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I've seen in several places (such as here) that when studying normal incidence of a plane EM wave on a dielectric the boundary condition

$$\mathbf{H}_{t1}=\mathbf{H}_{t2}$$

is applied. In words, the tangential component of $\mathbf{H}$ has to be conserved.

However, as far as I know, that is not true, as that boundary condition for the magnetic field depends on whether there is a surface density current in the boundary:

$$\mathbf{H}_{t1}-\mathbf{H}_{t2}=\mathbf{K}$$

If the surface current $\mathbf{K}$ is $0$, then the first equality is true. But why is the surface current assumed to be $0$?

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1 Answer 1

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$\mathbf{K}$ here is the free surface current that is already there on the surface as opposed to bound current induced by the incident field. Since there is no reason to expect a source for such a free current especially at optical frequencies we neglect it. This would not be the case if we were considering EM waves at the AC frequency reflecting from a current carrying conductor for example.

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