A point charge $q$ is placed between two parallel grounded conducting planes a distance $d$ apart.

1 The point charge $q$ a distance $a$ above the lower plane and a distance $b$ below the upper conductor has symmetrically located image charges. However, each image charge itself has an image in the opposite conductor. Show that an infinite number of image charges are necessary. What are the locations of these image charges?

2 Show that the total charge on each conductor cannot be found by this method as the resulting series is divergent.

Now consider a point charge $q$, a radial distance ${R}_{{0}}$ from the center of two concentric grounded conducting spheres of radii $R_{1}$ and $R_{2}$.

3 Show that an infinite number of image charges in each sphere are necessary where, if we denote the $n$-th image charge in the smaller sphere as $q_{n}$ a distance $b_{n}$ from the center and the $n$th image charge in the outer sphere as $q_{n}^{\prime}$ a distance $b_{n}^{\prime}$ from the center, then\begin{aligned}
& q_{n+1}=-\frac{R_{1}}{b_{n}^{\prime}} q_{n}^{\prime}, &\quad& q_{n+1}^{\prime}=-\frac{R_{2}}{b_{n}} q_{n} \\
& b_{n+1}=\frac{R_{1}^{2}}{b_{n}^{\prime}}, &\quad& b_{n+1}^{\prime}=\frac{R_{2}^{2}}{b_{n}}
\end{aligned}

4 Show that the equations in 3 can be simplified to\begin{gathered}
q_{n+1}-q_{n-1}\left(\frac{R_{1}}{R_{2}}\right)=0 \\
b_{n+1}-b_{n-1}\left(\frac{R_{1}}{R_{2}}\right)^{2}=0
\end{gathered}

5 Try power-law solutions$$ q_{n}=A \lambda^{n}, \quad b_{n}=B \alpha^{n} $$and find the characteristic values of $\lambda$ and $\alpha$ that satisfy the equations in 4.

6 Taking a linear combination of the solutions in 5, evaluate the unknown amplitude coefficients by substituting in values for $n=1$ and $n=2$. What are all the $q_{n}$ and $b_{n}$?

7 What is the total charge induced on the inner sphere?

Hint: $\displaystyle\sum_{n=1}^{\infty} a^{n}=a /(1-a)$ for $a<1$.

8 Using the solutions of 6 with the difference relations of 3, find $q_{n}^{\prime}$ and $b_{n}^{\prime}$.

9 Show that $\displaystyle\sum_{n=1}^{\infty} q_{n}^{\prime}$ is not a convergent series so that the total charge on the outer sphere cannot be found by this method.

10 Why must the total induced charge on both spheres be $-q$? What then is the total induced charge on the outer sphere?

11 Returning to our original problem in 1 and 2 of a point charge between parallel planes, let the radii of the spheres approach infinity such that the distances $$ d=R_{2}-R_{1}, \quad a=R_{2}-R_{0}, \quad b=R_{0}-R_{1} $$ remains finite. What is the total charge induced on each plane conductor?