Question

Charges $q$ and $-q$ are placed at points $A$ and $B$ respectively which are a distance $4L$ apart. $C$ is the mid - point between $A$ and $B$. The work done in moving a charge $+Q$ along the semi-circle $CRD$ is

• A. $\frac{-Qq}{12\pi {ϵ}_{0}L}$
• B. $\frac{-Qq}{4\pi {ϵ}_{0}L}$
• C. $\frac{Qq}{12\pi {ϵ}_{0}L}$
• D. $\frac{-Qq}{\pi {ϵ}_{0}L}$

Answer 1

The correct option is A $\frac{-Qq}{12\pi {ϵ}_{0}L}$


From data,
potential at point $C$ is $V_C=0$

potential at point $D$ is
$V_D=\frac{1}{4\pi {ϵ}_0}[\frac{q}{6L}-\frac{q}{2L}]=\frac{q}{4\pi ϵL}\left[\frac{1-3}{6}\right]=\frac{-q}{12\pi {ϵ}_{0}L}$
work done, $W_{C\to D}=Q[V_D-V_C]=Q[\frac{-q}{12\pi {ϵ}_{0}L}-0]=\frac{-Qq}{12\pi {ϵ}_{0}L}$