Question

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. $x^2$

Answer 1

The correct option is C

$x$

Let sphere has uniform charge density $\rho (=\frac{3Q}{4\pi R^3})$ and E is the electric field at distance X from
the centre of the sphere.
Applying Gauss law

$E.4\pi x^2=\frac{q}{{ϵ}_0}=\frac{\rho V^{\prime }}{{ϵ}_0}=\frac{\rho }{{ϵ}_0}\times \frac{4}{3}\pi x^3$
(v'= Volume of dotted sphere)
$\therefore E=\frac{\rho }{3{ϵ}_0}x\Rightarrow E\propto x$