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Consider the boundary between two magnetic materials with different relative permeabilities $\mu$. Using a small Gaussian pillbox at the surface and $\nabla \cdot \textbf{B} = 0 $, we can show that the normal component of the $\textbf{B}$-field is continuous across the boundary. However, what are the conditions for the $\textbf{H}$-field?

If we use the fact $\textbf{B} = \mu_0 \mu \textbf{H}$ surely we must conclude that the normal components of the $\textbf{H}$-field are discontinuous (since $\mu$ changes across the boundary). However, using $\nabla \cdot \textbf{H} = -\nabla \cdot \textbf{M}$ we find that the normal components of the $\textbf{H}$-field can be continuous if the normal components of $\textbf{M}$ are. How do we reconcile these two statements?

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The relation $B = \mu_0 \mu_r H$ is not a "fact", but an approximation that is valid for many materials over a wide range of conditions: so-called "linear magnetic media". But in a linear magnetic medium, we also have $M = \chi_m H = (\mu_r - 1)H$. So you are not free to set up a scenario where $M_\perp$ is continuous across the boundary, and the contradiction cannot arise. (The only exception is the case where $B_\perp = H_\perp = M_\perp = 0$.)

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  • $\begingroup$ So $B$ is determined first and we cannot choose what happens to $M$ (for linear media)? $\endgroup$
    – user246795
    Commented Mar 31, 2022 at 18:37
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    $\begingroup$ @user246795 Exactly, you do not have control over $M$ in a linear magnetic medium other than by controlling $B$. If you somehow had the ability to manipulate $M$ directly by manipulating the medium at a microscopic level, then $M$ would quickly just return to its equilibrium value of $\chi_m H$. $\endgroup$
    – Brian Bi
    Commented Mar 31, 2022 at 18:42
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Analogous to the "small pillbox" for Gauss' law, you use a small stripe area that overlaps the interface between the two materials for Ampere's law.

In the first case, the parts of the pillbox surface, that cross the material interface, are made smaller and smaller, so that their area integral over B does not contribute in the limit, and only the normal components of $B$ remain, and hence, have to be continuous.

In the second case also, the parts of the outline, that cross the material interface, are made smaller and smaller, so that only the line parts tangential to the interface of the materials remain, requiring that the tangential parts of $H$ must be continuous.

The only thing that is marginally different is, that you have the additional freedom of choosing the orientation of the stripe. That is of course necessary because there are two tangential components of $H$ while there is only one normal component of $B$.

So, there is no contradiction between the normal components of $H$ being discontinuous, and the tangential components of $H$ being continuous. It is simply an expression of the fact that the behavior of the normal component of $H$ at the interface does not follow from Maxwell's equations, but from the material law.

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"the H-field can be continuous if the normal components of M are." is true, and H is continuous where M is continuous, but M is discontinuous at the end of a magnet.

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