3039:, National Bureau of Standards Applied Mathematics Series 19, 1952. Look specifically on pages 228-263. The article by Chester Snow, "Magnetic Fields of Cylindrical Coils and Annular Coils" (National Bureau of Standards, Applied Mathematical Series 38, December 30, 1953), clearly shows the relationship between the free-space Green's function in cylindrical coordinates and the Q-function expression. Likewise, see another one of Snow's pieces of work, titled "Formulas for Computing Capacitance and Inductance", National Bureau of Standards Circular 544, September 10, 1954, pp 13–41. Indeed, not much has been published recently on the subject of toroidal functions and their applications in engineering or physics. However, a number of engineering applications do exist. One application was published; the article was written by J.P. Selvaggi, S. Salon, O. Kwon, and M.V.K. Chari, "Calculating the External Magnetic Field From Permanent Magnets in Permanent-Magnet Motors-An Alternative Method," IEEE Transactions on Magnetics, Vol. 40, No. 5, September 2004. These authors have done extensive work with Legendre functions of the second kind and half-integral degree or toroidal functions of zeroth order. They have solved numerous problems which exhibit circular cylindrical symmetry employing the toroidal functions.
3021:
25:
2344:. There are many expansions in terms of special functions for the Green's function. In the case of a boundary put at infinity with the boundary condition setting the solution to zero at infinity, then one has an infinite-extent Green's function. For the three-variable Laplace operator, one can for instance expand it in the rotationally invariant coordinate systems which allow
2735:
3271:
878:
3592:
2519:
3042:
The above expressions for the Green's function for the three-variable
Laplace operator are examples of single summation expressions for this Green's function. There are also single-integral expressions for this Green's function. Examples of these can be seen to exist in rotational cylindrical
2118:
770:
3016:{\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {x'} |}}={\sqrt {\frac {\pi }{2RR'(\chi ^{2}-1)^{1/2}}}}\sum _{m=-\infty }^{\infty }{\frac {\left(-1\right)^{m}}{\Gamma (m+1/2)}}P_{-{\frac {1}{2}}}^{m}{\left({\frac {\chi }{\sqrt {\chi ^{2}-1}}}\right)}e^{im(\varphi -\varphi ')}}
1203:
3050:
1442:
3368:
2324:
1646:
2351:
516:
963:
1550:
697:
3034:
in volume 18, 1947 pages 562-577 shows N.G. De Bruijn and C.J. Boukamp knew of the above relationship. In fact, virtually all the mathematics found in recent papers was already done by
Chester Snow. This is found in his book titled
2628:
42:
1977:
1958:
1108:
3601:
Green's function expansions exist in all of the rotationally invariant coordinate systems which are known to yield solutions to the three-variable
Laplace equation through the separation of variables technique.
1115:
775:
205:
741:
873:{\displaystyle {\begin{aligned}\mathbf {E} &=-\mathbf {\nabla } \phi (\mathbf {x} )\\{\boldsymbol {\nabla }}\cdot \mathbf {E} &={\frac {\rho (\mathbf {x} )}{\varepsilon _{0}}}\end{aligned}}}
1294:
1247:
564:
2217:
1350:
889:
2222:
1555:
2677:
425:
3266:{\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {x'} |}}=\int _{0}^{\infty }J_{0}{\left(k{\sqrt {R^{2}+{R'}^{2}-2RR'\cos(\varphi -\varphi ')}}\right)}e^{-k(z_{>}-z_{<})}\,dk,}
2156:
1692:
1471:
1345:
996:
618:
3587:{\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {x'} |}}={\frac {2}{\pi }}\int _{0}^{\infty }K_{0}{\left(k{\sqrt {R^{2}+{R'}^{2}-2RR'\cos(\varphi -\varphi ')}}\right)}\cos\,dk.}
3314:
1043:
613:
265:
2336:
in order to determine the potential function. Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable
420:
389:
327:
296:
2719:
586:
354:
232:
2514:{\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {x'} |}}={\frac {1}{\pi {\sqrt {RR'}}}}\sum _{m=-\infty }^{\infty }e^{im(\varphi -\varphi ')}Q_{m-{\frac {1}{2}}}(\chi )}
2328:
The free-space circular cylindrical Green's function (see below) is given in terms of the reciprocal distance between two points. The expression is derived in
Jackson's
1763:
89:
61:
3359:
2176:
1712:
68:
1772:
3334:
2524:
1314:
1048:
1016:
3030:, published by Howard Cohl and Joel Tohline. The above-mentioned formula is also known in the engineering community. For instance, a paper written in the
75:
2158:. The less than (greater than) notation means, take the primed or unprimed spherical radius depending on which is less than (greater than) the other. The
57:
3753:
152:
2341:
2113:{\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {x'} |}}=\sum _{l=0}^{\infty }{\frac {r_{<}^{l}}{r_{>}^{l+1}}}P_{l}(\cos \gamma ),}
1962:
Many expansion formulas are possible, given the algebraic expression for the Green's function. One of the most well-known of these, the
82:
3365:
of the difference of vertical heights whose kernel is given in terms of the order-zero modified Bessel function of the second kind as
108:
3617:
46:
3047:
in the difference of vertical heights whose kernel is given in terms of the order-zero Bessel function of the first kind as
998:
to this equation for an arbitrary charge distribution by temporarily considering the distribution created by a point charge
2683:
of the second kind, which is a toroidal harmonic. Here the expansion has been written in terms of cylindrical coordinates
1252:
1198:{\displaystyle -{\frac {\varepsilon _{0}}{q}}\mathbf {\nabla } ^{2}\phi (\mathbf {x} )=\delta (\mathbf {x} -\mathbf {x'} )}
3675:
3622:
1963:
146:
702:
3647:
3642:
1208:
525:
2181:
2633:
3657:
3361:. Similarly, the Green's function for the three-variable Laplace equation can be given as a Fourier integral
3652:
3637:
3607:
2345:
35:
2123:
1651:
1437:{\displaystyle \phi (\mathbf {x} )=\int G(\mathbf {x} ,\mathbf {x'} )\rho (\mathbf {x'} )\,d\mathbf {x} '}
1316:. Therefore, from the discussion above, if we can find the Green's function of this operator, we can find
958:{\displaystyle -\mathbf {\nabla } ^{2}\phi (\mathbf {x} )={\frac {\rho (\mathbf {x} )}{\varepsilon _{0}}}}
2319:{\displaystyle \cos \gamma =\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos(\varphi -\varphi ').}
1641:{\displaystyle G(\mathbf {x} ,\mathbf {x'} )=-{\frac {1}{4\pi \left|\mathbf {x} -\mathbf {x'} \right|}},}
1319:
970:
3627:
3612:
3276:
1766:
1460:
in three variables is given in terms of the reciprocal distance between two points and is known as the "
357:
883:
142:
391:
to a general system of this type can be written as an integral over a distribution of source given by
241:
3632:
2722:
1971:
1967:
1715:
744:
394:
363:
301:
270:
128:
2686:
3670:
1465:
1453:
1021:
591:
519:
138:
569:
3037:
Hypergeometric and
Legendre Functions with Applications to Integral Equations of Potential Theory
2337:
760:
332:
210:
1728:
511:{\displaystyle u(\mathbf {x} )=\int G(\mathbf {x} ,\mathbf {x'} )f(\mathbf {x'} )d\mathbf {x} '}
3730:
3044:
2680:
1725:
of the Green's function for the three-variable
Laplace operator, apart from the constant term
2161:
1697:
3722:
3362:
2729:
2333:
2332:. Using the Green's function for the three-variable Laplace operator, one can integrate the
1545:{\displaystyle \nabla ^{2}G(\mathbf {x} ,\mathbf {x'} )=\delta (\mathbf {x} -\mathbf {x'} )}
1457:
692:{\displaystyle \nabla ^{2}G(\mathbf {x} ,\mathbf {x'} )=\delta (\mathbf {x} -\mathbf {x'} )}
235:
3711:"A Compact Cylindrical Green's Function Expansion for the Solution of Potential Problems"
3339:
3319:
1299:
1001:
764:
756:
3747:
3026:
This formula was used in 1999 for astrophysical applications in a paper published in
1461:
3710:
759:. In such a system, the electric field is expressed as the negative gradient of the
134:
2732:
for toroidal harmonics we can obtain an alternative form of the Green's function
3597:
Rotationally invariant Green's functions for the three-variable
Laplace operator
24:
133:
is used to describe the response of a particular type of physical system to a
3734:
1953:{\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {x'} |}}=\left^{-{1}/{2}}.}
2623:{\displaystyle \chi ={\frac {R^{2}+{R'}^{2}+\left(z-z'\right)^{2}}{2RR'}}}
1103:{\displaystyle \rho (\mathbf {x} )=q\,\delta (\mathbf {x} -\mathbf {x'} )}
1694:
are the standard
Cartesian coordinates in a three-dimensional space, and
122:
3726:
1966:
for the three-variable
Laplace equation, is given in terms of the
3596:
18:
755:
One physical system of this type is a charge distribution in
131:) for the Laplacian (or Laplace operator) in three variables
200:{\displaystyle \nabla ^{2}u(\mathbf {x} )=f(\mathbf {x} )}
58:"Green's function for the three-variable Laplace equation"
2120:
which has been written in terms of spherical coordinates
1296:
will give the response of the system to the point charge
2178:
represents the angle between the two arbitrary vectors
1289:{\textstyle -{\frac {\varepsilon _{0}}{q}}\nabla ^{2}}
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in terms for a toroidal harmonic of the first kind.
736:{\displaystyle \delta (\mathbf {x} -\mathbf {x'} )}
49:. Unsourced material may be challenged and removed.
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566:describes the response of the system at the point
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1468:". That is to say, the solution of the equation
3709:Cohl, Howard S.; Tohline, Joel E. (1999-12-10).
1242:{\displaystyle G(\mathbf {x} ,\mathbf {x'} )}
559:{\displaystyle G(\mathbf {x} ,\mathbf {x'} )}
8:
2212:{\displaystyle (\mathbf {x} ,\mathbf {x'} )}
2672:{\displaystyle Q_{m-{\frac {1}{2}}}(\chi )}
141:arises in systems that can be described by
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1407:
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1352:
1329:
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1218:
1210:
1182:
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1157:
1145:
1140:
1128:
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1117:
1087:
1079:
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1058:
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1023:
1003:
980:
972:
947:
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914:
902:
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858:
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824:
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704:
676:
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543:
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527:
499:
482:
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365:
340:
334:
329:is the solution to the equation. Because
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246:
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218:
212:
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172:
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154:
109:Learn how and when to remove this message
3686:
298:is the source term of the system, and
882:Combining these expressions gives us
7:
2342:linear partial differential equation
2151:{\displaystyle (r,\theta ,\varphi )}
1687:{\displaystyle \mathbf {x} =(x,y,z)}
47:adding citations to reliable sources
3316:are the greater (lesser) variables
1444:for a general charge distribution.
1340:{\displaystyle \phi (\mathbf {x} )}
991:{\displaystyle \phi (\mathbf {x} )}
3432:
3309:{\displaystyle z_{>}(z_{<})}
3104:
2891:
2860:
2855:
2441:
2436:
2037:
1476:
1277:
1141:
898:
794:
623:
337:
215:
157:
14:
3698:(3rd ed.). pp. 125–127.
699:and the point source is given by
522:for Laplacian in three variables
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3384:
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3066:
2760:
2751:
2376:
2367:
2198:
2189:
2002:
1993:
1797:
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1656:
1619:
1610:
1575:
1566:
1531:
1522:
1501:
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1426:
1409:
1390:
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1361:
1330:
1228:
1219:
1184:
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1158:
1089:
1080:
1059:
1027:
981:
935:
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779:
722:
713:
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597:
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484:
465:
456:
436:
405:
374:
312:
281:
260:{\displaystyle \mathbb {R} ^{3}}
190:
173:
23:
2679:is the odd-half-integer degree
415:{\displaystyle f(\mathbf {x} )}
384:{\displaystyle u(\mathbf {x} )}
322:{\displaystyle u(\mathbf {x} )}
291:{\displaystyle f(\mathbf {x} )}
34:needs additional citations for
3754:Partial differential equations
3618:Prolate spheroidal coordinates
3571:
3568:
3551:
3545:
3528:
3511:
3402:
3379:
3303:
3290:
3248:
3222:
3200:
3183:
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3008:
2991:
2914:
2894:
2820:
2800:
2769:
2746:
2714:{\displaystyle (R,\varphi ,z)}
2708:
2690:
2666:
2660:
2508:
2502:
2474:
2457:
2385:
2362:
2310:
2293:
2206:
2185:
2145:
2127:
2104:
2092:
2011:
1988:
1806:
1783:
1752:
1743:
1681:
1663:
1583:
1562:
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1518:
1509:
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1404:
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1063:
1055:
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977:
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931:
919:
911:
850:
842:
809:
801:
767:in differential form applies:
730:
709:
686:
665:
656:
635:
553:
532:
492:
479:
473:
452:
440:
432:
409:
401:
378:
370:
316:
308:
285:
277:
194:
186:
177:
169:
16:Partial differential equations
1:
3648:Flat-disk cyclide coordinates
3643:Flat-ring cyclide coordinates
3623:Oblate spheroidal coordinates
1038:{\displaystyle \mathbf {x'} }
608:{\displaystyle \mathbf {x'} }
588:to a point source located at
147:partial differential equation
581:{\displaystyle \mathbf {x} }
3043:coordinates as an integral
349:{\displaystyle \nabla ^{2}}
227:{\displaystyle \nabla ^{2}}
3770:
3032:Journal of Applied Physics
1758:{\displaystyle -1/(4\pi )}
3715:The Astrophysical Journal
3696:Classical Electrodynamics
3028:The Astrophysical Journal
2330:Classical Electrodynamics
967:We can find the solution
1769:shall be referred to as
3658:Cap-cyclide coordinates
3638:Bispherical coordinates
3608:cylindrical coordinates
2346:separation of variables
2171:{\displaystyle \gamma }
1707:{\displaystyle \delta }
1448:Mathematical exposition
3653:Bi-cyclide coordinates
3588:
3355:
3330:
3310:
3267:
3017:
2864:
2715:
2673:
2624:
2515:
2445:
2320:
2213:
2172:
2152:
2114:
2041:
1954:
1759:
1708:
1688:
1642:
1546:
1438:
1341:
1310:
1290:
1243:
1199:
1104:
1039:
1012:
992:
959:
874:
737:
693:
609:
582:
560:
512:
416:
385:
350:
323:
292:
261:
228:
201:
137:. In particular, this
3628:Parabolic coordinates
3613:spherical coordinates
3589:
3356:
3331:
3311:
3268:
3018:
2841:
2716:
2674:
2625:
2516:
2422:
2321:
2214:
2173:
2153:
2115:
2021:
1955:
1767:Cartesian coordinates
1760:
1709:
1689:
1643:
1547:
1439:
1342:
1311:
1291:
1244:
1200:
1105:
1040:
1013:
993:
960:
875:
738:
694:
610:
583:
561:
513:
417:
386:
358:differential operator
351:
324:
293:
262:
229:
202:
127:Green's function (or
3633:Toroidal coordinates
3369:
3340:
3320:
3277:
3051:
2736:
2723:Toroidal coordinates
2721:. See for instance
2687:
2634:
2525:
2352:
2223:
2182:
2162:
2124:
1978:
1972:Legendre polynomials
1773:
1729:
1723:algebraic expression
1716:Dirac delta function
1698:
1652:
1556:
1472:
1351:
1320:
1300:
1253:
1209:
1116:
1049:
1022:
1002:
971:
890:
771:
745:Dirac delta function
703:
619:
592:
570:
526:
426:
395:
364:
333:
302:
271:
242:
211:
153:
129:fundamental solution
43:improve this article
3671:Newtonian potential
3436:
3108:
2944:
2079:
2058:
1968:generating function
1466:Newtonian potential
3584:
3422:
3354:{\displaystyle z'}
3351:
3326:
3306:
3263:
3094:
3013:
2920:
2711:
2669:
2620:
2511:
2338:coordinate systems
2316:
2209:
2168:
2148:
2110:
2059:
2044:
1950:
1755:
1704:
1684:
1638:
1542:
1434:
1337:
1306:
1286:
1239:
1195:
1100:
1035:
1008:
988:
955:
884:Poisson's equation
870:
868:
761:electric potential
733:
689:
605:
578:
556:
508:
412:
381:
346:
319:
288:
257:
224:
197:
149:(PDE) of the form
143:Poisson's equation
3676:Laplace expansion
3531:
3420:
3407:
3329:{\displaystyle z}
3203:
3089:
3045:Laplace transform
2973:
2972:
2936:
2918:
2839:
2838:
2774:
2728:Using one of the
2681:Legendre function
2656:
2618:
2498:
2420:
2417:
2390:
2348:. For instance:
2080:
2016:
1964:Laplace expansion
1811:
1633:
1309:{\displaystyle q}
1274:
1205:which shows that
1137:
1011:{\displaystyle q}
953:
864:
119:
118:
111:
93:
3761:
3739:
3738:
3706:
3700:
3699:
3691:
3593:
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3538:
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3527:
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3430:
3421:
3413:
3408:
3406:
3405:
3400:
3399:
3387:
3382:
3373:
3363:cosine transform
3360:
3358:
3357:
3352:
3350:
3335:
3333:
3332:
3327:
3315:
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3307:
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2812:
2811:
2799:
2781:
2780:
2775:
2773:
2772:
2767:
2766:
2754:
2749:
2740:
2730:Whipple formulae
2720:
2718:
2717:
2712:
2678:
2676:
2675:
2670:
2659:
2658:
2657:
2649:
2629:
2627:
2626:
2621:
2619:
2617:
2616:
2601:
2600:
2599:
2594:
2590:
2589:
2566:
2565:
2560:
2559:
2546:
2545:
2535:
2520:
2518:
2517:
2512:
2501:
2500:
2499:
2491:
2478:
2477:
2473:
2444:
2439:
2421:
2419:
2418:
2416:
2405:
2396:
2391:
2389:
2388:
2383:
2382:
2370:
2365:
2356:
2334:Poisson equation
2325:
2323:
2322:
2317:
2309:
2286:
2260:
2218:
2216:
2215:
2210:
2205:
2204:
2192:
2177:
2175:
2174:
2169:
2157:
2155:
2154:
2149:
2119:
2117:
2116:
2111:
2091:
2090:
2081:
2078:
2067:
2057:
2052:
2043:
2040:
2035:
2017:
2015:
2014:
2009:
2008:
1996:
1991:
1982:
1959:
1957:
1956:
1951:
1946:
1945:
1944:
1939:
1934:
1925:
1921:
1920:
1919:
1914:
1910:
1909:
1886:
1885:
1880:
1876:
1875:
1852:
1851:
1846:
1842:
1841:
1812:
1810:
1809:
1804:
1803:
1791:
1786:
1777:
1764:
1762:
1761:
1756:
1742:
1713:
1711:
1710:
1705:
1693:
1691:
1690:
1685:
1659:
1647:
1645:
1644:
1639:
1634:
1632:
1631:
1627:
1626:
1625:
1613:
1593:
1582:
1581:
1569:
1551:
1549:
1548:
1543:
1538:
1537:
1525:
1508:
1507:
1495:
1484:
1483:
1458:Laplace operator
1454:Green's function
1443:
1441:
1440:
1435:
1433:
1429:
1416:
1415:
1397:
1396:
1384:
1364:
1346:
1344:
1343:
1338:
1333:
1315:
1313:
1312:
1307:
1295:
1293:
1292:
1287:
1285:
1284:
1275:
1270:
1269:
1260:
1248:
1246:
1245:
1240:
1235:
1234:
1222:
1204:
1202:
1201:
1196:
1191:
1190:
1178:
1161:
1150:
1149:
1144:
1138:
1133:
1132:
1123:
1109:
1107:
1106:
1101:
1096:
1095:
1083:
1062:
1044:
1042:
1041:
1036:
1034:
1033:
1017:
1015:
1014:
1009:
997:
995:
994:
989:
984:
964:
962:
961:
956:
954:
952:
951:
942:
938:
926:
918:
907:
906:
901:
879:
877:
876:
871:
869:
865:
863:
862:
853:
849:
837:
828:
820:
808:
797:
782:
742:
740:
739:
734:
729:
728:
716:
698:
696:
695:
690:
685:
684:
672:
655:
654:
642:
631:
630:
614:
612:
611:
606:
604:
603:
587:
585:
584:
579:
577:
565:
563:
562:
557:
552:
551:
539:
520:Green's function
517:
515:
514:
509:
507:
503:
491:
490:
472:
471:
459:
439:
421:
419:
418:
413:
408:
390:
388:
387:
382:
377:
355:
353:
352:
347:
345:
344:
328:
326:
325:
320:
315:
297:
295:
294:
289:
284:
266:
264:
263:
258:
256:
255:
250:
236:Laplace operator
233:
231:
230:
225:
223:
222:
206:
204:
203:
198:
193:
176:
165:
164:
139:Green's function
114:
107:
103:
100:
94:
92:
51:
27:
19:
3769:
3768:
3764:
3763:
3762:
3760:
3759:
3758:
3744:
3743:
3742:
3708:
3707:
3703:
3693:
3692:
3688:
3684:
3667:
3662:
3599:
3560:
3520:
3497:
3473:
3471:
3458:
3452:
3448:
3437:
3392:
3377:
3367:
3366:
3343:
3338:
3337:
3318:
3317:
3293:
3280:
3275:
3274:
3238:
3225:
3211:
3192:
3169:
3145:
3143:
3130:
3124:
3120:
3109:
3074:
3059:
3049:
3048:
3000:
2980:
2956:
2946:
2890:
2872:
2868:
2867:
2819:
2803:
2792:
2785:
2759:
2744:
2734:
2733:
2685:
2684:
2637:
2632:
2631:
2609:
2602:
2582:
2575:
2571:
2570:
2552:
2550:
2537:
2536:
2523:
2522:
2479:
2466:
2446:
2409:
2400:
2375:
2360:
2350:
2349:
2302:
2279:
2253:
2221:
2220:
2197:
2180:
2179:
2160:
2159:
2122:
2121:
2082:
2001:
1986:
1976:
1975:
1902:
1895:
1891:
1890:
1868:
1861:
1857:
1856:
1834:
1827:
1823:
1822:
1821:
1817:
1816:
1796:
1781:
1771:
1770:
1727:
1726:
1696:
1695:
1650:
1649:
1618:
1608:
1604:
1597:
1574:
1554:
1553:
1530:
1500:
1475:
1470:
1469:
1452:The free-space
1450:
1424:
1408:
1389:
1349:
1348:
1318:
1317:
1298:
1297:
1276:
1261:
1251:
1250:
1227:
1207:
1206:
1183:
1139:
1124:
1114:
1113:
1088:
1047:
1046:
1026:
1020:
1019:
1000:
999:
969:
968:
943:
927:
896:
888:
887:
867:
866:
854:
838:
829:
813:
812:
783:
769:
768:
753:
721:
701:
700:
677:
647:
622:
617:
616:
596:
590:
589:
568:
567:
544:
524:
523:
498:
483:
464:
424:
423:
393:
392:
362:
361:
360:, the solution
336:
331:
330:
300:
299:
269:
268:
245:
240:
239:
214:
209:
208:
156:
151:
150:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
3767:
3765:
3757:
3756:
3746:
3745:
3741:
3740:
3727:10.1086/308062
3701:
3685:
3683:
3680:
3679:
3678:
3673:
3666:
3663:
3661:
3660:
3655:
3650:
3645:
3640:
3635:
3630:
3625:
3620:
3615:
3610:
3604:
3598:
3595:
3583:
3580:
3577:
3573:
3570:
3566:
3563:
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3536:
3530:
3526:
3523:
3519:
3516:
3513:
3510:
3507:
3503:
3500:
3496:
3493:
3490:
3485:
3479:
3476:
3470:
3465:
3461:
3455:
3451:
3444:
3440:
3434:
3429:
3425:
3419:
3416:
3411:
3404:
3398:
3395:
3390:
3386:
3381:
3376:
3349:
3346:
3325:
3305:
3300:
3296:
3292:
3287:
3283:
3262:
3259:
3256:
3250:
3245:
3241:
3237:
3232:
3228:
3224:
3221:
3218:
3214:
3208:
3202:
3198:
3195:
3191:
3188:
3185:
3182:
3179:
3175:
3172:
3168:
3165:
3162:
3157:
3151:
3148:
3142:
3137:
3133:
3127:
3123:
3116:
3112:
3106:
3101:
3097:
3093:
3086:
3080:
3077:
3072:
3068:
3063:
3058:
3010:
3006:
3003:
2999:
2996:
2993:
2990:
2987:
2983:
2977:
2971:
2968:
2963:
2959:
2954:
2949:
2942:
2935:
2932:
2927:
2923:
2916:
2913:
2909:
2905:
2902:
2899:
2896:
2893:
2887:
2882:
2878:
2875:
2871:
2862:
2857:
2854:
2851:
2848:
2844:
2834:
2830:
2826:
2822:
2818:
2815:
2810:
2806:
2802:
2798:
2795:
2791:
2788:
2784:
2778:
2771:
2765:
2762:
2757:
2753:
2748:
2743:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2668:
2665:
2662:
2655:
2652:
2647:
2644:
2640:
2615:
2612:
2608:
2605:
2598:
2593:
2588:
2585:
2581:
2578:
2574:
2569:
2564:
2558:
2555:
2549:
2544:
2540:
2533:
2530:
2510:
2507:
2504:
2497:
2494:
2489:
2486:
2482:
2476:
2472:
2469:
2465:
2462:
2459:
2456:
2453:
2449:
2443:
2438:
2435:
2432:
2429:
2425:
2415:
2412:
2408:
2403:
2399:
2394:
2387:
2381:
2378:
2373:
2369:
2364:
2359:
2315:
2312:
2308:
2305:
2301:
2298:
2295:
2292:
2289:
2285:
2282:
2278:
2275:
2272:
2269:
2266:
2263:
2259:
2256:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2208:
2203:
2200:
2195:
2191:
2187:
2167:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2109:
2106:
2103:
2100:
2097:
2094:
2089:
2085:
2077:
2074:
2071:
2066:
2062:
2056:
2051:
2047:
2039:
2034:
2031:
2028:
2024:
2020:
2013:
2007:
2004:
1999:
1995:
1990:
1985:
1949:
1943:
1938:
1933:
1929:
1924:
1918:
1913:
1908:
1905:
1901:
1898:
1894:
1889:
1884:
1879:
1874:
1871:
1867:
1864:
1860:
1855:
1850:
1845:
1840:
1837:
1833:
1830:
1826:
1820:
1815:
1808:
1802:
1799:
1794:
1790:
1785:
1780:
1754:
1751:
1748:
1745:
1741:
1737:
1734:
1703:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1658:
1637:
1630:
1624:
1621:
1616:
1612:
1607:
1603:
1600:
1596:
1591:
1588:
1585:
1580:
1577:
1572:
1568:
1564:
1561:
1541:
1536:
1533:
1528:
1524:
1520:
1517:
1514:
1511:
1506:
1503:
1498:
1494:
1490:
1487:
1482:
1478:
1449:
1446:
1432:
1428:
1423:
1419:
1414:
1411:
1406:
1403:
1400:
1395:
1392:
1387:
1383:
1379:
1376:
1373:
1370:
1367:
1363:
1359:
1356:
1336:
1332:
1328:
1325:
1305:
1283:
1279:
1273:
1268:
1264:
1258:
1238:
1233:
1230:
1225:
1221:
1217:
1214:
1194:
1189:
1186:
1181:
1177:
1173:
1170:
1167:
1164:
1160:
1156:
1153:
1148:
1143:
1136:
1131:
1127:
1121:
1112:In this case,
1099:
1094:
1091:
1086:
1082:
1078:
1075:
1071:
1068:
1065:
1061:
1057:
1054:
1032:
1029:
1007:
987:
983:
979:
976:
950:
946:
941:
937:
933:
930:
924:
921:
917:
913:
910:
905:
900:
895:
861:
857:
852:
848:
844:
841:
835:
832:
830:
827:
823:
819:
815:
814:
811:
807:
803:
800:
796:
792:
789:
786:
784:
781:
777:
776:
757:electrostatics
752:
749:
732:
727:
724:
719:
715:
711:
708:
688:
683:
680:
675:
671:
667:
664:
661:
658:
653:
650:
645:
641:
637:
634:
629:
625:
602:
599:
576:
555:
550:
547:
542:
538:
534:
531:
506:
502:
497:
494:
489:
486:
481:
478:
475:
470:
467:
462:
458:
454:
451:
448:
445:
442:
438:
434:
431:
411:
407:
403:
400:
380:
376:
372:
369:
343:
339:
318:
314:
310:
307:
287:
283:
279:
276:
254:
249:
221:
217:
196:
192:
188:
185:
182:
179:
175:
171:
168:
163:
159:
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
3766:
3755:
3752:
3751:
3749:
3736:
3732:
3728:
3724:
3721:(1): 86–101.
3720:
3716:
3712:
3705:
3702:
3697:
3690:
3687:
3681:
3677:
3674:
3672:
3669:
3668:
3664:
3659:
3656:
3654:
3651:
3649:
3646:
3644:
3641:
3639:
3636:
3634:
3631:
3629:
3626:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3606:
3605:
3603:
3594:
3581:
3578:
3575:
3564:
3561:
3557:
3554:
3548:
3542:
3539:
3534:
3524:
3521:
3517:
3514:
3508:
3505:
3501:
3498:
3494:
3491:
3488:
3483:
3477:
3474:
3468:
3463:
3459:
3453:
3449:
3442:
3438:
3427:
3423:
3417:
3414:
3409:
3396:
3388:
3374:
3364:
3347:
3344:
3323:
3298:
3294:
3285:
3281:
3260:
3257:
3254:
3243:
3239:
3235:
3230:
3226:
3219:
3216:
3212:
3206:
3196:
3193:
3189:
3186:
3180:
3177:
3173:
3170:
3166:
3163:
3160:
3155:
3149:
3146:
3140:
3135:
3131:
3125:
3121:
3114:
3110:
3099:
3095:
3091:
3078:
3070:
3056:
3046:
3040:
3038:
3033:
3029:
3024:
3004:
3001:
2997:
2994:
2988:
2985:
2981:
2975:
2969:
2966:
2961:
2957:
2952:
2947:
2940:
2933:
2930:
2925:
2921:
2911:
2907:
2903:
2900:
2897:
2885:
2880:
2876:
2873:
2869:
2852:
2849:
2846:
2842:
2832:
2828:
2824:
2816:
2813:
2808:
2804:
2796:
2793:
2789:
2786:
2782:
2776:
2763:
2755:
2741:
2731:
2726:
2724:
2705:
2702:
2699:
2696:
2693:
2682:
2663:
2653:
2650:
2645:
2642:
2638:
2613:
2610:
2606:
2603:
2596:
2591:
2586:
2583:
2579:
2576:
2572:
2567:
2562:
2556:
2553:
2547:
2542:
2538:
2531:
2528:
2505:
2495:
2492:
2487:
2484:
2480:
2470:
2467:
2463:
2460:
2454:
2451:
2447:
2433:
2430:
2427:
2423:
2413:
2410:
2406:
2401:
2397:
2392:
2379:
2371:
2357:
2347:
2343:
2339:
2335:
2331:
2326:
2313:
2306:
2303:
2299:
2296:
2290:
2287:
2283:
2280:
2276:
2273:
2270:
2267:
2264:
2261:
2257:
2254:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2201:
2193:
2165:
2142:
2139:
2136:
2133:
2130:
2107:
2101:
2098:
2095:
2087:
2083:
2075:
2072:
2069:
2064:
2060:
2054:
2049:
2045:
2032:
2029:
2026:
2022:
2018:
2005:
1997:
1983:
1973:
1969:
1965:
1960:
1947:
1941:
1936:
1931:
1927:
1922:
1916:
1911:
1906:
1903:
1899:
1896:
1892:
1887:
1882:
1877:
1872:
1869:
1865:
1862:
1858:
1853:
1848:
1843:
1838:
1835:
1831:
1828:
1824:
1818:
1813:
1800:
1792:
1778:
1768:
1765:expressed in
1749:
1746:
1739:
1735:
1732:
1724:
1719:
1717:
1701:
1678:
1675:
1672:
1669:
1666:
1660:
1635:
1628:
1622:
1614:
1605:
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1462:Newton kernel
1459:
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60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3718:
3714:
3704:
3695:
3689:
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2329:
2327:
1961:
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1451:
1111:
966:
881:
754:
356:is a linear
135:point source
126:
120:
105:
99:October 2012
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
1018:located at
765:Gauss's law
3682:References
751:Motivation
518:where the
69:newspapers
3735:0004-637X
3694:Jackson.
3558:−
3543:
3522:φ
3518:−
3515:φ
3509:
3489:−
3433:∞
3424:∫
3418:π
3389:−
3236:−
3217:−
3194:φ
3190:−
3187:φ
3181:
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3105:∞
3096:∫
3071:−
3002:φ
2998:−
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2926:−
2892:Γ
2874:−
2861:∞
2856:∞
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2814:−
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2756:−
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2646:−
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2442:∞
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2291:
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2277:
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2268:
2255:θ
2251:
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2230:
2219:given by
2166:γ
2143:φ
2137:θ
2102:γ
2099:
2038:∞
2023:∑
1998:−
1928:−
1900:−
1866:−
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1590:−
1527:−
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1477:∇
1402:ρ
1372:∫
1355:ϕ
1324:ϕ
1278:∇
1263:ε
1257:−
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1152:ϕ
1142:∇
1126:ε
1120:−
1085:−
1074:δ
1053:ρ
975:ϕ
945:ε
929:ρ
909:ϕ
899:∇
894:−
856:ε
840:ρ
822:⋅
818:∇
799:ϕ
795:∇
791:−
718:−
707:δ
674:−
663:δ
624:∇
447:∫
338:∇
216:∇
158:∇
3748:Category
3665:See also
3565:′
3525:′
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3397:′
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2006:′
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505:′
488:′
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1714:is the
234:is the
123:physics
83:scholar
3733:
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1464:" or "
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763:, and
743:, the
207:where
125:, the
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90:JSTOR
76:books
3731:ISSN
3336:and
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1249:for
145:, a
62:news
3723:doi
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3540:cos
3506:cos
3178:cos
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2274:sin
2265:sin
2248:cos
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2227:cos
2096:cos
1552:is
238:in
121:In
45:by
3750::
3729:.
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