4482:
4252:
3534:
4687:
54:
4477:{\displaystyle {\begin{aligned}{\frac {1}{\mu _{0}}}(\mathbf {\nabla } \times \mathbf {B} )&=\mathbf {\nabla } \times \left(\mathbf {H} +\mathbf {M} \right)\\&=\mathbf {\nabla } \times \mathbf {H} +\mathbf {J} _{\mathrm {M} }\\&=\mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {P} }+\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}+\mathbf {J} _{\mathrm {M} }.\end{aligned}}}
3341:
4493:
5682:
2778:. The displacement current is justified today because it serves several requirements of an electromagnetic theory: correct prediction of magnetic fields in regions where no free current flows; prediction of wave propagation of electromagnetic fields; and conservation of electric charge in cases where charge density is time-varying. For greater discussion see
4112:
3221:
4835:
2948:
3529:{\displaystyle \mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {D} }+\mathbf {J} _{\mathrm {M} }=\mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {P} }+\mathbf {J} _{\mathrm {M} }+\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}=\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\,,}
4682:{\displaystyle {\begin{aligned}{\frac {1}{\mu _{0}}}(\mathbf {\nabla } \times \mathbf {B} )&=\mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {bound} }+\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\\&=\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}},\end{aligned}}}
1535:
1362:
1824:
as their border. (Imagine a soap film on a wire loop, which can be deformed by blowing on the film). Which of those surfaces is to be chosen? If the loop does not lie in a single plane, for example, there is no one obvious choice. The answer is that it does not matter: in the magnetostatic case, the
2432:
In a dielectric the above contribution to displacement current is present too, but a major contribution to the displacement current is related to the polarization of the individual molecules of the dielectric material. Even though charges cannot flow freely in a dielectric, the charges in molecules
2002:
All current is fundamentally the same, microscopically. Nevertheless, there are often practical reasons for wanting to treat bound current differently from free current. For example, the bound current usually originates over atomic dimensions, and one may wish to take advantage of a simpler theory
1865:
When a material is magnetized (for example, by placing it in an external magnetic field), the electrons remain bound to their respective atoms, but behave as if they were orbiting the nucleus in a particular direction, creating a microscopic current. When the currents from all these atoms are put
3800:
2757:
2773:
Maxwell's original explanation for displacement current focused upon the situation that occurs in dielectric media. In the modern post-aether era, the concept has been extended to apply to situations with no material media present, for example, to the vacuum between the plates of a charging
3979:
4944:
3320:
3085:
4710:
2614:
3069:
2833:
1985:
3694:
2522:
2388:
988:
situation, to continuous steady currents flowing in a closed circuit. For systems with electric fields that change over time, the original law (as given in this section) must be modified to include a term known as
Maxwell's correction (see below).
2762:
The first term on the right hand side is present everywhere, even in a vacuum. It doesn't involve any actual movement of charge, but it nevertheless has an associated magnetic field, as if it were an actual current. Some authors apply the name
1428:
4235:
1250:
2207:
1795:: With the palm of the right-hand toward the area of integration, and the index-finger pointing along the direction of line-integration, the outstretched thumb points in the direction that must be chosen for the vector area
1583:
5030:
Ampère's electrodynamics: analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from
3705:
3074:
On the other hand, treating all charges on the same footing (disregarding whether they are bound or free charges), the generalized Ampère's equation, also called the
Maxwell–Ampère equation, is in integral form (see the
2667:
3968:
4107:{\displaystyle {\begin{aligned}\mathbf {J} _{\mathrm {bound} }&=\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}\\&=\mathbf {J} _{\mathrm {M} }+\mathbf {J} _{\mathrm {P} },\end{aligned}}}
1411:
3886:
3216:{\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{S}\left(\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\cdot \mathrm {d} \mathbf {S} }
4166:
4857:
3232:
2433:
can move a little under the influence of an electric field. The positive and negative charges in molecules separate under the applied field, causing an increase in the state of polarization, expressed as the
2088:
4830:{\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}={\frac {1}{c}}\iint _{S}\left(4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)\cdot \mathrm {d} \mathbf {S} ,}
2300:
2537:
2998:
2943:{\displaystyle \oint _{C}\mathbf {H} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{S}\left(\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}\right)\cdot \mathrm {d} \mathbf {S} }
933:
next to a wire carrying current turned so that the needle was perpendicular to the wire. He investigated and discovered the rules which govern the field around a straight current-carrying wire:
4498:
4257:
3984:
5049:
The "Ampère circuital law" is thus more properly termed the "Ampère–Maxwell law." It is named after Ampère because of his contributions to understanding electric current. Maxwell does not take
2770:
The second term on the right hand side is the displacement current as originally conceived by
Maxwell, associated with the polarization of the individual molecules of the dielectric material.
2132:
2254:
1913:
1791:, meaning the current passing through in one direction, minus the current in the other direction—but either direction could be chosen as positive. These ambiguities are resolved by the
3805:
Note that we are only dealing with the differential forms, not the integral forms, but that is sufficient since the differential and integral forms are equivalent in each case, by the
3629:
2403:
conceived of displacement current as a polarization current in the dielectric vortex sea, which he used to model the magnetic field hydrodynamically and mechanically. He added this
611:
2451:
2319:
584:
596:
969:
generalized these results and others into a single mathematical law. The original form of
Maxwell's circuital law, which he derived as early as 1855 in his paper
1850:
The electric current that arises in the simplest textbook situations would be classified as "free current"—for example, the current that passes through a wire or
1530:{\displaystyle \oint _{C}\mathbf {H} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{S}\mathbf {J} _{\mathrm {f} }\cdot \mathrm {d} \mathbf {S} =I_{\mathrm {f,enc} }}
1357:{\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}=\mu _{0}\iint _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =\mu _{0}I_{\mathrm {enc} }}
4181:
2790:
Next, the circuital equation is extended by including the polarization current, thereby remedying the limited applicability of the original circuital law.
2017:
and the free current only. For a detailed definition of free current and bound current, and the proof that the two formulations are equivalent, see the "
847:
3795:{\displaystyle {\frac {1}{\mu _{0}}}(\mathbf {\nabla } \times \mathbf {B} )=\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\,.}
2154:
2009:
and the microscopic current (which includes free, magnetization and polarization currents), is sometimes put into the equivalent form below in terms of
616:
5457:
1841:, with the same sign convention, is the same. In practice, one usually chooses the most convenient surface (with the given boundary) to integrate over.
1541:
2752:{\displaystyle \mathbf {J} _{\mathrm {D} }=\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}+{\frac {\partial \mathbf {P} }{\partial t}}\,.}
2212:
which is nonzero for a time-varying charge density. An example occurs in a capacitor circuit where time-varying charge densities exist on the plates.
626:
3335:
density as well as conduction and polarization current densities. That is, the current density on the right side of the Ampère–Maxwell equation is:
5059:
3918:
5561:
1368:
4939:{\displaystyle \mathbf {\nabla } \times \mathbf {B} ={\frac {1}{c}}\left(4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right).}
3839:
451:
3315:{\displaystyle \mathbf {\nabla } \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}
5042:
4123:
466:
88:
1889:, and when the bound charges move, the polarization changes, creating another contribution to the "bound current", the polarization current
981:
that produce them. It determines the magnetic field associated with a given current, or the current associated with a given magnetic field.
2408:
895:
5745:
5403:
1137:
of the magnetic field around some path (line integral) due to the current which passes through that enclosed path (surface integral).
5430:
1694:(i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve
346:
5664:
5645:
5598:
5571:
5544:
5524:
5467:
5440:
5413:
5386:
5359:
5332:
5296:
5269:
5153:
5126:
5099:
3619:
Proof that the formulations of the circuital law in terms of free current are equivalent to the formulations involving total current
2043:
78:
5376:
5286:
2609:{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} =\varepsilon _{0}\varepsilon _{\mathrm {r} }\mathbf {E} \,,}
5755:
3064:{\displaystyle \mathbf {\nabla } \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}\,.}
2265:
2029:
There are two important issues regarding the circuital law that require closer scrutiny. First, there is an issue regarding the
261:
5227:
5200:
840:
606:
83:
5722:
1866:
together, they create the same effect as a macroscopic current, circulating perpetually around the magnetized object. This
4986:
4976:
3607:
1719:(that is, a vector with magnitude equal to the area of the infinitesimal surface element, and direction normal to surface
1231:
621:
326:
5217:
5089:
2642:
1687:
486:
476:
461:
226:
93:
526:
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2528:
2003:
intended for larger dimensions. The result is that the more microscopic Ampère's circuital law, expressed in terms of
779:
654:
551:
446:
5143:
3602:
With the addition of the displacement current, Maxwell was able to hypothesize (correctly) that light was a form of
2099:
5686:
5116:
2034:
1980:{\displaystyle \mathbf {J} =\mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {M} }+\mathbf {J} _{\mathrm {P} }\,,}
1885:. When an electric field is applied, the positive and negative bound charges can separate over atomic distances in
970:
861:
279:
3806:
3689:{\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}}
2225:
1754:
There are a number of ambiguities in the above definitions that require clarification and a choice of convention.
1002:
5760:
2793:
Treating free charges separately from bound charges, the equation including
Maxwell's correction in terms of the
2306:
833:
794:
321:
311:
251:
246:
186:
3569:, the charge continuity issue with Ampère's original formulation is no longer a problem. Because of the term in
5701:
5054:
5050:
962:
958:
926:
869:
764:
644:
331:
31:
929:
discovered that an electric current creates a magnetic field around it, when he noticed that the needle of a
5750:
5740:
5657:
Physics for
Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.)
2517:{\displaystyle \mathbf {J} _{\mathrm {D} }={\frac {\partial }{\partial t}}\mathbf {D} (\mathbf {r} ,\,t)\,,}
2138:
1826:
1123:
953:
The strength of the field at any point is inversely proportional to the distance of the point from the wire.
769:
739:
171:
161:
156:
5617:
of the magnetization, so its divergence is zero and it does not contribute to the continuity equation. See
2383:{\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\,.}
997:
The original circuital law can be written in several different forms, which are all ultimately equivalent:
4981:
907:
591:
361:
136:
5618:
5169:
Oersted, H. C. (1820). "Experiments on the effect of a current of electricity on the magnetic needles".
4172:
3332:
1867:
1851:
1134:
1051:
873:
689:
376:
366:
316:
306:
53:
5244:
5484:
5074:
3558:
is the current density contribution actually due to movement of charges, both free and bound. Because
4991:
4961:
3813:
3603:
2779:
2652:
2445:
Both contributions to the displacement current are combined by defining the displacement current as:
2434:
2420:
2404:
2394:
1886:
1859:
1810:
1020:. Other units are possible, but rare. This section will use SI units, with cgs units discussed later.
899:
814:
714:
679:
431:
296:
196:
181:
116:
4956:
221:
5459:
Innovation in
Maxwell's Electromagnetic Theory: Molecular Vortices, Displacement Current, and Light
5324:
2400:
2310:
2145:
2030:
1834:
1001:
An "integral form" and a "differential form". The forms are exactly equivalent, and related by the
966:
887:
774:
754:
749:
556:
541:
426:
396:
291:
5506:
1830:
1742:
1112:
649:
389:
191:
151:
30:"Ampère's law" redirects here. For the law describing forces between current-carrying wires, see
5516:
5510:
5006:
2969:(also called "auxiliary magnetic field", "magnetic field intensity", or just "magnetic field"),
5028:
5660:
5641:
5594:
5567:
5540:
5534:
5520:
5463:
5436:
5409:
5382:
5355:
5328:
5292:
5265:
5259:
5223:
5196:
5190:
5149:
5122:
5095:
5038:
2629:
2215:
Second, there is an issue regarding the propagation of electromagnetic waves. For example, in
1767:
could go around the loop in either direction (clockwise or counterclockwise); the vector area
1080:
978:
911:
709:
1133:), and encloses the current. The mathematical statement of the law is a relation between the
4996:
2775:
1650:
1141:
1119:
1038:. These two forms use the total current density and free current density, respectively. The
957:
This sparked a great deal of research into the relation between electricity and magnetism.
930:
914:
881:
809:
724:
684:
674:
561:
516:
499:
416:
351:
121:
45:
5001:
1792:
1597:
974:
947:
If the direction of the current is reversed, the direction of the magnetic field reverses.
744:
669:
664:
531:
406:
371:
266:
231:
131:
5317:
784:
4230:{\displaystyle \mathbf {J} _{\mathrm {P} }={\frac {\partial \mathbf {P} }{\partial t}},}
5635:
4845:
4701:
2960:
2815:
2012:
1775:
1180:
1145:
1097:
1024:
985:
898:". In 1865 he generalized the equation to apply to time-varying currents by adding the
891:
877:
704:
699:
521:
411:
336:
286:
236:
209:
166:
141:
111:
104:
5711:
5692:
5734:
3892:
1855:
1712:
1683:
1631:
1144:, (which is the sum of both free current and bound current) the line integral of the
1093:
819:
804:
789:
729:
441:
356:
341:
256:
241:
146:
2429:, the displacement current is related to the time rate of change of electric field.
2037:
states that the divergence of the curl of a vector field must always be zero. Hence
1758:
First, three of these terms are associated with sign ambiguities: the line integral
5145:
Physics for
Scientists and Engineers: Foundations and Connections, Extended Version
2989:
1882:
1854:. In contrast, "bound current" arises in the context of bulk materials that can be
1155:
1101:
950:
The strength of the field is directly proportional to the magnitude of the current.
799:
694:
659:
601:
536:
421:
301:
176:
2202:{\displaystyle \nabla \cdot \mathbf {J} =-{\frac {\partial \rho }{\partial t}}\,,}
456:
5588:
5349:
4851:
The differential form of the equation (again, including
Maxwell's correction) is
1578:{\displaystyle \mathbf {\nabla } \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }}
17:
4966:
4704:
units, the integral form of the equation, including
Maxwell's correction, reads
1708:
1668:
961:
investigated the magnetic force between two current-carrying wires, discovering
719:
571:
401:
63:
2426:
2216:
938:
436:
1017:
759:
734:
546:
471:
68:
5053:
as a starting point in deriving any of his equations, although he mentions
3963:{\displaystyle {\frac {1}{\mu _{0}}}\mathbf {B} =\mathbf {H} +\mathbf {M} }
5681:
1406:{\displaystyle \mathbf {\nabla } \times \mathbf {B} =\mu _{0}\mathbf {J} }
5027:
Assis, André Koch Torres; Chaib, J. P. M. C; Ampère, André-Marie (2015).
3881:{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} \,.}
1225:
1013:
511:
506:
126:
1727:
by the right hand rule), see below for further explanation of the curve
5261:
Magnetic Fields: A comprehensive theoretical treatise for practical use
4161:{\displaystyle \mathbf {J} _{\mathrm {M} }=\nabla \times \mathbf {M} ,}
481:
1723:. The direction of the normal must correspond with the orientation of
5291:(Reprint of 1963 ed.). Courier-Dover Publications. p. 213.
4971:
1601:
1190:
1126:
566:
73:
5435:(Reprint of 1947 ed.). Courier Dover Publications. p. 83.
902:
term, resulting in the modern form of the law, sometimes called the
2661:
in the expression for displacement current, it has two components:
2805:-field is used because it includes the magnetization currents, so
1605:
1194:
944:
The magnetic field lines lie in a plane perpendicular to the wire.
5187:
H. A. M. Snelders, "Oersted's discovery of electromagnetism" in
5025:
Ampère never utilized the field concept in any of his works; cf.
2033:
for electrical charge. In vector calculus, the identity for the
4487:
Consequently, referring to the definition of the bound current:
2442:. A changing state of polarization is equivalent to a current.
2083:{\displaystyle \nabla \cdot (\nabla \times \mathbf {B} )=0\,,}
4240:
is the polarization current density. Taking the equation for
2407:
to Ampère's circuital law at equation 112 in his 1861 paper "
2025:
Shortcomings of the original formulation of the circuital law
5354:(3rd ed.). Pearson/Addison-Wesley. pp. 322–323.
2295:{\displaystyle \nabla \times \mathbf {B} =\mathbf {0} \,,}
5705:
5189:
Cunningham, Andrew Cunningham; Nicholas Jardine (1990).
2397:
must be added to the current term in the circuital law.
2093:
and so the original Ampère's circuital law implies that
1218:
Forms of the original circuital law written in SI units
1122:
passes through (again arbitrary but not closed—since no
5063:
vol. 2, part 4, ch. 2 (§§502-527) & 23 (§§845-866).
2786:
Extending the original law: the Ampère–Maxwell equation
2824:
1837:
imply that the flux through any surface with boundary
1179:). In terms of free current, the line integral of the
4860:
4713:
4496:
4255:
4184:
4126:
3982:
3921:
3842:
3708:
3632:
3344:
3235:
3088:
3001:
2836:
2670:
2540:
2454:
2322:
2268:
2228:
2157:
2102:
2046:
1916:
1544:
1431:
1371:
1253:
1092:
The integral form of the original circuital law is a
1999: the "free" or "conduction" current density.
1816:
Second, there are infinitely many possible surfaces
1802:. Also the current passing in the same direction as
5613:The magnetization current can be expressed as the
5316:
4938:
4829:
4681:
4476:
4229:
4160:
4106:
3962:
3880:
3794:
3688:
3599:, wave propagation in free space now is possible.
3528:
3314:
3215:
3063:
2942:
2751:
2608:
2516:
2382:
2294:
2248:
2201:
2126:
2082:
1979:
1577:
1529:
1405:
1356:
5716:The Ampere–Maxwell Equation; Displacement Current
5515:. Upper Saddle River, NJ: Prentice Hall. p.
3973:and the following relation to the bound current:
5381:. Mineola, NY: Dover Publications. p. 285.
965:. In the 1850s Scottish mathematical physicist
5724:A Dynamical Theory of the Electromagnetic Field
5719:(PDF file) by J. S. Kovacs for Project PHYSNET.
5587:Rogalski, Mircea S.; Palmer, Stuart B. (2006).
5560:Rogalski, Mircea S.; Palmer, Stuart B. (2006).
2393:To treat these situations, the contribution of
1787:is the net current passing through the surface
5462:. Cambridge University Press. pp. 96–98.
3623:In this proof, we will show that the equation
3610:for a discussion of this important discovery.
2127:{\displaystyle \nabla \cdot \mathbf {J} =0\,,}
973:based on an analogy to hydrodynamics, relates
5492:Philosophical Magazine and Journal of Science
984:The original circuital law only applies to a
841:
8:
2249:{\displaystyle \mathbf {J} =\mathbf {0} \,,}
1774:could point in either of the two directions
5408:. Cambridge University Press. p. 179.
5121:. Princeton University Press. p. 125.
5310:
5308:
5094:. Cambridge University Press. p. 83.
4695:
1107:(arbitrary but must be closed). The curve
848:
834:
52:
36:
5637:Introduction to Electrodynamics (3rd ed.)
4912:
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2669:
2602:
2597:
2590:
2589:
2579:
2567:
2559:
2553:
2541:
2539:
2510:
2503:
2495:
2487:
2472:
2462:
2461:
2456:
2453:
2376:
2360:
2354:
2346:
2337:
2329:
2321:
2288:
2283:
2275:
2267:
2242:
2237:
2229:
2227:
2195:
2175:
2164:
2156:
2120:
2109:
2101:
2076:
2062:
2045:
2018:
1973:
1966:
1965:
1960:
1949:
1948:
1943:
1932:
1931:
1926:
1917:
1915:
1568:
1567:
1562:
1553:
1545:
1543:
1508:
1507:
1495:
1490:
1480:
1479:
1474:
1467:
1455:
1450:
1442:
1436:
1430:
1398:
1392:
1380:
1372:
1370:
1341:
1340:
1330:
1318:
1313:
1305:
1299:
1289:
1277:
1272:
1264:
1258:
1252:
1006:
5118:Classical Electromagnetism in a Nutshell
5091:A Student's Guide to Maxwell's Equations
5037:. Montreal, QC: Apeiron. ch. 15 p. 221.
1878:is one contribution to "bound current".
1813:can also be used to determine the signs.
1216:
5060:A Treatise on Electricity and Magnetism
5018:
4738:
3113:
2861:
2311:continuity equation for electric charge
2309:, but to maintain consistency with the
2146:continuity equation for electric charge
1907:due to free and bound charges is then:
1456:
1278:
597:Electromagnetism and special relativity
44:
5536:Introduction to Electromagnetic Theory
5177:. London: Baldwin, Craddock, Joy: 273.
3900:, which has the following relation to
3821:, which has the following relation to
2988:is the enclosed conduction current or
1862:. (All materials can to some extent.)
920:
5539:. Jones & Bartlett. p. 204.
1881:The other source of bound current is
1162:is proportional to the total current
617:Maxwell equations in curved spacetime
27:Concept in classical electromagnetism
7:
5429:Slater, J. C.; Frank, N. H. (1969).
5402:Billingham, J.; King, A. C. (2006).
5219:Basic Electrical Engineering, Vol. 1
2144:But in general, reality follows the
4696:Ampère's circuital law in cgs units
941:encircle the current-carrying wire.
5148:. Cengage Learning. p. 1093.
4919:
4909:
4862:
4815:
4797:
4787:
4733:
4663:
4653:
4613:
4603:
4581:
4578:
4575:
4572:
4569:
4552:
4522:
4461:
4441:
4431:
4409:
4392:
4368:
4345:
4304:
4281:
4215:
4205:
4193:
4144:
4135:
4091:
4074:
4047:
4037:
4020:
4007:
4004:
4001:
3998:
3995:
3779:
3769:
3730:
3677:
3667:
3655:
3633:
3513:
3503:
3470:
3460:
3438:
3421:
3404:
3387:
3370:
3353:
3303:
3293:
3237:
3204:
3186:
3176:
3108:
3048:
3038:
3026:
3003:
2931:
2913:
2903:
2891:
2856:
2736:
2726:
2711:
2701:
2679:
2591:
2478:
2474:
2463:
2367:
2357:
2323:
2269:
2186:
2178:
2158:
2103:
2056:
2047:
1967:
1950:
1933:
1569:
1546:
1521:
1518:
1515:
1509:
1491:
1481:
1451:
1373:
1348:
1345:
1342:
1314:
1273:
25:
2137:i.e. that the current density is
1846:Free current versus bound current
1809:must be counted as positive. The
1619:is the free current density only,
5680:
5222:. Tata McGraw-Hill. p. 96.
4913:
4899:
4870:
4820:
4791:
4777:
4725:
4657:
4633:
4607:
4563:
4546:
4530:
4455:
4435:
4403:
4386:
4362:
4353:
4325:
4317:
4289:
4209:
4187:
4151:
4129:
4085:
4068:
4041:
4027:
3989:
3956:
3948:
3940:
3870:
3862:
3844:
3773:
3749:
3738:
3671:
3649:
3640:
3507:
3483:
3464:
3432:
3415:
3398:
3381:
3364:
3347:
3297:
3263:
3245:
3209:
3180:
3146:
3100:
3042:
3020:
3011:
2936:
2907:
2885:
2848:
2814:does not appear explicitly, see
2730:
2705:
2673:
2598:
2568:
2560:
2542:
2496:
2488:
2457:
2361:
2330:
2305:i.e. that the magnetic field is
2284:
2276:
2238:
2230:
2165:
2110:
2063:
1961:
1944:
1927:
1918:
1750:Ambiguities and sign conventions
1563:
1554:
1496:
1475:
1443:
1399:
1381:
1319:
1306:
1265:
1070:in non-magnetic materials where
5512:Introduction to Electrodynamics
5351:Introduction to Electrodynamics
5323:(3rd ed.). Wiley. p.
5247:. New York, Dover Publications.
5077:. New York, Dover Publications.
2992:density. In differential form,
2259:the circuital law implies that
921:Ampère's original circuital law
4972:Ampèrian magnetic dipole model
4534:
4518:
4293:
4277:
3742:
3726:
3699:is equivalent to the equation
2507:
2492:
2067:
2053:
1:
5483:Clerk Maxwell, James (1861).
5245:"On Faraday's Lines of Force"
5243:Clerk Maxwell, James (1890).
5073:Clerk Maxwell, James (1890).
4977:Electromagnetic wave equation
3608:electromagnetic wave equation
2655:. Substituting this form for
971:"On Faraday's Lines of Force"
894:in his 1861 published paper "
622:Relativistic electromagnetism
5634:Griffiths, David J. (1998).
5485:"On Physical Lines of Force"
5348:Griffiths, David J. (1999).
5315:Jackson, John David (1999).
5195:. CUP Archive. p. 228.
5192:Romanticism and the Sciences
5075:"On Physical Lines of Force"
2643:relative static permittivity
880:around a closed loop to the
5590:Advanced University Physics
5563:Advanced University Physics
5258:Knoepfel, Heinz E. (2000).
2977:electric displacement field
2767:to only this contribution.
2529:electric displacement field
1829:(see next section), so the
1197:, A·m) around closed curve
884:passing through the loop.
5777:
5593:. CRC Press. p. 251.
5566:. CRC Press. p. 267.
5456:Siegel, Daniel M. (2003).
4987:Faraday's law of induction
2418:
2409:On Physical Lines of Force
1901:The total current density
1171:passing through a surface
1050:fields are related by the
896:On Physical Lines of Force
862:classical electromagnetism
347:Liénard–Wiechert potential
29:
5746:Eponymous laws of physics
5319:Classical Electrodynamics
1245:-field and total current
1230:
1224:
1222:
1158:, T) around closed curve
925:In 1820 Danish physicist
868:(not to be confused with
612:Mathematical descriptions
322:Electromagnetic radiation
312:Electromagnetic induction
252:Magnetic vector potential
247:Magnetic scalar potential
5375:Owen, George E. (2003).
5285:Owen, George E. (2003).
5142:Katz, Debora M. (2016).
5088:Fleisch, Daniel (2008).
1634:around the closed curve
1423:-field and free current
1201:equals the free current
5727:Maxwell's paper of 1864
3891:Next, we introduce the
2953:(integral form), where
910:that form the basis of
162:Electrostatic induction
157:Electrostatic discharge
5704:) by Kirby Morgan for
5378:Electromagnetic Theory
5288:Electromagnetic Theory
4940:
4831:
4683:
4478:
4231:
4162:
4108:
3964:
3882:
3796:
3690:
3539:where current density
3530:
3316:
3226:In differential form,
3217:
3065:
2944:
2753:
2610:
2518:
2384:
2296:
2250:
2203:
2128:
2084:
1981:
1579:
1531:
1407:
1358:
1111:in turn bounds both a
866:Ampère's circuital law
592:Electromagnetic tensor
5655:Tipler, Paul (2004).
5619:magnetization current
5115:Garg, Anupam (2012).
4941:
4832:
4684:
4479:
4232:
4173:magnetization current
4163:
4109:
3965:
3893:magnetization density
3883:
3807:Kelvin–Stokes theorem
3797:
3691:
3531:
3333:magnetization current
3317:
3218:
3066:
2945:
2754:
2611:
2519:
2385:
2297:
2251:
2204:
2129:
2085:
1982:
1887:polarizable materials
1868:magnetization current
1657:bounded by the curve
1580:
1532:
1408:
1359:
1052:constitutive equation
1003:Kelvin–Stokes theorem
927:Hans Christian Ørsted
585:Covariant formulation
377:Synchrotron radiation
317:Electromagnetic pulse
307:Electromagnetic field
5689:at Wikimedia Commons
5533:Tai L. Chow (2006).
5264:. Wiley. p. 4.
5171:Annals of Philosophy
4992:Polarization density
4962:Displacement current
4858:
4711:
4692:as was to be shown.
4494:
4253:
4182:
4124:
3980:
3919:
3840:
3814:polarization density
3706:
3630:
3614:Proof of equivalence
3604:electromagnetic wave
3550:displacement current
3342:
3233:
3086:
2999:
2834:
2780:Displacement current
2765:displacement current
2668:
2653:polarization density
2538:
2452:
2435:polarization density
2421:Displacement current
2415:Displacement current
2405:displacement current
2395:displacement current
2320:
2266:
2226:
2155:
2100:
2044:
2035:divergence of a curl
1914:
1820:that have the curve
1811:right hand grip rule
1778:to the surface; and
1542:
1429:
1369:
1251:
900:displacement current
627:Stress–energy tensor
552:Reluctance (complex)
297:Displacement current
5756:Maxwell's equations
5507:Griffiths, David J.
4982:Maxwell's equations
2401:James Clerk Maxwell
2031:continuity equation
1835:continuity equation
1825:current density is
1715:element of surface
1219:
1023:Forms using either
967:James Clerk Maxwell
908:Maxwell's equations
888:James Clerk Maxwell
542:Magnetomotive force
427:Electromotive force
397:Alternating current
332:Jefimenko equations
292:Cyclotron radiation
5055:Ampère's force law
5051:Ampère's force law
4936:
4827:
4679:
4677:
4474:
4472:
4227:
4158:
4104:
4102:
3960:
3878:
3792:
3686:
3526:
3312:
3213:
3079:" section below):
3061:
2940:
2749:
2606:
2514:
2380:
2292:
2246:
2199:
2124:
2080:
1977:
1831:divergence theorem
1575:
1527:
1403:
1354:
1217:
1210:through a surface
1140:In terms of total
1016:, and those using
963:Ampère's force law
959:André-Marie Ampère
906:, which is one of
904:Ampère–Maxwell law
870:Ampère's force law
390:Electrical network
227:Gauss magnetic law
192:Static electricity
152:Electric potential
32:Ampère's force law
5714:
5695:
5685:Media related to
5659:. W. H. Freeman.
5640:. Prentice Hall.
5504:For example, see
5044:978-1-987980-03-5
4926:
4885:
4804:
4753:
4670:
4620:
4516:
4448:
4275:
4222:
4054:
3937:
3812:We introduce the
3786:
3724:
3684:
3520:
3477:
3310:
3193:
3055:
2920:
2743:
2718:
2630:electric constant
2485:
2374:
2352:
2193:
2021:" section below.
1653:over the surface
1588:
1587:
1124:three-dimensional
1081:magnetic constant
1009:" section below).
979:electric currents
890:derived it using
858:
857:
557:Reluctance (real)
527:Gyrator–capacitor
472:Resonant cavities
362:Maxwell equations
18:Ampère's law
16:(Redirected from
5768:
5761:Electromagnetism
5712:
5693:
5684:
5670:
5651:
5622:
5611:
5605:
5604:
5584:
5578:
5577:
5557:
5551:
5550:
5530:
5502:
5496:
5495:
5489:
5480:
5474:
5473:
5453:
5447:
5446:
5432:Electromagnetism
5426:
5420:
5419:
5399:
5393:
5392:
5372:
5366:
5365:
5345:
5339:
5338:
5322:
5312:
5303:
5302:
5282:
5276:
5275:
5255:
5249:
5248:
5240:
5234:
5233:
5213:
5207:
5206:
5185:
5179:
5178:
5166:
5160:
5159:
5139:
5133:
5132:
5112:
5106:
5105:
5085:
5079:
5078:
5070:
5064:
5048:
5036:
5023:
4997:Electric current
4945:
4943:
4942:
4937:
4932:
4928:
4927:
4925:
4917:
4916:
4907:
4902:
4886:
4878:
4873:
4865:
4843:
4836:
4834:
4833:
4828:
4823:
4818:
4810:
4806:
4805:
4803:
4795:
4794:
4785:
4780:
4764:
4763:
4754:
4746:
4741:
4736:
4728:
4723:
4722:
4688:
4686:
4685:
4680:
4678:
4671:
4669:
4661:
4660:
4651:
4649:
4648:
4636:
4625:
4621:
4619:
4611:
4610:
4601:
4599:
4598:
4586:
4585:
4584:
4566:
4557:
4556:
4555:
4549:
4533:
4525:
4517:
4515:
4514:
4502:
4483:
4481:
4480:
4475:
4473:
4466:
4465:
4464:
4458:
4449:
4447:
4439:
4438:
4429:
4427:
4426:
4414:
4413:
4412:
4406:
4397:
4396:
4395:
4389:
4377:
4373:
4372:
4371:
4365:
4356:
4348:
4337:
4333:
4329:
4328:
4320:
4307:
4292:
4284:
4276:
4274:
4273:
4261:
4245:
4236:
4234:
4233:
4228:
4223:
4221:
4213:
4212:
4203:
4198:
4197:
4196:
4190:
4167:
4165:
4164:
4159:
4154:
4140:
4139:
4138:
4132:
4113:
4111:
4110:
4105:
4103:
4096:
4095:
4094:
4088:
4079:
4078:
4077:
4071:
4059:
4055:
4053:
4045:
4044:
4035:
4030:
4012:
4011:
4010:
3992:
3969:
3967:
3966:
3961:
3959:
3951:
3943:
3938:
3936:
3935:
3923:
3911:
3905:
3899:
3887:
3885:
3884:
3879:
3873:
3865:
3860:
3859:
3847:
3832:
3826:
3820:
3801:
3799:
3798:
3793:
3787:
3785:
3777:
3776:
3767:
3765:
3764:
3752:
3741:
3733:
3725:
3723:
3722:
3710:
3695:
3693:
3692:
3687:
3685:
3683:
3675:
3674:
3665:
3660:
3659:
3658:
3652:
3643:
3598:
3597:
3595:
3594:
3588:
3585:
3568:
3557:
3547:
3535:
3533:
3532:
3527:
3521:
3519:
3511:
3510:
3501:
3499:
3498:
3486:
3478:
3476:
3468:
3467:
3458:
3456:
3455:
3443:
3442:
3441:
3435:
3426:
3425:
3424:
3418:
3409:
3408:
3407:
3401:
3392:
3391:
3390:
3384:
3375:
3374:
3373:
3367:
3358:
3357:
3356:
3350:
3330:
3321:
3319:
3318:
3313:
3311:
3309:
3301:
3300:
3291:
3289:
3288:
3279:
3278:
3266:
3261:
3260:
3248:
3240:
3222:
3220:
3219:
3214:
3212:
3207:
3199:
3195:
3194:
3192:
3184:
3183:
3174:
3172:
3171:
3162:
3161:
3149:
3144:
3143:
3129:
3128:
3116:
3111:
3103:
3098:
3097:
3070:
3068:
3067:
3062:
3056:
3054:
3046:
3045:
3036:
3031:
3030:
3029:
3023:
3014:
3006:
2987:
2974:
2966:
2958:
2949:
2947:
2946:
2941:
2939:
2934:
2926:
2922:
2921:
2919:
2911:
2910:
2901:
2896:
2895:
2894:
2888:
2877:
2876:
2864:
2859:
2851:
2846:
2845:
2820:
2813:
2804:
2798:
2776:vacuum capacitor
2758:
2756:
2755:
2750:
2744:
2742:
2734:
2733:
2724:
2719:
2717:
2709:
2708:
2699:
2697:
2696:
2684:
2683:
2682:
2676:
2660:
2650:
2640:
2627:
2615:
2613:
2612:
2607:
2601:
2596:
2595:
2594:
2584:
2583:
2571:
2563:
2558:
2557:
2545:
2523:
2521:
2520:
2515:
2499:
2491:
2486:
2484:
2473:
2468:
2467:
2466:
2460:
2441:
2389:
2387:
2386:
2381:
2375:
2373:
2365:
2364:
2355:
2353:
2351:
2350:
2338:
2333:
2301:
2299:
2298:
2293:
2287:
2279:
2255:
2253:
2252:
2247:
2241:
2233:
2208:
2206:
2205:
2200:
2194:
2192:
2184:
2176:
2168:
2133:
2131:
2130:
2125:
2113:
2089:
2087:
2086:
2081:
2066:
2016:
2008:
1998:
1986:
1984:
1983:
1978:
1972:
1971:
1970:
1964:
1955:
1954:
1953:
1947:
1938:
1937:
1936:
1930:
1921:
1906:
1897:
1877:
1840:
1823:
1819:
1808:
1801:
1790:
1786:
1773:
1766:
1740:
1734:
1730:
1726:
1722:
1718:
1706:
1697:
1693:
1681:
1666:
1660:
1656:
1651:surface integral
1648:
1637:
1629:
1618:
1595:
1584:
1582:
1581:
1576:
1574:
1573:
1572:
1566:
1557:
1549:
1536:
1534:
1533:
1528:
1526:
1525:
1524:
1499:
1494:
1486:
1485:
1484:
1478:
1472:
1471:
1459:
1454:
1446:
1441:
1440:
1422:
1412:
1410:
1409:
1404:
1402:
1397:
1396:
1384:
1376:
1363:
1361:
1360:
1355:
1353:
1352:
1351:
1335:
1334:
1322:
1317:
1309:
1304:
1303:
1294:
1293:
1281:
1276:
1268:
1263:
1262:
1244:
1220:
1213:
1209:
1200:
1186:
1178:
1174:
1170:
1161:
1151:
1132:
1120:electric current
1117:
1110:
1106:
1078:
1069:
1049:
1043:
1035:
1029:
993:Equivalent forms
915:electromagnetism
882:electric current
850:
843:
836:
517:Electric machine
500:Magnetic circuit
462:Parallel circuit
452:Network analysis
417:Electric current
352:London equations
197:Triboelectricity
187:Potential energy
56:
46:Electromagnetism
37:
21:
5776:
5775:
5771:
5770:
5769:
5767:
5766:
5765:
5731:
5730:
5706:Project PHYSNET
5677:
5667:
5654:
5648:
5633:
5630:
5628:Further reading
5625:
5612:
5608:
5601:
5586:
5585:
5581:
5574:
5559:
5558:
5554:
5547:
5532:
5527:
5505:
5503:
5499:
5487:
5482:
5481:
5477:
5470:
5455:
5454:
5450:
5443:
5428:
5427:
5423:
5416:
5401:
5400:
5396:
5389:
5374:
5373:
5369:
5362:
5347:
5346:
5342:
5335:
5314:
5313:
5306:
5299:
5284:
5283:
5279:
5272:
5257:
5256:
5252:
5242:
5241:
5237:
5230:
5216:Dhogal (1986).
5215:
5214:
5210:
5203:
5188:
5186:
5182:
5168:
5167:
5163:
5156:
5141:
5140:
5136:
5129:
5114:
5113:
5109:
5102:
5087:
5086:
5082:
5072:
5071:
5067:
5045:
5034:
5026:
5024:
5020:
5016:
5011:
5007:Stokes' theorem
5002:Vector calculus
4957:Biot–Savart law
4952:
4918:
4908:
4891:
4887:
4856:
4855:
4841:
4796:
4786:
4769:
4765:
4755:
4714:
4709:
4708:
4698:
4676:
4675:
4662:
4652:
4640:
4623:
4622:
4612:
4602:
4590:
4561:
4544:
4537:
4506:
4492:
4491:
4471:
4470:
4453:
4440:
4430:
4418:
4401:
4384:
4375:
4374:
4360:
4335:
4334:
4315:
4311:
4296:
4265:
4251:
4250:
4241:
4214:
4204:
4185:
4180:
4179:
4127:
4122:
4121:
4101:
4100:
4083:
4066:
4057:
4056:
4046:
4036:
4013:
3987:
3978:
3977:
3927:
3917:
3916:
3907:
3901:
3895:
3851:
3838:
3837:
3828:
3822:
3816:
3778:
3768:
3756:
3714:
3704:
3703:
3676:
3666:
3647:
3628:
3627:
3616:
3589:
3586:
3580:
3579:
3577:
3576:
3570:
3559:
3553:
3546:
3540:
3512:
3502:
3490:
3469:
3459:
3447:
3430:
3413:
3396:
3379:
3362:
3345:
3340:
3339:
3326:
3323:
3302:
3292:
3280:
3270:
3252:
3231:
3230:
3224:
3185:
3175:
3163:
3153:
3135:
3134:
3130:
3120:
3089:
3084:
3083:
3047:
3037:
3018:
2997:
2996:
2986:
2980:
2970:
2962:
2954:
2912:
2902:
2883:
2882:
2878:
2868:
2837:
2832:
2831:
2816:
2812:
2806:
2800:
2799:-field is (the
2794:
2788:
2735:
2725:
2710:
2700:
2688:
2671:
2666:
2665:
2656:
2646:
2639:
2633:
2626:
2620:
2585:
2575:
2549:
2536:
2535:
2531:is defined as:
2477:
2455:
2450:
2449:
2437:
2423:
2417:
2366:
2356:
2342:
2318:
2317:
2313:, we must have
2264:
2263:
2224:
2223:
2185:
2177:
2153:
2152:
2098:
2097:
2042:
2041:
2027:
2010:
2004:
1997:
1991:
1959:
1942:
1925:
1912:
1911:
1902:
1896:
1890:
1876:
1870:
1848:
1838:
1821:
1817:
1803:
1796:
1793:right-hand rule
1788:
1785:
1779:
1768:
1765:
1759:
1752:
1738:
1732:
1728:
1724:
1720:
1716:
1701:
1695:
1691:
1690:) of the curve
1674:
1664:
1658:
1654:
1647:
1641:
1635:
1628:
1622:
1617:
1611:
1598:current density
1591:
1561:
1540:
1539:
1503:
1473:
1463:
1432:
1427:
1426:
1418:
1388:
1367:
1366:
1336:
1326:
1295:
1285:
1254:
1249:
1248:
1240:
1211:
1208:
1202:
1198:
1182:
1176:
1172:
1169:
1163:
1159:
1147:
1130:
1129:is enclosed by
1115:
1108:
1104:
1090:
1077:
1071:
1065:
1055:
1045:
1039:
1036:magnetic fields
1031:
1025:
995:
975:magnetic fields
923:
854:
825:
824:
640:
632:
631:
587:
577:
576:
532:Induction motor
502:
492:
491:
407:Current density
392:
382:
381:
372:Poynting vector
282:
280:Electrodynamics
272:
271:
267:Right-hand rule
232:Magnetic dipole
222:Biot–Savart law
212:
202:
201:
137:Electric dipole
132:Electric charge
107:
35:
28:
23:
22:
15:
12:
11:
5:
5774:
5772:
5764:
5763:
5758:
5753:
5751:Magnetostatics
5748:
5743:
5741:Electrostatics
5733:
5732:
5729:
5728:
5720:
5709:
5690:
5676:
5675:External links
5673:
5672:
5671:
5665:
5652:
5646:
5629:
5626:
5624:
5623:
5606:
5599:
5579:
5572:
5552:
5545:
5525:
5497:
5475:
5468:
5448:
5441:
5421:
5414:
5394:
5387:
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5360:
5340:
5333:
5304:
5297:
5277:
5270:
5250:
5235:
5228:
5208:
5201:
5180:
5161:
5154:
5134:
5127:
5107:
5100:
5080:
5065:
5043:
5017:
5015:
5012:
5010:
5009:
5004:
4999:
4994:
4989:
4984:
4979:
4974:
4969:
4964:
4959:
4953:
4951:
4948:
4947:
4946:
4935:
4931:
4924:
4921:
4915:
4911:
4905:
4901:
4897:
4894:
4890:
4884:
4881:
4876:
4872:
4868:
4864:
4846:speed of light
4838:
4837:
4826:
4822:
4817:
4813:
4809:
4802:
4799:
4793:
4789:
4783:
4779:
4775:
4772:
4768:
4762:
4758:
4752:
4749:
4744:
4740:
4735:
4731:
4727:
4721:
4717:
4697:
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4689:
4674:
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4655:
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4631:
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4626:
4624:
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4609:
4605:
4597:
4593:
4589:
4583:
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4577:
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4565:
4560:
4554:
4548:
4543:
4540:
4538:
4536:
4532:
4528:
4524:
4520:
4513:
4509:
4505:
4500:
4499:
4485:
4484:
4469:
4463:
4457:
4452:
4446:
4443:
4437:
4433:
4425:
4421:
4417:
4411:
4405:
4400:
4394:
4388:
4383:
4380:
4378:
4376:
4370:
4364:
4359:
4355:
4351:
4347:
4343:
4340:
4338:
4336:
4332:
4327:
4323:
4319:
4314:
4310:
4306:
4302:
4299:
4297:
4295:
4291:
4287:
4283:
4279:
4272:
4268:
4264:
4259:
4258:
4238:
4237:
4226:
4220:
4217:
4211:
4207:
4201:
4195:
4189:
4171:is called the
4169:
4168:
4157:
4153:
4149:
4146:
4143:
4137:
4131:
4115:
4114:
4099:
4093:
4087:
4082:
4076:
4070:
4065:
4062:
4060:
4058:
4052:
4049:
4043:
4039:
4033:
4029:
4025:
4022:
4019:
4016:
4014:
4009:
4006:
4003:
4000:
3997:
3991:
3986:
3985:
3971:
3970:
3958:
3954:
3950:
3946:
3942:
3934:
3930:
3926:
3889:
3888:
3877:
3872:
3868:
3864:
3858:
3854:
3850:
3846:
3803:
3802:
3791:
3784:
3781:
3775:
3771:
3763:
3759:
3755:
3751:
3747:
3744:
3740:
3736:
3732:
3728:
3721:
3717:
3713:
3697:
3696:
3682:
3679:
3673:
3669:
3663:
3657:
3651:
3646:
3642:
3638:
3635:
3615:
3612:
3574:
3544:
3537:
3536:
3525:
3518:
3515:
3509:
3505:
3497:
3493:
3489:
3485:
3481:
3475:
3472:
3466:
3462:
3454:
3450:
3446:
3440:
3434:
3429:
3423:
3417:
3412:
3406:
3400:
3395:
3389:
3383:
3378:
3372:
3366:
3361:
3355:
3349:
3325:In both forms
3308:
3305:
3299:
3295:
3287:
3283:
3277:
3273:
3269:
3265:
3259:
3255:
3251:
3247:
3243:
3239:
3228:
3211:
3206:
3202:
3198:
3191:
3188:
3182:
3178:
3170:
3166:
3160:
3156:
3152:
3148:
3142:
3138:
3133:
3127:
3123:
3119:
3115:
3110:
3106:
3102:
3096:
3092:
3081:
3072:
3071:
3060:
3053:
3050:
3044:
3040:
3034:
3028:
3022:
3017:
3013:
3009:
3005:
2984:
2951:
2950:
2938:
2933:
2929:
2925:
2918:
2915:
2909:
2905:
2899:
2893:
2887:
2881:
2875:
2871:
2867:
2863:
2858:
2854:
2850:
2844:
2840:
2810:
2787:
2784:
2760:
2759:
2748:
2741:
2738:
2732:
2728:
2722:
2716:
2713:
2707:
2703:
2695:
2691:
2687:
2681:
2675:
2637:
2624:
2617:
2616:
2605:
2600:
2593:
2588:
2582:
2578:
2574:
2570:
2566:
2562:
2556:
2552:
2548:
2544:
2525:
2524:
2513:
2509:
2506:
2502:
2498:
2494:
2490:
2483:
2480:
2476:
2471:
2465:
2459:
2419:Main article:
2416:
2413:
2391:
2390:
2379:
2372:
2369:
2363:
2359:
2349:
2345:
2341:
2336:
2332:
2328:
2325:
2303:
2302:
2291:
2286:
2282:
2278:
2274:
2271:
2257:
2256:
2245:
2240:
2236:
2232:
2210:
2209:
2198:
2191:
2188:
2183:
2180:
2174:
2171:
2167:
2163:
2160:
2135:
2134:
2123:
2119:
2116:
2112:
2108:
2105:
2091:
2090:
2079:
2075:
2072:
2069:
2065:
2061:
2058:
2055:
2052:
2049:
2026:
2023:
1995:
1988:
1987:
1976:
1969:
1963:
1958:
1952:
1946:
1941:
1935:
1929:
1924:
1920:
1894:
1874:
1847:
1844:
1843:
1842:
1814:
1783:
1761:
1751:
1748:
1747:
1746:
1736:
1699:
1672:
1667:is the vector
1662:
1643:
1639:
1630:is the closed
1624:
1620:
1615:
1609:
1586:
1585:
1571:
1565:
1560:
1556:
1552:
1548:
1537:
1523:
1520:
1517:
1514:
1511:
1506:
1502:
1498:
1493:
1489:
1483:
1477:
1470:
1466:
1462:
1458:
1453:
1449:
1445:
1439:
1435:
1424:
1414:
1413:
1401:
1395:
1391:
1387:
1383:
1379:
1375:
1364:
1350:
1347:
1344:
1339:
1333:
1329:
1325:
1321:
1316:
1312:
1308:
1302:
1298:
1292:
1288:
1284:
1280:
1275:
1271:
1267:
1261:
1257:
1246:
1236:
1235:
1229:
1223:
1206:
1167:
1098:magnetic field
1089:
1086:
1085:
1084:
1075:
1063:
1021:
1010:
994:
991:
955:
954:
951:
948:
945:
942:
922:
919:
878:magnetic field
872:) relates the
856:
855:
853:
852:
845:
838:
830:
827:
826:
823:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
641:
638:
637:
634:
633:
630:
629:
624:
619:
614:
609:
607:Four-potential
604:
599:
594:
588:
583:
582:
579:
578:
575:
574:
569:
564:
559:
554:
549:
544:
539:
534:
529:
524:
522:Electric motor
519:
514:
509:
503:
498:
497:
494:
493:
490:
489:
484:
479:
477:Series circuit
474:
469:
464:
459:
454:
449:
447:Kirchhoff laws
444:
439:
434:
429:
424:
419:
414:
412:Direct current
409:
404:
399:
393:
388:
387:
384:
383:
380:
379:
374:
369:
367:Maxwell tensor
364:
359:
354:
349:
344:
339:
337:Larmor formula
334:
329:
324:
319:
314:
309:
304:
299:
294:
289:
287:Bremsstrahlung
283:
278:
277:
274:
273:
270:
269:
264:
259:
254:
249:
244:
239:
237:Magnetic field
234:
229:
224:
219:
213:
210:Magnetostatics
208:
207:
204:
203:
200:
199:
194:
189:
184:
179:
174:
169:
164:
159:
154:
149:
144:
142:Electric field
139:
134:
129:
124:
119:
114:
112:Charge density
108:
105:Electrostatics
103:
102:
99:
98:
97:
96:
91:
86:
81:
76:
71:
66:
58:
57:
49:
48:
42:
41:
40:Articles about
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5773:
5762:
5759:
5757:
5754:
5752:
5749:
5747:
5744:
5742:
5739:
5738:
5736:
5726:
5725:
5721:
5718:
5717:
5710:
5707:
5703:
5699:
5698:
5691:
5688:
5683:
5679:
5678:
5674:
5668:
5666:0-7167-0810-8
5662:
5658:
5653:
5649:
5647:0-13-805326-X
5643:
5639:
5638:
5632:
5631:
5627:
5620:
5616:
5610:
5607:
5602:
5600:1-58488-511-4
5596:
5592:
5591:
5583:
5580:
5575:
5573:1-58488-511-4
5569:
5565:
5564:
5556:
5553:
5548:
5546:0-7637-3827-1
5542:
5538:
5537:
5528:
5526:0-13-805326-X
5522:
5518:
5514:
5513:
5508:
5501:
5498:
5493:
5486:
5479:
5476:
5471:
5469:0-521-53329-5
5465:
5461:
5460:
5452:
5449:
5444:
5442:0-486-62263-0
5438:
5434:
5433:
5425:
5422:
5417:
5415:0-521-63450-4
5411:
5407:
5406:
5398:
5395:
5390:
5388:0-486-42830-3
5384:
5380:
5379:
5371:
5368:
5363:
5361:0-13-805326-X
5357:
5353:
5352:
5344:
5341:
5336:
5334:0-471-30932-X
5330:
5326:
5321:
5320:
5311:
5309:
5305:
5300:
5298:0-486-42830-3
5294:
5290:
5289:
5281:
5278:
5273:
5271:0-471-32205-9
5267:
5263:
5262:
5254:
5251:
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5157:
5155:9781337364300
5151:
5147:
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5128:9780691130187
5124:
5120:
5119:
5111:
5108:
5103:
5101:9781139468473
5097:
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4715:
4707:
4706:
4705:
4703:
4693:
4672:
4666:
4645:
4641:
4637:
4629:
4627:
4616:
4595:
4591:
4587:
4558:
4541:
4539:
4526:
4511:
4507:
4503:
4490:
4489:
4488:
4467:
4450:
4444:
4423:
4419:
4415:
4398:
4381:
4379:
4357:
4349:
4341:
4339:
4330:
4321:
4312:
4308:
4300:
4298:
4285:
4270:
4266:
4262:
4249:
4248:
4247:
4244:
4224:
4218:
4199:
4178:
4177:
4176:
4175:density, and
4174:
4155:
4147:
4141:
4120:
4119:
4118:
4097:
4080:
4063:
4061:
4050:
4031:
4023:
4017:
4015:
3976:
3975:
3974:
3952:
3944:
3932:
3928:
3924:
3915:
3914:
3913:
3910:
3904:
3898:
3894:
3875:
3866:
3856:
3852:
3848:
3836:
3835:
3834:
3831:
3825:
3819:
3815:
3810:
3808:
3789:
3782:
3761:
3757:
3753:
3745:
3734:
3719:
3715:
3711:
3702:
3701:
3700:
3680:
3661:
3644:
3636:
3626:
3625:
3624:
3621:
3620:
3613:
3611:
3609:
3605:
3600:
3593:
3584:
3573:
3567:
3563:
3556:
3551:
3543:
3523:
3516:
3495:
3491:
3487:
3479:
3473:
3452:
3448:
3444:
3427:
3410:
3393:
3376:
3359:
3338:
3337:
3336:
3334:
3329:
3322:
3306:
3285:
3281:
3275:
3271:
3267:
3257:
3253:
3249:
3241:
3227:
3223:
3200:
3196:
3189:
3168:
3164:
3158:
3154:
3150:
3140:
3136:
3131:
3125:
3121:
3117:
3104:
3094:
3090:
3080:
3078:
3058:
3051:
3032:
3015:
3007:
2995:
2994:
2993:
2991:
2983:
2978:
2973:
2968:
2965:
2957:
2927:
2923:
2916:
2897:
2879:
2873:
2869:
2865:
2852:
2842:
2838:
2830:
2829:
2828:
2826:
2822:
2819:
2809:
2803:
2797:
2791:
2785:
2783:
2781:
2777:
2771:
2768:
2766:
2746:
2739:
2720:
2714:
2693:
2689:
2685:
2664:
2663:
2662:
2659:
2654:
2649:
2644:
2636:
2631:
2623:
2603:
2586:
2580:
2576:
2572:
2564:
2554:
2550:
2546:
2534:
2533:
2532:
2530:
2511:
2504:
2500:
2481:
2469:
2448:
2447:
2446:
2443:
2440:
2436:
2430:
2428:
2422:
2414:
2412:
2410:
2406:
2402:
2398:
2396:
2377:
2370:
2347:
2343:
2339:
2334:
2326:
2316:
2315:
2314:
2312:
2308:
2289:
2280:
2272:
2262:
2261:
2260:
2243:
2234:
2222:
2221:
2220:
2218:
2213:
2196:
2189:
2181:
2172:
2169:
2161:
2151:
2150:
2149:
2147:
2142:
2140:
2121:
2117:
2114:
2106:
2096:
2095:
2094:
2077:
2073:
2070:
2059:
2050:
2040:
2039:
2038:
2036:
2032:
2024:
2022:
2020:
2015:
2014:
2007:
2000:
1994:
1974:
1956:
1939:
1922:
1910:
1909:
1908:
1905:
1899:
1893:
1888:
1884:
1879:
1873:
1869:
1863:
1861:
1857:
1853:
1845:
1836:
1832:
1828:
1815:
1812:
1807:
1800:
1794:
1782:
1777:
1772:
1764:
1757:
1756:
1755:
1749:
1744:
1737:
1714:
1713:infinitesimal
1710:
1705:
1700:
1689:
1685:
1684:infinitesimal
1680:
1679:
1673:
1670:
1663:
1652:
1646:
1640:
1633:
1632:line integral
1627:
1621:
1614:
1610:
1607:
1603:
1599:
1596:is the total
1594:
1590:
1589:
1558:
1550:
1538:
1512:
1504:
1500:
1487:
1468:
1464:
1460:
1447:
1437:
1433:
1425:
1421:
1416:
1415:
1393:
1389:
1385:
1377:
1365:
1337:
1331:
1327:
1323:
1310:
1300:
1296:
1290:
1286:
1282:
1269:
1259:
1255:
1247:
1243:
1238:
1237:
1233:
1227:
1221:
1215:
1205:
1196:
1192:
1188:
1185:
1175:(enclosed by
1166:
1157:
1153:
1150:
1143:
1138:
1136:
1128:
1125:
1121:
1114:
1103:
1099:
1095:
1094:line integral
1087:
1082:
1074:
1068:
1062:
1058:
1053:
1048:
1042:
1037:
1034:
1028:
1022:
1019:
1015:
1011:
1008:
1004:
1000:
999:
998:
992:
990:
987:
986:magnetostatic
982:
980:
976:
972:
968:
964:
960:
952:
949:
946:
943:
940:
937:The magnetic
936:
935:
934:
932:
928:
918:
916:
913:
909:
905:
901:
897:
893:
892:hydrodynamics
889:
885:
883:
879:
875:
871:
867:
863:
851:
846:
844:
839:
837:
832:
831:
829:
828:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
642:
636:
635:
628:
625:
623:
620:
618:
615:
613:
610:
608:
605:
603:
600:
598:
595:
593:
590:
589:
586:
581:
580:
573:
570:
568:
565:
563:
560:
558:
555:
553:
550:
548:
545:
543:
540:
538:
535:
533:
530:
528:
525:
523:
520:
518:
515:
513:
510:
508:
505:
504:
501:
496:
495:
488:
485:
483:
480:
478:
475:
473:
470:
468:
465:
463:
460:
458:
455:
453:
450:
448:
445:
443:
442:Joule heating
440:
438:
435:
433:
430:
428:
425:
423:
420:
418:
415:
413:
410:
408:
405:
403:
400:
398:
395:
394:
391:
386:
385:
378:
375:
373:
370:
368:
365:
363:
360:
358:
357:Lorentz force
355:
353:
350:
348:
345:
343:
340:
338:
335:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
284:
281:
276:
275:
268:
265:
263:
260:
258:
257:Magnetization
255:
253:
250:
248:
245:
243:
242:Magnetic flux
240:
238:
235:
233:
230:
228:
225:
223:
220:
218:
215:
214:
211:
206:
205:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
148:
147:Electric flux
145:
143:
140:
138:
135:
133:
130:
128:
125:
123:
120:
118:
115:
113:
110:
109:
106:
101:
100:
95:
92:
90:
87:
85:
84:Computational
82:
80:
77:
75:
72:
70:
67:
65:
62:
61:
60:
59:
55:
51:
50:
47:
43:
39:
38:
33:
19:
5723:
5715:
5697:Ampere's Law
5696:
5687:Ampere's law
5656:
5636:
5614:
5609:
5589:
5582:
5562:
5555:
5535:
5511:
5500:
5491:
5478:
5458:
5451:
5431:
5424:
5404:
5397:
5377:
5370:
5350:
5343:
5318:
5287:
5280:
5260:
5253:
5238:
5218:
5211:
5191:
5183:
5174:
5170:
5164:
5144:
5137:
5117:
5110:
5090:
5083:
5068:
5058:
5029:
5021:
4850:
4839:
4699:
4691:
4486:
4242:
4239:
4170:
4116:
3972:
3908:
3902:
3896:
3890:
3829:
3823:
3817:
3811:
3804:
3698:
3622:
3618:
3617:
3601:
3591:
3582:
3571:
3565:
3561:
3554:
3549:
3541:
3538:
3327:
3324:
3229:
3225:
3082:
3073:
2990:free current
2981:
2971:
2963:
2955:
2952:
2817:
2807:
2801:
2795:
2792:
2789:
2772:
2769:
2764:
2761:
2657:
2647:
2634:
2621:
2618:
2526:
2444:
2438:
2431:
2424:
2399:
2392:
2307:irrotational
2304:
2258:
2214:
2211:
2143:
2136:
2092:
2028:
2011:
2005:
2001:
1992:
1989:
1903:
1900:
1891:
1883:bound charge
1880:
1871:
1864:
1849:
1805:
1798:
1780:
1770:
1762:
1753:
1731:and surface
1703:
1688:differential
1677:
1676:
1644:
1625:
1612:
1592:
1419:
1241:
1232:Differential
1203:
1183:
1164:
1148:
1139:
1102:closed curve
1100:around some
1091:
1072:
1066:
1060:
1056:
1046:
1040:
1032:
1026:
1012:Forms using
996:
983:
956:
924:
903:
886:
865:
859:
602:Four-current
537:Linear motor
422:Electrolysis
302:Eddy current
262:Permeability
216:
182:Polarization
177:Permittivity
5405:Wave Motion
4967:Capacitance
1709:vector area
1686:element (a
1669:dot product
1604:per square
1135:circulation
1088:Explanation
939:field lines
874:circulation
572:Transformer
402:Capacitance
327:Faraday law
122:Coulomb law
64:Electricity
5735:Categories
5713:MISN-0-145
5694:MISN-0-138
5229:0074515861
5202:0521356857
5031:experience
2527:where the
2427:free space
2217:free space
2139:solenoidal
1856:magnetized
1827:solenoidal
1649:denotes a
1118:which the
1005:(see the "
639:Scientists
487:Waveguides
467:Resistance
437:Inductance
217:Ampère law
4920:∂
4910:∂
4896:π
4867:×
4863:∇
4812:⋅
4798:∂
4788:∂
4774:π
4757:∬
4730:⋅
4716:∮
4664:∂
4654:∂
4642:ε
4614:∂
4604:∂
4592:ε
4527:×
4523:∇
4508:μ
4442:∂
4432:∂
4420:ε
4350:×
4346:∇
4309:×
4305:∇
4286:×
4282:∇
4267:μ
4216:∂
4206:∂
4148:×
4145:∇
4048:∂
4038:∂
4024:×
4021:∇
3929:μ
3853:ε
3780:∂
3770:∂
3758:ε
3735:×
3731:∇
3716:μ
3678:∂
3668:∂
3637:×
3634:∇
3514:∂
3504:∂
3492:ε
3471:∂
3461:∂
3449:ε
3331:includes
3304:∂
3294:∂
3282:ε
3272:μ
3254:μ
3242:×
3238:∇
3201:⋅
3187:∂
3177:∂
3165:ε
3155:μ
3137:μ
3122:∬
3105:⋅
3091:∮
3049:∂
3039:∂
3008:×
3004:∇
2961:magnetic
2928:⋅
2914:∂
2904:∂
2870:∬
2853:⋅
2839:∮
2823:and also
2737:∂
2727:∂
2712:∂
2702:∂
2690:ε
2587:ε
2577:ε
2551:ε
2479:∂
2475:∂
2368:∂
2358:∂
2327:×
2324:∇
2273:×
2270:∇
2187:∂
2182:ρ
2179:∂
2173:−
2162:⋅
2159:∇
2107:⋅
2104:∇
2060:×
2057:∇
2051:⋅
2048:∇
1860:polarized
1745:operator.
1551:×
1547:∇
1488:⋅
1465:∬
1448:⋅
1434:∮
1390:μ
1378:×
1374:∇
1328:μ
1311:⋅
1297:∬
1287:μ
1270:⋅
1256:∮
1181:magnetic
1146:magnetic
1018:cgs units
912:classical
795:Steinmetz
725:Kirchhoff
710:Jefimenko
705:Hopkinson
690:Helmholtz
685:Heaviside
547:Permeance
432:Impedance
172:Insulator
167:Gauss law
117:Conductor
94:Phenomena
89:Textbooks
69:Magnetism
5702:PDF file
5509:(1999).
4950:See also
2219:, where
1226:Integral
1014:SI units
820:Wiechert
775:Poynting
665:Einstein
512:DC motor
507:AC motor
342:Lenz law
127:Electret
5057:in his
4844:is the
3596:
3578:
3548:is the
2975:is the
2959:is the
2651:is the
2628:is the
1858:and/or
1852:battery
1741:is the
1707:is the
1608:, A·m),
1602:amperes
1191:amperes
1142:current
1113:surface
1096:of the
1079:is the
931:compass
805:Thomson
780:Ritchie
770:Poisson
755:Neumann
750:Maxwell
745:Lorentz
740:Liénard
670:Faraday
655:Coulomb
482:Voltage
457:Ohm law
79:History
5663:
5644:
5597:
5570:
5543:
5523:
5466:
5439:
5412:
5385:
5358:
5331:
5295:
5268:
5226:
5199:
5152:
5125:
5098:
5041:
4840:where
4117:where
3606:. See
3552:, and
2979:, and
2821:-field
2645:, and
2619:where
1776:normal
1711:of an
1682:is an
1417:Using
1239:Using
1187:-field
1156:teslas
1152:-field
1127:volume
790:Singer
785:Savart
765:Ørsted
730:Larmor
720:Kelvin
675:Fizeau
645:Ampère
567:Stator
74:Optics
5488:(PDF)
5035:(PDF)
5014:Notes
3077:proof
2967:field
2019:proof
1990:with
1606:metre
1234:form
1228:form
1207:f,enc
1195:metre
1007:proof
876:of a
815:Weber
810:Volta
800:Tesla
715:Joule
700:Hertz
695:Henry
680:Gauss
562:Rotor
5661:ISBN
5642:ISBN
5615:curl
5595:ISBN
5568:ISBN
5541:ISBN
5531:and
5521:ISBN
5464:ISBN
5437:ISBN
5410:ISBN
5383:ISBN
5356:ISBN
5329:ISBN
5293:ISBN
5266:ISBN
5224:ISBN
5197:ISBN
5150:ISBN
5123:ISBN
5096:ISBN
5039:ISBN
3906:and
3827:and
3560:∇ ⋅
2825:Note
2641:the
1833:and
1743:curl
1600:(in
1193:per
1189:(in
1154:(in
1044:and
735:Lenz
660:Davy
650:Biot
5517:323
5325:238
4702:cgs
4700:In
2827:):
2425:In
2411:".
1784:enc
1739:∇ ×
1168:enc
1030:or
977:to
860:In
760:Ohm
5737::
5519:.
5490:.
5327:.
5307:^
5175:16
5173:.
4848:.
4246::
3912::
3833::
3809:.
3564:=
2782:.
2632:,
2148::
2141:.
1898:.
1214:.
1059:=
1054::
917:.
864:,
5708:.
5700:(
5669:.
5650:.
5621:.
5603:.
5576:.
5549:.
5529:.
5494:.
5472:.
5445:.
5418:.
5391:.
5364:.
5337:.
5301:.
5274:.
5232:.
5205:.
5158:.
5131:.
5104:.
5047:.
4934:.
4930:)
4923:t
4914:E
4904:+
4900:J
4893:4
4889:(
4883:c
4880:1
4875:=
4871:B
4842:c
4825:,
4821:S
4816:d
4808:)
4801:t
4792:E
4782:+
4778:J
4771:4
4767:(
4761:S
4751:c
4748:1
4743:=
4739:l
4734:d
4726:B
4720:C
4673:,
4667:t
4658:E
4646:0
4638:+
4634:J
4630:=
4617:t
4608:E
4596:0
4588:+
4582:d
4579:n
4576:u
4573:o
4570:b
4564:J
4559:+
4553:f
4547:J
4542:=
4535:)
4531:B
4519:(
4512:0
4504:1
4468:.
4462:M
4456:J
4451:+
4445:t
4436:E
4424:0
4416:+
4410:P
4404:J
4399:+
4393:f
4387:J
4382:=
4369:M
4363:J
4358:+
4354:H
4342:=
4331:)
4326:M
4322:+
4318:H
4313:(
4301:=
4294:)
4290:B
4278:(
4271:0
4263:1
4243:B
4225:,
4219:t
4210:P
4200:=
4194:P
4188:J
4156:,
4152:M
4142:=
4136:M
4130:J
4098:,
4092:P
4086:J
4081:+
4075:M
4069:J
4064:=
4051:t
4042:P
4032:+
4028:M
4018:=
4008:d
4005:n
4002:u
3999:o
3996:b
3990:J
3957:M
3953:+
3949:H
3945:=
3941:B
3933:0
3925:1
3909:H
3903:B
3897:M
3876:.
3871:P
3867:+
3863:E
3857:0
3849:=
3845:D
3830:D
3824:E
3818:P
3790:.
3783:t
3774:E
3762:0
3754:+
3750:J
3746:=
3743:)
3739:B
3727:(
3720:0
3712:1
3681:t
3672:D
3662:+
3656:f
3650:J
3645:=
3641:H
3592:t
3590:∂
3587:/
3583:E
3581:∂
3575:0
3572:ε
3566:ρ
3562:D
3555:J
3545:D
3542:J
3524:,
3517:t
3508:E
3496:0
3488:+
3484:J
3480:=
3474:t
3465:E
3453:0
3445:+
3439:M
3433:J
3428:+
3422:P
3416:J
3411:+
3405:f
3399:J
3394:=
3388:M
3382:J
3377:+
3371:D
3365:J
3360:+
3354:f
3348:J
3328:J
3307:t
3298:E
3286:0
3276:0
3268:+
3264:J
3258:0
3250:=
3246:B
3210:S
3205:d
3197:)
3190:t
3181:E
3169:0
3159:0
3151:+
3147:J
3141:0
3132:(
3126:S
3118:=
3114:l
3109:d
3101:B
3095:C
3075:"
3059:.
3052:t
3043:D
3033:+
3027:f
3021:J
3016:=
3012:H
2985:f
2982:J
2972:D
2964:H
2956:H
2937:S
2932:d
2924:)
2917:t
2908:D
2898:+
2892:f
2886:J
2880:(
2874:S
2866:=
2862:l
2857:d
2849:H
2843:C
2818:H
2811:M
2808:J
2802:H
2796:H
2747:.
2740:t
2731:P
2721:+
2715:t
2706:E
2694:0
2686:=
2680:D
2674:J
2658:D
2648:P
2638:r
2635:ε
2625:0
2622:ε
2604:,
2599:E
2592:r
2581:0
2573:=
2569:P
2565:+
2561:E
2555:0
2547:=
2543:D
2512:,
2508:)
2505:t
2501:,
2497:r
2493:(
2489:D
2482:t
2470:=
2464:D
2458:J
2439:P
2378:.
2371:t
2362:E
2348:2
2344:c
2340:1
2335:=
2331:B
2290:,
2285:0
2281:=
2277:B
2244:,
2239:0
2235:=
2231:J
2197:,
2190:t
2170:=
2166:J
2122:,
2118:0
2115:=
2111:J
2078:,
2074:0
2071:=
2068:)
2064:B
2054:(
2013:H
2006:B
1996:f
1993:J
1975:,
1968:P
1962:J
1957:+
1951:M
1945:J
1940:+
1934:f
1928:J
1923:=
1919:J
1904:J
1895:P
1892:J
1875:M
1872:J
1839:C
1822:C
1818:S
1806:S
1804:d
1799:S
1797:d
1789:S
1781:I
1771:S
1769:d
1763:C
1760:∮
1735:.
1733:S
1729:C
1725:C
1721:S
1717:S
1704:S
1702:d
1698:)
1696:C
1692:C
1678:l
1675:d
1671:,
1665:·
1661:,
1659:C
1655:S
1645:S
1642:∬
1638:,
1636:C
1626:C
1623:∮
1616:f
1613:J
1593:J
1570:f
1564:J
1559:=
1555:H
1522:c
1519:n
1516:e
1513:,
1510:f
1505:I
1501:=
1497:S
1492:d
1482:f
1476:J
1469:S
1461:=
1457:l
1452:d
1444:H
1438:C
1420:H
1400:J
1394:0
1386:=
1382:B
1349:c
1346:n
1343:e
1338:I
1332:0
1324:=
1320:S
1315:d
1307:J
1301:S
1291:0
1283:=
1279:l
1274:d
1266:B
1260:C
1242:B
1212:S
1204:I
1199:C
1184:H
1177:C
1173:S
1165:I
1160:C
1149:B
1131:S
1116:S
1109:C
1105:C
1083:.
1076:0
1073:μ
1067:H
1064:0
1061:μ
1057:B
1047:H
1041:B
1033:H
1027:B
849:e
842:t
835:v
34:.
20:)
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