# What is the meaning of a constant magnetic scalar potential?

Let a spherical shell of inner radius $a$ and outer radius $b$ have a uniform magnetization $\mathbf{M}=M\,\hat{\mathbf{z}}$,

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I've found that the magnetic scalar potential $\varphi^\star$ is given by \begin{align}\varphi^\star(r<a)&=\text{constant}\\ \varphi^\star(a\leq{r}\leq{b})&=\frac{M}{3}\left(r-\frac{a^3}{r^2}\right)\cos\theta\\ \varphi^\star(r>b)&=\frac{M}{3r^2}\left(b^3-a^3\right)\cos\theta\end{align} where $\theta$ is the polar angle. I guess my calculations are ok because when $a\to0$, the results for a sphere with uniform magnetization $\mathbf{M}=M\,\hat{\mathbf{z}}$ are recovered. Anyhow I got confused by the constant potential inside the shell, how can I interpret it?

If $\mathbf{B}=\mu_0(\mathbf{H}+\mathbf{M})$ and $\mathbf{H}=-\nabla\varphi^\star$, then there would be a uniform magnetic field inside the shell and still if I put a charged particle there, also a force upon it, so how is this potential different from any non-constant potential $\varphi^\star\propto{z}$, as the one inside the complete sphere (i.e. when $a\to0$)?

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A magnetic potential $\Phi_M$ tells you how to calculate the ${\bf H}$-field. If $\Phi_M$ is constant then it must be that ${\bf H}$ is zero. In a region where ${\bf M}$ is also zero then you must conclude finally that ${\bf B}$ is zero. When you say