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Consider a system of there charges $\frac{q}{3},\,\frac{q}{3}$ and $-\frac{2q}{3}$ placed at point $A, B$ and $C,$ respectively, as shown in the figure. Take $O$ to be the centre of the circle of radius $R$ and $\angle CAB\, = \,{60^o}$
The electric field at point $O$ is $\frac{q}{{8\pi { \in _0}{R^2}}}$ directed along the negative $x-$ axis
The Potential energy of the system is zero
The magnitude of the force between the charges at $C$ and $B$ is $\frac{{{q^2}}}{{54\pi { \in _0}{R^2}}}$
The potential at point $O$ is $\frac{q}{{12\pi { \in _0}R}}$
Solution
Field at $\mathrm{O}=\frac{\mathrm{K}(2 \mathrm{q} / 3)}{\mathrm{R}^{2}}$
Along $- \mathrm{ve}$ $\mathrm{x}$ – axis
$\mathrm{PE}$ of system $=\frac{+\mathrm{Kq}^{2}}{9 \times 2 \mathrm{R}}-\frac{2 \mathrm{Kq}^{2}}{9 \mathrm{R}}-\frac{2 \mathrm{Kq}^{2}}{9 \sqrt{3} \mathrm{R}}$
Force between $\mathrm{C}$ and $\mathrm{B}$ is
$\frac{{{\rm{K}}\frac{{2{{\rm{q}}^2}}}{9}}}{{3{{\rm{R}}^2}}} = \frac{1}{{54\pi { \in _0}}}\frac{{{{\rm{q}}^2}}}{{{{\rm{R}}^2}}}$
Potential at $\mathrm{O}=\mathrm{Zero}$
Option is $(3)$
