A charge Q is distributed over three concentric spherical shells of radii a , b , c a

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:

TestHub · Accepted Answer

Surface charge densities of all sphere is given as $\frac{q_{1}}{a^{2}} = \frac{q_{2}}{b^{2}} = \frac{q_{3}}{c^{2}} = c o n s t a n t$ , where $q_{1}$ , $q_{2}$ and $q_{3}$ is charge on sphere of radius $a$ , $b$ and $c$ respectively.

$q_{1} + q_{2} + q_{3} = Q$

$q_{1} = \frac{a^{2} (Q)}{a^{2} + b^{2} + c^{2}}$

$q_{2} = \frac{b^{2} (Q)}{a^{2} + b^{2} + c^{2}}$

$q_{3} = \frac{c^{2} (Q)}{a^{2} + b^{2} + c^{2}}$

Potential at a point where $r < a$ is given as $V = \frac{K q_{1}}{a} + \frac{K q_{2}}{b} + \frac{K q_{3}}{c}$ , where $K$ is a constant.

$V = \frac{1}{4 π ε_{0}} [Q] (\frac{a + b + c}{a^{2} + b^{2} + c^{2}})$

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