Show that the formula for the magnetic dipole moment derived in Example 11.1,

is consistent with the spherical multipole expansion of the vector potential derived in Section 11.4,

Transcribed Image Text:

Example 11.1 Let B(r) be the magnetic field produced by a current density j(r) that lies entirely inside a spherical volume V of radius R. Show that the magnetic moment of j(r) is 3 2μ0 m = Therefore, This problem is the magnetic analog of Example 4.1, which expressed the electric dipole moment of a charge distribution using a spherical average of the electric field produced by the charge. Solution: Assume first that j(r) does not lie entirely inside V. If we place the origin of coordinates at the center of V, the Biot-Savart law (10.15) gives d³r B(r)= Ho fa'r == E(r')= 4π V [a³r B = [ @³rs(³) × [d³r} The r integral is exactly the "electric" field E(r) due to a uniform charge density p(r) = 4.7 €0. From Gauss' law or otherwise, this is pr 4π 3€0 = pV 4 €0 d³r B(r). = = [d²rB®) = 1 [d²³²°*° x1(r^)=; fd³rr - 3V r' R. plo 4л Ja r'>R d³rj(r) x r r3