question_answer

For equal point charges Q each are placed in the \[xy\]plane at \[(0,2),(4,2),(4,-2)\text{ }and\text{ (0,-2)}.\] The work required to put a fifth change Q at the origin of the coordinate system will be:

A) \[\frac{{{Q}^{2}}}{4\pi {{\in }_{0}}}\left( 1+\frac{1}{\sqrt{3}} \right)\]

B) \[\frac{{{Q}^{2}}}{4\pi {{\in }_{0}}}\left( 1+\frac{1}{\sqrt{5}} \right)\]

C) \[\frac{{{Q}^{2}}}{2\sqrt{2}\pi {{\in }_{0}}}\]

D) \[\frac{{{Q}^{2}}}{4\pi {{\in }_{0}}}\]

Correct Answer: B

Solution :

\[W=VQ=\frac{1}{4\pi {{\varepsilon }_{0}}}{{Q}_{2}}\left[ \frac{1}{2}+\frac{1}{2}+\frac{1}{2\sqrt{5}}+\frac{1}{2\sqrt{5}} \right]\]

\[=\frac{1}{4\pi {{\varepsilon }_{0}}}{{Q}_{2}}\left[ \frac{1}{2}+\frac{1}{2}+\frac{1}{2\sqrt{5}} \right]\]

\[\therefore \]      \[W=\frac{{{Q}_{2}}}{4\pi {{\varepsilon }_{0}}}\left[ 1+\frac{1}{\sqrt{5}} \right].\]