Question

Three 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 $a+b=c$

$V_A=\frac{\sigma 4\pi a^2}{4\pi {ϵ}_{0}a}+\frac{(-\sigma )4\pi b^2}{4\pi {ϵ}_{0}b}+\frac{\sigma 4\pi c^2}{4\pi {ϵ}_{0}c}$

$=\frac{\sigma }{{ϵ}_0}(a-b+c)$

Similarly, $V_C=\frac{\sigma }{{ϵ}_0}(\frac{a^2}{c}-\frac{b^2}{c}+c)$

But $V_A=V_C$

$a-b+c=\frac{a^2-b^2}{c}+c$

Therefore, $a+b=c$