Applying Poisson Equation for Electrostatic Potential in a Spherical Shell

[KFC]

In many book I read, problems for electrostatic potential always lead to solving Poisson equation. I saw a problem about a spherical shell carrying some amount of charges uniformly on the surface with density $$\rho$$, and then someone put a small patch on the sphere. The patch is then made a constant potential $$V_0$$ on it and everywhere else on the shell has zero potential. I would like to find the potential everywhere inside the spherical shell. In this case, how can I apply Poisson equation to do that?

 

[Thaakisfox]

The Poisson equation always holds, and the given configuration is encoded in the boundary conditions. To solve a PDE you always have to give the boundary conditions. So in this particular problem, due to the spherical symmetry its best to operate in spherical coordinates, and then describe the given boundary conditions.

 

[JANm]

The Poisson equation always holds, and the given configuration is encoded in the boundary conditions. To solve a PDE you always have to give the boundary conditions. So in this particular problem, due to the spherical symmetry its best to operate in spherical coordinates, and then describe the given boundary conditions.

Do you mean the size of the patch compared to te radius of the sphere is to be known?
If in spherical co"ordinates north pole could be an answer. Would the use of Legendre polynomals work?

 

[Redbelly98]

It could be done using numerical methods, for example by relaxation methods. I'm pretty sure the book "Numerical Recipes in C" covers this, if you want to look up the details.