Three charges 'q' each are vertices of an equilateral triangle of side 'r'. How much charge should be placed at the centroid so that the system remains in equilibrium.

Explanation

Force on the charge .q. is `sqrt(3) (1)/(4pi in_(0)) .(q^(2))/(r^(2)) =F_(1)`
A charge .Q. is placed at its centroid. Force applied by this charge on .q. is `(1)/(4pi in_(0)) .(3Qq)/(r^(2)) =F_(2)`.
If `F_(1)=F_(2)` the system remains in equilibrium.

`sqrt(3) (1)/(4pi in_(0)) (q^(2))/(r^(2)) =(1)/(4pi in_(0)) (3Qq)/( r^(2)) `
`Q= -(q)/(sqrt(3))`