Question

One-fourth of a sphere of radius R is removed as shown in Fig. An electric field E exists parallel to the xy plane. Find the flux through the remaining curved part :

• A. $\pi R^{2}E$
• B. $\sqrt{2}\pi R^{2}E$
• C. $\pi R^{2}E/\sqrt{2}$
• D. none of these

Answer 1

Answer: (C)

$\pi R^{2}E/\sqrt{2}$

${\varphi }_{plain}+{\varphi }_{curve}=0$ or ${\varphi }_{plain}=-{\varphi }_{curve}$

${\overrightarrow{A}}_1=-\frac{\pi R^2}{2}\hat{i},{\overrightarrow{A}}_2=-\frac{\pi R^2}{2}\hat{j}$

$\overrightarrow{E}=Ecos{45}^{\circ }\hat{i}+Esin{45}^{\circ }\hat{j}$

$\frac{E}{\sqrt{2}}\hat{i}+\frac{E}{\sqrt{2}}\hat{j}$

$\varphi =\overrightarrow{E}\cdot ({\overrightarrow{A}}_1+{\overrightarrow{A}}_2)$

$\frac{-E}{\sqrt{2}}\frac{\pi R^2}{2}-\frac{E}{\sqrt{2}}\frac{\pi R^2}{2}=\frac{-\pi R^{2}E}{\sqrt{2}}$

Thus is the flux entering. So flux is $\frac{\pi R^{2}E}{\sqrt{2}}$