1644:
is available) then the magnetic field can be determined, at a position
1239:
966:
and assuming that charges are either fixed or move as a steady current
482:
1113:{\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0}
2756:
2304:
1869:
approach is useful. When the air gaps are large in comparison to the
805:
780:
592:
114:
1918:. Since the divergence of the magnetic flux density is always zero,
1019:
557:
552:
172:
1159:{\displaystyle \mathbf {\nabla } \times \mathbf {H} =\mathbf {J} }
2187:{\displaystyle \mathbf {B} =\mu _{0}(\mathbf {M} +\mathbf {H} ).}
1547:
but can provide a good approximation for slowly changing fields.
527:
2614:
2354:{\displaystyle \nabla ^{2}\Phi _{M}=\nabla \cdot \mathbf {M} .}
1851:
1835:
612:
119:
2534:
The
Feynman Lectures on Physics Vol. II Ch. 13: Magnetostatics
1317:, the second integral is a line integral around a closed loop
1834:
This technique works well for problems where the medium is a
2463:{\displaystyle \mathbf {J_{M}} =\nabla \times \mathbf {M} .}
937:
rapidly. Magnetostatics is widely used in applications of
921:(not changing with time). It is the magnetic analogue of
2496:
1957:
and the relation of the vector potential to current is:
1950:{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} ,}
1600:{\displaystyle \mathbf {r} =\mathbf {x} -\mathbf {x'} }
958:
Magnetostatics as a special case of
Maxwell's equations
1061:{\displaystyle \mathbf {\nabla } \cdot \mathbf {B} =0}
2430:
2400:
2369:
2313:
2282:
2238:
2202:
2143:
1963:
1924:
1895:
1658:
1617:
1570:
1527:
1487:
1457:
1421:
1399:
1365:
1343:
1323:
1276:
1256:
1173:
1134:
1075:
1038:
972:
988:, the equations separate into two equations for the
2462:
2413:
2386:
2353:
2295:
2268:
2222:
2186:
2092:
1949:
1903:
1824:
1636:
1599:
1535:
1509:
1465:
1443:
1407:
1378:
1351:
1329:
1287:
1262:
1225:
1158:
1112:
1060:
980:
2028:
2015:
1550:
2979:
1510:{\displaystyle \partial \mathbf {B} /\partial t}
1444:{\displaystyle \partial \mathbf {D} /\partial t}
2269:{\displaystyle \mathbf {H} =-\nabla \Phi _{M},}
1476:
2655:
2641:
2223:{\displaystyle \nabla \times \mathbf {H} =0.}
887:
2583:Introduction to the Theory of Ferromagnetism
2545:
2543:
2541:
2363:Thus, the divergence of the magnetization,
1914:The magnetic field can be derived from the
1877:becomes significant and usually requires a
2648:
2634:
2387:{\displaystyle \nabla \cdot \mathbf {M} ,}
1911:can be found from the magnetic potential.
1637:{\displaystyle \mathbf {J} (\mathbf {r} )}
894:
880:
98:
2307:. Substituting this in Gauss's law gives
71:Learn how and when to remove this message
2538:
1559:
1359:. The current going through the loop is
34:This article includes a list of general
2576:
2549:
1838:or air or some similar material with a
1250:, the first integral is over a surface
643:Electromagnetism and special relativity
2993:Electric and magnetic fields in matter
2980:
2846:Atomic, molecular, and optical physics
2490:
2629:
663:Maxwell equations in curved spacetime
20:
2117:Strongly magnetic materials (i.e.,
13:
2446:
2370:
2337:
2325:
2315:
2284:
2254:
2250:
2203:
2066:
1933:
1865:with relatively small air gaps, a
1801:
1555:
1501:
1488:
1435:
1422:
1217:
1214:
1211:
1193:
1136:
1095:
1040:
40:it lacks sufficient corresponding
14:
3009:
2607:
2556:(2nd ed.). New York: Wiley.
1295:. Where â with the cross denotes
2613:
2453:
2437:
2433:
2377:
2344:
2240:
2210:
2174:
2166:
2145:
2102:
2078:
2048:
2039:
2020:
2012:
1973:
1965:
1940:
1926:
1897:
1813:
1776:
1767:
1746:
1737:
1717:
1708:
1668:
1660:
1627:
1619:
1589:
1580:
1572:
1529:
1492:
1459:
1426:
1401:
1345:
1281:
1198:
1185:
1152:
1144:
1100:
1087:
1048:
974:
25:
2967:Timeline of physics discoveries
952:
2570:
2527:
2232:This has the general solution
2178:
2162:
2057:
2034:
1977:
1969:
1786:
1762:
1725:
1712:
1672:
1664:
1631:
1623:
1551:Solving for the magnetic field
1379:{\displaystyle I_{\text{enc}}}
1270:with oriented surface element
1:
2484:
1288:{\displaystyle d\mathbf {S} }
1238:Where â with the dot denotes
668:Relativistic electromagnetism
2550:Jackson, John David (1975).
1904:{\displaystyle \mathbf {B} }
1536:{\displaystyle \mathbf {E} }
1517:. Plugging this result into
1477:Re-introducing Faraday's law
1466:{\displaystyle \mathbf {J} }
1408:{\displaystyle \mathbf {J} }
1352:{\displaystyle \mathbf {l} }
981:{\displaystyle \mathbf {J} }
7:
2931:Quantum information science
2472:
1648:, from the currents by the
10:
3014:
2762:Classical electromagnetism
2521:10.1103/PhysRevB.65.140404
2106:
393:LiĂŠnardâWiechert potential
2939:
2876:
2804:
2720:
2692:
2664:
2553:Classical electrodynamics
2414:{\displaystyle \rho _{M}}
2296:{\displaystyle \Phi _{M}}
2133:that is primarily due to
1008:
1005:
658:Mathematical descriptions
368:Electromagnetic radiation
358:Electromagnetic induction
298:Magnetic vector potential
293:Magnetic scalar potential
2868:Condensed matter physics
1315:magnetic field intensity
208:Electrostatic induction
203:Electrostatic discharge
55:more precise citations.
2952:Nobel Prize in Physics
2814:Relativistic mechanics
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2355:
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1331:
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1114:
1062:
982:
638:Electromagnetic tensor
2957:Philosophy of physics
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2416:
2389:
2356:
2298:
2271:
2225:
2189:
2107:Further information:
2095:
1952:
1906:
1856:numerical integration
1848:air-core transformers
1840:relative permeability
1827:
1639:
1602:
1563:
1538:
1512:
1468:
1446:
1410:
1381:
1354:
1332:
1290:
1265:
1248:magnetic flux density
1228:
1161:
1115:
1063:
983:
913:in systems where the
631:Covariant formulation
423:Synchrotron radiation
363:Electromagnetic pulse
353:Electromagnetic field
2916:Mathematical physics
2622:at Wikimedia Commons
2428:
2398:
2367:
2311:
2280:
2236:
2200:
2141:
1961:
1922:
1893:
1842:of 1. This includes
1656:
1650:BiotâSavart equation
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1568:
1525:
1485:
1455:
1419:
1397:
1363:
1341:
1321:
1274:
1254:
1171:
1132:
1073:
1036:
970:
673:Stressâenergy tensor
598:Reluctance (complex)
343:Displacement current
2891:Atmospheric physics
2730:Classical mechanics
2658:branches of physics
2513:2002PhRvB..65n0404H
2109:Demagnetizing field
1545:Maxwell's equations
1391:Maxwell's equations
964:Maxwell's equations
588:Magnetomotive force
473:Electromotive force
443:Alternating current
378:Jefimenko equations
338:Cyclotron radiation
2947:History of physics
2460:
2411:
2384:
2351:
2293:
2266:
2220:
2184:
2090:
1947:
1901:
1887:magnetic potential
1844:air-core inductors
1822:
1634:
1609:
1597:
1533:
1521:finds a value for
1507:
1463:
1441:
1405:
1376:
1349:
1337:with line element
1327:
1285:
1260:
1223:
1156:
1110:
1058:
996:) and two for the
978:
941:such as models of
931:magnetic switching
436:Electrical network
273:Gauss magnetic law
238:Static electricity
198:Electric potential
2975:
2974:
2962:Physics education
2911:Materials science
2878:Interdisciplinary
2836:Quantum mechanics
2618:Media related to
2500:Physical Review B
2479:Darwin Lagrangian
2062:
2003:
1881:calculation. The
1797:
1698:
1415:term against the
1373:
1330:{\displaystyle C}
1263:{\displaystyle S}
1236:
1235:
904:
903:
603:Reluctance (real)
573:Gyratorâcapacitor
518:Resonant cavities
408:Maxwell equations
81:
80:
73:
3005:
2901:Chemical physics
2841:Particle physics
2767:Classical optics
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2333:
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2323:
2322:
2302:
2300:
2299:
2294:
2292:
2291:
2275:
2273:
2272:
2267:
2262:
2261:
2243:
2229:
2227:
2226:
2221:
2213:
2193:
2191:
2190:
2185:
2177:
2169:
2161:
2160:
2148:
2099:
2097:
2096:
2091:
2086:
2085:
2081:
2075:
2074:
2069:
2063:
2061:
2060:
2055:
2051:
2042:
2037:
2031:
2027:
2023:
2010:
2004:
2002:
1994:
1993:
1984:
1976:
1968:
1956:
1954:
1953:
1948:
1943:
1929:
1916:vector potential
1910:
1908:
1907:
1902:
1900:
1871:magnetic circuit
1867:magnetic circuit
1831:
1829:
1828:
1823:
1821:
1820:
1816:
1810:
1809:
1804:
1798:
1796:
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1789:
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1759:
1758:
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1753:
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1740:
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1711:
1705:
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1697:
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1663:
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1606:
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1603:
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1583:
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1534:
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1516:
1514:
1513:
1508:
1500:
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1312:
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1269:
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1266:
1261:
1232:
1230:
1229:
1224:
1222:
1221:
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1201:
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1165:
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1111:
1103:
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1059:
1051:
1043:
1003:
1002:
987:
985:
984:
979:
977:
943:magnetic storage
909:is the study of
896:
889:
882:
563:Electric machine
546:Magnetic circuit
508:Parallel circuit
498:Network analysis
463:Electric current
398:London equations
243:Triboelectricity
233:Potential energy
102:
92:Electromagnetism
83:
82:
76:
69:
65:
62:
56:
51:this article by
42:inline citations
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2831:Nuclear physics
2800:
2794:Non-equilibrium
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2588:Clarendon Press
2578:Aharoni, Amikam
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1890:
1889:. The value of
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1556:Current sources
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1338:
1322:
1319:
1318:
1308:
1305:current density
1280:
1275:
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1271:
1255:
1252:
1251:
1210:
1209:
1205:
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1192:
1184:
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1168:
1151:
1143:
1135:
1133:
1130:
1129:
1099:
1094:
1086:
1080:
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925:, where the
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648:Four-current
583:Linear motor
468:Electrolysis
348:Eddy current
308:Permeability
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228:Polarization
223:Permittivity
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2789:Statistical
2705:Theoretical
2682:Engineering
618:Transformer
448:Capacitance
373:Faraday law
168:Coulomb law
110:Electricity
53:introducing
2998:Potentials
2982:Categories
2906:Geophysics
2896:Biophysics
2740:Analytical
2693:Approaches
2563:047143132X
2485:References
1240:divergence
685:Scientists
533:Waveguides
513:Resistance
483:Inductance
263:Ampère law
36:references
2856:Molecular
2757:Acoustics
2750:Continuum
2745:Celestial
2735:Newtonian
2722:Classical
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2450:×
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2403:ρ
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2305:potential
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2255:Φ
2251:∇
2248:−
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2154:μ
2129:) have a
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2006:∫
2000:π
1987:μ
1937:×
1934:∇
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1742:−
1729:×
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1141:×
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1041:∇
935:alternate
841:Steinmetz
771:Kirchhoff
756:Jefimenko
751:Hopkinson
736:Helmholtz
731:Heaviside
593:Permeance
478:Impedance
218:Insulator
213:Gauss law
163:Conductor
140:Phenomena
135:Textbooks
115:Magnetism
2580:(1996).
2473:See also
2083:′
2053:′
2025:′
1875:fringing
1873:length,
1818:′
1781:′
1751:′
1722:′
1593:′
1020:Integral
915:currents
866:Wiechert
821:Poynting
711:Einstein
558:DC motor
553:AC motor
388:Lenz law
173:Electret
2940:Related
2824:General
2819:Special
2677:Applied
2509:Bibcode
1313:is the
1303:is the
1246:is the
927:charges
851:Thomson
826:Ritchie
816:Poisson
801:Neumann
796:Maxwell
791:Lorentz
786:LiĂŠnard
716:Faraday
701:Coulomb
528:Voltage
503:Ohm law
125:History
49:improve
2851:Atomic
2806:Modern
2656:Major
2594:
2560:
2276:where
1852:linear
1836:vacuum
1242:, and
919:steady
836:Singer
831:Savart
811:Ărsted
776:Larmor
766:Kelvin
721:Fizeau
691:Ampère
613:Stator
120:Optics
38:, but
1009:Form
1006:Name
992:(see
861:Weber
856:Volta
846:Tesla
761:Joule
746:Hertz
741:Henry
726:Gauss
608:Rotor
2777:Wave
2672:Pure
2592:ISBN
2558:ISBN
2111:and
1846:and
1307:and
1297:curl
917:are
781:Lenz
706:Davy
696:Biot
2772:Ray
2517:doi
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1199:l
1194:d
1186:H
1180:C
1153:J
1149:=
1145:H
1108:0
1105:=
1101:S
1096:d
1088:B
1082:S
1056:0
1053:=
1049:B
975:J
895:e
888:t
881:v
74:)
68:(
63:)
59:(
45:.
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