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 :

Solution:

Potential at point P, $V = \frac{kQ_{a}}{a} + \frac{kQ_{b}}{b} + \frac{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[\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] $
$\therefore$ correct answer is (2)

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