Question

A conducting sphere $A$ of radius a, with charge $Q$, is placed concentrically inside a conducting shell $B$ of radius b. $B$ is earthed. $C$ is the common center of $A$ and $B$. Study the following statements.

Which of the following statements are correct?

i. The potential at a distance $r$ from $C$, where $a\le r\le b,is\frac{1}{4\pi {\epsilon }_0}\left(\frac{Q}{r}\right)$

ii. The potential difference between $A$ and $B$ is $\frac{1}{4\pi {\epsilon }_0}Q(\frac{1}{a}-\frac{1}{b})$

iii. The potential at a distance $r$ from $C$, where $a\le r\le b,is\frac{1}{4\pi {\epsilon }_0}Q(\frac{1}{a}-\frac{1}{b})$

• A. Only (i) and (ii)
• B. Only (ii) and (iii)
• C. Only (i) and (iii)
• D. All the above
  

Answer 1

The correct option is B Only (ii) and (iii)

This system like a spherical shell capacitor. The charge $-Q$ will appear on the inner surface such that the charge inside the shell B is zero.

Consider Gaussian surface of radius $r$. Using Gauss's law the field in between the shells is

$E.4\pi r^2=\frac{Q_{en}}{{ϵ}_0}=\frac{Q}{{ϵ}_0}$

$E=\frac{Q}{4\pi {ϵ}_0{r}^2}$

As $E=-\frac{dV}{dr}$ so the potential between A and B as well as at r is

$V_a-V_b=V_r=-{\int }_b^{a}Edr=-\frac{Q}{4\pi {ϵ}_0}{\int }_b^a\frac{dr}{r}=\frac{Q}{4\pi {ϵ}_0}[\frac{1}{a}-\frac{1}{b}]$

thus correct option only (ii) and (iii)