question_answer

Two charges q and \[-3q\] are fixed on X-axis separated by distanced. Where should a third charge 2q be placed from A such that it will not experience any force?

 

A) \[\frac{d-\sqrt{3}d}{2}\]            

B) \[\frac{d+\sqrt{3}d}{2}\]

C) \[\frac{d+3d}{2}\]                    

D) \[\frac{d-3d}{2}\]

Correct Answer: B

Solution :

(b) \[\frac{d+\sqrt{3}d}{2}\] Let a charge 2q be placed at point P, at a distance I from A, where charge q is placed, as shown in figure. The charge 2q will not experience any force, when force of repulsion on it due to q is balanced by force of attraction on it due to 3q at B, where AB = (d) \[\frac{\left( 2q \right)\left( q \right)}{4\pi {{\varepsilon }_{0}}{{l}^{2}}}=\frac{\left( 2q \right)\,\left( 3q \right)}{4\pi {{\varepsilon }_{0}}{{\left( l+d \right)}^{2}}}\] \[{{\left( l+d \right)}^{2}}=3{{l}^{2}}\] Or \[2{{l}^{2}}-2ld-{{d}^{2}}=0\] \[l=\frac{2d\pm \sqrt{4{{d}^{2}}+8{{d}^{2}}}}{4}=\frac{d}{2}\pm \frac{\sqrt{3}\,d}{2}\] \[\Rightarrow l=\frac{d+\sqrt{3}\,d}{2}\]