Confusion about $B$-field being expressed in terms of magnetic scalar potential using auxiliary field $H$

Question

Potentially silly question, sorry! I understand that the $B$-field can be expressed as the curl of $A$, where $A$ is the magnetic vector potential, and so $B$ has no divergence. The auxiliary field $H$ however, is not necessarily divergenceless, and may have curl also (if free currents are present). I am confused about the following:

If there are no free currents present, then: $$ \nabla \times H = \vec{0} \rightarrow H = -\nabla \phi_m$$

where $\phi_m$ is a scalar magnetic potential. In linear media, $ \vec{B} = \mu \vec{H} $. Therefore:$$ \vec{B} = -\mu \nabla \phi_m$$ I am confused by this equation, because if B only has curl, how can it be expressed in terms of the grad of a scalar potential like this? Doesn't this expression imply that B has divergence, which is not possible? I am not sure where the error is coming from in this line of reasoning. Could someone please clarify?

Answer 1

Since $\nabla \times \nabla \phi=0$ whenever $\nabla \phi$ is differentiable $$\nabla \times \mathbf B = \nabla\times (\mu \nabla \phi)\\ =\mu \nabla \times \nabla \phi+\nabla\mu \times \nabla \phi\\ =\nabla\mu \times \nabla \phi$$

$$\nabla \cdot \mathbf B = \nabla\cdot(\mu \nabla \phi)\\ =\mu \nabla^2\phi+\nabla\mu \cdot \nabla \phi\\$$

If $\mu$ is constant in a region then there $\nabla \mu = 0$ and thus $\nabla \cdot \mathbf B =\mu \nabla^2\phi =0$ and we get the special case of Laplace's equation for the magnetic potential $\nabla ^2 \phi=0.$ Otherwise, in general, the only thing we know is that $\mu \nabla^2\phi+\nabla\mu \cdot \nabla \phi=0.$

Even if $\mu$ is "piece-wise" constant, that is $\nabla \mu = 0$ in discrete regions, you have at the boundary of those regions jump conditions: the normal component $B_n$ of $\mathbf B$ is continuous and the tangential component $H_t$ of $\mathbf H$ is continuous. These continuity conditions can be interpreted as surface sources generating a non-zero field within the constant $\mu$ regions.