Question

A charge $Q$ is distributed over three concentric spherical shells of radii $a,b,c(a<b<c)$ such that their surface charge densities are equal to one another. The total potential at a point at distance $r$ from their common centre, where $r<a$, would be

• A. $\frac{Q}{4\pi {ϵ}_0(a+b+c)}$
• B. $\frac{Q(a+b+c)}{4\pi {ϵ}_0(a^2+b^2+c^2)}$
• C. $\frac{Q}{12\pi {ϵ}_0}\frac{ab+bc+ca}{abc}$
• D. $\frac{Q}{4\pi {ϵ}_0}\frac{(a^2+b^2+c^2)}{(a^3+b^3+c^3)}$

Answer 1

The correct option is B $\frac{Q(a+b+c)}{4\pi {ϵ}_0(a^2+b^2+c^2)}$

Potential at point $P,V=\frac{k{Q}_a}{a}+\frac{k{Q}_b}{b}+\frac{k{Q}_c}{c}$
$\because Q_a:Q_b:Q_c::a^2:b^2:c^2$
$\{since{\sigma }_a={\sigma }_b={\sigma }_c\}$
$\therefore Q_a=\left[\frac{a^2}{a^2+b^2+c^2}\right]Q$
$Q_b=\left[\frac{b^2}{a^2+b^2+c^2}\right]Q$
$Q_c=\left[\frac{c^2}{a^2+b^2+c^2}\right]Q$
$V=\frac{Q}{4\pi {ϵ}_0}\left[\frac{(a+b+c)}{a^2+b^2+c^2}\right]$
$\therefore$ correct answer is $\left(2\right)$.