If J f = 0 everywhere, the curl of H vanishes (Eq. 6.19), and we can express H as the gr...

If J _f = 0 everywhere, the curl of H vanishes (Eq. 6.19), and we can express H as the gradient of a scalar potential W:

According to Eq. 6.23, then,

so W obeys Poisson’s equation, with ∇ ·M as the “source.” This opens up all the machinery of Chapter 3. As an example, find the field inside a uniformly magnetized sphere (Ex. 6.1) by separation of variables. [Hint: ∇ ·M = 0 everywhere except at the surface (r = R), so W satisfies Laplace’s equation in the regions r < R and r > R; use Eq. 3.65, and from Eq. 6.24 figure out the appropriate boundary condition on W.]

reference 3.65

Reference 6.24