Three equal point charges q are placed at the corners of an equilateral triangle. Another charge Q is placed at the centroid of the triangle. The system of charges will be in equilibrium if Q equals

The system be in equilibrium if the net force on charge q at one vertex due to charges q at the other two vertices is equal and opposite to the force due to charge Q at the centroid, i.e. (here a is the side of the triangle)

$-\frac{\sqrt{3}{q}^2}{4{\mathrm{\pi \epsilon }}_0{\mathrm{a}}^2}=\frac{Qq}{4{\mathrm{\pi \epsilon }}_0{\left({\displaystyle \frac{\mathrm{a}}{\sqrt{3}}}\right)}^2}$

which gives Q =  $-\frac{q}{\sqrt{3}}$.