Question

How do I calculate the surface potential of a conductor given a charge distribution?

Answer 1

So how do I calculate it. I have tried working out the conventional definition but since there is a charge density at the point we want to calculate the potential at, it turns out to be infinity. Now, i dont know how to calculate the sum of all other potentials of points except that of the point of calculation in an integral. Ive been stuck for days trying to do this.

Answer 2

HEY MATE ⭐⭐⭐⭐
HERE'S THE ANSWER ✌
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any conductor without cavities which contain charges(you can tweak this approach to satisfy those cases). Let us have a conducting surface SS enclosing a volume VV. The charges exist only at SS. Let the charge distribution be σ(r,θ,ϕ)σ(r,θ,ϕ).
Consider the following definition of potential at a point P∈VP∈V:

U(P)=∫Sσ(r,θ,ϕ).r.dθ.dϕ4πϵ0U(P)=∫Sσ(r,θ,ϕ).r.dθ.dϕ4πϵ0.

Using this formula appropriately and exploiting some other symmetries, you can find the potential an some interior point PP.
Since the metal is an equipotential, and since the potential suffers no discontinuity while crossing a charged surface, the same is the potential of the surface.
Or:

U(P)=U(X)U(P)=U(X) for any X∈S

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HOPE IT HELPS ☺☺☺☺☺