Question

A solid conducting sphere of radius R_{1} = 10 cm is placed concentrically inside a con-ducting spherical shell with inner radius R_{2} = 20 cm and outer radius R_{3} = 30 cm. The inner solid sphere is given a charge of +6 µC, and the outer shell is grounded (its potential is set to zero). Find the potential of the inner shell with respect to infinty.

Answer 1

270 kV

Answer 2

1. Identify the charge distribution: $+Q_1$ on the inner sphere, $-Q_1$ on the inner surface of the shell, $0$ on the outer surface of the grounded shell.

2. Determine the electric field: $E=0$ for $r<R_1$, $E = kQ_1/r^2$ for $R_1 < r < R_2$, $E=0$ for $r>R_2$.

3. Calculate the potential of the inner sphere ($V(R_1)$) relative to infinity by integrating $V(R_1) = -\int_{\infty}^{R_1} E dr$.

4. The integral simplifies to $V(R_1) = -\int_{R_2}^{R_1} (kQ_1/r^2) dr = kQ_1(1/R_1 - 1/R_2)$.

5. Substitute the given values $Q_1 = 6 \times 10^{-6}$ C, $R_1 = 0.1$ m, $R_2 = 0.2$ m, and $k = 9 \times 10^9$ N m$^2$/C$^2$ to get $V(R_1) = 270$ kV.