Question

Using the law of cosines, show that Eq. $\mathbf{3}\mathbf{.}\mathbf{17}$can be written as follows:

Whereandare the usual spherical polar coordinates, with the z axis along the

line through. In this form, it is obvious thaton the sphere,.

1. Find the induced surface charge on the sphere, as a function of. Integrate this to get the total induced charge. (What should it be?)

2. Calculate the energy of this configuration.

Short answer

1. The potential is zero, when.

1. The induces charge surface is.

2. The energy of this configuration is.

Step-by-step solution

Define functions

Write the expression for potential using law of cosines.

…… (1)

Here, and are the spherical polar coordinates, is the charge, is the permittivity for the free space and is the radius of the sphere.