Suppose that I have a perfect electrical conductor ($B=0$ inside conductor) in free space with a known magnetic field $\mathbf{B_s}$ outside of it, and no electric field. If I transform to a frame of reference moving non-relativistically with velocity $\mathbf{v_{s'}}$, I obtain an electric field $\mathbf{E_{s'}} = \mathbf{v_{s'}} \times \mathbf{B_s}$ outside the conductor. The boundary conditions imply that there is a surface charge density $\sigma$ in this frame on the conductor.
In general, \begin{equation}
\sigma = \epsilon \mathbf{E}.\mathbf{\hat{n}}|_{\rm surface} \tag{1}
\end{equation}
where $\mathbf{\hat{n}}$ is the normal unit vector to the surface of the conductor.
My question(s) :
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Whether the electric field vector $\mathbf{E}$ in $Eq. (1)$ is just the vector $\mathbf{E_{s'}}$ or the composite (or total) electric field vector $\mathbf{E_{s'}} + \mathbf{E_{c'}}$, where $\mathbf{E_{c'}}$ is the secondary field due to the surface charges?
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In both cases, how to calculate the total electric field outside the conductor and the surface charge, since I do not know $\mathbf{E_{c'}}$ and $\sigma$ apriori?
Best Answer
Your problem is just the standard problem of a conducting sphere which is solved n every EM textbook for a uniform E field. If the original B field is not uniform, a Legendre polynomial, spherical harmonic expansion is necessary. In any event the total field is the one that counts.