question_answer

Four charges equal to \[-\,Q\] are placed at the four corners of a square and a charge q is at its centre. If the system is in equilibrium the value of q is

A) \[-\frac{Q}{4}\left( 1+2\sqrt{2} \right)\]   

B) \[\frac{Q}{4}\left( 1+2\sqrt{2} \right)\]

C) \[-\frac{Q}{2}\left( 1+2\sqrt{2} \right)\]

D)        \[\frac{Q}{2}\left( 1+2\sqrt{2} \right)\]

Correct Answer: B

Solution :

[b] For the system to be in equilibrium, net force on each of the charges must be zero. Net for on any of - Q to be zero, we have \[\sqrt{2}k\frac{{{Q}^{2}}}{{{Q}^{2}}}+k\frac{{{Q}^{2}}}{2{{a}^{2}}}-\frac{kQq}{{{a}^{2}}/2}=0\] \[\therefore \] \[q=\frac{Q}{4}(1+2\sqrt{2})\]