Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson
Physics — E&M · magnetized-body-force
Problem
(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 JM=(∇×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(∇⋅M)Bed3x+∫S(M⋅n)Beda
where Be 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 ρM and σM. 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, ϕ0. 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.