3508:
7250:
6550:
1362:
3219:
66:
6921:
6271:
4111:
6761:
4516:
4175:
3899:
3503:{\displaystyle \nabla '\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)\equiv \left(\mathbf {e} _{x}{\frac {\partial }{\partial x'}}+\mathbf {e} _{y}{\frac {\partial }{\partial y'}}+\mathbf {e} _{z}{\frac {\partial }{\partial z'}}\right)\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)={\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}}
5731:
2641:
7245:{\displaystyle U_{ee}=J={\frac {1}{2}}\int \delta ^{3}\mathbf {r} '\int \delta ^{3}\mathbf {r} \left({\frac {(en(\mathbf {r} ))(en(\mathbf {r} '))}{|\mathbf {r} -\mathbf {r} '|}}\right)={\frac {1}{2}}\int \delta ^{3}\mathbf {r} '\int \delta ^{3}\mathbf {r} \left({\frac {\rho (\mathbf {r} )\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\right)}
4369:
5318:
3212:
6557:
6545:{\displaystyle n(\mathbf {r} )=\sum _{i}\int {\mathrm {d} }^{3}\mathbf {r} _{2}\cdots \int {\mathrm {d} }^{3}\mathbf {r} _{N}\,\Psi ^{*}(\mathbf {r} ,\mathbf {r} _{2},\dots ,\mathbf {r} _{i}=\mathbf {r} ',\dots ,\mathbf {r} _{N})\Psi (\mathbf {r} ,\mathbf {r} _{2},\dots ,\mathbf {r} _{i}=\mathbf {r} ',\dots ,\mathbf {r} _{N}).}
3648:
2066:
2981:
5497:
1198:
on an electron is so small (1.6⋅10 C) and there are so many of them in a macroscopic volume (there are about 10 conduction electrons in a cubic centimeter of copper) the continuous approximation is very accurate when applied to macroscopic volumes, and even microscopic volumes above the nanometer
2467:
3929:
1153:
charge distribution is an approximation, which becomes inaccurate at small length scales. A charge distribution is ultimately composed of individual charged particles separated by regions containing no charge. For example, the charge in an electrically charged metal object is made up of
5136:
2790:
6265:
3071:
5119:
4455:
1953:
3521:
2092:, and polarize other nearby dipoles tending to line them up, the net accumulation of charge from the orientation of the dipoles is the bound charge. They are called bound because they cannot be removed: in the dielectric material the charges are the
4205:
6916:
6756:{\displaystyle n(\mathbf {r} )=N\int {\mathrm {d} }^{3}\mathbf {r} _{2}\cdots \int {\mathrm {d} }^{3}\mathbf {r} _{N}\,\Psi ^{*}(\mathbf {r} ,\mathbf {r} _{2},\dots ,\mathbf {r} _{N})\Psi (\mathbf {r} ,\mathbf {r} _{2},\dots ,\mathbf {r} _{N}).}
5458:
924:(λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.
3843:
2863:
1218:, so the charge of an individual particle is not concentrated at a point but is 'smeared out' in space and acts like a true continuous charge distribution. This is the meaning of 'charge distribution' and 'charge density' used in
2458:
2309:
4995:
4106:{\displaystyle {\frac {\mathbf {P} \cdot \mathbf {\hat {n}} dS'}{|\mathbf {r} -\mathbf {r} '|}}-{\frac {1}{4\pi \varepsilon _{0}}}\iiint {\frac {\nabla '\cdot \mathbf {P} }{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r'} }
4606:
3049:
2673:
6144:
5726:{\displaystyle Q=\int _{R}d^{3}\mathbf {r} \sum _{i=1}^{N}\ q_{i}\delta (\mathbf {r} -\mathbf {r} _{i})=\sum _{i=1}^{N}\ q_{i}\int _{R}d^{3}\mathbf {r} \delta (\mathbf {r} -\mathbf {r} _{i})=\sum _{i=1}^{N}\ q_{i}}
6010:
2636:{\displaystyle \sigma _{b}={\frac {dq_{b}}{dS}}={\frac {d\mathbf {d} }{|\mathbf {s} |dS}}\cdot \mathbf {\hat {n}} ={\frac {d\mathbf {d} }{dV}}\cdot \mathbf {\hat {n}} =\mathbf {P} \cdot \mathbf {\hat {n}} \,.}
2224:
1691:
6822:
5819:
4171:
3895:
5015:
2174:
1770:
4747:
2838:
1852:
4844:
4378:
1481:
1140:
5313:{\displaystyle Q=\int _{R}d^{3}\mathbf {r} \,\rho _{q}=\int _{R}d^{3}\mathbf {r} \,q\delta (\mathbf {r} -\mathbf {r} _{0})=q\int _{R}d^{3}\mathbf {r} \,\delta (\mathbf {r} -\mathbf {r} _{0})=q}
6830:
1542:
1330:
1603:
5359:
1057:
1026:
1260:
1104:
995:
7748:
Razi
Epsztein, Evyatar Shaulsky, Nadir Dizge, David M Warsinger, Menachem Elimelech (2018). "Ionic Charge Density-Dependent Donnan Exclusion in Nanofiltration of Monovalent Anions".
6165:
623:
4364:{\displaystyle {\frac {\sigma _{b}dS'}{|\mathbf {r} -\mathbf {r} '|}}+{\frac {1}{4\pi \varepsilon _{0}}}\iiint {\frac {\rho _{b}}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r'} }
2368:
1282:
964:
4541:
4200:
3924:
3657:
4889:
3207:{\displaystyle \varphi ={\frac {1}{4\pi \varepsilon _{0}}}\iiint {\frac {(\mathbf {r} -\mathbf {r} ')\cdot \mathbf {P} }{|\mathbf {r} -\mathbf {r} '|^{3}}}d^{3}\mathbf {r'} }
596:
2384:
4676:
1194:, etc.) per unit volume. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. However, because the
2335:
2241:
5832:
4512:
608:
7791:
7302:
2061:{\displaystyle \langle \lambda _{q}\rangle ={\frac {Q}{\ell }}\,,\quad \langle \sigma _{q}\rangle ={\frac {Q}{S}}\,,\quad \langle \rho _{q}\rangle ={\frac {Q}{V}}\,.}
4927:
3643:{\displaystyle \varphi ={\frac {1}{4\pi \varepsilon _{0}}}\iiint \mathbf {P} \cdot \nabla '\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)d^{3}\mathbf {r'} }
6154:
5865:
to show "how a neutral current-bearing wire appears to carry a net charge density as observed in a moving frame." When a charge density is measured in a moving
2992:
6066:
4774:(to differ between the constant and non-constant densities), and so by the properties of an integral can be pulled outside of the integral resulting in:
859:
5940:
2842:
The negative sign arises due to the opposite signs on the charges in the dipoles, one end is within the volume of the object, the other at the surface.
628:
7418:
7921:
2179:
2976:{\displaystyle \varphi ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {(\mathbf {r} -\mathbf {r} ')\cdot \mathbf {d} }{|\mathbf {r} -\mathbf {r} '|^{3}}}}
638:
1635:
1887:
7252:
Note that this is excluding the exchange energy of the system, which is a purely quantum mechanical phenomenon, has to be calculated separately.
5762:
4546:
2129:
463:
1714:
7905:
7884:
7860:
7841:
7818:
7671:
7621:
7548:
7516:
4688:
2795:
478:
100:
7342:. The charge density of molecules impacts chemical and separation processes. For example, charge density influences metal-metal bonding and
1796:
7448:
4777:
7588:
1432:
7946:
5004:
As always, the integral of the charge density over a region of space is the charge contained in that region. The delta function has the
1493:
358:
7646:
7596:
7488:
7458:
7402:
90:
43:
2785:{\displaystyle q_{b}=\int \rho _{b}\,dV=-\oint _{S}\mathbf {P} \cdot {\hat {\mathbf {n} }}\,dS=-\int \nabla \cdot \mathbf {P} \,dV}
1554:
4128:
3852:
6766:
1064:
273:
852:
618:
95:
5836:
1112:
633:
338:
6048:
1867:
498:
488:
473:
238:
105:
6160:
For system of identical fermions, the number density is given as sum of probability density of each particle in :
538:
228:
4474:
791:
666:
563:
4849:
458:
1287:
5114:{\displaystyle \int _{R}d^{3}\mathbf {r} f(\mathbf {r} )\delta (\mathbf {r} -\mathbf {r} _{0})=f(\mathbf {r} _{0})}
4642:
932:
291:
5904:
1031:
845:
806:
333:
323:
263:
258:
198:
6260:{\displaystyle n(\mathbf {r} )=\sum _{i}\langle \psi |\delta ^{3}(\mathbf {r} -\mathbf {r} _{i}')|\psi \rangle }
1000:
7941:
4893:
The equivalent proofs for linear charge density and surface charge density follow the same arguments as above.
1215:
937:
5908:
4469:
for electricity; the volume integral of it is the free charge enclosed in a charged object - equal to the net
4450:{\displaystyle \sigma _{b}=\mathbf {P} \cdot \mathbf {\hat {n}} \,,\quad \rho _{b}=-\nabla \cdot \mathbf {P} }
2110:, i.e. when the charges are not moving and the resultant electric field is independent of time, or constitute
1234:
1078:
969:
776:
656:
343:
5892:
916:(σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m), at any point on a
781:
751:
183:
173:
168:
4639:, independent of position i.e. constant throughout the region of the material, the equation simplifies to:
7331:
4613:
2652:
2338:
1385:
1179:
603:
373:
148:
2344:
7785:
7688:
7335:
6911:{\displaystyle \langle \psi |U|\psi \rangle =\int V(\mathbf {r} )n(\mathbf {r} )\delta ^{3}\mathbf {r} }
4629:
4617:
1203:
901:
701:
388:
378:
328:
318:
65:
1265:
947:
4522:
4181:
3905:
1950:
The total charge divided by the length, surface area, or volume will be the average charge densities:
7757:
7714:
4921:
3652:
2648:
1389:
941:
826:
726:
691:
443:
308:
208:
193:
128:
5453:{\displaystyle \rho _{q}(\mathbf {r} )=\sum _{i=1}^{N}\ q_{i}\delta (\mathbf {r} -\mathbf {r} _{i})}
2849:
Derivation of bound surface and volume charge densities from internal dipole moments (bound charges)
233:
7373:
7359:
7327:
1155:
1072:
1063:, even though all real charge distributions are made up of discrete charged particles. Due to the
786:
766:
761:
568:
553:
438:
408:
303:
2318:
5866:
5850:
5842:
2853:
2667:
2371:
1163:
1146:
661:
401:
203:
163:
4487:
1361:
7901:
7880:
7856:
7837:
7814:
7773:
7730:
7667:
7642:
7617:
7592:
7544:
7512:
7484:
7454:
7398:
5914:
5862:
1207:
1195:
900:(symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the
721:
31:
7260:
5751:) the charge density can be expressed through the number of charge carriers per unit volume,
7936:
7765:
7722:
7343:
5861:
force of a current-bearing wire arises from this relative charge density. He used (p 260) a
4118:
2111:
1694:
1345:
1223:
1068:
873:
821:
736:
696:
686:
573:
528:
511:
428:
363:
133:
57:
39:
35:
7705:
R. J. Gillespie & P. L. A. Popelier (2001). "Chemical
Bonding and Molecular Geometry".
2079:
materials, the total charge of an object can be separated into "free" and "bound" charges.
1893:
Within the context of electromagnetism, the subscripts are usually dropped for simplicity:
7368:
7363:
7339:
5881:
5748:
3065:
1773:
1396:
1159:
1107:
881:
756:
681:
676:
543:
418:
383:
278:
243:
143:
3838:{\displaystyle \varphi ={\frac {1}{4\pi \varepsilon _{0}}}\iiint \leftd^{3}\mathbf {r'} }
796:
7761:
7718:
7895:
7870:
7830:
7347:
5858:
5854:
4466:
2660:
2453:{\displaystyle dq_{b}={\frac {d\mathbf {d} }{|\mathbf {s} |}}\cdot \mathbf {\hat {n}} }
2086:
1863:
1403:
1341:
1183:
917:
716:
711:
533:
423:
348:
298:
248:
221:
178:
153:
116:
17:
7930:
7874:
2986:
2378:
2304:{\displaystyle q_{b}={\frac {\mathbf {d} \cdot \mathbf {\hat {n}} }{|\mathbf {s} |}}}
2097:
1612:
1262:
whose square is proportional to the probability of finding the electron at any point
831:
816:
801:
741:
453:
368:
353:
268:
253:
158:
6918:
The electron-electron repulsion energy is thus derived under these conditions to be:
1369:
is the amount of charge per unit volume (three dimensional), surface charge density
7569:
5927:
5888:
2985:
For a continuous distribution, the material can be divided up into infinitely many
1859:
1484:
1426:
1228:
1171:
928:
889:
811:
706:
671:
613:
548:
433:
313:
188:
468:
7478:
5326:
discrete point-like charge carriers. The charge density of the system at a point
1858:
is to clarify that the density is for electric charge, not other densities like
1374:
1175:
731:
583:
413:
75:
4990:{\displaystyle \rho _{q}(\mathbf {r} )=q\delta (\mathbf {r} -\mathbf {r} _{0})}
5121:
so the delta function ensures that when the charge density is integrated over
2235:
2076:
1883:
448:
7769:
7726:
1219:
771:
746:
558:
483:
80:
7777:
7734:
1417:
The linear charge density is the ratio of an infinitesimal electric charge
7310:
is the kinetic and potential energy of electrons due to positive charges,
4601:{\displaystyle \mathbf {D} \cdot \mathbf {\hat {n}} dS=\iiint \rho _{f}dV}
3044:{\displaystyle d\mathbf {d} =\mathbf {P} dV=\mathbf {P} d^{3}\mathbf {r} }
7666:(3rd ed.). Cambridge: Cambridge University Press. pp. 443–453.
7350:, the charge density of ions influences their rejection by the membrane.
6055:. When the wavefunction is normalized - the average charge in the region
5846:
4917:
3511:
2093:
1337:
1187:
523:
518:
138:
6139:{\displaystyle Q=\int _{R}q|\psi (\mathbf {r} )|^{2}\,d^{3}\mathbf {r} }
5823:
Similar equations are used for the linear and surface charge densities.
1332:
is proportional to the charge density of the electron at any point. In
1178:
is composed of a cloud of free electrons moving randomly in space. The
7637:
Vanderlinde, Jack (2006). "11.1:The Four-potential and
Coulomb's Law".
1422:
905:
493:
2234:
The bound surface charge is the charge piled up at the surface of the
7419:"Physics 2: Electricity and Magnetism, Course Notes, Ch. 2, p. 15-16"
2315:
is the separation between the point charges constituting the dipole,
2226:
where subscripts "f" and "b" denote "free" and "bound" respectively.
1545:
893:
885:
578:
85:
1414:
Following are the definitions for continuous charge distributions.
1071:
of charge flows into or out of the volume. This is expressed by a
6005:{\displaystyle \rho _{q}(\mathbf {r} )=q|\psi (\mathbf {r} )|^{2}}
1360:
909:
4685:
Start with the definition of a continuous volume charge density:
4470:
4121:) and the potential due to the volume charge (volume integral):
2219:{\displaystyle \sigma =\sigma _{\text{f}}+\sigma _{\text{b}}\,.}
1333:
30:"Charge distribution" redirects here. This article is about the
1388:
between two point charges, the volume density of these is the
1191:
1167:
4465:
The free charge density serves as a useful simplification in
1686:{\displaystyle Q=\int _{L}\lambda _{q}(\mathbf {r} )\,d\ell }
1344:
which surround the atom or molecule, and are responsible for
7614:
An
Introduction to Tensor Calculus: Relativity and Cosmology
7338:; when the charge distribution moves, this corresponds to a
1340:
the charge of the electrons is distributed in clouds called
7511:(2nd ed.). Manchester Physics, John Wiley & Sons.
6817:{\textstyle \rho _{q}(\mathbf {r} )=q\cdot n(\mathbf {r} )}
5814:{\displaystyle \rho _{q}(\mathbf {r} )=qn(\mathbf {r} )\,.}
4166:{\displaystyle \varphi ={\frac {1}{4\pi \varepsilon _{0}}}}
3890:{\displaystyle \varphi ={\frac {1}{4\pi \varepsilon _{0}}}}
2238:, given by the dipole moment perpendicular to the surface:
1202:
At even smaller scales, of atoms and molecules, due to the
7853:
Physics for
Scientists and Engineers - with Modern Physics
6051:
i.e. probability per unit volume of a particle located at
4117:
which separates into the potential of the surface charge (
2169:{\displaystyle \rho =\rho _{\text{f}}+\rho _{\text{b}}\,.}
1365:
Continuous charge distribution. The volume charge density
2670:, the bound volume charge density within the material is
1166:
is caused by surface charges consisting of electrons and
1067:, the charge density in any volume can only change if an
7255:
Then, the energy is given using
Hartree-Fock method as:
1765:{\displaystyle Q=\int _{S}\sigma _{q}(\mathbf {r} )\,dS}
7543:(3rd ed.). Pearson Education, Dorling Kindersley.
4742:{\displaystyle Q=\int _{V}\rho _{q}(\mathbf {r} )\,dV.}
2833:{\displaystyle \rho _{b}=-\nabla \cdot \mathbf {P} \,.}
7641:. Springer Science & Business Media. p. 314.
6769:
4527:
4186:
3910:
1847:{\displaystyle Q=\int _{V}\rho _{q}(\mathbf {r} )\,dV}
1373:
is amount per unit surface area (circle) with outward
7263:
6924:
6833:
6560:
6274:
6168:
6069:
5943:
5765:
5500:
5362:
5139:
5018:
4930:
4852:
4839:{\displaystyle Q=\rho _{q,0}\int _{V}\,dV=\rho _{0}V}
4780:
4691:
4645:
4549:
4525:
4490:
4381:
4208:
4184:
4131:
3932:
3908:
3855:
3660:
3524:
3222:
3074:
2995:
2866:
2798:
2676:
2470:
2387:
2347:
2321:
2244:
2182:
2132:
1956:
1799:
1717:
1638:
1557:
1496:
1435:
1290:
1268:
1237:
1149:, which can be idealized as points, the concept of a
1115:
1081:
1034:
1003:
972:
950:
7450:
Physics for
Scientists and Engineers, Vol. 2, 9th Ed
4920:, the volume charge density can be expressed by the
1476:{\displaystyle \lambda _{q}={\frac {dQ}{d\ell }}\,,}
1135:{\displaystyle {\boldsymbol {J}}({\boldsymbol {x}})}
7564:French, A. (1968). "8:Relativity and electricity".
1870:, and prevent conflict with the many other uses of
1607:Integrating the definitions gives the total charge
7829:
7296:
7244:
6910:
6816:
6755:
6544:
6259:
6138:
6004:
5853:, so charge density will also depend on velocity.
5813:
5725:
5452:
5312:
5113:
4989:
4883:
4838:
4741:
4670:
4600:
4535:
4506:
4449:
4363:
4194:
4165:
4105:
3918:
3889:
3837:
3642:
3502:
3206:
3043:
2975:
2832:
2784:
2635:
2452:
2362:
2329:
2303:
2218:
2168:
2085:set up electric dipoles in response to an applied
2060:
1846:
1764:
1685:
1597:
1536:
1475:
1324:
1276:
1254:
1134:
1098:
1051:
1020:
989:
958:
5833:classical electromagnetism and special relativity
5330:is a sum of the charge densities for each charge
4564:
4409:
3950:
2623:
2600:
2560:
2460:and dividing by the differential surface element
2444:
2354:
2275:
1075:which links the rate of change of charge density
7314:is the electron electron interaction energy and
1182:in a conductor is equal to the number of mobile
6827:The potential energy of a system is written as:
5486:), ensures the integral of charge density over
3064:is the volume element, so the potential is the
1537:{\displaystyle \sigma _{q}={\frac {dQ}{dS}}\,,}
27:Electric charge per unit length, area or volume
1325:{\displaystyle |\psi ({\boldsymbol {x}})|^{2}}
7428:. Massachusetts Institute of Technology. 2007
5845:, the length of a segment of wire depends on
1598:{\displaystyle \rho _{q}={\frac {dQ}{dV}}\,,}
931:, charge density can vary with position. In
853:
8:
7790:: CS1 maint: multiple names: authors list (
7447:Serway, Raymond A.; Jewett, John W. (2013).
6856:
6834:
6254:
6196:
5735:If all charge carriers have the same charge
2038:
2025:
2004:
1991:
1970:
1957:
1483:similarly the surface charge density uses a
7662:Sakurai, Jun John; Napolitano, Jim (2021).
7583:Mould, Richard A. (2001). "Lorentz force".
7534:
7532:
7530:
7528:
2845:A more rigorous derivation is given below.
2106:are the excess charges which can move into
1214:a precise position but is represented by a
1052:{\displaystyle \lambda ({\boldsymbol {x}})}
7832:The Cambridge Handbook of Physics Formulas
1021:{\displaystyle \sigma ({\boldsymbol {x}})}
860:
846:
64:
48:
7713:(7). Oxford University Press: 4108–4116.
7334:. It is the principal source term of the
7262:
7230:
7221:
7212:
7207:
7193:
7178:
7169:
7160:
7154:
7138:
7131:
7114:
7099:
7090:
7081:
7076:
7059:
7035:
7020:
7011:
7005:
6989:
6982:
6965:
6929:
6923:
6903:
6897:
6885:
6871:
6848:
6840:
6832:
6806:
6783:
6774:
6768:
6741:
6736:
6720:
6715:
6706:
6691:
6686:
6670:
6665:
6656:
6647:
6642:
6636:
6631:
6624:
6618:
6617:
6604:
6599:
6592:
6586:
6585:
6567:
6559:
6530:
6525:
6506:
6496:
6491:
6475:
6470:
6461:
6446:
6441:
6422:
6412:
6407:
6391:
6386:
6377:
6368:
6363:
6357:
6352:
6345:
6339:
6338:
6325:
6320:
6313:
6307:
6306:
6296:
6281:
6273:
6246:
6234:
6229:
6220:
6211:
6202:
6190:
6175:
6167:
6131:
6125:
6120:
6114:
6109:
6100:
6089:
6080:
6068:
5996:
5991:
5982:
5971:
5957:
5948:
5942:
5807:
5799:
5779:
5770:
5764:
5717:
5704:
5693:
5677:
5672:
5663:
5652:
5646:
5636:
5626:
5613:
5602:
5586:
5581:
5572:
5560:
5547:
5536:
5527:
5521:
5511:
5499:
5441:
5436:
5427:
5415:
5402:
5391:
5376:
5367:
5361:
5295:
5290:
5281:
5274:
5269:
5263:
5253:
5234:
5229:
5220:
5210:
5205:
5199:
5189:
5176:
5171:
5166:
5160:
5150:
5138:
5102:
5097:
5078:
5073:
5064:
5050:
5039:
5033:
5023:
5017:
5001:is the position to calculate the charge.
4978:
4973:
4964:
4944:
4935:
4929:
4866:
4851:
4827:
4813:
4807:
4791:
4779:
4729:
4721:
4712:
4702:
4690:
4659:
4644:
4586:
4559:
4558:
4550:
4548:
4526:
4524:
4495:
4489:
4442:
4424:
4415:
4404:
4403:
4395:
4386:
4380:
4351:
4345:
4333:
4324:
4315:
4310:
4303:
4297:
4285:
4269:
4258:
4249:
4240:
4235:
4216:
4209:
4207:
4185:
4183:
4154:
4138:
4130:
4093:
4087:
4075:
4066:
4057:
4052:
4045:
4031:
4019:
4003:
3992:
3983:
3974:
3969:
3945:
3944:
3936:
3933:
3931:
3909:
3907:
3878:
3862:
3854:
3825:
3819:
3802:
3776:
3767:
3761:
3746:
3737:
3728:
3723:
3717:
3715:
3683:
3667:
3659:
3630:
3624:
3608:
3599:
3590:
3585:
3579:
3559:
3547:
3531:
3523:
3491:
3486:
3476:
3467:
3462:
3451:
3442:
3439:
3424:
3415:
3406:
3401:
3395:
3366:
3360:
3355:
3331:
3325:
3320:
3296:
3290:
3285:
3264:
3255:
3246:
3241:
3235:
3221:
3194:
3188:
3175:
3170:
3160:
3151:
3146:
3139:
3124:
3115:
3109:
3097:
3081:
3073:
3036:
3030:
3021:
3007:
2999:
2994:
2964:
2959:
2949:
2940:
2935:
2928:
2913:
2904:
2898:
2889:
2873:
2865:
2826:
2821:
2803:
2797:
2775:
2770:
2748:
2737:
2735:
2734:
2726:
2720:
2703:
2697:
2681:
2675:
2629:
2618:
2617:
2609:
2595:
2594:
2575:
2569:
2555:
2554:
2537:
2532:
2527:
2520:
2514:
2494:
2484:
2475:
2469:
2439:
2438:
2427:
2422:
2417:
2410:
2404:
2395:
2386:
2349:
2348:
2346:
2322:
2320:
2293:
2288:
2283:
2270:
2269:
2261:
2258:
2249:
2243:
2212:
2206:
2193:
2181:
2162:
2156:
2143:
2131:
2122:In terms of volume charge densities, the
2054:
2044:
2032:
2020:
2010:
1998:
1986:
1976:
1964:
1955:
1837:
1829:
1820:
1810:
1798:
1755:
1747:
1738:
1728:
1716:
1676:
1668:
1659:
1649:
1637:
1591:
1571:
1562:
1556:
1530:
1510:
1501:
1495:
1469:
1449:
1440:
1434:
1316:
1311:
1302:
1291:
1289:
1269:
1267:
1244:
1236:
1124:
1116:
1114:
1088:
1080:
1041:
1033:
1010:
1002:
979:
971:
951:
949:
2464:gives the bound surface charge density:
1255:{\displaystyle \psi ({\boldsymbol {x}})}
1099:{\displaystyle \rho ({\boldsymbol {x}})}
990:{\displaystyle \rho ({\boldsymbol {x}})}
7502:
7500:
7385:
1888:electrical resistivity and conductivity
1303:
1270:
1245:
1125:
1117:
1089:
1042:
1011:
980:
952:
609:Electromagnetism and special relativity
56:
7783:
7750:Environmental Science & Technology
7707:Environmental Science & Technology
1170:near the surface of objects, and the
7589:Springer Science & Business Media
7318:is the exchange energy of electrons.
6824:is considered as the charge density.
1544:and the volume charge density uses a
629:Maxwell equations in curved spacetime
7:
7897:McGraw Hill Encyclopaedia of Physics
7472:
7470:
7346:. For separation processes such as
5827:Charge density in special relativity
4751:Then, by definition of homogeneity,
1226:. An electron is represented by a
7616:. Courier Corporation. p. 74.
5899:Charge density in quantum mechanics
5462:The delta function for each charge
2363:{\displaystyle \mathbf {\hat {n}} }
7689:"The Hartree-Fock Method in Atoms"
7507:I.S. Grant; W.R. Phillips (2008).
7393:P.M. Whelan, M.J. Hodgson (1978).
7330:for electric current, and also in
7326:The charge density appears in the
6700:
6644:
6619:
6587:
6455:
6365:
6340:
6308:
6016:is the charge of the particle and
4492:
4436:
4035:
3792:
3701:
3568:
3372:
3368:
3337:
3333:
3302:
3298:
3224:
2815:
2764:
912:(C⋅m), at any point in a volume.
25:
7453:. Cengage Learning. p. 704.
2176:as for surface charge densities:
1410:is a point in the charged object.
1277:{\displaystyle {\boldsymbol {x}}}
959:{\displaystyle {\boldsymbol {x}}}
935:charge density is idealized as a
7900:(2nd ed.). VHC publishers.
7879:(2nd ed.). VHC publishers.
7639:Classical Electromagnetic Theory
7222:
7213:
7194:
7179:
7161:
7139:
7091:
7082:
7060:
7036:
7012:
6990:
6904:
6886:
6872:
6807:
6784:
6737:
6716:
6707:
6687:
6666:
6657:
6632:
6600:
6568:
6526:
6507:
6492:
6471:
6462:
6442:
6423:
6408:
6387:
6378:
6353:
6321:
6282:
6230:
6221:
6176:
6132:
6101:
5983:
5958:
5876:It turns out the charge density
5800:
5780:
5673:
5664:
5653:
5582:
5573:
5528:
5437:
5428:
5377:
5291:
5282:
5270:
5230:
5221:
5206:
5167:
5098:
5074:
5065:
5051:
5040:
4974:
4965:
4945:
4722:
4561:
4551:
4536:{\displaystyle {\scriptstyle S}}
4514:
4443:
4406:
4396:
4353:
4325:
4316:
4250:
4241:
4195:{\displaystyle {\scriptstyle S}}
4173:
4095:
4067:
4058:
4046:
3984:
3975:
3947:
3937:
3919:{\displaystyle {\scriptstyle S}}
3897:
3827:
3803:
3777:
3768:
3738:
3729:
3718:
3632:
3600:
3591:
3560:
3477:
3468:
3452:
3443:
3416:
3407:
3356:
3321:
3286:
3256:
3247:
3196:
3161:
3152:
3140:
3125:
3116:
3037:
3022:
3008:
3000:
2950:
2941:
2929:
2914:
2905:
2822:
2771:
2738:
2727:
2620:
2610:
2597:
2576:
2557:
2533:
2521:
2441:
2423:
2411:
2351:
2323:
2289:
2272:
2262:
1830:
1748:
1669:
933:classical electromagnetic theory
7851:P. A. Tipler, G. Mosca (2008).
7813:. Schaum Series, Mc Graw Hill.
7811:3000 Solved Problems in Physics
7541:Introduction to Electrodynamics
7395:Essential Principles of Physics
6554:Using symmetrization condition:
4884:{\displaystyle Q=V\rho _{q,0}.}
4419:
2024:
1990:
1905:. Other notations may include:
1697:of the surface charge density σ
1158:moving randomly in the metal's
1145:Since all charge is carried by
1065:conservation of electric charge
1061:continuous charge distributions
7923:- Spatial charge distributions
7836:. Cambridge University Press.
7483:. Cambridge University Press.
7477:Purcell, Edward (2011-09-22).
7273:
7267:
7231:
7208:
7202:
7189:
7183:
7175:
7100:
7077:
7071:
7068:
7055:
7046:
7043:
7040:
7032:
7023:
6959:
6953:
6944:
6938:
6890:
6882:
6876:
6868:
6849:
6841:
6811:
6803:
6788:
6780:
6747:
6703:
6697:
6653:
6572:
6564:
6536:
6458:
6452:
6374:
6286:
6278:
6247:
6243:
6217:
6203:
6180:
6172:
6110:
6105:
6097:
6090:
5992:
5987:
5979:
5972:
5962:
5954:
5804:
5796:
5784:
5776:
5683:
5660:
5592:
5569:
5447:
5424:
5381:
5373:
5301:
5278:
5240:
5217:
5108:
5093:
5084:
5061:
5055:
5047:
4984:
4961:
4949:
4941:
4726:
4718:
4334:
4311:
4259:
4236:
4076:
4053:
3993:
3970:
3845:using the divergence theorem:
3807:
3788:
3747:
3724:
3609:
3586:
3487:
3463:
3425:
3402:
3265:
3242:
3171:
3147:
3133:
3112:
2960:
2936:
2922:
2901:
2742:
2538:
2528:
2428:
2418:
2294:
2284:
1834:
1826:
1752:
1744:
1673:
1665:
1312:
1307:
1299:
1292:
1249:
1241:
1210:, a charged particle does not
1129:
1121:
1093:
1085:
1046:
1038:
1015:
1007:
984:
976:
920:on a two dimensional surface.
1:
7397:(2nd ed.). John Murray.
5837:relativistic electromagnetism
4671:{\displaystyle Q=V\rho _{0}.}
1776:of the volume charge density
1615:of the linear charge density
634:Relativistic electromagnetism
7362:relating charge density and
6049:probability density function
5490:returns the total charge in
4912:inside a region of 3d space
2330:{\displaystyle \mathbf {d} }
2071:Free, bound and total charge
1402:is a point to calculate the
4764:) is a constant denoted by
4475:electric displacement field
918:surface charge distribution
7963:
7947:Electromagnetic quantities
5902:
5830:
4901:For a single point charge
4628:For the special case of a
4624:Homogeneous charge density
4507:{\displaystyle \Phi _{D}=}
4480:emerging from the object:
1628:) over a line or 1d curve
359:Liénard–Wiechert potential
29:
7855:(6th ed.). Freeman.
7612:Lawden, Derek F. (2012).
7480:Electricity and Magnetism
5905:Density functional theory
2655:within the material, and
2108:electrostatic equilibrium
1611:of a region according to
624:Mathematical descriptions
334:Electromagnetic radiation
324:Electromagnetic induction
264:Magnetic vector potential
259:Magnetic scalar potential
7876:Encyclopaedia of Physics
7664:Modern quantum mechanics
6157:over 3d position space.
5887:transform together as a
5322:This can be extended to
1882:in electromagnetism for
1216:probability distribution
1059:are usually regarded as
7770:10.1021/acs.est.7b06400
7727:10.1021/acs.est.7b06400
7539:D.J. Griffiths (2007).
7297:{\displaystyle E=I+J-K}
5893:Lorentz transformations
5849:of observer because of
2856:due to a dipole moment
2653:electric dipole moments
174:Electrostatic induction
169:Electrostatic discharge
18:Electric charge density
7687:Littlejohn, Robert G.
7298:
7246:
6912:
6818:
6757:
6546:
6261:
6140:
6006:
5857:has described how the
5815:
5727:
5709:
5618:
5552:
5454:
5407:
5314:
5125:, the total charge in
5115:
4991:
4885:
4840:
4743:
4672:
4602:
4537:
4508:
4451:
4365:
4196:
4167:
4107:
3920:
3891:
3839:
3644:
3510:where ∇′ is the
3504:
3208:
3045:
2977:
2834:
2786:
2637:
2454:
2364:
2339:electric dipole moment
2331:
2305:
2220:
2170:
2118:Total charge densities
2062:
1848:
1766:
1687:
1599:
1538:
1477:
1425:) to an infinitesimal
1411:
1326:
1278:
1256:
1180:charge carrier density
1136:
1100:
1053:
1022:
991:
960:
914:Surface charge density
604:Electromagnetic tensor
7873:, G.L. Trigg (1991).
7336:electromagnetic field
7299:
7247:
6913:
6819:
6758:
6547:
6262:
6141:
6007:
5871:proper charge density
5831:Further information:
5816:
5728:
5689:
5598:
5532:
5455:
5387:
5315:
5116:
4992:
4886:
4841:
4744:
4673:
4618:constitutive relation
4603:
4538:
4509:
4452:
4366:
4197:
4168:
4108:
3921:
3892:
3840:
3645:
3505:
3209:
3046:
2978:
2835:
2787:
2638:
2455:
2365:
2332:
2306:
2221:
2171:
2063:
1849:
1767:
1688:
1600:
1539:
1478:
1364:
1327:
1279:
1257:
1204:uncertainty principle
1137:
1101:
1054:
1023:
992:
961:
944:function of position
922:Linear charge density
898:Volume charge density
597:Covariant formulation
389:Synchrotron radiation
329:Electromagnetic pulse
319:Electromagnetic field
38:. For other uses see
7894:C.B. Parker (1994).
7261:
6922:
6831:
6767:
6558:
6272:
6166:
6067:
5941:
5763:
5498:
5360:
5137:
5016:
4928:
4922:Dirac delta function
4850:
4778:
4689:
4643:
4547:
4523:
4488:
4379:
4206:
4182:
4129:
3930:
3906:
3853:
3658:
3653:Integrating by parts
3522:
3220:
3072:
2993:
2864:
2796:
2674:
2659:is the differential
2649:polarization density
2468:
2385:
2345:
2319:
2242:
2180:
2130:
1954:
1854:where the subscript
1797:
1715:
1636:
1555:
1494:
1433:
1390:polarization density
1288:
1266:
1235:
1156:conduction electrons
1113:
1079:
1032:
1001:
970:
966:, like a fluid, and
948:
639:Stress–energy tensor
564:Reluctance (complex)
309:Displacement current
7809:A. Halpern (1988).
7762:2018EnST...52.4108E
7719:2018EnST...52.4108E
7572:. pp. 229–265.
7374:Charge density wave
7360:Continuity equation
7332:Maxwell's Equations
7328:continuity equation
6242:
6155:integration measure
5909:Hartree–Fock method
4614:Maxwell's equations
4461:Free charge density
2126:charge density is:
1868:probability density
1147:subatomic particles
1073:continuity equation
554:Magnetomotive force
439:Electromotive force
409:Alternating current
344:Jefimenko equations
304:Cyclotron radiation
7566:Special Relativity
7426:MIT OpenCourseware
7294:
7242:
6908:
6814:
6753:
6542:
6301:
6257:
6228:
6195:
6136:
6002:
5867:frame of reference
5851:length contraction
5843:special relativity
5811:
5723:
5450:
5310:
5111:
4987:
4881:
4836:
4739:
4668:
4620:for more details.
4598:
4533:
4531:
4504:
4447:
4361:
4192:
4190:
4163:
4103:
3916:
3914:
3887:
3835:
3640:
3500:
3204:
3041:
2973:
2854:electric potential
2850:
2830:
2782:
2668:divergence theorem
2651:, i.e. density of
2633:
2450:
2372:unit normal vector
2360:
2327:
2301:
2216:
2166:
2058:
1844:
1762:
1683:
1595:
1534:
1473:
1412:
1357:Continuous charges
1322:
1274:
1252:
1164:Static electricity
1132:
1096:
1049:
1018:
987:
956:
402:Electrical network
239:Gauss magnetic law
204:Static electricity
164:Electric potential
7907:978-0-07-051400-3
7886:978-0-89573-752-6
7862:978-0-7167-8964-2
7843:978-0-521-57507-2
7820:978-0-07-025734-4
7673:978-1-108-47322-4
7623:978-0-486-13214-3
7550:978-81-7758-293-2
7518:978-0-471-92712-9
7236:
7122:
7105:
6973:
6292:
6186:
5937:) by the equation
5917:, charge density
5915:quantum mechanics
5863:Minkowski diagram
5712:
5621:
5555:
5410:
5008:for any function
5006:shifting property
4567:
4412:
4339:
4292:
4264:
4161:
4081:
4026:
3998:
3953:
3885:
3786:
3752:
3690:
3614:
3554:
3498:
3430:
3384:
3349:
3314:
3270:
3182:
3104:
3068:over the object:
2971:
2896:
2848:
2745:
2626:
2603:
2589:
2563:
2549:
2509:
2447:
2433:
2357:
2299:
2278:
2209:
2196:
2159:
2146:
2112:electric currents
2052:
2018:
1984:
1707:) over a surface
1589:
1528:
1467:
1208:quantum mechanics
1196:elementary charge
880:is the amount of
870:
869:
569:Reluctance (real)
539:Gyrator–capacitor
484:Resonant cavities
374:Maxwell equations
32:physical quantity
16:(Redirected from
7954:
7911:
7890:
7866:
7847:
7835:
7828:G. Woan (2010).
7824:
7796:
7795:
7789:
7781:
7756:(7): 4108–4116.
7745:
7739:
7738:
7702:
7696:
7695:
7693:
7684:
7678:
7677:
7659:
7653:
7652:
7634:
7628:
7627:
7609:
7603:
7602:
7585:Basic Relativity
7580:
7574:
7573:
7561:
7555:
7554:
7536:
7523:
7522:
7509:Electromagnetism
7504:
7495:
7494:
7474:
7465:
7464:
7444:
7438:
7437:
7435:
7433:
7423:
7415:
7409:
7408:
7390:
7344:hydrogen bonding
7303:
7301:
7300:
7295:
7251:
7249:
7248:
7243:
7241:
7237:
7235:
7234:
7229:
7225:
7216:
7211:
7205:
7201:
7197:
7182:
7170:
7164:
7159:
7158:
7146:
7142:
7136:
7135:
7123:
7115:
7110:
7106:
7104:
7103:
7098:
7094:
7085:
7080:
7074:
7067:
7063:
7039:
7021:
7015:
7010:
7009:
6997:
6993:
6987:
6986:
6974:
6966:
6937:
6936:
6917:
6915:
6914:
6909:
6907:
6902:
6901:
6889:
6875:
6852:
6844:
6823:
6821:
6820:
6815:
6810:
6787:
6779:
6778:
6762:
6760:
6759:
6754:
6746:
6745:
6740:
6725:
6724:
6719:
6710:
6696:
6695:
6690:
6675:
6674:
6669:
6660:
6652:
6651:
6641:
6640:
6635:
6629:
6628:
6623:
6622:
6609:
6608:
6603:
6597:
6596:
6591:
6590:
6571:
6551:
6549:
6548:
6543:
6535:
6534:
6529:
6514:
6510:
6501:
6500:
6495:
6480:
6479:
6474:
6465:
6451:
6450:
6445:
6430:
6426:
6417:
6416:
6411:
6396:
6395:
6390:
6381:
6373:
6372:
6362:
6361:
6356:
6350:
6349:
6344:
6343:
6330:
6329:
6324:
6318:
6317:
6312:
6311:
6300:
6285:
6266:
6264:
6263:
6258:
6250:
6238:
6233:
6224:
6216:
6215:
6206:
6194:
6179:
6145:
6143:
6142:
6137:
6135:
6130:
6129:
6119:
6118:
6113:
6104:
6093:
6085:
6084:
6046:
6028:
6011:
6009:
6008:
6003:
6001:
6000:
5995:
5986:
5975:
5961:
5953:
5952:
5820:
5818:
5817:
5812:
5803:
5783:
5775:
5774:
5732:
5730:
5729:
5724:
5722:
5721:
5710:
5708:
5703:
5682:
5681:
5676:
5667:
5656:
5651:
5650:
5641:
5640:
5631:
5630:
5619:
5617:
5612:
5591:
5590:
5585:
5576:
5565:
5564:
5553:
5551:
5546:
5531:
5526:
5525:
5516:
5515:
5459:
5457:
5456:
5451:
5446:
5445:
5440:
5431:
5420:
5419:
5408:
5406:
5401:
5380:
5372:
5371:
5355:
5319:
5317:
5316:
5311:
5300:
5299:
5294:
5285:
5273:
5268:
5267:
5258:
5257:
5239:
5238:
5233:
5224:
5209:
5204:
5203:
5194:
5193:
5181:
5180:
5170:
5165:
5164:
5155:
5154:
5120:
5118:
5117:
5112:
5107:
5106:
5101:
5083:
5082:
5077:
5068:
5054:
5043:
5038:
5037:
5028:
5027:
4996:
4994:
4993:
4988:
4983:
4982:
4977:
4968:
4948:
4940:
4939:
4897:Discrete charges
4890:
4888:
4887:
4882:
4877:
4876:
4845:
4843:
4842:
4837:
4832:
4831:
4812:
4811:
4802:
4801:
4748:
4746:
4745:
4740:
4725:
4717:
4716:
4707:
4706:
4677:
4675:
4674:
4669:
4664:
4663:
4608:
4607:
4605:
4604:
4599:
4591:
4590:
4569:
4568:
4560:
4554:
4543:
4542:
4540:
4539:
4534:
4532:
4518:
4517:
4513:
4511:
4510:
4505:
4500:
4499:
4456:
4454:
4453:
4448:
4446:
4429:
4428:
4414:
4413:
4405:
4399:
4391:
4390:
4371:
4370:
4368:
4367:
4362:
4360:
4359:
4350:
4349:
4340:
4338:
4337:
4332:
4328:
4319:
4314:
4308:
4307:
4298:
4293:
4291:
4290:
4289:
4270:
4265:
4263:
4262:
4257:
4253:
4244:
4239:
4233:
4232:
4221:
4220:
4210:
4202:
4201:
4199:
4198:
4193:
4191:
4177:
4176:
4172:
4170:
4169:
4164:
4162:
4160:
4159:
4158:
4139:
4119:surface integral
4113:
4112:
4110:
4109:
4104:
4102:
4101:
4092:
4091:
4082:
4080:
4079:
4074:
4070:
4061:
4056:
4050:
4049:
4041:
4032:
4027:
4025:
4024:
4023:
4004:
3999:
3997:
3996:
3991:
3987:
3978:
3973:
3967:
3966:
3955:
3954:
3946:
3940:
3934:
3926:
3925:
3923:
3922:
3917:
3915:
3901:
3900:
3896:
3894:
3893:
3888:
3886:
3884:
3883:
3882:
3863:
3844:
3842:
3841:
3836:
3834:
3833:
3824:
3823:
3814:
3810:
3806:
3798:
3787:
3785:
3784:
3780:
3771:
3762:
3757:
3753:
3751:
3750:
3745:
3741:
3732:
3727:
3721:
3716:
3707:
3691:
3689:
3688:
3687:
3668:
3649:
3647:
3646:
3641:
3639:
3638:
3629:
3628:
3619:
3615:
3613:
3612:
3607:
3603:
3594:
3589:
3580:
3574:
3563:
3555:
3553:
3552:
3551:
3532:
3509:
3507:
3506:
3501:
3499:
3497:
3496:
3495:
3490:
3484:
3480:
3471:
3466:
3460:
3459:
3455:
3446:
3440:
3435:
3431:
3429:
3428:
3423:
3419:
3410:
3405:
3396:
3390:
3386:
3385:
3383:
3382:
3367:
3365:
3364:
3359:
3350:
3348:
3347:
3332:
3330:
3329:
3324:
3315:
3313:
3312:
3297:
3295:
3294:
3289:
3275:
3271:
3269:
3268:
3263:
3259:
3250:
3245:
3236:
3230:
3213:
3211:
3210:
3205:
3203:
3202:
3193:
3192:
3183:
3181:
3180:
3179:
3174:
3168:
3164:
3155:
3150:
3144:
3143:
3132:
3128:
3119:
3110:
3105:
3103:
3102:
3101:
3082:
3063:
3050:
3048:
3047:
3042:
3040:
3035:
3034:
3025:
3011:
3003:
2982:
2980:
2979:
2974:
2972:
2970:
2969:
2968:
2963:
2957:
2953:
2944:
2939:
2933:
2932:
2921:
2917:
2908:
2899:
2897:
2895:
2894:
2893:
2874:
2839:
2837:
2836:
2831:
2825:
2808:
2807:
2791:
2789:
2788:
2783:
2774:
2747:
2746:
2741:
2736:
2730:
2725:
2724:
2702:
2701:
2686:
2685:
2642:
2640:
2639:
2634:
2628:
2627:
2619:
2613:
2605:
2604:
2596:
2590:
2588:
2580:
2579:
2570:
2565:
2564:
2556:
2550:
2548:
2541:
2536:
2531:
2525:
2524:
2515:
2510:
2508:
2500:
2499:
2498:
2485:
2480:
2479:
2459:
2457:
2456:
2451:
2449:
2448:
2440:
2434:
2432:
2431:
2426:
2421:
2415:
2414:
2405:
2400:
2399:
2374:to the surface.
2369:
2367:
2366:
2361:
2359:
2358:
2350:
2336:
2334:
2333:
2328:
2326:
2310:
2308:
2307:
2302:
2300:
2298:
2297:
2292:
2287:
2281:
2280:
2279:
2271:
2265:
2259:
2254:
2253:
2225:
2223:
2222:
2217:
2211:
2210:
2207:
2198:
2197:
2194:
2175:
2173:
2172:
2167:
2161:
2160:
2157:
2148:
2147:
2144:
2067:
2065:
2064:
2059:
2053:
2045:
2037:
2036:
2019:
2011:
2003:
2002:
1985:
1977:
1969:
1968:
1853:
1851:
1850:
1845:
1833:
1825:
1824:
1815:
1814:
1789:) over a volume
1771:
1769:
1768:
1763:
1751:
1743:
1742:
1733:
1732:
1695:surface integral
1692:
1690:
1689:
1684:
1672:
1664:
1663:
1654:
1653:
1604:
1602:
1601:
1596:
1590:
1588:
1580:
1572:
1567:
1566:
1543:
1541:
1540:
1535:
1529:
1527:
1519:
1511:
1506:
1505:
1482:
1480:
1479:
1474:
1468:
1466:
1458:
1450:
1445:
1444:
1331:
1329:
1328:
1323:
1321:
1320:
1315:
1306:
1295:
1283:
1281:
1280:
1275:
1273:
1261:
1259:
1258:
1253:
1248:
1224:chemical bonding
1141:
1139:
1138:
1133:
1128:
1120:
1105:
1103:
1102:
1097:
1092:
1069:electric current
1058:
1056:
1055:
1050:
1045:
1027:
1025:
1024:
1019:
1014:
996:
994:
993:
988:
983:
965:
963:
962:
957:
955:
874:electromagnetism
862:
855:
848:
529:Electric machine
512:Magnetic circuit
474:Parallel circuit
464:Network analysis
429:Electric current
364:London equations
209:Triboelectricity
199:Potential energy
68:
58:Electromagnetism
49:
36:electromagnetism
21:
7962:
7961:
7957:
7956:
7955:
7953:
7952:
7951:
7942:Electric charge
7927:
7926:
7918:
7908:
7893:
7887:
7869:
7863:
7850:
7844:
7827:
7821:
7808:
7805:
7803:Further reading
7800:
7799:
7782:
7747:
7746:
7742:
7704:
7703:
7699:
7691:
7686:
7685:
7681:
7674:
7661:
7660:
7656:
7649:
7636:
7635:
7631:
7624:
7611:
7610:
7606:
7599:
7582:
7581:
7577:
7563:
7562:
7558:
7551:
7538:
7537:
7526:
7519:
7506:
7505:
7498:
7491:
7476:
7475:
7468:
7461:
7446:
7445:
7441:
7431:
7429:
7421:
7417:
7416:
7412:
7405:
7392:
7391:
7387:
7382:
7369:Ionic potential
7364:current density
7356:
7340:current density
7324:
7259:
7258:
7220:
7206:
7192:
7171:
7165:
7150:
7137:
7127:
7089:
7075:
7058:
7022:
7016:
7001:
6988:
6978:
6925:
6920:
6919:
6893:
6829:
6828:
6770:
6765:
6764:
6735:
6714:
6685:
6664:
6643:
6630:
6616:
6598:
6584:
6556:
6555:
6524:
6505:
6490:
6469:
6440:
6421:
6406:
6385:
6364:
6351:
6337:
6319:
6305:
6270:
6269:
6207:
6164:
6163:
6121:
6108:
6076:
6065:
6064:
6019:
6017:
5990:
5944:
5939:
5938:
5925:
5911:
5901:
5882:current density
5839:
5829:
5766:
5761:
5760:
5749:electron charge
5739:(for electrons
5713:
5671:
5642:
5632:
5622:
5580:
5556:
5517:
5507:
5496:
5495:
5485:
5467:
5435:
5411:
5363:
5358:
5357:
5347:
5345:
5335:
5289:
5259:
5249:
5228:
5195:
5185:
5172:
5156:
5146:
5135:
5134:
5096:
5072:
5029:
5019:
5014:
5013:
4972:
4931:
4926:
4925:
4911:
4899:
4862:
4848:
4847:
4823:
4803:
4787:
4776:
4775:
4773:
4759:
4708:
4698:
4687:
4686:
4683:
4655:
4641:
4640:
4638:
4632:charge density
4626:
4582:
4545:
4544:
4521:
4520:
4519:
4515:
4491:
4486:
4485:
4484:
4463:
4458:
4420:
4382:
4377:
4376:
4352:
4341:
4323:
4309:
4299:
4281:
4274:
4248:
4234:
4225:
4212:
4211:
4204:
4203:
4180:
4179:
4178:
4174:
4150:
4143:
4127:
4126:
4125:
4094:
4083:
4065:
4051:
4034:
4033:
4015:
4008:
3982:
3968:
3959:
3935:
3928:
3927:
3904:
3903:
3902:
3898:
3874:
3867:
3851:
3850:
3849:
3826:
3815:
3791:
3775:
3766:
3736:
3722:
3711:
3700:
3699:
3695:
3679:
3672:
3656:
3655:
3631:
3620:
3598:
3584:
3575:
3567:
3543:
3536:
3520:
3519:
3485:
3475:
3461:
3450:
3441:
3414:
3400:
3391:
3375:
3371:
3354:
3340:
3336:
3319:
3305:
3301:
3284:
3283:
3279:
3254:
3240:
3231:
3223:
3218:
3217:
3195:
3184:
3169:
3159:
3145:
3123:
3111:
3093:
3086:
3070:
3069:
3066:volume integral
3052:
3026:
2991:
2990:
2958:
2948:
2934:
2912:
2900:
2885:
2878:
2862:
2861:
2799:
2794:
2793:
2716:
2693:
2677:
2672:
2671:
2581:
2571:
2526:
2516:
2501:
2490:
2486:
2471:
2466:
2465:
2416:
2406:
2391:
2383:
2382:
2343:
2342:
2317:
2316:
2282:
2260:
2245:
2240:
2239:
2232:
2202:
2189:
2178:
2177:
2152:
2139:
2128:
2127:
2120:
2073:
2028:
1994:
1960:
1952:
1951:
1945:
1938:
1931:
1924:
1917:
1910:
1816:
1806:
1795:
1794:
1784:
1774:volume integral
1734:
1724:
1713:
1712:
1702:
1655:
1645:
1634:
1633:
1623:
1581:
1573:
1558:
1553:
1552:
1520:
1512:
1497:
1492:
1491:
1459:
1451:
1436:
1431:
1430:
1397:Position vector
1359:
1354:
1310:
1286:
1285:
1264:
1263:
1233:
1232:
1184:charge carriers
1160:crystal lattice
1111:
1110:
1108:current density
1077:
1076:
1030:
1029:
999:
998:
968:
967:
946:
945:
882:electric charge
866:
837:
836:
652:
644:
643:
599:
589:
588:
544:Induction motor
514:
504:
503:
419:Current density
404:
394:
393:
384:Poynting vector
294:
292:Electrodynamics
284:
283:
279:Right-hand rule
244:Magnetic dipole
234:Biot–Savart law
224:
214:
213:
149:Electric dipole
144:Electric charge
119:
47:
28:
23:
22:
15:
12:
11:
5:
7960:
7958:
7950:
7949:
7944:
7939:
7929:
7928:
7925:
7924:
7917:
7916:External links
7914:
7913:
7912:
7906:
7891:
7885:
7867:
7861:
7848:
7842:
7825:
7819:
7804:
7801:
7798:
7797:
7740:
7697:
7679:
7672:
7654:
7647:
7629:
7622:
7604:
7597:
7575:
7556:
7549:
7524:
7517:
7496:
7489:
7466:
7459:
7439:
7410:
7403:
7384:
7383:
7381:
7378:
7377:
7376:
7371:
7366:
7355:
7352:
7348:nanofiltration
7323:
7320:
7293:
7290:
7287:
7284:
7281:
7278:
7275:
7272:
7269:
7266:
7240:
7233:
7228:
7224:
7219:
7215:
7210:
7204:
7200:
7196:
7191:
7188:
7185:
7181:
7177:
7174:
7168:
7163:
7157:
7153:
7149:
7145:
7141:
7134:
7130:
7126:
7121:
7118:
7113:
7109:
7102:
7097:
7093:
7088:
7084:
7079:
7073:
7070:
7066:
7062:
7057:
7054:
7051:
7048:
7045:
7042:
7038:
7034:
7031:
7028:
7025:
7019:
7014:
7008:
7004:
7000:
6996:
6992:
6985:
6981:
6977:
6972:
6969:
6964:
6961:
6958:
6955:
6952:
6949:
6946:
6943:
6940:
6935:
6932:
6928:
6906:
6900:
6896:
6892:
6888:
6884:
6881:
6878:
6874:
6870:
6867:
6864:
6861:
6858:
6855:
6851:
6847:
6843:
6839:
6836:
6813:
6809:
6805:
6802:
6799:
6796:
6793:
6790:
6786:
6782:
6777:
6773:
6752:
6749:
6744:
6739:
6734:
6731:
6728:
6723:
6718:
6713:
6709:
6705:
6702:
6699:
6694:
6689:
6684:
6681:
6678:
6673:
6668:
6663:
6659:
6655:
6650:
6646:
6639:
6634:
6627:
6621:
6615:
6612:
6607:
6602:
6595:
6589:
6583:
6580:
6577:
6574:
6570:
6566:
6563:
6541:
6538:
6533:
6528:
6523:
6520:
6517:
6513:
6509:
6504:
6499:
6494:
6489:
6486:
6483:
6478:
6473:
6468:
6464:
6460:
6457:
6454:
6449:
6444:
6439:
6436:
6433:
6429:
6425:
6420:
6415:
6410:
6405:
6402:
6399:
6394:
6389:
6384:
6380:
6376:
6371:
6367:
6360:
6355:
6348:
6342:
6336:
6333:
6328:
6323:
6316:
6310:
6304:
6299:
6295:
6291:
6288:
6284:
6280:
6277:
6256:
6253:
6249:
6245:
6241:
6237:
6232:
6227:
6223:
6219:
6214:
6210:
6205:
6201:
6198:
6193:
6189:
6185:
6182:
6178:
6174:
6171:
6134:
6128:
6124:
6117:
6112:
6107:
6103:
6099:
6096:
6092:
6088:
6083:
6079:
6075:
6072:
5999:
5994:
5989:
5985:
5981:
5978:
5974:
5970:
5967:
5964:
5960:
5956:
5951:
5947:
5926:is related to
5921:
5900:
5897:
5859:magnetic field
5855:Anthony French
5828:
5825:
5810:
5806:
5802:
5798:
5795:
5792:
5789:
5786:
5782:
5778:
5773:
5769:
5720:
5716:
5707:
5702:
5699:
5696:
5692:
5688:
5685:
5680:
5675:
5670:
5666:
5662:
5659:
5655:
5649:
5645:
5639:
5635:
5629:
5625:
5616:
5611:
5608:
5605:
5601:
5597:
5594:
5589:
5584:
5579:
5575:
5571:
5568:
5563:
5559:
5550:
5545:
5542:
5539:
5535:
5530:
5524:
5520:
5514:
5510:
5506:
5503:
5481:
5465:
5449:
5444:
5439:
5434:
5430:
5426:
5423:
5418:
5414:
5405:
5400:
5397:
5394:
5390:
5386:
5383:
5379:
5375:
5370:
5366:
5341:
5333:
5309:
5306:
5303:
5298:
5293:
5288:
5284:
5280:
5277:
5272:
5266:
5262:
5256:
5252:
5248:
5245:
5242:
5237:
5232:
5227:
5223:
5219:
5216:
5213:
5208:
5202:
5198:
5192:
5188:
5184:
5179:
5175:
5169:
5163:
5159:
5153:
5149:
5145:
5142:
5110:
5105:
5100:
5095:
5092:
5089:
5086:
5081:
5076:
5071:
5067:
5063:
5060:
5057:
5053:
5049:
5046:
5042:
5036:
5032:
5026:
5022:
4986:
4981:
4976:
4971:
4967:
4963:
4960:
4957:
4954:
4951:
4947:
4943:
4938:
4934:
4909:
4898:
4895:
4880:
4875:
4872:
4869:
4865:
4861:
4858:
4855:
4835:
4830:
4826:
4822:
4819:
4816:
4810:
4806:
4800:
4797:
4794:
4790:
4786:
4783:
4768:
4755:
4738:
4735:
4732:
4728:
4724:
4720:
4715:
4711:
4705:
4701:
4697:
4694:
4682:
4679:
4667:
4662:
4658:
4654:
4651:
4648:
4636:
4625:
4622:
4610:
4609:
4597:
4594:
4589:
4585:
4581:
4578:
4575:
4572:
4566:
4563:
4557:
4553:
4530:
4503:
4498:
4494:
4462:
4459:
4445:
4441:
4438:
4435:
4432:
4427:
4423:
4418:
4411:
4408:
4402:
4398:
4394:
4389:
4385:
4373:
4372:
4358:
4355:
4348:
4344:
4336:
4331:
4327:
4322:
4318:
4313:
4306:
4302:
4296:
4288:
4284:
4280:
4277:
4273:
4268:
4261:
4256:
4252:
4247:
4243:
4238:
4231:
4228:
4224:
4219:
4215:
4189:
4157:
4153:
4149:
4146:
4142:
4137:
4134:
4115:
4114:
4100:
4097:
4090:
4086:
4078:
4073:
4069:
4064:
4060:
4055:
4048:
4044:
4040:
4037:
4030:
4022:
4018:
4014:
4011:
4007:
4002:
3995:
3990:
3986:
3981:
3977:
3972:
3965:
3962:
3958:
3952:
3949:
3943:
3939:
3913:
3881:
3877:
3873:
3870:
3866:
3861:
3858:
3832:
3829:
3822:
3818:
3813:
3809:
3805:
3801:
3797:
3794:
3790:
3783:
3779:
3774:
3770:
3765:
3760:
3756:
3749:
3744:
3740:
3735:
3731:
3726:
3720:
3714:
3710:
3706:
3703:
3698:
3694:
3686:
3682:
3678:
3675:
3671:
3666:
3663:
3637:
3634:
3627:
3623:
3618:
3611:
3606:
3602:
3597:
3593:
3588:
3583:
3578:
3573:
3570:
3566:
3562:
3558:
3550:
3546:
3542:
3539:
3535:
3530:
3527:
3494:
3489:
3483:
3479:
3474:
3470:
3465:
3458:
3454:
3449:
3445:
3438:
3434:
3427:
3422:
3418:
3413:
3409:
3404:
3399:
3394:
3389:
3381:
3378:
3374:
3370:
3363:
3358:
3353:
3346:
3343:
3339:
3335:
3328:
3323:
3318:
3311:
3308:
3304:
3300:
3293:
3288:
3282:
3278:
3274:
3267:
3262:
3258:
3253:
3249:
3244:
3239:
3234:
3229:
3226:
3201:
3198:
3191:
3187:
3178:
3173:
3167:
3163:
3158:
3154:
3149:
3142:
3138:
3135:
3131:
3127:
3122:
3118:
3114:
3108:
3100:
3096:
3092:
3089:
3085:
3080:
3077:
3039:
3033:
3029:
3024:
3020:
3017:
3014:
3010:
3006:
3002:
2998:
2967:
2962:
2956:
2952:
2947:
2943:
2938:
2931:
2927:
2924:
2920:
2916:
2911:
2907:
2903:
2892:
2888:
2884:
2881:
2877:
2872:
2869:
2847:
2829:
2824:
2820:
2817:
2814:
2811:
2806:
2802:
2781:
2778:
2773:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2744:
2740:
2733:
2729:
2723:
2719:
2715:
2712:
2709:
2706:
2700:
2696:
2692:
2689:
2684:
2680:
2661:volume element
2632:
2625:
2622:
2616:
2612:
2608:
2602:
2599:
2593:
2587:
2584:
2578:
2574:
2568:
2562:
2559:
2553:
2547:
2544:
2540:
2535:
2530:
2523:
2519:
2513:
2507:
2504:
2497:
2493:
2489:
2483:
2478:
2474:
2446:
2443:
2437:
2430:
2425:
2420:
2413:
2409:
2403:
2398:
2394:
2390:
2379:infinitesimals
2356:
2353:
2325:
2296:
2291:
2286:
2277:
2274:
2268:
2264:
2257:
2252:
2248:
2231:
2228:
2215:
2205:
2201:
2192:
2188:
2185:
2165:
2155:
2151:
2142:
2138:
2135:
2119:
2116:
2087:electric field
2072:
2069:
2057:
2051:
2048:
2043:
2040:
2035:
2031:
2027:
2023:
2017:
2014:
2009:
2006:
2001:
1997:
1993:
1989:
1983:
1980:
1975:
1972:
1967:
1963:
1959:
1943:
1936:
1929:
1922:
1915:
1908:
1864:number density
1843:
1840:
1836:
1832:
1828:
1823:
1819:
1813:
1809:
1805:
1802:
1780:
1761:
1758:
1754:
1750:
1746:
1741:
1737:
1731:
1727:
1723:
1720:
1698:
1682:
1679:
1675:
1671:
1667:
1662:
1658:
1652:
1648:
1644:
1641:
1619:
1594:
1587:
1584:
1579:
1576:
1570:
1565:
1561:
1533:
1526:
1523:
1518:
1515:
1509:
1504:
1500:
1472:
1465:
1462:
1457:
1454:
1448:
1443:
1439:
1404:electric field
1358:
1355:
1353:
1350:
1346:chemical bonds
1319:
1314:
1309:
1305:
1301:
1298:
1294:
1272:
1251:
1247:
1243:
1240:
1131:
1127:
1123:
1119:
1095:
1091:
1087:
1084:
1048:
1044:
1040:
1037:
1017:
1013:
1009:
1006:
986:
982:
978:
975:
954:
878:charge density
868:
867:
865:
864:
857:
850:
842:
839:
838:
835:
834:
829:
824:
819:
814:
809:
804:
799:
794:
789:
784:
779:
774:
769:
764:
759:
754:
749:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
653:
650:
649:
646:
645:
642:
641:
636:
631:
626:
621:
619:Four-potential
616:
611:
606:
600:
595:
594:
591:
590:
587:
586:
581:
576:
571:
566:
561:
556:
551:
546:
541:
536:
534:Electric motor
531:
526:
521:
515:
510:
509:
506:
505:
502:
501:
496:
491:
489:Series circuit
486:
481:
476:
471:
466:
461:
459:Kirchhoff laws
456:
451:
446:
441:
436:
431:
426:
424:Direct current
421:
416:
411:
405:
400:
399:
396:
395:
392:
391:
386:
381:
379:Maxwell tensor
376:
371:
366:
361:
356:
351:
349:Larmor formula
346:
341:
336:
331:
326:
321:
316:
311:
306:
301:
299:Bremsstrahlung
295:
290:
289:
286:
285:
282:
281:
276:
271:
266:
261:
256:
251:
249:Magnetic field
246:
241:
236:
231:
225:
222:Magnetostatics
220:
219:
216:
215:
212:
211:
206:
201:
196:
191:
186:
181:
176:
171:
166:
161:
156:
154:Electric field
151:
146:
141:
136:
131:
126:
124:Charge density
120:
117:Electrostatics
115:
114:
111:
110:
109:
108:
103:
98:
93:
88:
83:
78:
70:
69:
61:
60:
54:
53:
52:Articles about
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7959:
7948:
7945:
7943:
7940:
7938:
7935:
7934:
7932:
7922:
7920:
7919:
7915:
7909:
7903:
7899:
7898:
7892:
7888:
7882:
7878:
7877:
7872:
7868:
7864:
7858:
7854:
7849:
7845:
7839:
7834:
7833:
7826:
7822:
7816:
7812:
7807:
7806:
7802:
7793:
7787:
7779:
7775:
7771:
7767:
7763:
7759:
7755:
7751:
7744:
7741:
7736:
7732:
7728:
7724:
7720:
7716:
7712:
7708:
7701:
7698:
7690:
7683:
7680:
7675:
7669:
7665:
7658:
7655:
7650:
7648:1-4020-2700-1
7644:
7640:
7633:
7630:
7625:
7619:
7615:
7608:
7605:
7600:
7598:0-387-95210-1
7594:
7590:
7586:
7579:
7576:
7571:
7567:
7560:
7557:
7552:
7546:
7542:
7535:
7533:
7531:
7529:
7525:
7520:
7514:
7510:
7503:
7501:
7497:
7492:
7490:9781107013605
7486:
7482:
7481:
7473:
7471:
7467:
7462:
7460:9781133954149
7456:
7452:
7451:
7443:
7440:
7427:
7420:
7414:
7411:
7406:
7404:0-7195-3382-1
7400:
7396:
7389:
7386:
7379:
7375:
7372:
7370:
7367:
7365:
7361:
7358:
7357:
7353:
7351:
7349:
7345:
7341:
7337:
7333:
7329:
7321:
7319:
7317:
7313:
7309:
7304:
7291:
7288:
7285:
7282:
7279:
7276:
7270:
7264:
7256:
7253:
7238:
7226:
7217:
7198:
7186:
7172:
7166:
7155:
7151:
7147:
7143:
7132:
7128:
7124:
7119:
7116:
7111:
7107:
7095:
7086:
7064:
7052:
7049:
7029:
7026:
7017:
7006:
7002:
6998:
6994:
6983:
6979:
6975:
6970:
6967:
6962:
6956:
6950:
6947:
6941:
6933:
6930:
6926:
6898:
6894:
6879:
6865:
6862:
6859:
6853:
6845:
6837:
6825:
6800:
6797:
6794:
6791:
6775:
6771:
6750:
6742:
6732:
6729:
6726:
6721:
6711:
6692:
6682:
6679:
6676:
6671:
6661:
6648:
6637:
6625:
6613:
6610:
6605:
6593:
6581:
6578:
6575:
6561:
6552:
6539:
6531:
6521:
6518:
6515:
6511:
6502:
6497:
6487:
6484:
6481:
6476:
6466:
6447:
6437:
6434:
6431:
6427:
6418:
6413:
6403:
6400:
6397:
6392:
6382:
6369:
6358:
6346:
6334:
6331:
6326:
6314:
6302:
6297:
6293:
6289:
6275:
6267:
6251:
6239:
6235:
6225:
6212:
6208:
6199:
6191:
6187:
6183:
6169:
6161:
6158:
6156:
6152:
6149:
6126:
6122:
6115:
6094:
6086:
6081:
6077:
6073:
6070:
6062:
6058:
6054:
6050:
6044:
6040:
6036:
6032:
6026:
6022:
6015:
5997:
5976:
5968:
5965:
5949:
5945:
5936:
5932:
5929:
5924:
5920:
5916:
5910:
5906:
5898:
5896:
5894:
5891:vector under
5890:
5886:
5883:
5879:
5874:
5872:
5869:it is called
5868:
5864:
5860:
5856:
5852:
5848:
5844:
5838:
5834:
5826:
5824:
5821:
5808:
5793:
5790:
5787:
5771:
5767:
5758:
5754:
5750:
5746:
5742:
5738:
5733:
5718:
5714:
5705:
5700:
5697:
5694:
5690:
5686:
5678:
5668:
5657:
5647:
5643:
5637:
5633:
5627:
5623:
5614:
5609:
5606:
5603:
5599:
5595:
5587:
5577:
5566:
5561:
5557:
5548:
5543:
5540:
5537:
5533:
5522:
5518:
5512:
5508:
5504:
5501:
5493:
5489:
5484:
5480:
5476:
5472:
5468:
5460:
5442:
5432:
5421:
5416:
5412:
5403:
5398:
5395:
5392:
5388:
5384:
5368:
5364:
5354:
5351:= 1, 2, ...,
5350:
5344:
5340:
5336:
5329:
5325:
5320:
5307:
5304:
5296:
5286:
5275:
5264:
5260:
5254:
5250:
5246:
5243:
5235:
5225:
5214:
5211:
5200:
5196:
5190:
5186:
5182:
5177:
5173:
5161:
5157:
5151:
5147:
5143:
5140:
5132:
5128:
5124:
5103:
5090:
5087:
5079:
5069:
5058:
5044:
5034:
5030:
5024:
5020:
5011:
5007:
5002:
5000:
4979:
4969:
4958:
4955:
4952:
4936:
4932:
4923:
4919:
4915:
4908:
4904:
4896:
4894:
4891:
4878:
4873:
4870:
4867:
4863:
4859:
4856:
4853:
4833:
4828:
4824:
4820:
4817:
4814:
4808:
4804:
4798:
4795:
4792:
4788:
4784:
4781:
4771:
4767:
4763:
4758:
4754:
4749:
4736:
4733:
4730:
4713:
4709:
4703:
4699:
4695:
4692:
4680:
4678:
4665:
4660:
4656:
4652:
4649:
4646:
4635:
4631:
4623:
4621:
4619:
4615:
4595:
4592:
4587:
4583:
4579:
4576:
4573:
4570:
4555:
4528:
4501:
4496:
4483:
4482:
4481:
4479:
4476:
4472:
4468:
4460:
4457:
4439:
4433:
4430:
4425:
4421:
4416:
4400:
4392:
4387:
4383:
4356:
4346:
4342:
4329:
4320:
4304:
4300:
4294:
4286:
4282:
4278:
4275:
4271:
4266:
4254:
4245:
4229:
4226:
4222:
4217:
4213:
4187:
4155:
4151:
4147:
4144:
4140:
4135:
4132:
4124:
4123:
4122:
4120:
4098:
4088:
4084:
4071:
4062:
4042:
4038:
4028:
4020:
4016:
4012:
4009:
4005:
4000:
3988:
3979:
3963:
3960:
3956:
3941:
3911:
3879:
3875:
3871:
3868:
3864:
3859:
3856:
3848:
3847:
3846:
3830:
3820:
3816:
3811:
3799:
3795:
3781:
3772:
3763:
3758:
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3742:
3733:
3712:
3708:
3704:
3696:
3692:
3684:
3680:
3676:
3673:
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3664:
3661:
3654:
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3621:
3616:
3604:
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3581:
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3525:
3518:coordinates,
3517:
3513:
3492:
3481:
3472:
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3447:
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3379:
3376:
3361:
3351:
3344:
3341:
3326:
3316:
3309:
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3237:
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3189:
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3062:
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3055:
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3012:
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2988:
2987:infinitesimal
2983:
2965:
2954:
2945:
2925:
2918:
2909:
2890:
2886:
2882:
2879:
2875:
2870:
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2804:
2800:
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2767:
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2731:
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2203:
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2163:
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2115:
2113:
2109:
2105:
2101:
2099:
2096:bound to the
2095:
2091:
2088:
2084:
2083:Bound charges
2080:
2078:
2070:
2068:
2055:
2049:
2046:
2041:
2033:
2029:
2021:
2015:
2012:
2007:
1999:
1995:
1987:
1981:
1978:
1973:
1965:
1961:
1948:
1946:
1939:
1932:
1925:
1918:
1911:
1904:
1900:
1896:
1891:
1889:
1885:
1881:
1877:
1873:
1869:
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1857:
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1838:
1821:
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1800:
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1677:
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1631:
1627:
1622:
1618:
1614:
1613:line integral
1610:
1605:
1592:
1585:
1582:
1577:
1574:
1568:
1563:
1559:
1551:
1547:
1531:
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1463:
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1437:
1428:
1424:
1420:
1415:
1409:
1405:
1401:
1398:
1394:
1391:
1387:
1386:dipole moment
1383:
1379:
1376:
1372:
1368:
1363:
1356:
1351:
1349:
1347:
1343:
1339:
1335:
1317:
1296:
1284:in space, so
1238:
1231:
1230:
1225:
1221:
1217:
1213:
1209:
1205:
1200:
1197:
1193:
1189:
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1177:
1173:
1169:
1165:
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1157:
1152:
1148:
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1109:
1082:
1074:
1070:
1066:
1062:
1035:
1004:
973:
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940:
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930:
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911:
907:
903:
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887:
883:
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875:
863:
858:
856:
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843:
841:
840:
833:
830:
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810:
808:
805:
803:
800:
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793:
790:
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785:
783:
780:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
654:
648:
647:
640:
637:
635:
632:
630:
627:
625:
622:
620:
617:
615:
612:
610:
607:
605:
602:
601:
598:
593:
592:
585:
582:
580:
577:
575:
572:
570:
567:
565:
562:
560:
557:
555:
552:
550:
547:
545:
542:
540:
537:
535:
532:
530:
527:
525:
522:
520:
517:
516:
513:
508:
507:
500:
497:
495:
492:
490:
487:
485:
482:
480:
477:
475:
472:
470:
467:
465:
462:
460:
457:
455:
454:Joule heating
452:
450:
447:
445:
442:
440:
437:
435:
432:
430:
427:
425:
422:
420:
417:
415:
412:
410:
407:
406:
403:
398:
397:
390:
387:
385:
382:
380:
377:
375:
372:
370:
369:Lorentz force
367:
365:
362:
360:
357:
355:
352:
350:
347:
345:
342:
340:
337:
335:
332:
330:
327:
325:
322:
320:
317:
315:
312:
310:
307:
305:
302:
300:
297:
296:
293:
288:
287:
280:
277:
275:
272:
270:
269:Magnetization
267:
265:
262:
260:
257:
255:
254:Magnetic flux
252:
250:
247:
245:
242:
240:
237:
235:
232:
230:
227:
226:
223:
218:
217:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
185:
182:
180:
177:
175:
172:
170:
167:
165:
162:
160:
159:Electric flux
157:
155:
152:
150:
147:
145:
142:
140:
137:
135:
132:
130:
127:
125:
122:
121:
118:
113:
112:
107:
104:
102:
99:
97:
96:Computational
94:
92:
89:
87:
84:
82:
79:
77:
74:
73:
72:
71:
67:
63:
62:
59:
55:
51:
50:
45:
41:
37:
33:
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7896:
7875:
7852:
7831:
7810:
7786:cite journal
7753:
7749:
7743:
7710:
7706:
7700:
7682:
7663:
7657:
7638:
7632:
7613:
7607:
7584:
7578:
7570:W. W. Norton
7565:
7559:
7540:
7508:
7479:
7449:
7442:
7430:. Retrieved
7425:
7413:
7394:
7388:
7325:
7315:
7311:
7307:
7305:
7257:
7254:
6826:
6553:
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5934:
5930:
5928:wavefunction
5922:
5918:
5912:
5889:four-current
5884:
5877:
5875:
5870:
5840:
5822:
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5752:
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5470:
5469:in the sum,
5463:
5461:
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5338:
5337:at position
5331:
5327:
5323:
5321:
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5009:
5005:
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4906:
4905:at position
4902:
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2665:
2656:
2644:
2461:
2376:
2312:
2233:
2230:Bound charge
2123:
2121:
2107:
2104:Free charges
2103:
2102:
2089:
2082:
2081:
2074:
1949:
1941:
1934:
1927:
1920:
1913:
1906:
1902:
1898:
1894:
1892:
1879:
1875:
1871:
1860:mass density
1855:
1790:
1786:
1781:
1777:
1708:
1704:
1699:
1693:similarly a
1629:
1625:
1620:
1616:
1608:
1606:
1549:
1488:
1485:surface area
1427:line element
1418:
1416:
1413:
1407:
1399:
1392:
1381:
1377:
1370:
1366:
1229:wavefunction
1227:
1211:
1201:
1172:space charge
1150:
1144:
1060:
936:
929:mass density
926:
921:
913:
897:
890:surface area
877:
871:
614:Four-current
549:Linear motor
434:Electrolysis
314:Eddy current
274:Permeability
194:Polarization
189:Permittivity
123:
7871:R.G. Lerner
7432:December 3,
7322:Application
4630:homogeneous
4467:Gauss's law
1375:unit normal
1352:Definitions
1176:vacuum tube
584:Transformer
414:Capacitance
339:Faraday law
134:Coulomb law
76:Electricity
7931:Categories
7380:References
5903:See also:
4916:, like an
2666:Using the
2236:dielectric
2077:dielectric
1884:wavelength
1421:(SI unit:
1151:continuous
938:continuous
908:per cubic
904:system in
651:Scientists
499:Waveguides
479:Resistance
449:Inductance
229:Ampère law
7289:−
7218:−
7187:ρ
7173:ρ
7152:δ
7148:∫
7129:δ
7125:∫
7087:−
7003:δ
6999:∫
6980:δ
6976:∫
6895:δ
6863:∫
6857:⟩
6854:ψ
6838:ψ
6835:⟨
6798:⋅
6772:ρ
6730:…
6701:Ψ
6680:…
6649:∗
6645:Ψ
6614:∫
6611:⋯
6582:∫
6519:…
6485:…
6456:Ψ
6435:…
6401:…
6370:∗
6366:Ψ
6335:∫
6332:⋯
6303:∫
6294:∑
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6252:ψ
6226:−
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6200:ψ
6197:⟨
6188:∑
6095:ψ
6078:∫
6029:| =
5977:ψ
5946:ρ
5768:ρ
5691:∑
5669:−
5658:δ
5634:∫
5600:∑
5578:−
5567:δ
5534:∑
5509:∫
5433:−
5422:δ
5389:∑
5365:ρ
5287:−
5276:δ
5251:∫
5226:−
5215:δ
5187:∫
5174:ρ
5148:∫
5070:−
5059:δ
5021:∫
4970:−
4959:δ
4933:ρ
4864:ρ
4825:ρ
4805:∫
4789:ρ
4710:ρ
4700:∫
4657:ρ
4584:ρ
4580:∭
4565:^
4556:⋅
4493:Φ
4440:⋅
4437:∇
4434:−
4422:ρ
4410:^
4401:⋅
4384:σ
4321:−
4301:ρ
4295:∭
4283:ε
4279:π
4246:−
4214:σ
4152:ε
4148:π
4133:φ
4063:−
4043:⋅
4036:∇
4029:∭
4017:ε
4013:π
4001:−
3980:−
3951:^
3942:⋅
3876:ε
3872:π
3857:φ
3800:⋅
3793:∇
3773:−
3759:−
3734:−
3709:⋅
3702:∇
3693:∭
3681:ε
3677:π
3662:φ
3596:−
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3565:⋅
3557:∭
3545:ε
3541:π
3526:φ
3473:−
3448:−
3412:−
3373:∂
3369:∂
3338:∂
3334:∂
3303:∂
3299:∂
3277:≡
3252:−
3225:∇
3157:−
3137:⋅
3121:−
3107:∭
3095:ε
3091:π
3076:φ
2946:−
2926:⋅
2910:−
2887:ε
2883:π
2868:φ
2819:⋅
2816:∇
2813:−
2801:ρ
2768:⋅
2765:∇
2762:∫
2759:−
2743:^
2732:⋅
2718:∮
2714:−
2695:ρ
2691:∫
2624:^
2615:⋅
2601:^
2592:⋅
2561:^
2552:⋅
2473:σ
2445:^
2436:⋅
2355:^
2276:^
2267:⋅
2204:σ
2191:σ
2184:σ
2154:ρ
2141:ρ
2134:ρ
2094:electrons
2039:⟩
2030:ρ
2026:⟨
2005:⟩
1996:σ
1992:⟨
1982:ℓ
1971:⟩
1962:λ
1958:⟨
1818:ρ
1808:∫
1736:σ
1726:∫
1681:ℓ
1657:λ
1647:∫
1560:ρ
1499:σ
1464:ℓ
1438:λ
1338:molecules
1297:ψ
1239:ψ
1220:chemistry
1188:electrons
1083:ρ
1036:λ
1005:σ
974:ρ
884:per unit
807:Steinmetz
737:Kirchhoff
722:Jefimenko
717:Hopkinson
702:Helmholtz
697:Heaviside
559:Permeance
444:Impedance
184:Insulator
179:Gauss law
129:Conductor
106:Phenomena
101:Textbooks
81:Magnetism
7778:29510032
7735:29510032
7354:See also
7227:′
7199:′
7144:′
7096:′
7065:′
6995:′
6512:′
6428:′
6240:′
5847:velocity
5346:, where
4918:electron
4375:that is
4357:′
4330:′
4255:′
4230:′
4099:′
4072:′
4039:′
3989:′
3964:′
3831:′
3796:′
3782:′
3743:′
3705:′
3636:′
3605:′
3572:′
3516:r′
3512:gradient
3482:′
3457:′
3421:′
3380:′
3345:′
3310:′
3261:′
3228:′
3200:′
3166:′
3130:′
3061:r′
2989:dipoles
2955:′
2919:′
1548:element
1487:element
1408:r′
1342:orbitals
1106:and the
906:coulombs
832:Wiechert
787:Poynting
677:Einstein
524:DC motor
519:AC motor
354:Lenz law
139:Electret
7937:Density
7758:Bibcode
7715:Bibcode
6153:is the
6047:is the
4473:of the
3514:in the
2792:hence:
2647:is the
2377:Taking
2370:is the
2337:is the
1384:is the
1199:level.
817:Thomson
792:Ritchie
782:Poisson
767:Neumann
762:Maxwell
757:Lorentz
752:Liénard
682:Faraday
667:Coulomb
494:Voltage
469:Ohm law
91:History
44:density
7904:
7883:
7859:
7840:
7817:
7776:
7733:
7670:
7645:
7620:
7595:
7547:
7515:
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7457:
7401:
7306:Where
6763:where
6146:where
6018:|
6012:where
5759:), by
5747:, the
5711:
5620:
5554:
5409:
4997:where
3216:Since
3051:where
2643:where
2311:where
2098:nuclei
1772:and a
1546:volume
1028:, and
942:scalar
894:volume
886:length
802:Singer
797:Savart
777:Ørsted
742:Larmor
732:Kelvin
687:Fizeau
657:Ampère
579:Stator
86:Optics
40:charge
7692:(PDF)
7422:(PDF)
4681:Proof
2124:total
1947:etc.
1334:atoms
1174:in a
927:Like
910:meter
892:, or
827:Weber
822:Volta
812:Tesla
727:Joule
712:Hertz
707:Henry
692:Gauss
574:Rotor
7902:ISBN
7881:ISBN
7857:ISBN
7838:ISBN
7815:ISBN
7792:link
7774:PMID
7731:PMID
7668:ISBN
7643:ISBN
7618:ISBN
7593:ISBN
7545:ISBN
7513:ISBN
7485:ISBN
7455:ISBN
7434:2017
7399:ISBN
5907:and
5880:and
5835:and
4846:so,
4616:and
4612:See
4471:flux
2860:is:
2852:The
1336:and
1222:and
1212:have
1192:ions
1168:ions
747:Lenz
672:Davy
662:Biot
42:and
7766:doi
7723:doi
5913:In
5841:In
5743:= −
5129:is
4772:, 0
2075:In
1793:,
1632:,
1429:,
1206:of
1162:.
896:.
872:In
772:Ohm
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7933::
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7784:{{
7772:.
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7527:^
7499:^
7469:^
7424:.
6063:is
6059:∈
6033:*(
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5477:−
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4924::
3056:=
3054:dV
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2657:dV
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2341:,
2114:.
2100:.
1940:,
1933:,
1926:,
1919:,
1912:,
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1890:.
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1378:n̂
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1190:,
1142:.
997:,
902:SI
888:,
876:,
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7889:.
7865:.
7846:.
7823:.
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7780:.
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7316:K
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7292:K
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7280:I
7277:=
7274:]
7271:n
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7203:)
7195:r
7190:(
7184:)
7180:r
7176:(
7167:(
7162:r
7156:3
7140:r
7133:3
7120:2
7117:1
7112:=
7108:)
7101:|
7092:r
7083:r
7078:|
7072:)
7069:)
7061:r
7056:(
7053:n
7050:e
7047:(
7044:)
7041:)
7037:r
7033:(
7030:n
7027:e
7024:(
7018:(
7013:r
7007:3
6991:r
6984:3
6971:2
6968:1
6963:=
6960:]
6957:n
6954:[
6951:J
6948:=
6945:]
6942:n
6939:[
6934:e
6931:e
6927:U
6905:r
6899:3
6891:)
6887:r
6883:(
6880:n
6877:)
6873:r
6869:(
6866:V
6860:=
6850:|
6846:U
6842:|
6812:)
6808:r
6804:(
6801:n
6795:q
6792:=
6789:)
6785:r
6781:(
6776:q
6751:.
6748:)
6743:N
6738:r
6733:,
6727:,
6722:2
6717:r
6712:,
6708:r
6704:(
6698:)
6693:N
6688:r
6683:,
6677:,
6672:2
6667:r
6662:,
6658:r
6654:(
6638:N
6633:r
6626:3
6620:d
6606:2
6601:r
6594:3
6588:d
6579:N
6576:=
6573:)
6569:r
6565:(
6562:n
6540:.
6537:)
6532:N
6527:r
6522:,
6516:,
6508:r
6503:=
6498:i
6493:r
6488:,
6482:,
6477:2
6472:r
6467:,
6463:r
6459:(
6453:)
6448:N
6443:r
6438:,
6432:,
6424:r
6419:=
6414:i
6409:r
6404:,
6398:,
6393:2
6388:r
6383:,
6379:r
6375:(
6359:N
6354:r
6347:3
6341:d
6327:2
6322:r
6315:3
6309:d
6298:i
6290:=
6287:)
6283:r
6279:(
6276:n
6248:|
6244:)
6236:i
6231:r
6222:r
6218:(
6213:3
6204:|
6192:i
6184:=
6181:)
6177:r
6173:(
6170:n
6151:r
6148:d
6133:r
6127:3
6123:d
6116:2
6111:|
6106:)
6102:r
6098:(
6091:|
6087:q
6082:R
6074:=
6071:Q
6061:R
6057:r
6053:r
6045:)
6043:r
6041:(
6039:ψ
6037:)
6035:r
6031:ψ
6027:)
6025:r
6023:(
6021:ψ
6014:q
5998:2
5993:|
5988:)
5984:r
5980:(
5973:|
5969:q
5966:=
5963:)
5959:r
5955:(
5950:q
5935:r
5933:(
5931:ψ
5923:q
5919:ρ
5885:J
5878:ρ
5809:.
5805:)
5801:r
5797:(
5794:n
5791:q
5788:=
5785:)
5781:r
5777:(
5772:q
5757:r
5755:(
5753:n
5745:e
5741:q
5737:q
5719:i
5715:q
5706:N
5701:1
5698:=
5695:i
5687:=
5684:)
5679:i
5674:r
5665:r
5661:(
5654:r
5648:3
5644:d
5638:R
5628:i
5624:q
5615:N
5610:1
5607:=
5604:i
5596:=
5593:)
5588:i
5583:r
5574:r
5570:(
5562:i
5558:q
5549:N
5544:1
5541:=
5538:i
5529:r
5523:3
5519:d
5513:R
5505:=
5502:Q
5492:R
5488:R
5483:i
5479:r
5475:r
5473:(
5471:δ
5466:i
5464:q
5448:)
5443:i
5438:r
5429:r
5425:(
5417:i
5413:q
5404:N
5399:1
5396:=
5393:i
5385:=
5382:)
5378:r
5374:(
5369:q
5353:N
5349:i
5343:i
5339:r
5334:i
5332:q
5328:r
5324:N
5308:q
5305:=
5302:)
5297:0
5292:r
5283:r
5279:(
5271:r
5265:3
5261:d
5255:R
5247:q
5244:=
5241:)
5236:0
5231:r
5222:r
5218:(
5212:q
5207:r
5201:3
5197:d
5191:R
5183:=
5178:q
5168:r
5162:3
5158:d
5152:R
5144:=
5141:Q
5131:q
5127:R
5123:R
5109:)
5104:0
5099:r
5094:(
5091:f
5088:=
5085:)
5080:0
5075:r
5066:r
5062:(
5056:)
5052:r
5048:(
5045:f
5041:r
5035:3
5031:d
5025:R
5010:f
4999:r
4985:)
4980:0
4975:r
4966:r
4962:(
4956:q
4953:=
4950:)
4946:r
4942:(
4937:q
4914:R
4910:0
4907:r
4903:q
4879:.
4874:0
4871:,
4868:q
4860:V
4857:=
4854:Q
4834:V
4829:0
4821:=
4818:V
4815:d
4809:V
4799:0
4796:,
4793:q
4785:=
4782:Q
4770:q
4766:ρ
4762:r
4760:(
4757:q
4753:ρ
4737:.
4734:V
4731:d
4727:)
4723:r
4719:(
4714:q
4704:V
4696:=
4693:Q
4666:.
4661:0
4653:V
4650:=
4647:Q
4637:0
4634:ρ
4596:V
4593:d
4588:f
4577:=
4574:S
4571:d
4562:n
4552:D
4529:S
4502:=
4497:D
4478:D
4444:P
4431:=
4426:b
4417:,
4407:n
4397:P
4393:=
4388:b
4354:r
4347:3
4343:d
4335:|
4326:r
4317:r
4312:|
4305:b
4287:0
4276:4
4272:1
4267:+
4260:|
4251:r
4242:r
4237:|
4227:S
4223:d
4218:b
4188:S
4156:0
4145:4
4141:1
4136:=
4096:r
4089:3
4085:d
4077:|
4068:r
4059:r
4054:|
4047:P
4021:0
4010:4
4006:1
3994:|
3985:r
3976:r
3971:|
3961:S
3957:d
3948:n
3938:P
3912:S
3880:0
3869:4
3865:1
3860:=
3828:r
3821:3
3817:d
3812:]
3808:)
3804:P
3789:(
3778:r
3769:r
3764:1
3755:)
3748:|
3739:r
3730:r
3725:|
3719:P
3713:(
3697:[
3685:0
3674:4
3670:1
3665:=
3633:r
3626:3
3622:d
3617:)
3610:|
3601:r
3592:r
3587:|
3582:1
3577:(
3561:P
3549:0
3538:4
3534:1
3529:=
3493:3
3488:|
3478:r
3469:r
3464:|
3453:r
3444:r
3437:=
3433:)
3426:|
3417:r
3408:r
3403:|
3398:1
3393:(
3388:)
3377:z
3362:z
3357:e
3352:+
3342:y
3327:y
3322:e
3317:+
3307:x
3292:x
3287:e
3281:(
3273:)
3266:|
3257:r
3248:r
3243:|
3238:1
3233:(
3197:r
3190:3
3186:d
3177:3
3172:|
3162:r
3153:r
3148:|
3141:P
3134:)
3126:r
3117:r
3113:(
3099:0
3088:4
3084:1
3079:=
3058:d
3038:r
3032:3
3028:d
3023:P
3019:=
3016:V
3013:d
3009:P
3005:=
3001:d
2997:d
2966:3
2961:|
2951:r
2942:r
2937:|
2930:d
2923:)
2915:r
2906:r
2902:(
2891:0
2880:4
2876:1
2871:=
2858:d
2828:.
2823:P
2810:=
2805:b
2780:V
2777:d
2772:P
2756:=
2753:S
2750:d
2739:n
2728:P
2722:S
2711:=
2708:V
2705:d
2699:b
2688:=
2683:b
2679:q
2645:P
2631:.
2621:n
2611:P
2607:=
2598:n
2586:V
2583:d
2577:d
2573:d
2567:=
2558:n
2546:S
2543:d
2539:|
2534:s
2529:|
2522:d
2518:d
2512:=
2506:S
2503:d
2496:b
2492:q
2488:d
2482:=
2477:b
2442:n
2429:|
2424:s
2419:|
2412:d
2408:d
2402:=
2397:b
2393:q
2389:d
2352:n
2324:d
2313:s
2295:|
2290:s
2285:|
2273:n
2263:d
2256:=
2251:b
2247:q
2214:.
2208:b
2200:+
2195:f
2187:=
2164:.
2158:b
2150:+
2145:f
2137:=
2090:E
2056:.
2050:V
2047:Q
2042:=
2034:q
2022:,
2016:S
2013:Q
2008:=
2000:q
1988:,
1979:Q
1974:=
1966:q
1944:V
1942:ρ
1937:S
1935:ρ
1930:L
1928:ρ
1923:v
1921:ρ
1916:s
1914:ρ
1909:ℓ
1907:ρ
1903:ρ
1899:σ
1895:λ
1880:ρ
1876:σ
1872:λ
1856:q
1842:V
1839:d
1835:)
1831:r
1827:(
1822:q
1812:V
1804:=
1801:Q
1791:V
1787:r
1785:(
1782:q
1778:ρ
1760:S
1757:d
1753:)
1749:r
1745:(
1740:q
1730:S
1722:=
1719:Q
1709:S
1705:r
1703:(
1700:q
1678:d
1674:)
1670:r
1666:(
1661:q
1651:L
1643:=
1640:Q
1630:C
1626:r
1624:(
1621:q
1617:λ
1609:Q
1593:,
1586:V
1583:d
1578:Q
1575:d
1569:=
1564:q
1532:,
1525:S
1522:d
1517:Q
1514:d
1508:=
1503:q
1471:,
1461:d
1456:Q
1453:d
1447:=
1442:q
1423:C
1400:r
1393:P
1382:d
1371:σ
1367:ρ
1318:2
1313:|
1308:)
1304:x
1300:(
1293:|
1271:x
1250:)
1246:x
1242:(
1186:(
1130:)
1126:x
1122:(
1118:J
1094:)
1090:x
1086:(
1047:)
1043:x
1039:(
1016:)
1012:x
1008:(
985:)
981:x
977:(
953:x
861:e
854:t
847:v
46:.
20:)
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