(a) Starting from the force equation (5.12) and the fact that a magnetization M inside a volume V bounded by a surface S is equivalent to a volume current density J_M = (Ñ × M) and a surface current density (M × n), show that in the absence of macroscopic conduction currents the total magnetic force on the body can be written

F = – ∫_v (Ñ • M)B_e d³x + ∫_s(M • n)B_eda

Where B_e is the applied magnetic induction (not including that of the body in question). The force is now expressed in terms of the effective charge densities ρ_M and σ_M. If the distribution of magnetization is not discontinuous, the surface can be at infinity and the force given by just the volume integral.

(b) A sphere of radius R with uniform magnetization has its center at the origin of coordinates and its direction of magnetization making spherical angles θ₀, ф₀. If the external magnetic field is the same as in Problem 5.11, use the expression of part a to evaluate the components of the force acting on the sphere.