A point charge $Q$ is a distance $D$ above a ground plane. Directly below is the center of a small conducting sphere of radius $R$ that rests on the plane.

1 Find the first image charges and their positions in the sphere and in the plane.

2 Now find the next image of each induced in the other. Show that two sets of image charges are induced on the sphere where each obey the difference equations:$$
q_{n+1}=\frac{q_{n} R}{2 R-b_{n}}, \quad b_{n+1}=\frac{R^{2}}{2 R-b_{n}}
$$

3 Eliminating the $b_{n}$, show that the governing difference equation is$$ \frac{1}{q_{n+1}}-\frac{2}{q_{n}}+\frac{1}{q_{n-1}}=0 $$Guess solutions of the form$$ P_{n}=1 / q_{n}=A \lambda^{n} $$and find the allowed values of $\lambda$ that satisfy the difference equation.

Hint: For double roots of $\lambda$ the total solution is of the form $P_{n}=\left(A_{1}+A_{2} n\right) \lambda^{n}$.

4 Find all the image charges and their positions in the sphere and in the plane.

5 Write the total charge induced on the sphere in the form$$
q_{T}=\sum_{n=1}^{\infty} \frac{A}{1-a n^{2}}
$$What are $A$ and $a$?

6 We wish to generalize this problem to that of a sphere resting on the ground plane with an applied field $\vec{E}=-E_{0}\hat z$ at infinity. What must the ratio $Q / D^{2}$ be, such that as $Q$ and $D$ become infinite the field far from the sphere in the $\theta=\pi / 2$ plane is $-E_{0} \hat z$?

7 In this limit what is the total charge induced on the sphere?

Hint:$$\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\pi^{2} / 6.$$