Question

Electric field in a certain region is acting radially outward and variation of electric field with distance from origin of field is represented below. Slope of the straight line given in the graph is $\text{A}$.


The charge contained in a sphere of radius $a$ centred at the origin of field will be given by

• A. $\pi {\epsilon }_{o}A{a}^2$
• B. $\pi {\epsilon }_{o}A{a}^3$
• C. $4\pi {\epsilon }_{o}A{a}^2$
• D. $4\pi {\epsilon }_{o}A{a}^3$

Answer 1

The correct option is D $4\pi {\epsilon }_{o}A{a}^3$

Slope of the straight line passing through origin is $A$. Thus equation relating $E$ and $r$ will be

$E=Ar....\left(1\right)$

Let us consider a gaussian sphere of radius $a$ centered origin of field.

Since $\overrightarrow{E}$ is radially outwards, applying Gaussian law,

$\varphi =\overrightarrow{E}.d\overrightarrow{A}=EA=\frac{q_{in}}{{\epsilon }_o}$

$\Rightarrow EA=\frac{q_{in}}{{\epsilon }_o}$

Substituting the values, we get

$Ar\times (4\times \pi r^2)=\frac{q_{in}}{{\epsilon }_o}$

For the surface $r=a$,

$Aa\times (4\times \pi a^2)=\frac{q_{in}}{{\epsilon }_o}$

$\therefore q_{in}=4\pi {\epsilon }_{o}A{a}^3$

Hence, the correct option is (b).