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A non-conducting spherical ball of radius R contains a spherically symmetric charge with volume charge density ${ρ=kr}^{n}$ where r is the distance from the center of the ball, n and k are constants of appropriate dimension. What should be $n$ such that the electric field inside the ball is directly proportional to square of distance from the centre?

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Detailed Solution

$E . (4 {πR}^{2}) = \frac{\int_{0}^{R} ρ . 4 {πr}^{2} dr}{ε_{0}}$

given that, electric field inside the ball is directly proportional to square of distance from the centre

$K r^{2} . 4 π r^{2} = \frac{k r^{n + 3}}{(n + 3) . ε_{0}}$

$\Rightarrow n = 1$

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