Question

Let a total charge $2Q$ be distributed in a sphere of radius $R$, with the charge density given by $\rho (r) = kr$, where r is the distance from the centre. Two charges $A$ and $B$, of $-Q$ each, are placed on diametrically opposite points, at equal distance, a, from the centre. If $A$ and $B$ do not experience any force, then :

• A. $a = \frac{3R}{2^{1/4}}$
• B. $a = R / \sqrt{3}$
• C. $a = 8^{-1/4 R}$
• D. $a = 2^{-1/4} R$

Answer 1

Answer: (C)

$a = 8^{-1/4 R}$

Answer 2

$E 4\pi a^2 = \frac{\int_0^a \, kr 4 \pi r^2 dr}{\epsilon}$
$E = \frac{k 4 \pi a^4}{4 \times 4 \pi \epsilon_0}$
$2Q = \int_0^R kr 4 \pi r^2 dr$
$k = \frac{2Q}{\pi R^4}$
$QE = \frac{1}{4 \pi \epsilon_0} \frac{QQ}{(2a)^2}$
$R = a8^{1/4}$