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Continuity of perpendicular component of the magnetic field

Maxwell's fourth equation states that div B = 0. Use this law to show that the
component of B perpendicular to the surface is continuous (i.e., the same just below
and just above the surface), regardless of the surface current Js(r, t).

Solution Preview

The volume integral of div ยท B over a box equals, by Gauss' theorem the integral of B dot dO over the surface of the box. Here the surface element dO points in the direction of the outward normal. So, we have:

Maxwell's fourth equation thus implies:

Integral of B dot dO = 0

Take a box consisting of two sides of area A ...

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