Question

An electric field prevailing in a region depends only on X and Y coordinates according to an equation, $\overrightarrow{E}=b\frac{X\hat{i}+Y\hat{j}}{X^2+Y^2}$ where b is a constant. Find the flux passing through a sphere of radius r whose center is on the origin of the coordinate system.

• A. $\pi rb$
• B. $2\pi br$
• C. $6\pi rb$
• D. $4\pi br$

Answer 1

The correct option is D $4\pi br$

$\overrightarrow{E}=b\frac{X\hat{i}+Y\hat{j}}{X^2+Y^2}$

here, $\overrightarrow{r}=X\hat{i}+Y\hat{j}$

$\left|\overrightarrow{r}\right|=X^2+Y^2$

$\begin{array}{c}\overrightarrow{E}=\frac{b}{\left|r\right|}.\hat{r}\\ =\frac{br}{r^2}.\hat{r}\\ \therefore \frac{KQ}{r^2}=\frac{br}{r^2}\\ \Rightarrow Q=\frac{br}{K}=br4\pi E_0\\ \therefore \varphi =\frac{Q}{E_0}=4\pi br\end{array}$

$\therefore$ Option $D$ is correct.