A non-conducting spherical ball of radius R contains a spherically symmetric charge with volume charge density ρ = krn, where r is the distance from the center of the ball and n is a constant. What should be n such that the electric field inside the ball is directly proportional to square of distance from the centre.

Engineering

Physics

Charge and Coulombs Law

Electric Flux and Gauss Law and its Applications

Calculus

Question

A non-conducting spherical ball of radius R contains a spherically symmetric charge with volume charge density ρ = krⁿ, where r is the distance from the center of the ball and n is a constant. 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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By Gauss law
$E \times 4 {πr}^{2} = \frac{k \int r^{n} \times 4 {πr}^{2} dr}{\in_{0}} = \frac{4 πk}{\in_{0}} \frac{r^{x + 3}}{x + 3}$
$E = \frac{k}{(n + 3) \in_{0}} (r^{n + 1})$
n + 1 = 2 ⇒ n = 1