# How to define the b.c.s for the EM field of a perfectly contucting surface?

If you have a 3 dimensional perfectly conducting body the conditions at the boundary for the EM field is as follows:

$$\boldsymbol{E}_{\parallel} = 0, B_{\perp} = 0, E_{\perp} = \frac{\sigma}{\epsilon_0}, \boldsymbol{B}_{\parallel} = \mu_0 \boldsymbol{j} \times \boldsymbol{\hat{n}}$$

where $$\sigma$$ and $$\boldsymbol{j}$$ are the surface charge and current density at the boundary respectively. $$\parallel$$ / $$\perp$$ denotes the component parallel/perpendicular to the surface of the body, with normal vector $$\boldsymbol{\hat{n}}$$. The derivations of these boundary conditions comes from assuming that the EM field vanishes inside the conductor and using Maxwell's equations. But when one assume that the body is a planar conducting surface, there is no longer a finite volume in which the EM field vanish. If you for instance take a conducting box, and then study the limit where the height goes to zero, then the top and bottom surface charge and current densities can no longer be separated, and they are unified in one overall surface carge and current density. Can one still assume that these conditions are true, or should they be modified in some way? Could it be possible that these conditions are instead only valid for the differences between the fields on each side of the surface?

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