Question

A square plate of side 'a' is kept below an infinite line charge of charge per unit length $\lambda$ (as shown in figure). Electric flux associated with the plate is $\frac{3\lambda a}{n\pi {ϵ}_o}ta{n}^{-1}\left(\frac{1}{2}\right).$
Here, n is

Answer 1

${\varphi }_{net}$ at a distance 'a' from the wire can be given by :
${\varphi }_{net}=\frac{\lambda a}{{ϵ}_o}$

As it is clear from the diagram that the we have to calculate electric flux for an angular range of $2\theta$, which can be given by:
$\varphi =\frac{{\varphi }_{net}}{2\pi }.2\theta$

Now, $tan\theta =\frac{a/2}{a}=\frac{1}{2}$
$\therefore \varphi =\frac{{\varphi }_{net}}{2\pi }.2ta{n}^{-1}\left(\frac{1}{2}\right)=\frac{\lambda a}{\pi {ϵ}_0}.ta{n}^{-1}\left(\frac{1}{2}\right)$
Hence, by comparison of the expression given with this, we get n = 3