Three point charges `q` are placed on vertices of an equilateral triangle of side length `a`. The charge `Q` that should be kept at centroid of triangle such that the system is in equilibrium is equal to

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))`