Charges +q and −q are placed at points A and B respectively which are a distance 2L apart, C is the midpoint between A and B. The work done in moving a charge +Q along the semicircle CRD is ?

Charges +q and −q are placed at points A and B respectively which are a distance 2L apart, C is the midpoint between A and B. The work done in moving a charge +Q along the semicircle CRD is ?

\( \text{(a)} \ \frac{qQ}{2\pi \varepsilon_0 L} \)

\( \text{(b)} \ \frac{qQ}{6\pi \varepsilon_0 L} \)

\( \text{(c)} \ -\frac{qQ}{6\pi \varepsilon_0 L} \)

\( \text{(d)} \ \frac{qQ}{4\pi \varepsilon_0 L} \)

Answer :  \( \text{(c)} \ -\frac{qQ}{6\pi \varepsilon_0 L} \)

\( \text{Work} = \text{change in P.E.} \)

\( W = \Delta U \)

\( = U_f - U_i \)

\( = \left[ \frac{kQq}{3L} - \frac{kQq}{L} \right] - \left[ \frac{kQq}{L} - \frac{kQq}{L} \right] \)

\( = -\frac{2}{3} \cdot \frac{kQq}{L} \)

\( = -\frac{2}{3} \cdot \frac{1}{4\pi \varepsilon_0} \cdot \frac{Qq}{L} \)

\( = -\frac{qQ}{6\pi \varepsilon_0 L} \)