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Consider a system of three charges $\frac{q}{3},\frac{q}{3}$ and $\frac{-2q}{3}$ placed at points $A,B$ 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 $\frac{q}{{8\pi {\varepsilon _0}{R^2}}}$ 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 $\frac{{{q^2}}}{{54\pi {\varepsilon _0}{R^2}}}$
The potential at point $O$ is $\frac{q}{{12\pi {\varepsilon _0}R}}$
Solution
Three charges $=\frac{q}{3}, \frac{q}{3}, \frac{-2 q}{3}$
Radius of circle $=\mathrm{R}$
Angle $\mathrm{CAB}=60^{\circ}$
Now, the electric force between the charges at $\mathrm{C}$ and $\mathrm{B}$ $\vec{F}=\frac{1}{4 \pi \varepsilon_{0}} \frac{\frac{q}{3} \times\left(\frac{-2 q}{3}\right)}{(R \sqrt{3})^{2}}$
$\vec{F}=\frac{-q^{2}}{54 \pi \varepsilon_{0} R^{2}}$
$F=\frac{q^{2}}{54 \pi \varepsilon_{0} R^{2}}$
Hence, the magnitude of the force between the charges at $\mathrm{C}$ and $\mathrm{B}$ is $\frac{q^{2}}{54 \pi \varepsilon_{0} R^{2}}$
