question_answer

Two identical thin rings each of radius R meters are coaxially placed at a distance R meters apart. If \[{{Q}_{1}}\]coulomb and \[{{Q}_{2}}\]coulomb are respectively the charges uniformly spread on the two rings, the work done in moving a charge q from the center of one ring to that of other is

A) zero                  

B)        \[\frac{q\left( {{Q}_{1}}-{{Q}_{2}} \right)\left( \sqrt{2}-1 \right)}{\sqrt{2}.4\pi {{\varepsilon }_{0}}R}\]

C) \[\frac{q\sqrt{2}\left( {{Q}_{1}}+{{Q}_{2}} \right)}{4\pi {{\varepsilon }_{0}}R}\]                                   

D) \[\frac{q\left( {{Q}_{1}}+{{Q}_{2}} \right)\left( \sqrt{2}+1 \right)}{\sqrt{2}.4\pi {{\varepsilon }_{0}}R}\]

Correct Answer: B

Solution :

[b] Work done \[{{W}_{21}}=\left( {{V}_{1}}-{{V}_{2}} \right)q\] \[V=\frac{1}{4\pi {{\in }_{0}}}\left[ \frac{{{Q}_{1}}}{R}+\frac{{{Q}_{2}}}{\sqrt{2}R} \right]\] and \[{{V}_{2}}=\frac{1}{4\pi {{\in }_{0}}}\left[ \frac{{{Q}_{2}}}{R}+\frac{{{Q}_{1}}}{\sqrt{2}R} \right]\] \[\text{Thus,}{{\text{W}}_{21}}=\frac{q\left( {{Q}_{1}}-{{Q}_{2}} \right)\left( \sqrt{2}-1 \right)}{\sqrt{2}.4\pi {{\in }_{0}}R}.\]