Charge and Conducting Sphere II

A charge is placed at a distance R from the center of a grounded conducting sphere of radius a < R (see Figure 1.1). Show that the force acting

Figure 1.1

on the charge is proportional to 1/R³ when R >> a and 1/(R – a)² = 1/δ² when R = a  δ, 0 < δ << a.

SOLUTION

We found the expression for the force acting on the charge in the presence of an insulated conducting sphere. For the grounded sphere, the potential on the surface of the sphere is zero, and therefore there is only one image charge

at a distance

from the center of the sphere. The force on can be computed as the force between it and the image charge e':

Now, if R >> a,

If R = a + δ, then

We may look at this case in a different way, for R = a + δ, the sphere looks like an infinite conducting plane. So the force should not differ much from the force between the charge and the image charge –e.