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.)

Question

michele_laino · Answer

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:
|dw:1465322175485:dw|
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