Q. A sphere of radius $R$ has a uniform distribution of electric charge in its volume. At a distance $x$ from its centre, for $x < R,$ the electric field is directly proportional to
A) $\frac{1}{x^{2}}$
B) $\frac{1}{x}$
C) $x$
D) $x^{2}$
Explanation
(c) Let sphere has uniform charge density $\rho\left( = \frac{3Q}{4\pi R^{3}} \right)$ and $E$ is the electric field at distance $x$ from the centre of the sphere
Applying Gauss law
$E.4\pi x^{2} = \frac{q}{\varepsilon_{0}} = \frac{\rho V'}{\varepsilon_{0}} = \frac{\rho}{\varepsilon_{0}} \times \frac{4}{3}\pi x^{3}$
[$V^{'} =$ Volume of dotted sphere]
$\therefore E = \frac{\rho}{3\varepsilon_{0}}x \Rightarrow E \propto x$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.