ps1sol1 - a given surface current K , the magnetic moment m...

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Supplement to Solution of Problem 3 of Problem Set 1 Besides the method shown in the posted solution, there are at least two other ways of Fnding the magnetic moment of the sphere. The Frst is simply to note that the magnetic Feld B outside the sphere is the sum of two parts: the applied Feld H 0 = H 0 ˆ z , and a dipolar Feld B dip = p B 0 a 3 cos θ 2 r 3 P . (1) This second term is precisely the magnetic Feld produced by a magnetic moment m = a 3 2 H 0 ˆ z (2) (Note that H and B are equal outside the sphere, in Gaussian units.) The second method is messier but more direct. The applied magnetic Feld induces a screening current K on the surface of the sphere. K is related to the discontinuity of B across the surface. Since B = 0 inside the sphere, K is given by K = c 4 π B 0 × ˆ n = c 4 π B 0 sin θ ˆ φ (3) where ˆ n is a unit vector normal to the sphere (i. e., in the radial direction), θ is the polar angle, and ˆ φ is a unit vector in the azimuthal direction. ±or
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Unformatted text preview: a given surface current K , the magnetic moment m is given by (in gaussian units) m = 1 2 c i r × K d 2 r, (4) where the integral is over the surface of the sphere. Now r × K = rK ˆ θ , where ˆ θ is a unit vector in the θ direction. However, only the z component of m is non-zero; the transverse parts cancel out. So we can replace ˆ θ by ( ˆ θ · ˆ z )ˆ z = − cos θ ˆ z , and m becomes m = − ˆ z B 1 2 c c 4 π i r cos θd 2 r. (5) Writing d 2 r = r 2 sin θdθdφ , and setting r = a, we get m = − ˆ z B a 3 8 π i π cos θ sin θdθ i π dφ. (6) 1 Carrying out the two integrals, we get m = − B a 3 2 ˆ z (7) as obtained by the other two methods. (Note that B = H .) 2...
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This note was uploaded on 07/28/2011 for the course PHYSICS 880.06 taught by Professor Stroud during the Fall '10 term at Ohio State.

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ps1sol1 - a given surface current K , the magnetic moment m...

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