- #1

- 242

- 37

- Homework Statement
- *The sphere is metal, so charges effectively lie on the surface.

In terms of the external charge q, radius of the sphere R and the distance d (distance between external charge and centre of sphere)

determine the following electrostatic energies:

a) the electrostatic energy of the interaction between charge q and the induced

charges on the sphere;

b) the electrostatic energy of the interaction among the induced charges on the

sphere;

c) the total electrostatic energy of the interaction in the system.

- Relevant Equations
- Method of image charges gives the following:

The metal sphere can be replaced with a charge q' = -q* R/d,

at a distance

d' = R^2/d from the centre of the sphere.

Energy can be found by integrating -Fdx.

Let us attempt part C first, which is to find the total energy of the entire system.

I can definitely find an expression for the force, as given by Coulomb's Law. However, why should I integrate this force from infinity to d, where d is the distance of the external charge to the centre of the sphere?

I was thinking that the upper limit of the integration should be d - d', which is the distance between the external charge and the image charge. OR, If you consider the limiting case where R --> infinity, the external charge is effectively facing a flat plate, and so the upper limit of integration should be d - R. Why is this wrong?

For part (a), the electrostatic energy between the external charge and the induced charges is apparently just the energy between the external charge and the image charge, as given by kqq'/(d-d'). Why is this true? Aren't the formulas for energy inapplicable for regions within the sphere, as the image charge only replicates conditions outside the sphere?

Any help would be greatly appreciated. Thank you!

Last edited: