Question

The electric field in a region is radially outwards and has a magnitude $E=Kr$. The charge contained in a sphere of radius $a$ is :

• A. $K4\pi {\epsilon }_0{a}^2$
• B. $K\frac{4}{3}\pi {\epsilon }_0{a}^3$
• C. $K4\pi {\epsilon }_0{a}^3$
• D. none of these

Answer 1

The correct option is C $K4\pi {\epsilon }_0{a}^3$

Using Gauss's law the electric field at distance $a$ from the center of sphere is

$E\left(a\right).4\pi a^2=\frac{Q}{{ϵ}_0}$

as $E\left(r\right)=Kr\Rightarrow E\left(a\right)=Ka$

thus, $Ka.4\pi a^2=\frac{Q}{{ϵ}_0}$

or $Q=4\pi {ϵ}_{0}K{a}^3$