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. \[\therefore \] \[\Sigma U=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{qQ}{r/2}+\frac{qQ}{r/2}+\frac{Q\times Q}{r} \right]\] \[0=\frac{2qQ}{r}+\frac{2qQ}{r}+\frac{{{Q}^{2}}}{r}\] \[\Rightarrow \] \[4q=-Q\] \[\Rightarrow \] \[q=-\frac{Q}{4}\]