Question

What is the electric flux $(\int \overrightarrow{E}\cdot d\hat{a})$ through a quarter-cylinder of height H (as shown in te figure) due to an infinitely long line charge along the axis of the cylinder with a charge density of Q?

• A. $\frac{4H}{Q{\epsilon }_0}$
• B. $\frac{H{\epsilon }_0}{4Q}$
• C. $\frac{HQ}{{\epsilon }_0}$
• D. $\frac{HQ}{4{\epsilon }_0}$

Answer 1

The correct option is D $\frac{HQ}{4{\epsilon }_0}$

Ans : (d)

Electric field intensity $\left(\overrightarrow{E}\right)at\rho$ distance due to infinite long line having line charge dnesity Q is

$\overrightarrow{E}=\frac{Q}{2\pi {\epsilon }_0\rho }{\hat{a}}_{\rho }$

$\int \overrightarrow{E}\cdot \overrightarrow{da}=\iint \frac{Q}{2\pi {\epsilon }_0\rho }{\hat{a}}_{\rho }\cdot \rho d\varphi dz{\hat{a}}_{\rho }$

$=\frac{Q}{2\pi {\epsilon }_0}\underset{<\pi /2>}{\int }d\varphi \underset{z=0}{\overset{H}{\int }}dz$

$=\frac{Q}{2\pi {\epsilon }_0}\left(\frac{\pi }{2}\right)H=\frac{HQ}{4{\epsilon }_0}$