Question

How can I prove that the tangential component of electrostatic field is continuous from one side of a charged surface to another? (By that I mean prove, not foolish assumptions.)

Answer 1

it is suffice to start from this equation of Maxweel:
\[{\text{curl }}{\mathbf{E}} = 0\]
which holds afor a static electric field. Next we have to consider this loop:

so we get this equation:
\[{E_{T1}}dl - {E_{T2}}dl = 0\]
where \(dl\) is the length of the longest dimensions of the rectangle. We can neglect the contributions from the littlest dimensions