In the proof for the uniqueness theorem for potential in electrostatics, we assume that the volume V contains only volume charge distributions and hence the usage of 'ρ' as volume charge density. I wish to know how this theorem is valid even for regions containing surface charge distributions, where 'ρ' wouldn't be very logical to use in regions where charge is distributed over surfaces. Are there any additional boundary conditions to be satisfied in such situations? Is there a more general proof involving surface charge densities too?
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$\begingroup$ If you want to prove uniqueness only, by linearity it is enough to prove it for 0 charge distribution. $\endgroup$– Peter KravchukCommented Jun 8 at 8:13
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For surface charge, $\rho$ could still be used, for instance: $\rho=\delta(x\!−\!x_0)\,f(y,z)$ describes a surface charge distribution on the surface $x=x_0$, with variation $f(y,z)$ in the other two directions.
Of course physically speaking a surface charge layer always has some finite thickness so that would avoid the whole problem, but still it can be done with $\delta$-functions, which often is mathematically more convenient than to keep track of this very small layer thickness.
answered Jun 8 at 8:10