Four equal positive charges, each of value Q, are arranged at the four corners of a square of diagonal 2a. A small body of mass m carrying a unit positive charge is placed at height h above the center of the square.
What should be the value of Q in order that this body may be in equilibrium?

Explanation

The situation is shown in figure (a)

Force experienced by unit positive charge placed at P due to a charge Q at A is given by
`F=(K(Qxx1))/((h^(2)+a^(2)/(2)))`
Similarly, equal forces act on unit positive charge at P due to charge at B, C and D. When these forces are resolved in horizontal and vertical directions, the horizontal component `(F sin theta)` cancel each other and the net vertical force is `4F costheta`.
Thus net upward force
`=(4KQ)/((h^(2)+a^(2)/(2))).costheta`
For the equilibrium of unit positive charge at P, Upward force = Weight of unit charge
`(4KQ)/((h^(2)+a^(2)/(2))).costheta=mg`
From figure (b)
`costheta={(h)/(sqrt(h^(2)+a^(2)//2))}`
or `(4KQh)/((h^(2)+a^(2)/(2))^(3//2))=mg`
or `Q=(mg)/(4Kh)((h^(2)+a^(2))/(2))^(3//2)`