question_answer

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

A) zero

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

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

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

Correct Answer: B

Solution :

[b] Work  \[=q\,({{V}_{A}}-{{V}_{B}})\] \[=\frac{q\,\left( {{Q}_{1}}-{{Q}_{2}} \right)\,\left( \sqrt{2}-1 \right)}{4\pi {{\varepsilon }_{0}}\sqrt{2}R.}\] (\[\because \] potential at A = potential due to \[{{Q}_{1}}+\]potential due to \[{{Q}_{2}}\])