Question

A rectangular plane of dimension $a \times 2a$ is placed at distance $a$ from the infinite line charge of charge density $\lambda$ such that line $OP$ is normal to the plane of rectangle as shown. The net electric flux passing through this plane is

Answer 1

$\frac{2\lambda a}{\pi\epsilon_0} \arctan\left(\frac{1}{2}\right)$

Answer 2

The electric field due to an infinite line charge is radial. The flux through the rectangular plane is calculated by integrating the dot product of the electric field vector and the area vector over the surface. The plane is positioned such that its normal is along the x-axis, and the line charge is along the y-axis. Thus, only the x-component of the electric field contributes to the flux. The integral is set up for the given dimensions and solved using standard integration techniques. Alternatively, the problem can be solved by considering the angular distribution of flux from an infinite line charge, where the flux through a segment of length $L$ spanning an angle $\Delta\theta$ is $\frac{\lambda L \Delta\theta}{2\pi\epsilon_0}$.