The question of how a receiving antenna works has been asked on this site before, such as here How does a receiving antenna get an induced electric current? and here How does a receiving antenna work?. I understand the basic principle that the external EM field from the transmitting antenna causes electrons to move in the receiving antenna, creating a current.
My question though is in the title. How can an external electric field, as in the form of a radio or other EM wave, induce a current in a wire when the $E$ field is always zero in there? In terms of physical laws and math how can I calculate the current as a function of time if I know the external fields as a function of time?
I wanted to try to calculate the potential difference between two points of a wire from the external field using
$$\varepsilon=\int_{\text{start point}}^{\text{end point}}\mathbf{E}\cdot\text{d}\mathbf{l}$$
but that assumes that the electric field in the wire is as it would be if the wire weren't there and the waves were propagating through vacuum...