1000:
3332:
4485:
2886:
38:
4710:
4229:
1759:
4492:
2623:
5418:
4480:{\displaystyle \nabla \times \mathbf {B} ={\frac {\mu _{0}}{4\pi }}\nabla \iiint _{V}d^{3}{\boldsymbol {\ell }}\,\mathbf {J} ({\boldsymbol {\ell }})\cdot \nabla \left({\frac {1}{|\mathbf {r} -{\boldsymbol {\ell }}|}}\right)-{\frac {\mu _{0}}{4\pi }}\iiint _{V}d^{3}{\boldsymbol {\ell }}\,\mathbf {J} ({\boldsymbol {\ell }})\nabla ^{2}\left({\frac {1}{|\mathbf {r} -{\boldsymbol {\ell }}|}}\right)}
3902:
4188:
4705:{\displaystyle {\begin{aligned}\nabla \left({\frac {1}{|\mathbf {r} -{\boldsymbol {\ell }}|}}\right)&=-\nabla _{\boldsymbol {\ell }}\left({\frac {1}{|\mathbf {r} -{\boldsymbol {\ell }}|}}\right),\\\nabla ^{2}\left({\frac {1}{|\mathbf {r} -{\boldsymbol {\ell }}|}}\right)&=-4\pi \delta (\mathbf {r} -{\boldsymbol {\ell }})\end{aligned}}}
2881:{\displaystyle {\begin{aligned}\mathbf {E} &={\frac {q}{4\pi \varepsilon _{0}}}{\frac {1-\beta ^{2}}{\left(1-\beta ^{2}\sin ^{2}\theta \right)^{3/2}}}{\frac {\mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}\\\mathbf {H} &=\mathbf {v} \times \mathbf {D} \\\mathbf {B} &={\frac {1}{c^{2}}}\mathbf {v} \times \mathbf {E} \end{aligned}}}
3753:
4021:
1638:
2428:
1274:
2588:
2146:
4056:
3276:
3165:
3306:
the "Biot–Savart law for a point charge" due to its close resemblance to the standard Biot–Savart law. However, this language is misleading as the Biot–Savart law applies only to steady currents and a point charge moving in space does not constitute a steady current.
3533:
The electric current equation can be viewed as a convective current of electric charge that involves linear motion. By analogy, the magnetic equation is an inductive current involving spin. There is no linear motion in the inductive current along the direction of the
3909:
1506:
2294:
2466:
1946:
3897:{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\iiint _{V}d^{3}{\boldsymbol {\ell }}\,\mathbf {J} ({\boldsymbol {\ell }})\times {\frac {\mathbf {r} -{\boldsymbol {\ell }}}{|\mathbf {r} -{\boldsymbol {\ell }}|^{3}}}}
1718:, since stationary electric currents can only flow around closed paths when they are bounded. However, the law also applies to infinitely long wires (this concept was used in the definition of the SI unit of electric current—the
1151:
2014:
2003:
The formulations given above work well when the current can be approximated as running through an infinitely-narrow wire. If the conductor has some thickness, the proper formulation of the Biot–Savart law (again in
1417:
4885:
components from a moving frame of reference, where there is only an electric field of some distribution of charges, into a stationary frame of reference, in which these charges move. 2) the use of the method of
2628:
4872:
outside the system of continuously distributed currents is emphasized, and then the magnetic field is calculated through the curl from the vector potential. Another approach involves a general solution of the
3169:
4778:
4826:
4183:{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\nabla \times \iiint _{V}d^{3}{\boldsymbol {\ell }}\,{\frac {\mathbf {J} ({\boldsymbol {\ell }})}{|\mathbf {r} -{\boldsymbol {\ell }}|}}}
3065:
1747:). Holding that point fixed, the line integral over the path of the electric current is calculated to find the total magnetic field at that point. The application of this law implicitly relies on the
3435:
was a weighted vorticity that was weighted for the density of the vortex sea. Maxwell considered magnetic permeability μ to be a measure of the density of the vortex sea. Hence the relationship,
4497:
3699:
3513:
3626:
is the perpendicular distance between the point and the vortex line. This is similar to the magnetic field produced on a plane by an infinitely long straight thin wire normal to the plane.
3472:
3616:
2290:
1674:
1065:
2923:
1029:
1446:
1302:
1977:
1831:
1468:
1354:
1102:
3541:
In aerodynamics the induced air currents form solenoidal rings around a vortex axis. Analogy can be made that the vortex axis is playing the role that electric current plays in
3034:
2176:
1705:
4881:
for the electromagnetic force acting from one charged particle on another particle. Two other ways of deriving the Biot–Savart law include: 1) Lorentz transformation of the
4016:{\displaystyle {\frac {\mathbf {r} -{\boldsymbol {\ell }}}{|\mathbf {r} -{\boldsymbol {\ell }}|^{3}}}=-\nabla \left({\frac {1}{|\mathbf {r} -{\boldsymbol {\ell }}|}}\right)}
4224:
4051:
3304:
3007:
2461:
2247:
2198:
1745:
1490:
3556:
lines form solenoidal rings around the source electric current, whereas in aerodynamics, the air currents (velocity) form solenoidal rings around the source vortex axis.
2959:
1979:
is the unit vector of along the center-line of the loop (and the loop is taken to be centered at the origin). Loops such as the one described appear in devices like the
3399:
5017:
4874:
4829:
1633:{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int _{C}{\frac {I\,d{\boldsymbol {\ell }}\times \mathbf {{\hat {r}}'} }{|\mathbf {r'} |^{2}}}}
3379:
3356:
2221:
1806:
2979:
1826:
1783:
1374:
1322:
5286:
Daniel Zile and James
Overdui. Derivation of the Biot-Savart Law from Coulomb’s Law and Implications for Gravity. APS April Meeting 2014, abstract id. D1.033.
595:
2423:{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\iiint _{V}\ dV{\frac {\mathbf {J} \times \mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}}
1269:{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int _{C}{\frac {I\,d{\boldsymbol {\ell }}\times \mathbf {r'} }{|\mathbf {r'} |^{3}}}}
568:
3559:
Hence in electromagnetism, the vortex plays the role of 'effect' whereas in aerodynamics, the vortex plays the role of 'cause'. Yet when we look at the
2583:{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}I\int _{C}{\frac {d{\boldsymbol {\ell }}\times \mathbf {r'} }{|\mathbf {r'} |^{3}}}}
983:
In the following equations, it is assumed that the medium is not magnetic (e.g., vacuum). This allows for straightforward derivation of magnetic field
2141:{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\iiint _{V}\ {\frac {(\mathbf {J} \,dV)\times \mathbf {r} '}{|\mathbf {r} '|^{3}}}}
4737:
580:
5182:
4949:
5253:
See
Jackson, page 178–79 or Griffiths p. 222–24. The presentation in Griffiths is particularly thorough, with all the details spelled out.
1991:
spacecraft propulsion system. Calculation of the magnetic field at points off the center line requires more complex mathematics involving
2925:
is the unit vector pointing from the current (non-retarded) position of the particle to the point at which the field is being measured,
831:
600:
3632:
1379:
3482:
4787:
3444:
610:
3538:
vector. The magnetic inductive current represents lines of force. In particular, it represents lines of inverse square law force.
3580:
1129:
which does not change with time and the charge neither accumulates nor depletes at any point. The law is a physical example of a
435:
3545:. This puts the air currents of aerodynamics (fluid velocity field) into the equivalent role of the magnetic induction vector
5376:
5335:
5151:
5046:
4877:
for the vector potential in the case of constant currents. The magnetic field can also be calculated as a consequence of the
450:
445:
72:
4860:
Initially, the Biot–Savart law was discovered experimentally, then this law was derived in different ways theoretically. In
460:
3271:{\displaystyle \mathbf {B} ={\mu _{0} \over 4\pi }q{\mathbf {v} \times {\hat {\mathbf {r} }}' \over |\mathbf {r} '|^{2}}}
5463:
4997:
3160:{\displaystyle \mathbf {E} ={\frac {q}{4\pi \varepsilon _{0}}}\ {\frac {\mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}}
5404:
5387:
5366:
5356:
5314:
5116:
4972:
3315:
The Biot–Savart law can be used in the calculation of magnetic responses even at the atomic or molecular level, e.g.
62:
4197:
4024:
330:
5422:
5273:
David Tong. Lectures on
Electromagnetism. University of Cambridge, Part IB and Part II Mathematical Tripos (2015).
5382:
Physics for
Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008,
5057:
The superposition principle holds for the electric and magnetic fields because they are the solution to a set of
4861:
245:
824:
590:
67:
1725:
To apply the equation, the point in space where the magnetic field is to be calculated is arbitrarily chosen (
1006:
944:. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current.
605:
310:
5041:
Electromagnetism (2nd
Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008,
2259:
1643:
1426:
1282:
1034:
5396:, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
5058:
4845:
4193:
3736:
1951:
1451:
1337:
960:
470:
210:
77:
4849:
3740:
2894:
956:
200:
763:
638:
535:
510:
430:
3629:
This is a limiting case of the formula for vortex segments of finite length (similar to a finite wire):
3420:
application, the roles of vorticity and current are reversed in comparison to the magnetic application.
3722:
263:
5186:
5458:
4869:
3323:, provided that the current density can be obtained from a quantum mechanical calculation or theory.
1070:
817:
778:
305:
295:
235:
230:
170:
4841:
3012:
2154:
1683:
964:
315:
4207:
4034:
3287:
2990:
2444:
2230:
2181:
1728:
1473:
748:
250:
4904:
4878:
3331:
3320:
1748:
999:
628:
155:
145:
140:
5108:
5102:
2930:
753:
723:
5062:
4882:
3037:
2613:
575:
345:
120:
5076:
1941:{\displaystyle \mathbf {B} ={\mu _{0}IR^{2} \over 2(x^{2}+R^{2})^{3/2}}{\hat {\mathbf {x} }},}
5453:
5361:
Essential
Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray,
3712:
3384:
673:
360:
350:
300:
290:
5133:
4914:
4731:
4719:
1330:
798:
698:
663:
415:
280:
180:
165:
100:
1988:
37:
8:
4944:
4909:
1420:
758:
738:
733:
540:
525:
410:
380:
275:
5263:
The
Feynman Lectures on Physics Vol. II Ch. 14: The Magnetic Field in Various Situations
5206:
See the cautionary footnote in
Griffiths p. 219 or the discussion in Jackson p. 175–176.
3361:
3338:
2203:
1788:
5371:
The
Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010,
5351:
Electricity and Modern
Physics (2nd Edition), G.A.G. Bennet, Edward Arnold (UK), 1974,
5304:
4887:
4865:
3718:
3526:
was seen as a kind of magnetic current of vortices aligned in their axial planes, with
2964:
1992:
1811:
1768:
1359:
1307:
968:
633:
373:
175:
135:
5432:
455:
5400:
5383:
5372:
5362:
5352:
5331:
5310:
5217:
5147:
5112:
5042:
4993:
4968:
4926:
4840:-magnetostatic situation, the Biot–Savart law ceases to be true (it is superseded by
1708:
1500:
1125:(for example due to a wire). A steady (or stationary) current is a continual flow of
693:
3577:, for a vortex line of infinite length, the induced velocity at a point is given by
5139:
3281:
3041:
2599:
1119:
941:
899:
859:
849:
793:
708:
668:
658:
545:
500:
483:
400:
335:
105:
29:
5262:
3474:
was essentially a rotational analogy to the linear electric current relationship,
2250:
1126:
728:
653:
648:
515:
390:
355:
215:
115:
5287:
972:
963:. When magnetostatics does not apply, the Biot–Savart law should be replaced by
768:
5393:
5274:
4781:
4727:
3423:
In Maxwell's 1861 paper 'On Physical Lines of Force', magnetic field strength
3316:
2617:
2224:
1980:
1108:
952:
948:
937:
688:
683:
505:
395:
320:
270:
220:
193:
150:
125:
95:
88:
5143:
1828:
along the center line of the loop, the magnetic field vector at that point is:
5447:
3728:
1755:
of the field created by each infinitesimal section of the wire individually.
1325:
1130:
803:
788:
773:
713:
425:
340:
325:
240:
225:
130:
3709:
are the (signed) angles between the point and the two ends of the segment.
1715:
1142:
783:
678:
643:
585:
520:
440:
405:
285:
160:
3713:
The Biot–Savart law, Ampère's circuital law, and Gauss's law for magnetism
3417:
3406:
1998:
1677:
703:
555:
385:
47:
3563:
lines in isolation, we see exactly the aerodynamic scenario insomuch as
1758:
5399:
McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994,
3045:
1752:
420:
5325:
3542:
3428:
1137:
in which the electric currents flow (e.g. the wire). The equation in
743:
718:
530:
52:
2606:
1984:
495:
490:
110:
4196:. Next, taking the curl of both sides, using the formula for the
845:
465:
4192:
Since the divergence of a curl is always zero, this establishes
1751:
for magnetic fields, i.e. the fact that the magnetic field is a
1412:{\displaystyle \mathbf {r'} =\mathbf {r} -{\boldsymbol {\ell }}}
994:
5417:
4773:{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} }
3410:
3062:, the electric field and magnetic field can be approximated as
1719:
550:
57:
868:
4821:{\displaystyle \partial \mathbf {E} /\partial t=\mathbf {0} }
1765:
For example, consider the magnetic field of a loop of radius
3571:
is the circumferential velocity as in Maxwell's 1861 paper.
928:
888:
3735:
as calculated from the Biot–Savart law will always satisfy
919:
911:
879:
871:
5018:"Electromagnetic Field Theory: A Problem Solving Approach"
5437:
3358:) induced at a point P by an element of vortex filament (
908:
1107:
The Biot–Savart law is used for computing the resultant
4780:
i.e. Ampère's circuital law. (Due to the assumption of
2005:
1138:
1999:
Electric current density (throughout conductor volume)
4790:
4740:
4495:
4232:
4210:
4059:
4037:
3912:
3756:
3635:
3583:
3485:
3447:
3387:
3364:
3341:
3290:
3172:
3068:
3036:. Alternatively, these can be derived by considering
3015:
2993:
2967:
2933:
2897:
2626:
2469:
2447:
2297:
2262:
2233:
2206:
2184:
2157:
2017:
1954:
1834:
1814:
1791:
1771:
1731:
1686:
1646:
1509:
1476:
1454:
1429:
1382:
1362:
1340:
1310:
1285:
1154:
1073:
1037:
1009:
916:
876:
3530:
being the circumferential velocity of the vortices.
2593:
2590:
i.e., the current can be taken out of the integral.
925:
905:
885:
865:
5249:
5247:
5245:
5243:
5241:
4962:
4868:outside the static distribution of charges and the
3284:in 1888. Some authors call the above equation for
1995:that require numerical solution or approximations.
1470:
to the point at which the field is being computed (
922:
902:
882:
862:
4864:, at first, the similarity of expressions for the
4820:
4772:
4704:
4479:
4218:
4182:
4045:
4015:
3896:
3693:
3610:
3507:
3466:
3393:
3373:
3350:
3310:
3298:
3270:
3159:
3028:
3001:
2973:
2953:
2917:
2880:
2582:
2455:
2437:In the special case of a uniform constant current
2422:
2284:
2241:
2215:
2192:
2170:
2140:
1971:
1940:
1820:
1800:
1777:
1739:
1699:
1668:
1632:
1484:
1462:
1440:
1411:
1368:
1348:
1316:
1296:
1268:
1096:
1059:
1023:
5065:, where the current is one of the "source terms".
3115:
2905:
2757:
2377:
2271:
1655:
1587:
1046:
5445:
5238:
3694:{\displaystyle v={\frac {\Gamma }{4\pi r}}\left}
5171:(4th ed.). Pearson Higher Ed. p. 800.
3508:{\displaystyle \mathbf {J} =\rho \mathbf {v} ,}
2253:vector in that volume (in SI in units of A/m).
2178:is the vector from dV to the observation point
4987:
5288:https://doi.org/10.1103/BAPS.2014.APRIL.D1.33
5081:Journal of the British Interplanetary Society
3467:{\displaystyle \mathbf {B} =\mu \mathbf {H} }
995:Electric currents (along a closed curve/wire)
825:
5275:http://www.damtp.cam.ac.uk/user/tong/em.html
4967:(3rd ed.). New York: Wiley. Chapter 5.
4950:Random House Webster's Unabridged Dictionary
3409:theory to calculate the velocity induced by
3326:
3048:form) in the source charge's inertial frame.
975:, who discovered this relationship in 1820.
5183:"Magnetic Field from a Moving Point Charge"
5096:
5094:
3611:{\displaystyle v={\frac {\Gamma }{2\pi r}}}
2432:
832:
818:
36:
5436:by Orilla McHarris and Peter Signell for
5302:
5135:Classical Electromagnetism via Relativity
5100:
4855:
4722:), using the fact that the divergence of
4410:
4297:
4132:
3823:
2083:
1567:
1212:
5392:Encyclopaedia of Physics (2nd Edition),
5107:(3rd ed.). Prentice Hall. pp.
5091:
5074:
4990:The Classical Theory of Fields: Volume 2
3330:
1757:
1328:element of the wire in the direction of
998:
5323:
4992:(4th ed.). Butterworth-Heinemann.
4988:Landau, L. D.; Lifshitz, E. M. (1980).
4691:
4644:
4587:
4558:
4527:
4461:
4420:
4406:
4344:
4307:
4293:
4168:
4145:
4128:
3997:
3945:
3925:
3875:
3855:
3833:
3819:
2531:
1572:
1456:
1434:
1405:
1342:
1290:
1217:
1118:in 3D-space generated by a filamentary
1024:{\displaystyle Id{\boldsymbol {\ell }}}
1017:
987:, while the fundamental vector here is
581:Electromagnetism and special relativity
5446:
5166:
5131:
3280:These equations were first derived by
2616:give the following expression for the
2285:{\displaystyle \mathbf {{\hat {r}}'} }
1669:{\displaystyle \mathbf {{\hat {r}}'} }
1441:{\displaystyle d{\boldsymbol {\ell }}}
1297:{\displaystyle d{\boldsymbol {\ell }}}
1060:{\displaystyle \mathbf {{\hat {r}}'} }
947:The Biot–Savart law is fundamental to
1972:{\displaystyle {\hat {\mathbf {x} }}}
1463:{\displaystyle {\boldsymbol {\ell }}}
1349:{\displaystyle {\boldsymbol {\ell }}}
1324:whose magnitude is the length of the
601:Maxwell equations in curved spacetime
5169:Physics for Scientists and Engineers
5087:: 306–323 – via bis-space.com.
5015:
5011:
5009:
4053:, this equation can be rewritten as
3405:The Biot–Savart law is also used in
2918:{\displaystyle \mathbf {\hat {r}} '}
18:Important law of classical magnetism
5215:
4920:
4489:Finally, plugging in the relations
3750:Starting with the Biot–Savart law:
13:
5433:The Ampère–Laplace–Biot–Savart Law
5345:
5068:
4804:
4791:
4741:
4726:is zero (due to the assumption of
4611:
4554:
4500:
4428:
4317:
4269:
4233:
4101:
3970:
3644:
3622:is the strength of the vortex and
3592:
3388:
14:
5475:
5410:
5006:
2594:Point charge at constant velocity
1714:The integral is usually around a
1707:. The symbols in boldface denote
5416:
5330:(2nd ed.). Addison-Wesley.
4814:
4795:
4766:
4748:
4683:
4636:
4579:
4519:
4453:
4412:
4336:
4299:
4240:
4212:
4200:, and again using the fact that
4160:
4137:
4069:
4061:
4039:
3989:
3937:
3917:
3867:
3847:
3825:
3766:
3758:
3498:
3487:
3460:
3449:
3292:
3245:
3222:
3210:
3174:
3134:
3112:
3070:
3018:
2995:
2902:
2870:
2862:
2833:
2824:
2816:
2804:
2776:
2754:
2632:
2557:
2540:
2479:
2471:
2449:
2397:
2374:
2362:
2307:
2299:
2268:
2235:
2186:
2160:
2115:
2098:
2079:
2027:
2019:
1959:
1925:
1836:
1733:
1689:
1652:
1607:
1584:
1519:
1511:
1478:
1397:
1385:
1243:
1226:
1164:
1156:
1133:, being evaluated over the path
1081:
1043:
936:) is an equation describing the
898:
858:
5327:The Feynman Lectures on Physics
5309:(3rd ed.). Prentice Hall.
5306:Introduction to Electrodynamics
5280:
5267:
5256:
5209:
5200:
5175:
5160:
5104:Introduction to Electrodynamics
4862:The Feynman Lectures on Physics
4226:, we eventually get the result
3427:was directly equated with pure
3335:The figure shows the velocity (
3311:Magnetic responses applications
1097:{\displaystyle |\mathbf {r'} |}
5125:
5051:
5035:
4981:
4956:
4938:
4695:
4679:
4649:
4631:
4592:
4574:
4532:
4514:
4466:
4448:
4424:
4416:
4349:
4331:
4311:
4303:
4173:
4155:
4149:
4141:
4073:
4065:
4002:
3984:
3951:
3932:
3881:
3862:
3837:
3829:
3770:
3762:
3731:situation, the magnetic field
3255:
3239:
3226:
3144:
3128:
2786:
2770:
2567:
2551:
2483:
2475:
2407:
2391:
2311:
2303:
2125:
2109:
2090:
2075:
2031:
2023:
1963:
1929:
1901:
1874:
1617:
1601:
1523:
1515:
1253:
1237:
1168:
1160:
1090:
1075:
1:
5296:
5059:linear differential equations
4734:, the result turns out to be
3029:{\displaystyle \mathbf {r} '}
2171:{\displaystyle \mathbf {r'} }
1700:{\displaystyle \mathbf {r'} }
606:Relativistic electromagnetism
5303:Griffiths, David J. (1998).
5218:"On Physical Lines of Force"
5101:Griffiths, David J. (1998).
4963:Jackson, John David (1999).
4219:{\displaystyle \mathbf {r} }
4046:{\displaystyle \mathbf {r} }
3299:{\displaystyle \mathbf {B} }
3002:{\displaystyle \mathbf {v} }
2456:{\displaystyle \mathbf {B} }
2242:{\displaystyle \mathbf {J} }
2193:{\displaystyle \mathbf {r} }
1740:{\displaystyle \mathbf {r} }
1485:{\displaystyle \mathbf {r} }
1003:Shown are the directions of
7:
4893:
4875:inhomogeneous wave equation
4027:, as well as the fact that
3519:is electric charge density.
3477:Electric convection current
1304:is a vector along the path
978:
953:magnetostatic approximation
10:
5480:
3906:Substituting the relation
3723:vector calculus identities
3716:
3439:Magnetic induction current
2954:{\displaystyle \beta =v/c}
331:Liénard–Wiechert potential
5464:Eponymous laws of physics
5324:Feynman, Richard (2005).
5144:10.1007/978-1-4899-6559-2
5132:Rosser, W. G. V. (1968).
4965:Classical Electrodynamics
4898:
4870:magnetic vector potential
4846:Gauss's law for magnetism
4830:displacement current term
4194:Gauss's law for magnetism
3737:Gauss's law for magnetism
3327:Aerodynamics applications
3321:magnetic susceptibilities
2961:is the speed in units of
967:. The law is named after
961:Gauss's law for magnetism
955:and consistent with both
596:Mathematical descriptions
306:Electromagnetic radiation
296:Electromagnetic induction
236:Magnetic vector potential
231:Magnetic scalar potential
5167:Knight, Randall (2017).
5077:"Mathematics of Magsail"
4932:
3552:In electromagnetism the
2433:Constant uniform current
2256:In terms of unit vector
940:generated by a constant
5075:Freeland, R.M. (2015).
4879:Lorentz transformations
4828:, so there is no extra
3567:is the vortex axis and
3394:{\displaystyle \Gamma }
2598:In the case of a point
1808:For a point a distance
1749:superposition principle
1423:from the wire element (
146:Electrostatic induction
141:Electrostatic discharge
4883:electromagnetic tensor
4856:Theoretical background
4822:
4774:
4706:
4481:
4220:
4184:
4047:
4025:product rule for curls
4017:
3898:
3741:Ampère's circuital law
3695:
3612:
3509:
3468:
3402:
3395:
3375:
3352:
3300:
3272:
3161:
3038:Lorentz transformation
3030:
3003:
2975:
2955:
2919:
2882:
2584:
2457:
2424:
2286:
2243:
2217:
2194:
2172:
2142:
1973:
1942:
1822:
1802:
1779:
1762:
1741:
1701:
1670:
1634:
1486:
1464:
1442:
1413:
1370:
1350:
1318:
1298:
1270:
1104:
1098:
1061:
1025:
957:Ampère's circuital law
576:Electromagnetic tensor
5016:Zhan, Marcus (2003).
4842:Jefimenko's equations
4823:
4775:
4730:), and performing an
4707:
4482:
4221:
4185:
4048:
4018:
3899:
3696:
3613:
3549:in electromagnetism.
3510:
3469:
3396:
3376:
3353:
3334:
3301:
3273:
3162:
3031:
3004:
2987:is the angle between
2976:
2956:
2920:
2883:
2605:moving at a constant
2585:
2458:
2441:, the magnetic field
2425:
2287:
2244:
2218:
2195:
2173:
2143:
1974:
1943:
1823:
1803:
1780:
1761:
1742:
1722:—until 20 May 2019).
1702:
1671:
1635:
1487:
1465:
1443:
1414:
1371:
1351:
1319:
1299:
1271:
1109:magnetic flux density
1099:
1062:
1026:
1002:
965:Jefimenko's equations
951:. It is valid in the
569:Covariant formulation
361:Synchrotron radiation
301:Electromagnetic pulse
291:Electromagnetic field
5425:at Wikimedia Commons
4915:Pierre-Simon Laplace
4788:
4738:
4732:integration by parts
4720:Dirac delta function
4493:
4230:
4208:
4057:
4035:
3910:
3754:
3633:
3581:
3483:
3445:
3385:
3362:
3339:
3288:
3170:
3066:
3013:
2991:
2965:
2931:
2895:
2624:
2620:and magnetic field:
2467:
2445:
2295:
2260:
2231:
2204:
2182:
2155:
2015:
1952:
1832:
1812:
1789:
1769:
1729:
1684:
1644:
1507:
1474:
1452:
1427:
1380:
1360:
1338:
1331:conventional current
1308:
1283:
1152:
1071:
1035:
1007:
611:Stress–energy tensor
536:Reluctance (complex)
281:Displacement current
5063:Maxwell's equations
4910:James Clerk Maxwell
4888:retarded potentials
4204:does not depend on
4031:does not depend on
3317:chemical shieldings
2614:Maxwell's equations
1785:carrying a current
1421:displacement vector
1356:is a point on path
1067:, and the value of
526:Magnetomotive force
411:Electromotive force
381:Alternating current
316:Jefimenko equations
276:Cyclotron radiation
5138:. pp. 29–42.
4905:André-Marie Ampère
4866:electric potential
4850:Maxwell–Ampère law
4832:in Ampère's law.)
4818:
4770:
4702:
4700:
4477:
4216:
4180:
4043:
4013:
3894:
3748:
3719:Curl (mathematics)
3691:
3608:
3505:
3464:
3403:
3391:
3374:{\displaystyle dl}
3371:
3351:{\displaystyle dV}
3348:
3296:
3268:
3157:
3026:
2999:
2971:
2951:
2915:
2878:
2876:
2580:
2453:
2420:
2282:
2239:
2216:{\displaystyle dV}
2213:
2190:
2168:
2138:
1993:elliptic integrals
1969:
1938:
1818:
1801:{\displaystyle I.}
1798:
1775:
1763:
1737:
1697:
1666:
1630:
1482:
1460:
1438:
1409:
1366:
1346:
1314:
1294:
1266:
1105:
1094:
1057:
1021:
969:Jean-Baptiste Biot
374:Electrical network
211:Gauss magnetic law
176:Static electricity
136:Electric potential
5430:
5421:Media related to
5377:978-0-521-57507-2
5337:978-0-8053-9045-2
5225:Wikimedia commons
5153:978-1-4899-6258-4
5047:978-0-471-92712-9
4945:"Biot–Savart law"
4927:Darwin Lagrangian
4654:
4597:
4537:
4471:
4383:
4354:
4267:
4178:
4099:
4007:
3962:
3892:
3796:
3746:
3658:
3606:
3575:In two dimensions
3266:
3229:
3201:
3155:
3118:
3104:
3100:
2974:{\displaystyle c}
2908:
2859:
2797:
2760:
2745:
2666:
2578:
2509:
2418:
2380:
2351:
2337:
2274:
2136:
2071:
2057:
1966:
1932:
1919:
1821:{\displaystyle x}
1778:{\displaystyle R}
1709:vector quantities
1658:
1628:
1590:
1549:
1503:. Alternatively:
1501:magnetic constant
1369:{\displaystyle C}
1317:{\displaystyle C}
1264:
1194:
1049:
842:
841:
541:Reluctance (real)
511:Gyrator–capacitor
456:Resonant cavities
346:Maxwell equations
5471:
5459:Electromagnetism
5428:
5420:
5341:
5320:
5291:
5284:
5278:
5271:
5265:
5260:
5254:
5251:
5236:
5235:
5233:
5231:
5222:
5213:
5207:
5204:
5198:
5197:
5195:
5194:
5185:. Archived from
5179:
5173:
5172:
5164:
5158:
5157:
5129:
5123:
5122:
5109:222–224, 435–440
5098:
5089:
5088:
5072:
5066:
5055:
5049:
5039:
5033:
5032:
5030:
5028:
5013:
5004:
5003:
4985:
4979:
4978:
4960:
4954:
4942:
4921:Electromagnetism
4852:are still true.
4827:
4825:
4824:
4819:
4817:
4803:
4798:
4779:
4777:
4776:
4771:
4769:
4764:
4763:
4751:
4717:
4711:
4709:
4708:
4703:
4701:
4694:
4686:
4659:
4655:
4653:
4652:
4647:
4639:
4634:
4625:
4619:
4618:
4602:
4598:
4596:
4595:
4590:
4582:
4577:
4568:
4562:
4561:
4542:
4538:
4536:
4535:
4530:
4522:
4517:
4508:
4486:
4484:
4483:
4478:
4476:
4472:
4470:
4469:
4464:
4456:
4451:
4442:
4436:
4435:
4423:
4415:
4409:
4404:
4403:
4394:
4393:
4384:
4382:
4374:
4373:
4364:
4359:
4355:
4353:
4352:
4347:
4339:
4334:
4325:
4310:
4302:
4296:
4291:
4290:
4281:
4280:
4268:
4266:
4258:
4257:
4248:
4243:
4225:
4223:
4222:
4217:
4215:
4189:
4187:
4186:
4181:
4179:
4177:
4176:
4171:
4163:
4158:
4152:
4148:
4140:
4134:
4131:
4126:
4125:
4116:
4115:
4100:
4098:
4090:
4089:
4080:
4072:
4064:
4052:
4050:
4049:
4044:
4042:
4022:
4020:
4019:
4014:
4012:
4008:
4006:
4005:
4000:
3992:
3987:
3978:
3963:
3961:
3960:
3959:
3954:
3948:
3940:
3935:
3929:
3928:
3920:
3914:
3903:
3901:
3900:
3895:
3893:
3891:
3890:
3889:
3884:
3878:
3870:
3865:
3859:
3858:
3850:
3844:
3836:
3828:
3822:
3817:
3816:
3807:
3806:
3797:
3795:
3787:
3786:
3777:
3769:
3761:
3700:
3698:
3697:
3692:
3690:
3686:
3659:
3657:
3643:
3621:
3617:
3615:
3614:
3609:
3607:
3605:
3591:
3514:
3512:
3511:
3506:
3501:
3490:
3473:
3471:
3470:
3465:
3463:
3452:
3431:(spin), whereas
3400:
3398:
3397:
3392:
3380:
3378:
3377:
3372:
3357:
3355:
3354:
3349:
3305:
3303:
3302:
3297:
3295:
3282:Oliver Heaviside
3277:
3275:
3274:
3269:
3267:
3265:
3264:
3263:
3258:
3252:
3248:
3242:
3236:
3235:
3231:
3230:
3225:
3220:
3213:
3207:
3202:
3200:
3192:
3191:
3182:
3177:
3166:
3164:
3163:
3158:
3156:
3154:
3153:
3152:
3147:
3141:
3137:
3131:
3125:
3124:
3120:
3119:
3111:
3106:
3102:
3101:
3099:
3098:
3097:
3078:
3073:
3061:
3035:
3033:
3032:
3027:
3025:
3021:
3008:
3006:
3005:
3000:
2998:
2986:
2980:
2978:
2977:
2972:
2960:
2958:
2957:
2952:
2947:
2924:
2922:
2921:
2916:
2914:
2910:
2909:
2901:
2887:
2885:
2884:
2879:
2877:
2873:
2865:
2860:
2858:
2857:
2845:
2836:
2827:
2819:
2807:
2798:
2796:
2795:
2794:
2789:
2783:
2779:
2773:
2767:
2766:
2762:
2761:
2753:
2748:
2746:
2744:
2743:
2739:
2730:
2726:
2719:
2718:
2709:
2708:
2687:
2686:
2685:
2669:
2667:
2665:
2664:
2663:
2644:
2635:
2600:charged particle
2589:
2587:
2586:
2581:
2579:
2577:
2576:
2575:
2570:
2564:
2563:
2554:
2548:
2547:
2546:
2534:
2525:
2523:
2522:
2510:
2508:
2500:
2499:
2490:
2482:
2474:
2462:
2460:
2459:
2454:
2452:
2429:
2427:
2426:
2421:
2419:
2417:
2416:
2415:
2410:
2404:
2400:
2394:
2388:
2387:
2386:
2382:
2381:
2373:
2365:
2359:
2349:
2348:
2347:
2338:
2336:
2328:
2327:
2318:
2310:
2302:
2291:
2289:
2288:
2283:
2281:
2280:
2276:
2275:
2267:
2248:
2246:
2245:
2240:
2238:
2222:
2220:
2219:
2214:
2199:
2197:
2196:
2191:
2189:
2177:
2175:
2174:
2169:
2167:
2166:
2147:
2145:
2144:
2139:
2137:
2135:
2134:
2133:
2128:
2122:
2118:
2112:
2106:
2105:
2101:
2082:
2073:
2069:
2068:
2067:
2058:
2056:
2048:
2047:
2038:
2030:
2022:
1978:
1976:
1975:
1970:
1968:
1967:
1962:
1957:
1947:
1945:
1944:
1939:
1934:
1933:
1928:
1923:
1920:
1918:
1917:
1916:
1912:
1899:
1898:
1886:
1885:
1869:
1868:
1867:
1855:
1854:
1844:
1839:
1827:
1825:
1824:
1819:
1807:
1805:
1804:
1799:
1784:
1782:
1781:
1776:
1746:
1744:
1743:
1738:
1736:
1706:
1704:
1703:
1698:
1696:
1695:
1675:
1673:
1672:
1667:
1665:
1664:
1660:
1659:
1651:
1639:
1637:
1636:
1631:
1629:
1627:
1626:
1625:
1620:
1614:
1613:
1604:
1598:
1597:
1596:
1592:
1591:
1583:
1575:
1562:
1560:
1559:
1550:
1548:
1540:
1539:
1530:
1522:
1514:
1491:
1489:
1488:
1483:
1481:
1469:
1467:
1466:
1461:
1459:
1447:
1445:
1444:
1439:
1437:
1418:
1416:
1415:
1410:
1408:
1400:
1392:
1391:
1375:
1373:
1372:
1367:
1355:
1353:
1352:
1347:
1345:
1323:
1321:
1320:
1315:
1303:
1301:
1300:
1295:
1293:
1275:
1273:
1272:
1267:
1265:
1263:
1262:
1261:
1256:
1250:
1249:
1240:
1234:
1233:
1232:
1220:
1207:
1205:
1204:
1195:
1193:
1185:
1184:
1175:
1167:
1159:
1103:
1101:
1100:
1095:
1093:
1088:
1087:
1078:
1066:
1064:
1063:
1058:
1056:
1055:
1051:
1050:
1042:
1030:
1028:
1027:
1022:
1020:
942:electric current
935:
934:
931:
930:
927:
924:
921:
918:
914:
913:
910:
907:
904:
895:
894:
891:
890:
887:
884:
881:
878:
874:
873:
870:
867:
864:
850:electromagnetism
834:
827:
820:
501:Electric machine
484:Magnetic circuit
446:Parallel circuit
436:Network analysis
401:Electric current
336:London equations
181:Triboelectricity
171:Potential energy
40:
30:Electromagnetism
21:
20:
5479:
5478:
5474:
5473:
5472:
5470:
5469:
5468:
5444:
5443:
5438:Project PHYSNET
5423:Biot-Savart law
5413:
5348:
5346:Further reading
5338:
5317:
5299:
5294:
5285:
5281:
5272:
5268:
5261:
5257:
5252:
5239:
5229:
5227:
5220:
5216:Maxwell, J. C.
5214:
5210:
5205:
5201:
5192:
5190:
5181:
5180:
5176:
5165:
5161:
5154:
5130:
5126:
5119:
5099:
5092:
5073:
5069:
5056:
5052:
5040:
5036:
5026:
5024:
5014:
5007:
5000:
4986:
4982:
4975:
4961:
4957:
4943:
4939:
4935:
4923:
4901:
4896:
4858:
4834:
4813:
4799:
4794:
4789:
4786:
4785:
4765:
4759:
4755:
4747:
4739:
4736:
4735:
4713:
4699:
4698:
4690:
4682:
4660:
4648:
4643:
4635:
4630:
4629:
4624:
4620:
4614:
4610:
4607:
4606:
4591:
4586:
4578:
4573:
4572:
4567:
4563:
4557:
4553:
4543:
4531:
4526:
4518:
4513:
4512:
4507:
4503:
4496:
4494:
4491:
4490:
4465:
4460:
4452:
4447:
4446:
4441:
4437:
4431:
4427:
4419:
4411:
4405:
4399:
4395:
4389:
4385:
4375:
4369:
4365:
4363:
4348:
4343:
4335:
4330:
4329:
4324:
4320:
4306:
4298:
4292:
4286:
4282:
4276:
4272:
4259:
4253:
4249:
4247:
4239:
4231:
4228:
4227:
4211:
4209:
4206:
4205:
4172:
4167:
4159:
4154:
4153:
4144:
4136:
4135:
4133:
4127:
4121:
4117:
4111:
4107:
4091:
4085:
4081:
4079:
4068:
4060:
4058:
4055:
4054:
4038:
4036:
4033:
4032:
4001:
3996:
3988:
3983:
3982:
3977:
3973:
3955:
3950:
3949:
3944:
3936:
3931:
3930:
3924:
3916:
3915:
3913:
3911:
3908:
3907:
3885:
3880:
3879:
3874:
3866:
3861:
3860:
3854:
3846:
3845:
3843:
3832:
3824:
3818:
3812:
3808:
3802:
3798:
3788:
3782:
3778:
3776:
3765:
3757:
3755:
3752:
3751:
3725:
3715:
3664:
3660:
3647:
3642:
3634:
3631:
3630:
3619:
3595:
3590:
3582:
3579:
3578:
3497:
3486:
3484:
3481:
3480:
3459:
3448:
3446:
3443:
3442:
3386:
3383:
3382:
3363:
3360:
3359:
3340:
3337:
3336:
3329:
3313:
3291:
3289:
3286:
3285:
3259:
3254:
3253:
3244:
3243:
3238:
3237:
3221:
3219:
3218:
3217:
3209:
3208:
3206:
3193:
3187:
3183:
3181:
3173:
3171:
3168:
3167:
3148:
3143:
3142:
3133:
3132:
3127:
3126:
3110:
3109:
3108:
3107:
3105:
3093:
3089:
3082:
3077:
3069:
3067:
3064:
3063:
3053:
3042:Coulomb's force
3017:
3016:
3014:
3011:
3010:
2994:
2992:
2989:
2988:
2984:
2966:
2963:
2962:
2943:
2932:
2929:
2928:
2900:
2899:
2898:
2896:
2893:
2892:
2875:
2874:
2869:
2861:
2853:
2849:
2844:
2837:
2832:
2829:
2828:
2823:
2815:
2808:
2803:
2800:
2799:
2790:
2785:
2784:
2775:
2774:
2769:
2768:
2752:
2751:
2750:
2749:
2747:
2735:
2731:
2714:
2710:
2704:
2700:
2693:
2689:
2688:
2681:
2677:
2670:
2668:
2659:
2655:
2648:
2643:
2636:
2631:
2627:
2625:
2622:
2621:
2596:
2571:
2566:
2565:
2556:
2555:
2550:
2549:
2539:
2538:
2530:
2526:
2524:
2518:
2514:
2501:
2495:
2491:
2489:
2478:
2470:
2468:
2465:
2464:
2448:
2446:
2443:
2442:
2435:
2411:
2406:
2405:
2396:
2395:
2390:
2389:
2372:
2371:
2370:
2369:
2361:
2360:
2358:
2343:
2339:
2329:
2323:
2319:
2317:
2306:
2298:
2296:
2293:
2292:
2266:
2265:
2264:
2263:
2261:
2258:
2257:
2251:current density
2234:
2232:
2229:
2228:
2205:
2202:
2201:
2185:
2183:
2180:
2179:
2159:
2158:
2156:
2153:
2152:
2149:
2129:
2124:
2123:
2114:
2113:
2108:
2107:
2097:
2096:
2078:
2074:
2072:
2063:
2059:
2049:
2043:
2039:
2037:
2026:
2018:
2016:
2013:
2012:
2001:
1958:
1956:
1955:
1953:
1950:
1949:
1924:
1922:
1921:
1908:
1904:
1900:
1894:
1890:
1881:
1877:
1870:
1863:
1859:
1850:
1846:
1845:
1843:
1835:
1833:
1830:
1829:
1813:
1810:
1809:
1790:
1787:
1786:
1770:
1767:
1766:
1732:
1730:
1727:
1726:
1688:
1687:
1685:
1682:
1681:
1650:
1649:
1648:
1647:
1645:
1642:
1641:
1621:
1616:
1615:
1606:
1605:
1600:
1599:
1582:
1581:
1580:
1579:
1571:
1563:
1561:
1555:
1551:
1541:
1535:
1531:
1529:
1518:
1510:
1508:
1505:
1504:
1498:
1477:
1475:
1472:
1471:
1455:
1453:
1450:
1449:
1433:
1428:
1425:
1424:
1404:
1396:
1384:
1383:
1381:
1378:
1377:
1361:
1358:
1357:
1341:
1339:
1336:
1335:
1309:
1306:
1305:
1289:
1284:
1281:
1280:
1277:
1257:
1252:
1251:
1242:
1241:
1236:
1235:
1225:
1224:
1216:
1208:
1206:
1200:
1196:
1186:
1180:
1176:
1174:
1163:
1155:
1153:
1150:
1149:
1089:
1080:
1079:
1074:
1072:
1069:
1068:
1041:
1040:
1039:
1038:
1036:
1033:
1032:
1016:
1008:
1005:
1004:
997:
981:
915:
901:
897:
875:
861:
857:
854:Biot–Savart law
848:, specifically
838:
809:
808:
624:
616:
615:
571:
561:
560:
516:Induction motor
486:
476:
475:
391:Current density
376:
366:
365:
356:Poynting vector
266:
264:Electrodynamics
256:
255:
251:Right-hand rule
216:Magnetic dipole
206:Biot–Savart law
196:
186:
185:
121:Electric dipole
116:Electric charge
91:
19:
12:
11:
5:
5477:
5467:
5466:
5461:
5456:
5442:
5441:
5426:
5412:
5411:External links
5409:
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5034:
5005:
4999:978-0750627689
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4854:
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4802:
4797:
4793:
4782:magnetostatics
4768:
4762:
4758:
4754:
4750:
4746:
4743:
4728:magnetostatics
4697:
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4678:
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4342:
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4275:
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4265:
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4252:
4246:
4242:
4238:
4235:
4214:
4198:curl of a curl
4175:
4170:
4166:
4162:
4157:
4151:
4147:
4143:
4139:
4130:
4124:
4120:
4114:
4110:
4106:
4103:
4097:
4094:
4088:
4084:
4078:
4075:
4071:
4067:
4063:
4041:
4023:and using the
4011:
4004:
3999:
3995:
3991:
3986:
3981:
3976:
3972:
3969:
3966:
3958:
3953:
3947:
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3641:
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3504:
3500:
3496:
3493:
3489:
3478:
3475:
3462:
3458:
3455:
3451:
3440:
3390:
3381:) of strength
3370:
3367:
3347:
3344:
3328:
3325:
3312:
3309:
3294:
3262:
3257:
3251:
3247:
3241:
3234:
3228:
3224:
3216:
3212:
3205:
3199:
3196:
3190:
3186:
3180:
3176:
3151:
3146:
3140:
3136:
3130:
3123:
3117:
3114:
3096:
3092:
3088:
3085:
3081:
3076:
3072:
3050:
3049:
3024:
3020:
2997:
2982:
2970:
2950:
2946:
2942:
2939:
2936:
2926:
2913:
2907:
2904:
2872:
2868:
2864:
2856:
2852:
2848:
2843:
2840:
2838:
2835:
2831:
2830:
2826:
2822:
2818:
2814:
2811:
2809:
2806:
2802:
2801:
2793:
2788:
2782:
2778:
2772:
2765:
2759:
2756:
2742:
2738:
2734:
2729:
2725:
2722:
2717:
2713:
2707:
2703:
2699:
2696:
2692:
2684:
2680:
2676:
2673:
2662:
2658:
2654:
2651:
2647:
2642:
2639:
2637:
2634:
2630:
2629:
2618:electric field
2595:
2592:
2574:
2569:
2562:
2559:
2553:
2545:
2542:
2537:
2533:
2529:
2521:
2517:
2513:
2507:
2504:
2498:
2494:
2488:
2485:
2481:
2477:
2473:
2451:
2434:
2431:
2414:
2409:
2403:
2399:
2393:
2385:
2379:
2376:
2368:
2364:
2357:
2354:
2346:
2342:
2335:
2332:
2326:
2322:
2316:
2313:
2309:
2305:
2301:
2279:
2273:
2270:
2237:
2225:volume element
2212:
2209:
2188:
2165:
2162:
2132:
2127:
2121:
2117:
2111:
2104:
2100:
2095:
2092:
2089:
2086:
2081:
2077:
2066:
2062:
2055:
2052:
2046:
2042:
2036:
2033:
2029:
2025:
2021:
2010:
2000:
1997:
1981:Helmholtz coil
1965:
1961:
1937:
1931:
1927:
1915:
1911:
1907:
1903:
1897:
1893:
1889:
1884:
1880:
1876:
1873:
1866:
1862:
1858:
1853:
1849:
1842:
1838:
1817:
1797:
1794:
1774:
1735:
1694:
1691:
1663:
1657:
1654:
1624:
1619:
1612:
1609:
1603:
1595:
1589:
1586:
1578:
1574:
1570:
1566:
1558:
1554:
1547:
1544:
1538:
1534:
1528:
1525:
1521:
1517:
1513:
1496:
1480:
1458:
1436:
1432:
1407:
1403:
1399:
1395:
1390:
1387:
1365:
1344:
1313:
1292:
1288:
1260:
1255:
1248:
1245:
1239:
1231:
1228:
1223:
1219:
1215:
1211:
1203:
1199:
1192:
1189:
1183:
1179:
1173:
1170:
1166:
1162:
1158:
1147:
1092:
1086:
1083:
1077:
1054:
1048:
1045:
1019:
1015:
1012:
996:
993:
980:
977:
949:magnetostatics
938:magnetic field
840:
839:
837:
836:
829:
822:
814:
811:
810:
807:
806:
801:
796:
791:
786:
781:
776:
771:
766:
761:
756:
751:
746:
741:
736:
731:
726:
721:
716:
711:
706:
701:
696:
691:
686:
681:
676:
671:
666:
661:
656:
651:
646:
641:
636:
631:
625:
622:
621:
618:
617:
614:
613:
608:
603:
598:
593:
591:Four-potential
588:
583:
578:
572:
567:
566:
563:
562:
559:
558:
553:
548:
543:
538:
533:
528:
523:
518:
513:
508:
506:Electric motor
503:
498:
493:
487:
482:
481:
478:
477:
474:
473:
468:
463:
461:Series circuit
458:
453:
448:
443:
438:
433:
431:Kirchhoff laws
428:
423:
418:
413:
408:
403:
398:
396:Direct current
393:
388:
383:
377:
372:
371:
368:
367:
364:
363:
358:
353:
351:Maxwell tensor
348:
343:
338:
333:
328:
323:
321:Larmor formula
318:
313:
308:
303:
298:
293:
288:
283:
278:
273:
271:Bremsstrahlung
267:
262:
261:
258:
257:
254:
253:
248:
243:
238:
233:
228:
223:
221:Magnetic field
218:
213:
208:
203:
197:
194:Magnetostatics
192:
191:
188:
187:
184:
183:
178:
173:
168:
163:
158:
153:
148:
143:
138:
133:
128:
126:Electric field
123:
118:
113:
108:
103:
98:
96:Charge density
92:
89:Electrostatics
87:
86:
83:
82:
81:
80:
75:
70:
65:
60:
55:
50:
42:
41:
33:
32:
26:
25:
24:Articles about
17:
9:
6:
4:
3:
2:
5476:
5465:
5462:
5460:
5457:
5455:
5452:
5451:
5449:
5439:
5435:
5434:
5427:
5424:
5419:
5415:
5414:
5406:
5405:0-07-051400-3
5402:
5398:
5395:
5391:
5389:
5388:0-7167-8964-7
5385:
5381:
5378:
5374:
5370:
5368:
5367:0-7195-3382-1
5364:
5360:
5358:
5357:0-7131-2459-8
5354:
5350:
5349:
5339:
5333:
5329:
5328:
5322:
5318:
5316:0-13-805326-X
5312:
5308:
5307:
5301:
5300:
5289:
5283:
5276:
5270:
5264:
5259:
5250:
5248:
5246:
5244:
5242:
5226:
5219:
5212:
5203:
5189:on 2009-06-19
5188:
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5128:
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3799:
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3738:
3734:
3730:
3729:magnetostatic
3724:
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3704:
3687:
3683:
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3677:
3674:
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2690:
2682:
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2053:
2050:
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2040:
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2009:
2007:
1996:
1994:
1990:
1986:
1982:
1935:
1913:
1909:
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1197:
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1177:
1171:
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1132:
1131:line integral
1128:
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1117:
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454:
452:
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442:
439:
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434:
432:
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427:
426:Joule heating
424:
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419:
417:
414:
412:
409:
407:
404:
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397:
394:
392:
389:
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382:
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370:
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349:
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344:
342:
341:Lorentz force
339:
337:
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332:
329:
327:
324:
322:
319:
317:
314:
312:
309:
307:
304:
302:
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294:
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282:
279:
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274:
272:
269:
268:
265:
260:
259:
252:
249:
247:
244:
242:
241:Magnetization
239:
237:
234:
232:
229:
227:
226:Magnetic flux
224:
222:
219:
217:
214:
212:
209:
207:
204:
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199:
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195:
190:
189:
182:
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167:
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162:
159:
157:
154:
152:
149:
147:
144:
142:
139:
137:
134:
132:
131:Electric flux
129:
127:
124:
122:
119:
117:
114:
112:
109:
107:
104:
102:
99:
97:
94:
93:
90:
85:
84:
79:
76:
74:
71:
69:
68:Computational
66:
64:
61:
59:
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51:
49:
46:
45:
44:
43:
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35:
34:
31:
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5454:Aerodynamics
5431:
5326:
5305:
5282:
5269:
5258:
5228:. Retrieved
5224:
5211:
5202:
5191:. Retrieved
5187:the original
5177:
5168:
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5037:
5025:. Retrieved
5021:
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3411:vortex lines
3404:
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2255:
2150:
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1764:
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1716:closed curve
1713:
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1419:is the full
1329:
1326:differential
1278:
1148:
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1114:at position
1111:
1106:
988:
984:
982:
973:Félix Savart
946:
853:
843:
586:Four-current
521:Linear motor
406:Electrolysis
286:Eddy current
246:Permeability
205:
166:Polarization
161:Permittivity
15:
5394:R.G. Lerner
5230:25 December
5022:ocw.mit.edu
3418:aerodynamic
3407:aerodynamic
2008:units) is:
1678:unit vector
1448:) at point
556:Transformer
386:Capacitance
311:Faraday law
106:Coulomb law
48:Electricity
5448:Categories
5429:MISN-0-125
5297:References
5193:2009-09-30
3717:See also:
3046:four-force
1987:, and the
1753:vector sum
1143:teslas (T)
623:Scientists
471:Waveguides
451:Resistance
421:Inductance
201:Ampère law
5061:, namely
4844:), while
4805:∂
4792:∂
4757:μ
4745:×
4742:∇
4692:ℓ
4688:−
4677:δ
4674:π
4668:−
4645:ℓ
4641:−
4612:∇
4588:ℓ
4584:−
4559:ℓ
4555:∇
4551:−
4528:ℓ
4524:−
4501:∇
4462:ℓ
4458:−
4429:∇
4421:ℓ
4407:ℓ
4387:∭
4380:π
4367:μ
4361:−
4345:ℓ
4341:−
4318:∇
4315:⋅
4308:ℓ
4294:ℓ
4274:∭
4270:∇
4264:π
4251:μ
4237:×
4234:∇
4169:ℓ
4165:−
4146:ℓ
4129:ℓ
4109:∭
4105:×
4102:∇
4096:π
4083:μ
3998:ℓ
3994:−
3971:∇
3968:−
3946:ℓ
3942:−
3926:ℓ
3922:−
3876:ℓ
3872:−
3856:ℓ
3852:−
3841:×
3834:ℓ
3820:ℓ
3800:∭
3793:π
3780:μ
3681:
3675:−
3669:
3652:π
3645:Γ
3600:π
3593:Γ
3543:magnetism
3495:ρ
3457:μ
3429:vorticity
3389:Γ
3227:^
3215:×
3198:π
3185:μ
3116:^
3091:ε
3087:π
2935:β
2906:^
2867:×
2821:×
2758:^
2724:θ
2721:
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2378:^
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2272:^
2094:×
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1964:^
1930:^
1848:μ
1656:^
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1577:×
1573:ℓ
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1457:ℓ
1435:ℓ
1406:ℓ
1402:−
1343:ℓ
1291:ℓ
1222:×
1218:ℓ
1198:∫
1191:π
1178:μ
1047:^
1018:ℓ
779:Steinmetz
709:Kirchhoff
694:Jefimenko
689:Hopkinson
674:Helmholtz
669:Heaviside
531:Permeance
416:Impedance
156:Insulator
151:Gauss law
101:Conductor
78:Phenomena
73:Textbooks
53:Magnetism
4894:See also
4848:and the
3250:′
3233:′
3139:′
3122:′
3023:′
2912:′
2781:′
2764:′
2607:velocity
2561:′
2544:′
2402:′
2384:′
2278:′
2164:′
2120:′
2103:′
1985:solenoid
1693:′
1662:′
1611:′
1594:′
1389:′
1247:′
1230:′
1085:′
1053:′
979:Equation
804:Wiechert
759:Poynting
649:Einstein
496:DC motor
491:AC motor
326:Lenz law
111:Electret
5027:July 3,
4718:is the
4712:(where
3416:In the
2249:is the
2223:is the
1989:Magsail
1676:is the
1499:is the
1492:), and
1376:, and
1127:charges
1120:current
846:physics
789:Thomson
764:Ritchie
754:Poisson
739:Neumann
734:Maxwell
729:Lorentz
724:Liénard
654:Faraday
639:Coulomb
466:Voltage
441:Ohm law
63:History
5403:
5386:
5375:
5365:
5355:
5334:
5313:
5150:
5115:
5045:
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4971:
4899:People
3701:where
3618:where
3515:where
3103:
2888:where
2350:
2227:, and
2151:where
2070:
1983:, the
1948:where
1720:Ampere
1640:where
1279:where
1141:units
852:, the
774:Singer
769:Savart
749:Ørsted
714:Larmor
704:Kelvin
659:Fizeau
629:Ampère
551:Stator
58:Optics
5221:(PDF)
4933:Notes
4836:In a
3747:Proof
3727:In a
3052:When
799:Weber
794:Volta
784:Tesla
699:Joule
684:Hertz
679:Henry
664:Gauss
546:Rotor
5401:ISBN
5384:ISBN
5373:ISBN
5363:ISBN
5353:ISBN
5332:ISBN
5311:ISBN
5232:2011
5148:ISBN
5113:ISBN
5043:ISBN
5029:2022
4994:ISBN
4969:ISBN
3739:and
3721:and
3705:and
3044:(in
3009:and
971:and
959:and
719:Lenz
644:Davy
634:Biot
5140:doi
4838:non
3678:cos
3666:cos
3319:or
3040:of
2981:and
2712:sin
2463:is
1680:of
1145:is
929:ɑːr
896:or
889:ɑːr
844:In
744:Ohm
5450::
5290:.
5277:.
5240:^
5223:.
5146:.
5111:.
5093:^
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5079:.
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5008:^
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2006:SI
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1139:SI
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912:oʊ
872:oʊ
869:iː
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886:v
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