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

Answer 1

Answer: (B)

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