91:
1703:
1695:
99:
1429:
2158:
1659:
907:
1856:
653:
2405:. The potential outside the grounded sphere will be determined only by the distribution of charge outside the sphere and will be independent of the charge distribution inside the sphere. If we assume for simplicity (without loss of generality) that the inner charge lies on the z-axis, then the induced charge density will be simply a function of the
78:, the electric field is uniquely determined if the total charge on each conductor is given. Possessing knowledge of either the electric potential or the electric field and the corresponding boundary conditions we can swap the charge distribution we are considering for one with a configuration that is easier to analyze, so long as it satisfies
478:
2623:
2866:
is a bit more complicated. If the dipole is pictured as two large charges separated by a small distance, then the image of the dipole will not only have the charges modified by the above procedure, but the distance between them will be modified as well. Following the above procedure, it is found that
1537:) will develop a bound polarization charge. It can be shown that the resulting electric field inside the dielectric containing the particle is modified in a way that can be described by an image charge inside the other dielectric. Inside the other dielectric, however, the image charge is not present.
2153:{\displaystyle 4\pi \varepsilon _{0}V(\mathbf {r} )={\frac {q}{|\mathbf {r} _{1}|}}+{\frac {(-qR/p)}{|\mathbf {r} _{2}|}}={\frac {q}{\sqrt {r^{2}+p^{2}-2\mathbf {r} \cdot \mathbf {p} }}}+{\frac {(-qR/p)}{\sqrt {r^{2}+{\frac {R^{4}}{p^{2}}}-{\frac {2R^{2}}{p^{2}}}\mathbf {r} \cdot \mathbf {p} }}}}
638:
902:{\displaystyle {\begin{aligned}Q_{t}&=\int _{0}^{2\pi }\int _{0}^{\infty }\sigma \left(\rho \right)\,\rho \,d\rho \,d\theta \\&={\frac {-qa}{2\pi }}\int _{0}^{2\pi }d\theta \int _{0}^{\infty }{\frac {\rho \,d\rho }{\left(\rho ^{2}+a^{2}\right)^{3/2}}}\\&=-q\end{aligned}}}
295:
2418:
3356:
2746:
1316:
1648:
258:(i.e., above the conducting plate), and satisfies the boundary condition that the potential along the plate must be zero. This situation is equivalent to the original setup, and so the force on the real charge can now be calculated with
1414:
1192:
1103:
3750:
493:
3031:
3156:
2364:
1715:
The method of images may be applied to a sphere as well. In fact, the case of image charges in a plane is a special case of the case of images for a sphere. Referring to the figure, we wish to find the potential
1650:. It may not even have the same sign, if the charge is placed inside the stronger dielectric material (charges are repelled away from regions of lower dielectric constant). This can be seen from the formula.
658:
473:{\displaystyle V\left(\rho ,\varphi ,z\right)={\frac {1}{4\pi \varepsilon _{0}}}\left({\frac {q}{\sqrt {\rho ^{2}+\left(z-a\right)^{2}}}}+{\frac {-q}{\sqrt {\rho ^{2}+\left(z+a\right)^{2}}}}\right)\,}
2968:
2844:
1826:
3584:
2781:, the potential outside of the sphere is given by the sum of the potentials of the charge and its image charge inside the sphere. Just as in the first case, the image charge will have charge −
2618:{\displaystyle \sigma (\theta )=\varepsilon _{0}\left.{\frac {\partial V}{\partial r}}\right|_{r=R}={\frac {-q\left(R^{2}-p^{2}\right)}{4\pi R\left(R^{2}+p^{2}-2pR\cos \theta \right)^{3/2}}}}
1750:(For the opposite case, the potential outside a sphere due to a charge outside the sphere, the method is applied in a similar way). In the figure, this is represented by the green point. Let
3465:
2383:
be valid outside the sphere, since the image charge does not actually exist, but is rather "standing in" for the surface charge densities induced on the sphere by the inner charge at
3645:
3209:
920:, considering that the dipole field decreases at the cube of the distance at large distances, and the therefore total flux of the field though an infinitely large sphere vanishes.
2634:
1204:
3257:
3241:
1535:
1508:
43:. The name originates from the replacement of certain elements in the original layout with fictitious charges, which replicates the boundary conditions of the problem (see
3510:
2907:
2775:
2403:
1848:
1748:
1020:
230:
1563:
975:
151:
256:
3604:
3406:
3379:
1568:
1327:
2846:. The potential inside the sphere will be dependent only upon the true charge distribution inside the sphere. Unlike the first case the integral will be of value −
927:, a conducting plane below multiple point charges can be replaced by the mirror images of each of the charges individually, with no other modifications necessary.
181:
2885:
1481:
2379:), the potential vanishes. The potential inside the sphere is thus given by the above expression for the potential of the two charges. This potential will
1108:
1025:
633:{\displaystyle \sigma =-\varepsilon _{0}\left.{\frac {\partial V}{\partial z}}\right|_{z=0}={\frac {-qa}{2\pi \left(\rho ^{2}+a^{2}\right)^{3/2}}}}
3650:
2169:
3791:
55:
3876:
3925:
2976:
3042:
1754:
be the point charge of this point. The image of this charge with respect to the grounded sphere is shown in red. It has a charge of
4045:
3842:
2916:
2792:
1774:
3515:
3945:
4011:
3906:
3411:
3997:
1022:
with equal magnitude and direction rotated azimuthally by π. That is, a dipole moment with
Cartesian components
3941:
1440:
3801:
2741:{\displaystyle Q_{t}=\int _{0}^{\pi }d\theta \int _{0}^{2\pi }d\phi \,\,\sigma (\theta )R^{2}\sin \theta =-q}
1311:{\displaystyle F=-{\frac {1}{4\pi \varepsilon _{0}}}{\frac {3p^{2}}{16a^{4}}}\left(1+\cos ^{2}\theta \right)}
647:
charge induced on the conducting plane will be the integral of the charge density over the entire plane, so:
44:
3609:
3351:{\displaystyle \Phi '(r,\theta ,\phi )={\frac {R}{r}}\,\Phi {\left({\frac {R^{2}}{r}},\theta ,\phi \right)}}
3173:
3888:
3786:
66:
on all boundaries are specified. Alternatively, application of this corollary to the differential form of
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48:
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4079:
3214:
1513:
1486:
3470:
2890:
2758:
2386:
1831:
1731:
4084:
924:
286:
1690:
from the origin. The potential produced by the two charges is zero on the surface of the sphere.
1771:
and lies on a line connecting the center of the sphere and the inner charge at vector position
1643:{\textstyle q'={\frac {\varepsilon _{1}-\varepsilon _{2}}{\varepsilon _{1}+\varepsilon _{2}}}q}
484:
3892:
1409:{\displaystyle \tau =-{\frac {1}{4\pi \varepsilon _{0}}}{\frac {p^{2}}{16a^{3}}}\sin 2\theta }
984:
194:
94:
The field of a positive charge above a flat conducting surface, found by the method of images.
90:
3244:
2406:
942:
118:
79:
62:
is uniquely determined if both the charge density throughout the region and the value of the
235:
187:-plane. To simplify this problem, we may replace the plate of equipotential with a charge −
4055:
3951:
3589:
3384:
3364:
154:
3166:
The method of images for a sphere leads directly to the method of inversion. If we have a
1702:
8:
3963:
3959:
160:
1543:
3988:
3898:
3826:
3771:
2870:
1466:
111:
The simplest example of method of image charges is that of a point charge, with charge
63:
1459:
Similar to the conducting plane, the case of a planar interface between two different
4041:
4007:
3902:
3838:
3167:
2751:
Note that the reciprocal problem is also solved by this method. If we have a charge
1662:
Diagram illustrating the image method for
Laplace's equation for a sphere of radius
3781:
3766:
3761:
259:
22:
3467:, then the image potential will be the result of a series of charges of magnitude
4025:
3984:
1187:{\displaystyle (-p\sin \theta \cos \phi ,-p\sin \theta \sin \phi ,p\cos \theta )}
3247:
of the position, then the image of this harmonic function in a sphere of radius
1321:
and a torque in the plane perpendicular to the dipole and the conducting plane,
4029:
4003:
1098:{\displaystyle (p\sin \theta \cos \phi ,p\sin \theta \sin \phi ,p\cos \theta )}
917:
265:
The potential at any point in space, due to these two point charges of charge +
232:. This arrangement will produce the same electric field at any point for which
40:
4073:
3992:
3796:
1698:
Field lines outside a grounded sphere for a charge placed outside the sphere.
67:
2628:
The total charge on the sphere may be found by integrating over all angles:
82:
in the region of interest and assumes the correct values at the boundaries.
2970:(i.e. the same as for the simple charge) and will have a simple charge of:
1694:
1510:, then the interface (with the dielectric that has the dielectric constant
98:
1828:. It can be seen that the potential at a point specified by radius vector
1674:
from the origin, the red point is the image of that point, having charge −
54:
The validity of the method of image charges rests upon a corollary of the
4062:
3834:
3745:{\displaystyle \rho '(r,\theta ,\phi )=(R/r)\rho (R^{2}/r,\theta ,\phi )}
1428:
4021:
2359:{\displaystyle V(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\left}
1460:
102:
Method of images for an electric dipole moment in a conducting plane
4059:
Berkeley
Physics Course, Vol-2: Electricity and Magnetism (2nd ed.)
4037:
3026:{\displaystyle q'={\frac {R\mathbf {p} \cdot \mathbf {M} }{p^{3}}}}
1706:
Several surfaces require an infinite series of point image charges.
74:
surrounded by conductors and containing a specified charge density
3647:, then the image potential will be the result of a charge density
3151:{\displaystyle \mathbf {M} '=\left({\frac {R}{p}}\right)^{3}\left}
1850:
due to both charges alone is given by the sum of the potentials:
1658:
2369:
and it can be seen that on the surface of the sphere (i.e. when
2863:
1419:
912:
The total charge induced on the plane turns out to be simply −
1483:
is placed in the dielectric that has the dielectric constant
3776:
2449:
518:
4020:
2163:
Multiplying through on the rightmost expression yields:
58:, which states that the electric potential in a volume
1571:
3653:
3612:
3592:
3518:
3473:
3414:
3387:
3367:
3260:
3217:
3176:
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1111:
1028:
987:
945:
656:
496:
298:
238:
197:
163:
121:
3983:
3965:
2963:{\displaystyle \left(R^{2}/p^{2}\right)\mathbf {p} }
2839:{\displaystyle \left(R^{2}/p^{2}\right)\mathbf {p} }
1821:{\displaystyle \left(R^{2}/p^{2}\right)\mathbf {p} }
1653:
3579:{\displaystyle (R^{2}/r_{i},\theta _{i},\phi _{i})}
977:above an infinite grounded conducting plane in the
85:
3744:
3639:
3598:
3578:
3504:
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224:
175:
145:
18:Calculation technique for classical electrostatics
3887:
4071:
1724:, centered at the origin, due to a point charge
4034:Electrodynamics of Continuous Media 2nd Edition
1540:Unlike the case of the metal, the image charge
2913:will have an image located at vector position
3460:{\displaystyle (r_{i},\theta _{i},\phi _{i})}
2857:
1682:, lying outside the sphere at a distance of
1565:is not exactly opposite to the real charge:
487:on the grounded plane is therefore given by
1463:media can be considered. If a point charge
1420:Reflection in a dielectric planar interface
3381:arises from a set of charges of magnitude
930:
3831:Introduction to Electrodynamics (4th ed.)
3825:
3792:Uniqueness theorem for Poisson's equation
3303:
2697:
2696:
1194:. The dipole experiences a force in the
821:
736:
729:
725:
469:
1701:
1693:
1657:
97:
89:
3940:
3921:
3872:
2789:and will be located at vector position
2777:outside of a grounded sphere of radius
935:The image of an electric dipole moment
4072:
3640:{\displaystyle \rho (r,\theta ,\phi )}
3204:{\displaystyle \Phi (r,\theta ,\phi )}
3161:
1670:lying inside the sphere at a distance
3958:
3857:
39:) is a basic problem-solving tool in
4000:, Mainly Electromagnetism and Matter
1423:
923:Because electric fields satisfy the
4054:
3586:. It follows that if the potential
13:
3977:
3891:; Samarskii, Alexander A. (1963).
3593:
3368:
3304:
3262:
3177:
2909:lying inside the sphere of radius
2462:
2454:
1710:
810:
706:
531:
523:
14:
4096:
3894:Equations of Mathematical Physics
1654:Reflection in a conducting sphere
1105:will have in image dipole moment
916:. This can also be seen from the
3123:
3115:
3107:
3093:
3048:
3006:
2998:
2956:
2895:
2832:
2763:
2391:
2344:
2336:
2264:
2256:
2180:
2143:
2135:
2027:
2019:
1962:
1906:
1883:
1836:
1814:
1736:
1427:
106:
86:Reflection in a conducting plane
3236:{\displaystyle r,\theta ,\phi }
3915:
3881:
3866:
3851:
3819:
3739:
3706:
3700:
3686:
3680:
3662:
3634:
3616:
3573:
3519:
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3415:
3287:
3269:
3198:
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3111:
2707:
2701:
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2425:
2184:
2176:
2060:
2040:
1973:
1956:
1950:
1930:
1917:
1900:
1887:
1879:
1666:. The green point is a charge
1181:
1112:
1092:
1029:
1009:
988:
964:
946:
219:
198:
140:
122:
1:
3807:
3802:Surface equivalence principle
3606:arises from a charge density
1530:{\displaystyle \epsilon _{2}}
1503:{\displaystyle \epsilon _{1}}
981:-plane is a dipole moment at
45:Dirichlet boundary conditions
3787:Schwarz reflection principle
3505:{\displaystyle Rq_{i}/r_{i}}
2902:{\displaystyle \mathbf {p} }
2867:a dipole with dipole moment
2770:{\displaystyle \mathbf {p} }
2398:{\displaystyle \mathbf {p} }
1843:{\displaystyle \mathbf {r} }
1743:{\displaystyle \mathbf {p} }
1720:a grounded sphere of radius
7:
3998:Feynman Lectures on Physics
3755:
262:between two point charges.
49:Neumann boundary conditions
10:
4101:
3970:Cambridge University Press
3934:
183:) conducting plate in the
20:
3947:Classical Electrodynamics
3251:about the origin will be
3812:
3036:and a dipole moment of:
1015:{\displaystyle (0,0,-a)}
225:{\displaystyle (0,0,-a)}
37:method of mirror charges
2858:Electric dipole moments
1728:the sphere at position
970:{\displaystyle (0,0,a)}
931:Electric dipole moments
925:superposition principle
287:cylindrical coordinates
146:{\displaystyle (0,0,a)}
70:shows that in a volume
29:method of image charges
3746:
3641:
3600:
3580:
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1707:
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1559:
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1477:
1410:
1312:
1188:
1099:
1016:
971:
903:
634:
485:surface charge density
474:
252:
251:{\displaystyle z>0}
226:
177:
147:
103:
95:
3952:John Wiley & Sons
3747:
3642:
3601:
3599:{\displaystyle \Phi }
3581:
3507:
3462:
3403:
3401:{\displaystyle q_{i}}
3376:
3374:{\displaystyle \Phi }
3353:
3245:spherical coordinates
3238:
3206:
3153:
3028:
2965:
2904:
2882:
2864:electric point dipole
2841:
2772:
2743:
2620:
2400:
2361:
2155:
1845:
1823:
1745:
1705:
1697:
1661:
1645:
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1532:
1505:
1478:
1411:
1313:
1189:
1100:
1017:
972:
904:
635:
475:
253:
227:
178:
148:
101:
93:
3651:
3610:
3590:
3516:
3471:
3412:
3385:
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3258:
3215:
3174:
3043:
2977:
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2419:
2387:
2170:
1857:
1832:
1775:
1732:
1569:
1544:
1514:
1487:
1467:
1328:
1205:
1198:direction, given by
1109:
1026:
985:
943:
654:
494:
296:
236:
195:
161:
119:
4026:Lifshitz, Evgeny M.
3989:Leighton, Robert B.
3889:Tikhonov, Andrey N.
3827:Griffiths, David J.
3162:Method of inversion
2887:at vector position
2755:at vector position
2689:
2665:
814:
793:
710:
695:
285:-axis, is given in
176:{\displaystyle V=0}
31:(also known as the
4056:Purcell, Edward M.
4030:Pitaevskii, Lev P.
3899:Dover Publications
3772:Divergence theorem
3742:
3637:
3596:
3576:
3502:
3457:
3398:
3371:
3348:
3233:
3201:
3148:
3023:
2960:
2899:
2877:
2836:
2767:
2738:
2672:
2651:
2615:
2395:
2356:
2150:
1840:
1818:
1740:
1708:
1700:
1692:
1640:
1558:{\displaystyle q'}
1555:
1527:
1500:
1473:
1439:. You can help by
1406:
1308:
1184:
1095:
1012:
967:
899:
897:
800:
776:
696:
678:
630:
470:
248:
222:
173:
153:above an infinite
143:
104:
96:
80:Poisson's equation
64:electric potential
56:uniqueness theorem
4047:978-0-7506-2634-7
3844:978-0-321-85656-2
3361:If the potential
3328:
3301:
3168:harmonic function
3141:
3072:
3021:
2880:{\displaystyle M}
2613:
2469:
2412:and is given by:
2349:
2348:
2314:
2269:
2268:
2213:
2148:
2147:
2132:
2100:
2032:
2031:
1978:
1922:
1635:
1476:{\displaystyle q}
1457:
1456:
1392:
1363:
1274:
1240:
877:
774:
643:In addition, the
628:
538:
462:
461:
406:
405:
353:
4092:
4080:Electromagnetism
4066:
4051:
4017:
3985:Feynman, Richard
3973:
3955:
3942:Jackson, John D.
3928:
3919:
3913:
3912:
3885:
3879:
3870:
3864:
3855:
3849:
3848:
3823:
3782:Gaussian surface
3762:Kelvin transform
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2862:The image of an
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1997:
1988:
1984:
1979:
1977:
1976:
1971:
1970:
1965:
1959:
1953:
1946:
1928:
1923:
1921:
1920:
1915:
1914:
1909:
1903:
1894:
1886:
1875:
1874:
1849:
1847:
1846:
1841:
1839:
1827:
1825:
1824:
1819:
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1812:
1808:
1807:
1806:
1797:
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1791:
1770:
1761:
1749:
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1649:
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1641:
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1609:
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1525:
1509:
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1499:
1498:
1482:
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1452:
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1415:
1413:
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1407:
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1364:
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1341:
1317:
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1314:
1309:
1307:
1303:
1296:
1295:
1275:
1273:
1272:
1271:
1258:
1257:
1256:
1243:
1241:
1239:
1238:
1237:
1218:
1193:
1191:
1190:
1185:
1104:
1102:
1101:
1096:
1021:
1019:
1018:
1013:
976:
974:
973:
968:
908:
906:
905:
900:
898:
882:
878:
876:
875:
871:
862:
858:
857:
856:
844:
843:
828:
816:
813:
808:
792:
784:
775:
773:
765:
754:
746:
724:
709:
704:
694:
686:
670:
669:
639:
637:
636:
631:
629:
627:
626:
625:
621:
612:
608:
607:
606:
594:
593:
571:
560:
555:
554:
543:
539:
537:
529:
521:
515:
514:
479:
477:
476:
471:
468:
464:
463:
460:
459:
454:
450:
431:
430:
421:
420:
412:
407:
404:
403:
398:
394:
375:
374:
365:
361:
354:
352:
351:
350:
331:
326:
322:
257:
255:
254:
249:
231:
229:
228:
223:
182:
180:
179:
174:
152:
150:
149:
144:
33:method of images
23:Method of images
4100:
4099:
4095:
4094:
4093:
4091:
4090:
4089:
4070:
4069:
4048:
4014:
3980:
3978:Further reading
3960:Jeans, James H.
3937:
3932:
3931:
3920:
3916:
3909:
3901:. p. 354.
3886:
3882:
3871:
3867:
3856:
3852:
3845:
3837:. p. 121.
3824:
3820:
3815:
3810:
3758:
3719:
3713:
3709:
3692:
3654:
3652:
3649:
3648:
3611:
3608:
3607:
3591:
3588:
3587:
3567:
3563:
3554:
3550:
3541:
3537:
3532:
3526:
3522:
3517:
3514:
3513:
3496:
3492:
3487:
3481:
3477:
3472:
3469:
3468:
3448:
3444:
3435:
3431:
3422:
3418:
3413:
3410:
3409:
3392:
3388:
3386:
3383:
3382:
3366:
3363:
3362:
3319:
3315:
3313:
3312:
3308:
3307:
3293:
3261:
3259:
3256:
3255:
3216:
3213:
3212:
3175:
3172:
3171:
3164:
3135:
3131:
3122:
3114:
3106:
3102:
3100:
3092:
3088:
3084:
3078:
3064:
3060:
3059:
3047:
3046:
3044:
3041:
3040:
3015:
3011:
3005:
2997:
2993:
2991:
2980:
2978:
2975:
2974:
2955:
2944:
2940:
2935:
2929:
2925:
2924:
2920:
2918:
2915:
2914:
2894:
2892:
2889:
2888:
2872:
2869:
2868:
2860:
2831:
2820:
2816:
2811:
2805:
2801:
2800:
2796:
2794:
2791:
2790:
2762:
2760:
2757:
2756:
2714:
2710:
2681:
2676:
2660:
2655:
2642:
2638:
2636:
2633:
2632:
2602:
2598:
2566:
2562:
2553:
2549:
2548:
2544:
2543:
2533:
2521:
2517:
2508:
2504:
2503:
2499:
2492:
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2475:
2461:
2453:
2451:
2448:
2447:
2441:
2437:
2420:
2417:
2416:
2390:
2388:
2385:
2384:
2370:
2343:
2335:
2323:
2319:
2308:
2304:
2297:
2293:
2287:
2283:
2282:
2280:
2274:
2263:
2255:
2243:
2239:
2230:
2226:
2220:
2219:
2215:
2206:
2202:
2195:
2190:
2179:
2171:
2168:
2167:
2142:
2134:
2126:
2122:
2115:
2111:
2107:
2105:
2094:
2090:
2084:
2080:
2078:
2069:
2065:
2052:
2039:
2037:
2026:
2018:
2006:
2002:
1993:
1989:
1983:
1972:
1966:
1961:
1960:
1955:
1954:
1942:
1929:
1927:
1916:
1910:
1905:
1904:
1899:
1898:
1893:
1882:
1870:
1866:
1858:
1855:
1854:
1835:
1833:
1830:
1829:
1813:
1802:
1798:
1793:
1787:
1783:
1782:
1778:
1776:
1773:
1772:
1759:
1755:
1735:
1733:
1730:
1729:
1713:
1656:
1628:
1624:
1615:
1611:
1610:
1603:
1599:
1590:
1586:
1585:
1583:
1572:
1570:
1567:
1566:
1547:
1545:
1542:
1541:
1521:
1517:
1515:
1512:
1511:
1494:
1490:
1488:
1485:
1484:
1468:
1465:
1464:
1453:
1447:
1444:
1437:needs expansion
1422:
1385:
1381:
1377:
1371:
1367:
1365:
1356:
1352:
1345:
1340:
1329:
1326:
1325:
1291:
1287:
1280:
1276:
1267:
1263:
1259:
1252:
1248:
1244:
1242:
1233:
1229:
1222:
1217:
1206:
1203:
1202:
1110:
1107:
1106:
1027:
1024:
1023:
986:
983:
982:
944:
941:
940:
933:
896:
895:
880:
879:
867:
863:
852:
848:
839:
835:
834:
830:
829:
817:
815:
809:
804:
785:
780:
766:
755:
753:
744:
743:
714:
705:
700:
687:
682:
671:
665:
661:
657:
655:
652:
651:
617:
613:
602:
598:
589:
585:
584:
580:
579:
572:
561:
559:
544:
530:
522:
520:
517:
516:
510:
506:
495:
492:
491:
455:
440:
436:
435:
426:
422:
413:
411:
399:
384:
380:
379:
370:
366:
360:
359:
355:
346:
342:
335:
330:
306:
302:
297:
294:
293:
237:
234:
233:
196:
193:
192:
162:
159:
158:
120:
117:
116:
109:
88:
25:
19:
12:
11:
5:
4098:
4088:
4087:
4085:Electrostatics
4082:
4068:
4067:
4052:
4046:
4022:Landau, Lev D.
4018:
4012:
4004:Addison-Wesley
3993:Sands, Matthew
3979:
3976:
3975:
3974:
3956:
3936:
3933:
3930:
3929:
3914:
3907:
3880:
3865:
3850:
3843:
3817:
3816:
3814:
3811:
3809:
3806:
3805:
3804:
3799:
3794:
3789:
3784:
3779:
3774:
3769:
3764:
3757:
3754:
3741:
3738:
3735:
3732:
3729:
3726:
3722:
3716:
3712:
3708:
3705:
3702:
3699:
3695:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
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3667:
3664:
3660:
3657:
3636:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3595:
3575:
3570:
3566:
3562:
3557:
3553:
3549:
3544:
3540:
3535:
3529:
3525:
3521:
3499:
3495:
3490:
3484:
3480:
3476:
3456:
3451:
3447:
3443:
3438:
3434:
3430:
3425:
3421:
3417:
3395:
3391:
3370:
3359:
3358:
3345:
3341:
3338:
3335:
3332:
3327:
3322:
3318:
3311:
3306:
3300:
3297:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3267:
3264:
3232:
3229:
3226:
3223:
3220:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3163:
3160:
3159:
3158:
3146:
3138:
3134:
3129:
3125:
3121:
3117:
3113:
3109:
3105:
3099:
3095:
3091:
3087:
3081:
3076:
3071:
3068:
3063:
3058:
3054:
3050:
3034:
3033:
3018:
3014:
3008:
3004:
3000:
2996:
2990:
2986:
2983:
2958:
2953:
2947:
2943:
2938:
2932:
2928:
2923:
2897:
2876:
2859:
2856:
2834:
2829:
2823:
2819:
2814:
2808:
2804:
2799:
2765:
2749:
2748:
2737:
2734:
2731:
2728:
2725:
2722:
2717:
2713:
2709:
2706:
2703:
2700:
2695:
2692:
2687:
2684:
2679:
2675:
2671:
2668:
2663:
2658:
2654:
2650:
2645:
2641:
2626:
2625:
2609:
2605:
2601:
2596:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2569:
2565:
2561:
2556:
2552:
2547:
2542:
2539:
2536:
2530:
2524:
2520:
2516:
2511:
2507:
2502:
2498:
2495:
2489:
2484:
2481:
2478:
2473:
2467:
2464:
2459:
2456:
2450:
2444:
2440:
2436:
2433:
2430:
2427:
2424:
2393:
2367:
2366:
2354:
2346:
2342:
2338:
2334:
2331:
2326:
2322:
2318:
2311:
2307:
2300:
2296:
2290:
2286:
2278:
2273:
2266:
2262:
2258:
2254:
2251:
2246:
2242:
2238:
2233:
2229:
2224:
2218:
2209:
2205:
2201:
2198:
2194:
2189:
2186:
2182:
2178:
2175:
2161:
2160:
2145:
2141:
2137:
2129:
2125:
2118:
2114:
2110:
2104:
2097:
2093:
2087:
2083:
2077:
2072:
2068:
2062:
2059:
2055:
2051:
2048:
2045:
2042:
2036:
2029:
2025:
2021:
2017:
2014:
2009:
2005:
2001:
1996:
1992:
1987:
1982:
1975:
1969:
1964:
1958:
1952:
1949:
1945:
1941:
1938:
1935:
1932:
1926:
1919:
1913:
1908:
1902:
1897:
1892:
1889:
1885:
1881:
1878:
1873:
1869:
1865:
1862:
1838:
1816:
1811:
1805:
1801:
1796:
1790:
1786:
1781:
1738:
1712:
1709:
1655:
1652:
1639:
1631:
1627:
1623:
1618:
1614:
1606:
1602:
1598:
1593:
1589:
1582:
1578:
1575:
1553:
1550:
1524:
1520:
1497:
1493:
1472:
1455:
1454:
1448:September 2013
1434:
1432:
1421:
1418:
1417:
1416:
1405:
1402:
1399:
1396:
1388:
1384:
1380:
1374:
1370:
1359:
1355:
1351:
1348:
1344:
1339:
1336:
1333:
1319:
1318:
1306:
1302:
1299:
1294:
1290:
1286:
1283:
1279:
1270:
1266:
1262:
1255:
1251:
1247:
1236:
1232:
1228:
1225:
1221:
1216:
1213:
1210:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
966:
963:
960:
957:
954:
951:
948:
932:
929:
910:
909:
894:
891:
888:
885:
883:
881:
874:
870:
866:
861:
855:
851:
847:
842:
838:
833:
827:
824:
820:
812:
807:
803:
799:
796:
791:
788:
783:
779:
772:
769:
764:
761:
758:
752:
749:
747:
745:
742:
739:
735:
732:
728:
723:
720:
717:
713:
708:
703:
699:
693:
690:
685:
681:
677:
674:
672:
668:
664:
660:
659:
641:
640:
624:
620:
616:
611:
605:
601:
597:
592:
588:
583:
578:
575:
570:
567:
564:
558:
553:
550:
547:
542:
536:
533:
528:
525:
519:
513:
509:
505:
502:
499:
481:
480:
467:
458:
453:
449:
446:
443:
439:
434:
429:
425:
419:
416:
410:
402:
397:
393:
390:
387:
383:
378:
373:
369:
364:
358:
349:
345:
341:
338:
334:
329:
325:
321:
318:
315:
312:
309:
305:
301:
247:
244:
241:
221:
218:
215:
212:
209:
206:
203:
200:
172:
169:
166:
142:
139:
136:
133:
130:
127:
124:
108:
105:
87:
84:
41:electrostatics
17:
9:
6:
4:
3:
2:
4097:
4086:
4083:
4081:
4078:
4077:
4075:
4064:
4060:
4057:
4053:
4049:
4043:
4039:
4035:
4031:
4027:
4023:
4019:
4015:
4013:0-201-51003-0
4009:
4005:
4001:
3999:
3994:
3990:
3986:
3982:
3981:
3971:
3967:
3966:
3961:
3957:
3953:
3949:
3948:
3943:
3939:
3938:
3927:
3923:
3918:
3910:
3908:0-486-66422-8
3904:
3900:
3896:
3895:
3890:
3884:
3878:
3874:
3869:
3863:
3859:
3854:
3846:
3840:
3836:
3832:
3828:
3822:
3818:
3803:
3800:
3798:
3797:Image antenna
3795:
3793:
3790:
3788:
3785:
3783:
3780:
3778:
3775:
3773:
3770:
3768:
3767:Coulomb's law
3765:
3763:
3760:
3759:
3753:
3736:
3733:
3730:
3727:
3724:
3720:
3714:
3710:
3703:
3697:
3693:
3689:
3683:
3677:
3674:
3671:
3668:
3665:
3658:
3655:
3631:
3628:
3625:
3622:
3619:
3613:
3568:
3564:
3560:
3555:
3551:
3547:
3542:
3538:
3533:
3527:
3523:
3512:at positions
3497:
3493:
3488:
3482:
3478:
3474:
3449:
3445:
3441:
3436:
3432:
3428:
3423:
3419:
3408:at positions
3393:
3389:
3343:
3339:
3336:
3333:
3330:
3325:
3320:
3316:
3309:
3298:
3295:
3290:
3284:
3281:
3278:
3275:
3272:
3265:
3254:
3253:
3252:
3250:
3246:
3230:
3227:
3224:
3221:
3218:
3195:
3192:
3189:
3186:
3183:
3169:
3144:
3136:
3132:
3119:
3103:
3097:
3089:
3085:
3079:
3074:
3069:
3066:
3061:
3056:
3052:
3039:
3038:
3037:
3016:
3012:
3002:
2994:
2988:
2984:
2981:
2973:
2972:
2971:
2951:
2945:
2941:
2936:
2930:
2926:
2921:
2912:
2874:
2865:
2855:
2853:
2849:
2827:
2821:
2817:
2812:
2806:
2802:
2797:
2788:
2784:
2780:
2754:
2735:
2732:
2729:
2726:
2723:
2720:
2715:
2711:
2704:
2698:
2693:
2690:
2685:
2682:
2677:
2673:
2669:
2666:
2661:
2656:
2652:
2648:
2643:
2639:
2631:
2630:
2629:
2607:
2603:
2599:
2594:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2567:
2563:
2559:
2554:
2550:
2545:
2540:
2537:
2534:
2528:
2522:
2518:
2514:
2509:
2505:
2500:
2496:
2493:
2487:
2482:
2479:
2476:
2471:
2465:
2457:
2442:
2438:
2434:
2428:
2422:
2415:
2414:
2413:
2411:
2408:
2382:
2377:
2373:
2352:
2340:
2332:
2329:
2324:
2320:
2316:
2309:
2305:
2298:
2294:
2288:
2284:
2276:
2271:
2260:
2252:
2249:
2244:
2240:
2236:
2231:
2227:
2222:
2216:
2207:
2203:
2199:
2196:
2192:
2187:
2173:
2166:
2165:
2164:
2139:
2127:
2123:
2116:
2112:
2108:
2102:
2095:
2091:
2085:
2081:
2075:
2070:
2066:
2057:
2053:
2049:
2046:
2043:
2034:
2023:
2015:
2012:
2007:
2003:
1999:
1994:
1990:
1985:
1980:
1967:
1947:
1943:
1939:
1936:
1933:
1924:
1911:
1895:
1890:
1876:
1871:
1867:
1863:
1860:
1853:
1852:
1851:
1809:
1803:
1799:
1794:
1788:
1784:
1779:
1769:
1765:
1758:
1753:
1727:
1723:
1719:
1711:Point charges
1704:
1696:
1689:
1685:
1681:
1677:
1673:
1669:
1665:
1660:
1651:
1637:
1629:
1625:
1621:
1616:
1612:
1604:
1600:
1596:
1591:
1587:
1580:
1576:
1573:
1551:
1548:
1538:
1522:
1518:
1495:
1491:
1470:
1462:
1451:
1442:
1438:
1435:This section
1433:
1430:
1426:
1425:
1403:
1400:
1397:
1394:
1386:
1382:
1378:
1372:
1368:
1357:
1353:
1349:
1346:
1342:
1337:
1334:
1331:
1324:
1323:
1322:
1304:
1300:
1297:
1292:
1288:
1284:
1281:
1277:
1268:
1264:
1260:
1253:
1249:
1245:
1234:
1230:
1226:
1223:
1219:
1214:
1211:
1208:
1201:
1200:
1199:
1197:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1006:
1003:
1000:
997:
994:
991:
980:
961:
958:
955:
952:
949:
938:
928:
926:
921:
919:
915:
892:
889:
886:
884:
872:
868:
864:
859:
853:
849:
845:
840:
836:
831:
825:
822:
818:
805:
801:
797:
794:
789:
786:
781:
777:
770:
767:
762:
759:
756:
750:
748:
740:
737:
733:
730:
726:
721:
718:
715:
711:
701:
697:
691:
688:
683:
679:
675:
673:
666:
662:
650:
649:
648:
646:
622:
618:
614:
609:
603:
599:
595:
590:
586:
581:
576:
573:
568:
565:
562:
556:
551:
548:
545:
540:
534:
526:
511:
507:
503:
500:
497:
490:
489:
488:
486:
465:
456:
451:
447:
444:
441:
437:
432:
427:
423:
417:
414:
408:
400:
395:
391:
388:
385:
381:
376:
371:
367:
362:
356:
347:
343:
339:
336:
332:
327:
323:
319:
316:
313:
310:
307:
303:
299:
292:
291:
290:
288:
284:
280:
276:
272:
268:
263:
261:
260:Coulomb's law
245:
242:
239:
216:
213:
210:
207:
204:
201:
191:, located at
190:
186:
170:
167:
164:
156:
137:
134:
131:
128:
125:
115:, located at
114:
107:Point charges
100:
92:
83:
81:
77:
73:
69:
65:
61:
57:
52:
50:
46:
42:
38:
34:
30:
24:
16:
4058:
4033:
3996:
3964:
3946:
3922:Jackson 1962
3917:
3897:. New York:
3893:
3883:
3873:Jackson 1962
3868:
3853:
3830:
3821:
3360:
3248:
3170:of position
3165:
3035:
2910:
2861:
2851:
2847:
2786:
2782:
2778:
2752:
2750:
2627:
2409:
2380:
2375:
2371:
2368:
2162:
1767:
1763:
1756:
1751:
1725:
1721:
1717:
1714:
1687:
1683:
1679:
1675:
1671:
1667:
1663:
1539:
1458:
1445:
1441:adding to it
1436:
1320:
1195:
978:
936:
934:
922:
913:
911:
644:
642:
482:
282:
278:
274:
270:
266:
264:
188:
184:
112:
110:
75:
71:
59:
53:
36:
32:
28:
26:
15:
4063:McGraw-Hill
2407:polar angle
918:Gauss's law
4074:Categories
4036:. London:
3924:, p.
3858:Jeans 1908
3808:References
1461:dielectric
68:Gauss' Law
21:See also:
3737:ϕ
3731:θ
3704:ρ
3678:ϕ
3672:θ
3656:ρ
3632:ϕ
3626:θ
3614:ρ
3594:Φ
3565:ϕ
3552:θ
3446:ϕ
3433:θ
3369:Φ
3340:ϕ
3334:θ
3305:Φ
3285:ϕ
3279:θ
3263:Φ
3231:ϕ
3225:θ
3196:ϕ
3190:θ
3178:Φ
3120:⋅
3090:−
3003:⋅
2733:−
2727:θ
2724:
2705:θ
2699:σ
2694:ϕ
2686:π
2674:∫
2670:θ
2662:π
2653:∫
2591:θ
2588:
2573:−
2538:π
2515:−
2494:−
2463:∂
2455:∂
2439:ε
2429:θ
2423:σ
2341:⋅
2330:−
2272:−
2261:⋅
2250:−
2204:ε
2200:π
2140:⋅
2103:−
2044:−
2024:⋅
2013:−
1934:−
1868:ε
1864:π
1626:ε
1613:ε
1601:ε
1597:−
1588:ε
1519:ϵ
1492:ϵ
1404:θ
1398:
1354:ε
1350:π
1338:−
1332:τ
1301:θ
1298:
1231:ε
1227:π
1215:−
1179:θ
1176:
1164:ϕ
1161:
1155:θ
1152:
1143:−
1137:ϕ
1134:
1128:θ
1125:
1116:−
1090:θ
1087:
1075:ϕ
1072:
1066:θ
1063:
1051:ϕ
1048:
1042:θ
1039:
1004:−
890:−
837:ρ
826:ρ
819:ρ
811:∞
802:∫
798:θ
790:π
778:∫
771:π
757:−
741:θ
734:ρ
727:ρ
719:ρ
712:σ
707:∞
698:∫
692:π
680:∫
587:ρ
577:π
563:−
532:∂
524:∂
508:ε
504:−
498:σ
424:ρ
415:−
389:−
368:ρ
344:ε
340:π
314:φ
308:ρ
214:−
4038:Elsevier
4032:(1960).
3995:(1989).
3962:(1908).
3944:(1962).
3829:(2013).
3756:See also
3659:′
3266:′
3243:are the
3053:′
2985:′
1577:′
1552:′
155:grounded
3935:Sources
3835:Pearson
281:on the
157:(i.e.:
4044:
4010:
3905:
3877:p. 111
3862:p. 186
3841:
3211:where
2410:θ
1726:inside
1718:inside
3813:Notes
645:total
273:and −
4042:ISBN
4008:ISBN
3903:ISBN
3839:ISBN
3777:Flux
483:The
277:at −
269:at +
243:>
35:and
27:The
2721:sin
2585:cos
2381:not
1762:= −
1443:.
1395:sin
1289:cos
1173:cos
1158:sin
1149:sin
1131:cos
1122:sin
1084:cos
1069:sin
1060:sin
1045:cos
1036:sin
939:at
289:as
51:).
47:or
4076::
4061:.
4040:.
4028:;
4024:;
4006:.
4002:.
3991:;
3987:;
3968:.
3950:.
3926:35
3875:,
3860:,
3833:.
3752:.
2854:.
2848:qR
2783:qR
2374:=
1764:qR
1676:qR
1379:16
1261:16
979:xy
185:xy
4065:.
4050:.
4016:.
3972:.
3954:.
3911:.
3847:.
3740:)
3734:,
3728:,
3725:r
3721:/
3715:2
3711:R
3707:(
3701:)
3698:r
3694:/
3690:R
3687:(
3684:=
3681:)
3675:,
3669:,
3666:r
3663:(
3635:)
3629:,
3623:,
3620:r
3617:(
3574:)
3569:i
3561:,
3556:i
3548:,
3543:i
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3520:(
3498:i
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3483:i
3479:q
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3455:)
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3442:,
3437:i
3429:,
3424:i
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3416:(
3394:i
3390:q
3344:)
3337:,
3331:,
3326:r
3321:2
3317:R
3310:(
3299:r
3296:R
3291:=
3288:)
3282:,
3276:,
3273:r
3270:(
3249:R
3228:,
3222:,
3219:r
3199:)
3193:,
3187:,
3184:r
3181:(
3145:]
3137:2
3133:p
3128:)
3124:M
3116:p
3112:(
3108:p
3104:2
3098:+
3094:M
3086:[
3080:3
3075:)
3070:p
3067:R
3062:(
3057:=
3049:M
3017:3
3013:p
3007:M
2999:p
2995:R
2989:=
2982:q
2957:p
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2946:2
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2922:(
2911:R
2896:p
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2833:p
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2822:2
2818:p
2813:/
2807:2
2803:R
2798:(
2787:p
2785:/
2779:R
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2736:q
2730:=
2716:2
2712:R
2708:)
2702:(
2691:d
2683:2
2678:0
2667:d
2657:0
2649:=
2644:t
2640:Q
2608:2
2604:/
2600:3
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2582:R
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2576:2
2568:2
2564:p
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2529:)
2523:2
2519:p
2510:2
2506:R
2501:(
2497:q
2488:=
2483:R
2480:=
2477:r
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2466:r
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2443:0
2435:=
2432:)
2426:(
2392:p
2376:R
2372:r
2353:]
2345:p
2337:r
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2325:2
2321:R
2317:+
2310:2
2306:R
2299:2
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2265:p
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2245:2
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2232:2
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2217:[
2208:0
2197:4
2193:1
2188:=
2185:)
2181:r
2177:(
2174:V
2144:p
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2124:p
2117:2
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2109:2
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2058:p
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2041:(
2035:+
2028:p
2020:r
2016:2
2008:2
2004:p
2000:+
1995:2
1991:r
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1981:=
1974:|
1968:2
1963:r
1957:|
1951:)
1948:p
1944:/
1940:R
1937:q
1931:(
1925:+
1918:|
1912:1
1907:r
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1622:+
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1581:=
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1523:2
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1269:4
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989:(
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887:=
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846:+
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832:(
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751:=
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716:(
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676:=
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582:(
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72:V
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