Question

The concentric spherical metallic shells, $A,B$ and $C$ of radii $a,b$ and $c$ $(a<b<c)$ have charge densities $\sigma ,-\sigma$ and $\sigma$ respectively. If the shells $A$ and $C$ are at the same potential, then the relation between $a,b$ and $c$ is:

• A. $a+b+c=0$
• B. $a-c=b$
• C. $a+b=c$
• D. $a=b-c$

Answer 1

The correct option is C $a+b=c$


Potential of outer shell $C$
$V_C=\frac{K(\sigma \times 4\pi a^2)}{c}-\frac{K(\sigma \times 4\pi b^2)}{c}+\frac{K(\sigma \times 4\pi c^2)}{c}$
Now, potential of inner shell $\left(A\right)$
$V_A=\frac{K(\sigma \times 4\pi a^2)}{a}-\frac{K(\sigma \times 4\pi b^2)}{b}+\frac{K(\sigma \times 4\pi c^2)}{c}$
Now, $V_A=V_C$
$\Rightarrow K\left(4\pi \sigma \right)(a-b+c)=K\left(4\pi \sigma \right)[\frac{a^2}{c}-\frac{b^2}{c}+c]$
$\Rightarrow a-b+c=\frac{(a^2-b^2)}{c}+c$
$\Rightarrow (a-b)=\left[\frac{(a-b)(a+b)}{c}\right]$
$\Rightarrow a+b=c$