What is the Potential on a Spherical Shell Due to an External Point Charge?

[maverick280857]

Potential at ANY point on a spherical shell due to external point charge

Hi everyone

Here is another problem:

A point charge q is placed at a distance of r from the center of an uncharged conducting sphere of radius R (< r). Find the potential at any point on the sphere.

I know the answer is
$$\frac{1}{4\pi\epsilon_0}\frac{q}{r}$$
but I want to do it rigourously. Any suggestions?

Thanks,

Vivek

 

[maverick280857]

There is a somewhat similar problem on Page 124 of Classical Electrodynamics by Griffiths. It uses the method of images. The method used to "construct" an image charge is particularly interesting. I would be grateful if someone could point a resource on the internet (or a book reference) where I can learn how to apply the method of images to relatively simple problems.

I have not had any hardcore experience with mathematics of the kind required for a rigourous treatment (PDEs, laguerre/legendre polynomials, etc.) so a relatively simplified treatment would be appreciated (as I have little time to read at present).

EDIT: (cf Page 115 Griffths): The value of V at a point P is the average value of V over a spherical surface of radius R centered at P:

$$V(P) = \frac{1}{4\pi R^2}\oint_{sphere}V da$$

Am I right in thinking that this and the example (consequence) mentioned below answer my original question?

Thanks
Cheers
vivek

 

[Crosson]

This is the definition of electrostatic potential:

$$V(P) = -\int_{\infty}^P \vec{E} \cdot \vec{ds}$$

Since you know E everywhere outside the sphere (using the integral form of Gauss's law), this integral is easy to compute.

If you would instead prefer, solve laplace's equation in spherical coordinates with angular symmetry i.e. only the dr terms are non zero.