MAGNETISM 233 Airty have discontinuities there, and they are equal in amount and of opposite sign. For an external point the potential is The immediate interpretation of this is that the potential is the same as that due to a normally magnetized layer on the surface of the body whose strength at dS is <f> ; in other words, qua external action, every lamellar magnet may bo replaced by a complex shell on its surface. There is, however, another way of looking at the result. Since A, B, C are derivable from a potential <f>, the difference between the values of <f> at any two points is simply the value of the line integral f (A.dx + ~Bdy + Cdz) along any path between those points. Hence if the tangential component of magnetization be given in direction and magnitude all over any surface, the value of <f>, a constant jrris, is given all over that surface. We conclude therefore that, if a body be lamellarly magnetized, and we know the tangential component of its magnetization all over its surface, its external action is deter mined ; l for a constant c added to ^ will simply add to the surface integral /yc C M Sector
- >oten-
- ial of
.amellar nagnet. which, being c times the whole solid angle subtended by the surface at any external point, vanishes. For another very interesting proof of this result, see Thomson, Reprint, p. 398 sq. The vector potential of a lamellar magnet may be expressed by means of the formula; F = <f<i) dy (40). These formula; furnish an immediate proof of the theorems of Thomson above stated. Poten- By means of the last of them, it has been shown in the article
- ial ELECTRICITY 2 that the vector potential of a simple magnetic shell
energy of can, as might be expected, be expressed by means of a line integral magnetic taken round its boundary ; in the same place it has also been shown ihell,&c. that the potential energy of such a shell in a magnetic field reduces to a similar line integral ; and that the mutual potential energy of two such shells reduces to a double line integral taken round their boundaries. Approxi- An important approximate expression for the potential of a uate magnet at a point P, whose distance r from some chosen point in
- xpres- the magnet is great compared with the greatest linear dimension of
>ion for the magnet, may be obtained as follows. Let the coordinates of P magnetic -with respect to the chosen point and any axes through it be , 77, ; poten- and let the coordinates of any point in the body referred to the same and axes be x, y, z. Also let D= | (|- let r = (f 2 + i) 2 + O, s = (z 2 +?/ + z 2 t = x + i,y+&. Then the potential U at (x, y, z) of a unit pole placed at ({, rj, ) is given by 1 t - 3/rV by a well-known theorem, where Uj, TJ 2 , U 3 , &c., are spherical harmonics of degrees 1, 2, 3, &c., in x, y, 2, and - 2, - 3, - 4, &c., i 1 It 1. C- Now, by the theorem of mutual potential energy, the potential V of the magnet at ({, TJ, is the potential energy "of the magnet in the field due to a unit pole at (, 77, C); hence by (17) V = dx , + V, (41), where Vj arises from Uj, V 2 from U 2 , and so on. Vj, V 2 , V 3 , &c., will be spherical harmonics in , j, ^ of the most general kind, involving essentially 3, 5, 7, ... 2i + l constants respectively, their degrees being - 2, - 3, ... -i respectively. These con stants will, however, depend in each case on a larger number of integrals taken throughout the magnetized body, thus the constants in V,- will depend upon f(i+l)(i + 2) integrals. 3 There is no diffi- 1 Sir W. Thomson, Reprint of Papers on Electricity and Magnetism, p. 502. 2 Vol. viii. p. 60.
- This fact is a further proof, if that were v;infed, that the magnetization of a
boitv cannot foe determined from its external action. culty in writing down these terms except the length of the formula . Putting we get (42), = Kw , fjfdd o = K?i L =fffAxdv , M -fffVajdv , N -fffCzdv = &c., K = &c. J (2L - M - c. &c It may be shown 4 that, in tho most general case, if we take tho axis of x parallel to the magnetic axis, and the origin at the point {(2L-M-N)/2K, K/K, Q/KJ-, 5 and turn the axes about an angle tan - 1 P/(M - N), tho above reduces to Y = 3 - + -JS - - * V &c. . . . (44). An interesting particular case is that in which the magnet is symmetrical with respect to the three coordinate planes. If wo take its axis to be in the axis of x, then, since all the integrals L, M, N, P, Q, R vanish, V 2 disappears, of the next set only >dv . (45) remain, and we get (46). The potential to the same degree of approximation of a positive and negative pole of strength /*, placed on the magnetic axis at distances +L and - L from the origin (centre of symmetry), is ~ 1 If we attempt now to find /j. and L, so that the two magnetic Ideal systems may be equivalent, we find different values for L for represen- different positions of the external point. If, however, the magnet tative be symmetrical about its axis, so that A.j = A 3 , then the expression magnet. for V reduces to V _ v -3 (47), wethenget2,uL = K,and2,uL 3 = 3(Ai- A 2 ), whence L 2 = 3(Aj-A 2 )/K. In other words, in the case of a magnet which is symmetrical about its axis and also about an equatorial plane, we can represent tho external action by means of a fixed ideal magnet, provided higher powers of the ratio of the greatest linear dimension of the magnet to the distance of tlie point considered than the fourth can be neglected. It is to be observed, however, that if A^<A 2 tho length of the ideal representative magnet will be imaginary. 6 A convergent series for tho mutual potential energy of two Series for magnets M and M may be obtained from the sextuple integral of mutual (19). Let the origin be a fixed point in M, and let the coor- potential dinatea of a fixed point in M with reference to a set of axes fixed energy, in be |, rj, ; further, let x, y, z and x , ?/ , z be the coordinates of any elements dv and dv in M and M , the axes being in the former case the system already indicated, in the latter a parallel system through ; then, if r denote ({ 3 + iJ a +C S )*i aml 5 i> 5 v 5 a stand for .* A A we have rf{ d-n dC where U n where W lt V 2 , W 3 , are spherical harmonics in |, 77, C of degrees - 2, - 3, - 4, &c. If we neglect all the terms except those of the first order, which amounts to supposing M and M infinitely small, we get , j , s ) JL If 1( 6. 2 be the angles between the axes of M and M and the line , , Y and / its -econdavy axes. It should be observed, however, tl.nt UiU is n "ot nc^aril -the .nld.llc poiut of the line joining the ^^ positive and negative magnetism. On this subject see apupei by IJeltr 1-otenx.iale Ma K netieo," Ann. <l. Matfm Wl. 6 C/. Kiecke in I ogg. Ann., p. 149, 1873 ; and Hied. An*., p. 8, 18
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