Question

Consider a hollow sphere of radius $R$, Its quarter part is cut and kept that geometrical centre lies at origin as shown in figure.
$x-y$ axis lies in plane of paper and $z$-axis is $\perp$ to the plane of the plane the paper. Sphere is kept in a manner such that plane portions of sphere lies in $x-z$ and $y-z$ plane. Uniform electric field $\overrightarrow{E}=E_0\hat{i}+2E_0\hat{j}+2E_0\hat{k}$ $\left(N/c\right)$ lies in the complete space Electric flux coming out of the curved surface of sphere is:

• A. $\frac{3}{2}\pi R^2{E}_0$
• B. $\frac{5}{2}\pi R^2{E}_0$
• C. $2\pi R^2{E}_0$
• D. $Zero$

Answer 1

Answer: (A)

$\frac{3}{2}\pi R^2{E}_0$

Magnitude of electric flux entering the XZ plane surface and YZ plane will be equal to electric flux coming out of curved surface.

Flux through XZ plane,

Area vector for XZ plane, $A=-\frac{\pi R^2}{2}\hat{j}$

${\varphi }_1=E.A=(E_0\hat{i}+2E_0\hat{j}+2E_0\hat{k}).(-\frac{\pi R^2}{2}\hat{j})$

${\varphi }_1=-\pi R^2{E}_0$

Flux through YZ plane,

Area vector for YZ plane, $A=-\frac{\pi R^2}{2}\hat{i}$

${\varphi }_2=E.A=(E_0\hat{i}+2E_0\hat{j}+2E_0\hat{k}).(-\frac{\pi R^2}{2}\hat{i})$

${\varphi }_2=-\frac{\pi R^2{E}_0}{2}$

Total flux, $\varphi ={\varphi }_1+{\varphi }_2=\frac{3\pi R^2{E}_0}{2}$