1.Consider a point charge in front of an insulated, charged conducting sphere of radius R. If the sphere is to contain a charge Q, what is the potential outside the sphere?

Initially, assume that the sphere is grounded. If the charge q is at the position  with respect to the centre of the sphere, an image charge  located at  satisfies the required boundary conditions. In order that the sphere has a total charge Q, we now disconnect it from the ground and add  amount of charge to the sphere, which will be uniformly distributed over the surface. The potential at  is then given by,

2. A conducting sphere of radius R has a charge Q.  A charge q is located at a distance 3R from the centre of the sphere. Calculate the potential at a distance R/2 from the centre of the sphere along the line joining the centre with the charge q.

The image of q in the sphere is a charge  at  from the centre.  Since the sphere is a conductor all parts of it are equipotential.  As the overall charge is Q, it can be looked upon as an image charge  at R/3 which cancels the potential due to the charge q outside and a charge at the centre equal to . The potential of the sphere due to this is .

3. Two point charges q each are at a distance  d from each other. What should be the minimum radius R of a grounded sphere that can be put with its centre at the mid point of the line joining the two charges so that the mutual repulsion between the point charges is  compensated? Assume d >> R.

Each of the point charge gives an image charge within the sphere with charge  at a distance  from the centre towards the charge.

 

These two image charges, having opposite sign to that of the original charges, attract each of them. The distance of these image charges from either of the charges is . The net attraction due to these charges must cancel the repulsive force . Thus, we have,

Simplifying, we get,

Since b>> R, as a first approximation we can neglect all terms other than the first term on either sides of this equation, and get,  which gives

4. Two spheres, each of radius R contain identical charge Q. The spheres are separated by a negligible distance. Locate all image charges.

The problem is very similar to Problem 4 of the tutorial assignment with the difference that both the charges being +Q, the image charges alternate in sign.  Taking d=2R, we have, for image on the left sphere,

etc.  The total induced charge is .