Chapter 8, Problem 8.45P

1. Beginning with the definition, of the scalar magnetic potential. H = ∇Ă—V, and using Eq. (25), show that in a region having permanent magnetization M and zero current a Poisson equation for magnetic potential can be developed:

${\nabla }^2{V}_m=\frac{p_x}{p_y}$

where Pm = μ₀ V. M is an equivalent magnetic charge density. What happens when M is uniform? (b) Using the definition of the vector magnetic potential, B = V x A. and Eq. (25). How that when a permanent magnetization exists. and with zero current:

$\nabla \times \nabla \times A={\mu }_0{J}_{rq}$

where Jeq = V x M in analogy to Eq. (57) in Chapter 7.