Question

A very long uniformly charged wire oriented along the axis of a circular ring of radius R rests on its center with one of the ends(as shown in Fig. The linear charge density on the wire is $\lambda$ Evaluate the flux of the vector $\overrightarrow{E}$ across the circle area.

• A. $\frac{\lambda R}{2{\epsilon }_0}$
• B. $\frac{5\lambda R}{2{\epsilon }_0}$
• C. $\frac{\lambda R}{9{\epsilon }_0}$
• D. $\frac{3\lambda R}{2{\epsilon }_0}$

Answer 1

The correct option is A $\frac{\lambda R}{2{\epsilon }_0}$

The component of electric field $E_y$ will not contribute to flux as angle between $E_y$ and $\overrightarrow{A}$ is ${90}^{\circ }$. Hence, flux due to $E_x$ is
$d\varphi =E_{x}2\pi ydy$
$\varphi -\int d\varphi ={\int }_0^R\frac{\lambda }{4\pi \epsilon y}2\pi ydy$
$\frac{\lambda }{2{\epsilon }_0}{\int }_0^{R}dy=\frac{\lambda R}{2{\epsilon }_0}$