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A charge $Q$ is distributed over three concentric spherical shell 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
$\frac{Q}{{12\pi \,{ \in _0}}}\frac{{ab + bc + ca}}{{abc}}$
$\frac{{Q\,\left( {{a^2} + {b^2} + {c^2}} \right)}}{{4\pi \,{ \in _0}\,\left( {{a^3} + {b^3} + {c^3}} \right)\,}}$
$\frac{Q}{{4\pi \,{ \in _0}\,\left( {a + b + c} \right)\,}}$
$\frac{{Q\,\left( {a + b + c} \right)}}{{4\pi \,{ \in _0}\,\left( {{a^2} + {b^2} + {c^2}} \right)\,\,}}$
Solution
$\mathrm{Q}_{1}+\mathrm{q}_{2}+\mathrm{Q}_{3}=\mathrm{Q}………(1)$
$\frac{\mathrm{Q}_{1}}{4 \pi \mathrm{a}^{2}}=\frac{\mathrm{Q}_{2}}{4 \pi \mathrm{b}^{2}}=\frac{\mathrm{Q}_{3}}{4 \pi \mathrm{c}^{2}}=\mathrm{k}………(2)$
Subs. $Q_{1}, Q_{2}, Q_{3}$ in $( 1)$
$k=\frac{Q}{4 \pi\left(a^{2}+b^{2}+c^{2}\right)}$
$v=\frac{k Q_{1}}{a}+\frac{k Q_{2}}{b}+\frac{k Q_{3}}{c}$
