Question

A small conducting spherical shell with inner radius $a$ and outer radius $b$ is concentric with a larger conducting spherical shell with inner radius $c$ and outer radius $d$ as shown in the figure. The inner shell has total charge $+2q$ and outer shell has charge $+4q$. The graph of radial component of electric field ($E$) as a function of distance ($r$) from centre of shell will be:

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

The shell with inner radius $a$ has a total charge $+2q$, which will appear at its outer surface.
Assume an imaginary Gaussian surface is formed at distance $r$ from the centre of the shell and $\overrightarrow{E}$ is electric field at $r$.

Case I: When $r<a$
According to Gauss law
$\oint \overrightarrow{E}.\overrightarrow{dA}=\frac{q_{en}}{{ϵ}_0}$ ($dA$= area of Gaussian surface enclosing the charge $q_{en}$)
Thus, for any spherical Gaussian surface inside the inner shell, charge enclosed will be zero.
i.e $q_{en}=0$
$\therefore E=0$

Case II: When $a<r<b$
again, $q_{en}=0$
By Gauss law
$\therefore E=0$ in between $(a<r<b)$

Case III: When $b<r<c$
$q_{en}=+2q$
By Gauss law,
$\Rightarrow \oint \overrightarrow{E}.\overrightarrow{dA}=\frac{q_{in}}{{ϵ}_0}=\frac{2q}{{ϵ}_0}$
$\Rightarrow E\left(4\pi r^2\right)=\frac{2q}{{ϵ}_0}$
$\therefore E=\frac{2q}{4\pi {ϵ}_0{r}^2}$
Thus, the graph of $E$ vs $r$ will be hyperbolic in this region.

Case IV: When $c<r<d$,
$-2q$ charge will be induced at the innermost surface of the outer shell due to the charge ($+2q$) at the outer surface on the inner shell.


$\Rightarrow q_{en}=-2q+2q$
$\Rightarrow q_{en}=0$
By Gauss law
$E=0$

Case V: When $r>d$:
$q_{en}=(+2q)+\left(4q\right)=+6q$
By Gauss law
$\oint \overrightarrow{E}.\overrightarrow{dA}=\frac{q_{in}}{{ϵ}_0}$
$\Rightarrow E=\frac{6q}{4\pi {ϵ}_0{r}^2}$
Thus graph of $E$ vs $r$ will be hyperbola in this region.
$\therefore$ option (a) is the best representation for variation of electric field.

$\begin{array}{c}\text{Why this question}?\\ \text{Tip: In problems involving conducting shells, always try to}\\ \text{visualize a spherical Gaussian surface about centre and focus}\\ \text{on the}{q}_{en}\text{to find E at varying radius}r.\end{array}$