Intro:
In magnetostatics, working in c.g.s, $\vec{H}=\vec{B}-\frac{4\pi}c \vec{M}$.
We also have $$\vec{\nabla}\times\vec{B}=\frac{4\pi}c\vec{j}$$ $$ \vec{\nabla}\times\vec{H}=\frac{4\pi}c\vec{j}_f$$ $$ \vec{\nabla}\times\vec{M}=\frac{1}c\vec{j}_M$$ $$\vec{\nabla}\cdot\vec{B}=0$$
When $\vec{j}_f=0$, we have $ \vec{\nabla}\times\vec{H}=0$ so we can define a scalar magnetic potential, $\Psi_M$, fulfilling $\vec{H}=-\vec{\nabla}\Psi_M$.
Now, $$\vec{\nabla}\cdot\vec{H}=\vec{\nabla}\cdot\vec{B}-\frac{4\pi}c\vec{\nabla}\cdot\vec{M}=-\frac{4\pi}c\vec{\nabla}\cdot\vec{M}$$ or $$\nabla^2 \Psi_M=\frac{4\pi}c\vec{\nabla}\cdot\vec{M}$$ So I understand that only when $\vec{M}$ is constant, we get the Laplace equation which we like and know how to solve.
My problem:
Say we have a ball with radius $a$ and permeability $\mu$ in an external magnetic field $\vec{B}=B_0\hat{z}$, and we want to find $\vec{B}$ and $\vec{H}$ everywhere.
The solution in my study material uses the Laplace equation, with the magnetic potential as defined above.
However, there is in fact a magnetization $\vec{M}$ induced in the ball due to the external magnetic field. How do I know, before solving anything, that it is constant so the use of Laplace equation is justified?
Or in general, when is the use of Laplace equation justified as we come to solve a problem in magnetostatics? How can we predict $\vec{M}$ is constant before finding it rigorously?
