Question

In the arrangement shown in figure, three concentric conducting shells are shown. The charge on the shell of radius $b$ is $q_0$. If the innermost and outermost shells are connected to the earth, find their charge densities and the potentials on the shell of radius $b$ in terms of $a$ and $q_0$. Given that $a:b:c=1:2:4$.

• A. ${\sigma }_{inner}=\frac{q_0}{12\pi a^2},{\sigma }_{outer}=\frac{q_0}{36\pi a^2},V_0=\frac{q_0}{12\pi {\epsilon }_{0}a}$
• B. ${\sigma }_{inner}=\frac{q_0}{12\pi a^2},{\sigma }_{outer}=\frac{q_0}{36\pi a^2},V_0=\frac{q_0}{24\pi {\epsilon }_{0}a}$
• C. ${\sigma }_{inner}=\frac{q_0}{12\pi a^2},{\sigma }_{outer}=\frac{q_0}{16\pi a^2},V_0=\frac{q_0}{36\pi {\epsilon }_{0}a}$
• D. ${\sigma }_{inner}=\frac{q_0}{12\pi a^2},{\sigma }_{outer}=\frac{q_0}{96\pi a^2},V_0=\frac{q_0}{24\pi {\epsilon }_{0}a}$

Answer 1

The correct option is D ${\sigma }_{inner}=\frac{q_0}{12\pi a^2},{\sigma }_{outer}=\frac{q_0}{96\pi a^2},V_0=\frac{q_0}{24\pi {\epsilon }_{0}a}$

The charge on the shell of radius $b=q_0$

relevant equations $V=\frac{K_q}{r}=\frac{{Kq}_0}{r}$

${\sigma }_{inner}=\frac{q}{{ϵ}_0}$ (using Gauss's law)

$=\frac{q}{4\pi {ϵ}_0\times 3}\times \frac{1}{a^2}$ $K=3$

$=\frac{q_0}{12\pi a^2}$

${\sigma }_{outer}=\frac{q_0}{4\pi a^2}\times \frac{1}{24}$ $K=24$

$=\frac{q_0}{96\pi a^2}$

$V_0=\frac{q_0}{4\pi {ϵ}_0}\frac{1}{\sigma \times a}$ ($\because$ $K=6$)

$=\frac{q_0}{24\pi {ϵ}_{0}a}$