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 :

Answer 1

Let charge on spherical shell of radius a is $Q_a$

charge on spherical shell of radius a is $Q_b$

charge on spherical shell of radius a is $Q_c$

now, potential at the surface of spherical shell A , $V_A=\frac{kQ_b}{a}$

potential at the surface of spherical shell B , $V_B=\frac{kQ_b}{a}$

potential at the surface of spherical shell C , $V_C=\frac{kQ_c}{c}$

Let $\sigma$ is the surface charge density of all of them.

then, charge = $\sigma \textbf{area}$

so, $Q_a:Q_b:Q_c=a^2:b^2:c^2$

so, $Q_a=\frac{a^2}{a^2+b^2+c^2}q$

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

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

so, total potential , V = $V_A+V_B+V_C$

= $\frac{kq}{a^2+b^2+c^2}(a+b+c)$ This is required answer.