Classical Electrodynamics · Problem 5.21

Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson

Physics — E&M · permanent-magnetization-energy

Problem

A magnetostatic field is due entirely to a localized distribution of permanent magnetization. (a) Show that ∫B⋅H d3x=0\int \mathbf{B} \cdot \mathbf{H}\, d^3x = 0 provided the integral is taken over all space. (b) From the potential energy (5.72) of a dipole in an external field, show that for a continuous distribution of permanent magnetization the magnetostatic energy can be written W=μ02∫H⋅H d3x=−μ02∫M⋅H d3xW = \frac{\mu_0}{2}\int \mathbf{H} \cdot \mathbf{H}\, d^3x = -\frac{\mu_0}{2}\int \mathbf{M} \cdot \mathbf{H}\, d^3x apart from an additive constant, which is independent of the orientation or position of the various constituent magnetized bodies.

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