Question

A simple pendulum of length $l$ and bob mass $m$ is hanging in front of a large non-conducting sheet having surface charge density $\sigma$. If suddenly a charge $+q$ is given to the bob & it is released from the position shown in figure. The maximum angle through which the string is deflected from vertical is $x{\tan }^{-1}\left(\frac{\sigma q_0}{2{\epsilon }_{0}mg}\right)$. Find $x$.

Answer 1

The force on bob due to the sheet is $F=qE=q\frac{\sigma }{2{ϵ}_0}$.
For maximum deflected from vertical , $mg\sin \theta =qE\cos \theta =q\frac{\sigma }{2{ϵ}_0}\cos \theta$
or $\tan \theta =\frac{\sigma q}{2{ϵ}_{0}mg}\Rightarrow \theta ={\tan }^{-1}\left(\frac{\sigma q}{2{ϵ}_{0}mg}\right)$
thus, $x=1$