Consider a system of three charges $\frac{\mathrm{q}}{3}, \frac{\mathrm{q}}{3}$ and $-\frac{2 \mathrm{q}}{3}$ placed at points $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$, respectively, as shown in the figure,
Take $\mathrm{O}$ to be the centre of the circle of radius $\mathrm{R}$ and angle $\mathrm{CAB}=60^{\circ}$
Figure:$Image$ 
IIT 2008, Advanced
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$\mathrm{F}_{\mathrm{BC}}=\frac{1}{4 \pi \varepsilon_0} \frac{\left(\frac{\mathrm{q}}{3}\right)\left(\frac{2 \mathrm{q}}{3}\right)}{(\mathrm{R} \sqrt{3})^2}=\frac{\mathrm{q}^2}{54 \pi \varepsilon_0 \mathrm{R}^2}$
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