question_answer

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

A) \[\frac{qQ}{2\pi {{\varepsilon }_{0}}L}\]

B) \[\frac{qQ}{6\pi {{\varepsilon }_{0}}L}\]

C) \[-\frac{qQ}{6\pi {{\varepsilon }_{0}}L}\]          

D) \[\frac{qQ}{4\pi {{\varepsilon }_{0}}L}\]

Correct Answer: C

Solution :

[c]: From figure, \[AC=L,BC=L,BD=BC=L\] \[AD=AB+5D=2L+L=3L\] Potential at C is given by \[{{V}_{C}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{q}{AC}+\frac{(-q)}{BC} \right]=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{q}{L}-\frac{q}{L} \right]=0\] Potential at D is given by \[{{V}_{D}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{q}{AD}+\frac{(-q)}{BD} \right]=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{q}{3L}-\frac{q}{L} \right]\] \[=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{q}{L}\left[ \frac{3}{3}-1 \right]=\frac{-q}{6\pi {{\varepsilon }_{0}}L}\] Work done in moving charge +Q along the semicircle CRD is given by or\[{{K}_{1}}=\frac{5.5}{55}MeV\]