question_answer

A charge q is lying at mid-point of the line joining the two similar charges Q. The system will be in equilibrium, if the value of q is:

A) \[\frac{Q}{2}\]                                  

B) \[-\frac{Q}{2}\]

C) \[\frac{Q}{4}\]                                  

D) \[-\frac{Q}{4}\]

Correct Answer: D

Solution :

                 When two charges\[{{q}_{1}}\]and\[{{q}_{2}}\]are situated in vacuum at a distance r metre, then the electric potential energy of the system is \[U=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{{{q}_{1}}{{q}_{2}}}{r}joule\] In order that system be in equilibrium, the potential energy of the system should be zero. \[\therefore \]    \[\Sigma U=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{qQ}{r/2}+\frac{qQ}{r/2}+\frac{Q\times Q}{r} \right]\]                 \[0=\frac{2qQ}{r}+\frac{2qQ}{r}+\frac{{{Q}^{2}}}{r}\] \[\Rightarrow \]               \[4q=-Q\] \[\Rightarrow \]               \[q=-\frac{Q}{4}\]