A uniformly magnetized and conducting sphere of radius R and total magnetic moment m=4πMR3/3 rotates about its magnetization axis with angular speed ω. In the steady state no current flows in the conductor. The motion is nonrelativistic; the sphere has no excess charge on it.
(a) By considering Ohm's law in the moving conductor, show that the motion induces an electric field and a uniform volume charge density in the conductor,
ρ=−mω/πc2R3.
(b) Because the sphere is electrically neutral, there is no monopole electric field outside. Use symmetry arguments to show that the lowest possible electric multipolarity is quadrupole. Show that only a quadrupole field exists outside and that the quadrupole moment tensor has nonvanishing components,
Q33=−4mωR2/3c2,Q11=Q22=−Q33/2.
(c) By considering the radial electric fields inside and outside the sphere, show that the necessary surface-charge density σ(θ) is
σ(θ)=4πR21⋅3c24mω⋅[1−25P2(cosθ)]
(d) The rotating sphere serves as a unipolar induction device if a stationary circuit is attached by a slip ring to the pole and a sliding contact to the equator. Show that the line integral of the electric field from the equator contact to the pole contact (by any path) is E=μ0mω/4πR.
[See Landau and Lifshitz, Electrodynamics of Continuous Media, p. 221, for an alternative discussion of this electromotive force.]