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 speed of rotation is slow compared to the speed of light and the sphere has no excess charge on it. a) (5pts) Consider the Lorentz force due the magnetic field inside the sphere on a mobile charge carrier q inside the sphere. This will tend to push the charge outward. In order for there not to be any net current, there must be a compensating electric field. Find this electric field. b) (5pts) There is another way to find this electric field. In the frame of reference rotating along with the sphere, there is only a magnetic field B but no Lorentz force or corresponding electric field because in that frame the sphere is at rest. However, what is the electric field in the laboratory frame? c) (5pts) Show that this electric field corresponds to a constant charge density ρ = −mω/(πc2R3 ). d) (5pts) Because the sphere is electrically neutral, so the constant charge density of part (c) inside the sphere must be compensated by a surface charge density. To find this let’s first find what type of electric field can be present outside the rotating sphere. Use symmetry arguments to explain why there is no monopole electric field or dipole field and why the lowest order field must an azimuthal quadrupole field. The electrostatic potential produced by this quadrupole outside the sphere is thus proportional to Y20(θ, φ). Find the corresponding radial and tangential θ-component of the electric field outside. Matching it to the field inside, find the quadrupole moment. e) (5pts) Usin the discontinuity in the normal component of the electric field, find the surface charge density. f) (5pts) The tangential component of the electric field on the surface, means that there is a net electromotive force (EMF) between the poles and the equator of the sphere. Find this induced EMF.