Question

Charges $+q$ and $-q$ are placed at points $\text{A}$ and $\text{B}$ respectively which are at a distance $2L$ apart and $\text{C}$ is the mid point between $\text{A}$ and $\text{B}$. The workdone in moving a charge $+Q$ along the semicircle $\text{CRD}$ with $B$ as centre is

• A. $\frac{qQ}{4\pi {\epsilon }_{0}L}$
• B. $\frac{qQ}{6\pi {\epsilon }_{0}L}$
• C. $\frac{qQ}{2\pi {\epsilon }_{0}L}$
• D. $-\frac{qQ}{6\pi {\epsilon }_{0}L}$

Answer 1

The correct option is D $-\frac{qQ}{6\pi {\epsilon }_{0}L}$




From the figure, we get

Potential at $\text{C}=V_C=0$

Let, Potential at $\text{D}=V_D$

$V_D=\frac{1}{4\pi {\epsilon }_0}[\frac{q}{3L}-\frac{q}{L}]$

$=-\frac{2q}{4\pi {\epsilon }_0\left[3L\right]}$

Potential difference between $V_D$ and $V_C$ is,

$V_D-V_C=-\frac{2q}{4\pi {\epsilon }_0\left(3L\right)}-0=\frac{-q}{6\pi {\epsilon }_{0}L}$

The workdone $=Q[V_D-V_C]$

$=-\frac{qQ}{6\pi {\epsilon }_{0}L}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, (d) is the correct answer.