# Q. 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 :

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Solution:

## Potential at point P, $V = \dfrac{kQ_{a}}{a} + \dfrac{kQ_{b}}{b} + \dfrac{kQ_{c}}{c}$ $\because Q_{a} : Q_{b} : Q_{c} : : a^{2} :b^{2} :c^{2}$ $\left\{\text{since} \sigma_{a} = \sigma_{b} =\sigma_{c}\right\}$ $\therefore Q_{a} = \left[\dfrac{a^{2}}{a^{2}+b^{2}+c^{2}}\right]Q$ $Q_{b} = \left[\dfrac{b^{2}}{a^{2}+ b^{2} +c^{2}}\right]Q$ $Q_{c} = \left[\dfrac{c^{2}}{a^{2}+b^{2} +c^{2}}\right]Q$ $V = \dfrac{Q}{4 \pi\in_{0}} \left[\dfrac{\left(a+b+c\right)}{a^{2}+b^{2}+c^{2}}\right]$ $\therefore$ correct answer is (2)

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