Problem 3.65

A system consists of two concentric conducting spheres, with the inside sphere of radius $a$ carrying a positive charge $q_{1}$ What charge $q_{2}$ has to be deposited on the outside sphere of radius $b$ to reduce the potential of the inside sphere to zero? How does the potential $\varphi$ depend in this case on a distance $r$ from the centre of the system? Draw the approximate plot of this dependence.

q_{2}=-\frac{b}{a} q_{1} ; \quad \varphi=\frac{q_{1}}{4 \pi \varepsilon_{0}} \times\left\{\begin{array}{l} 1 / r-1 / a \text { if } a \leqslant r \leqslant b \\ (1-b / a) r \text { if } r \geqslant b \end{array}\right.

Conductors and Dielectrics in an Electric Field