question_answer

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

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

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

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

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

Correct Answer: D

Solution :

                Key Idea: Work done is equal to change ii potential energy.                 In Ist case, when charge +Q is situated at C.                                 Electric potential energy of system                 \[{{U}_{1}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{(q)\,(-q)}{2L}+\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{(-q)\,Q}{L}+\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{qQ}{L}\]                 In IInd case, when charge +Q is moved from C to D.                                 Electric potential energy of system in that case                 \[{{U}_{2}}=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{(q)\,(-q)}{2L}+\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{qQ}{3L}\]                 \[+\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{(-q)\,(Q)}{L}\]                               \[\therefore \]  Work done = \[\Delta U={{U}_{2}}-{{U}_{1}}\]                 \[=\left( -\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}^{2}}}{2L}+\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{qQ}{3L}-\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{qQ}{L} \right)\]                 \[-\left( -\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}^{2}}}{2L}-\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{qQ}{L}+\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{qQ}{L} \right)\]                 \[=\frac{qQ}{4\pi {{\varepsilon }_{0}}}.\left[ \frac{1}{3L}-\frac{1}{L} \right]=\frac{qQ}{4\pi {{\varepsilon }_{0}}}\frac{(1-3)}{3L}\]                 \[=\frac{-2qQ}{12\pi {{\varepsilon }_{0}}L}=-\frac{qQ}{6\pi {{\varepsilon }_{0}}L}\]