A charge q is placed at the center of the line joining two equal charges Q. Show that the system of three charges will be in equilibrium if q = -Q/4.
Here,
OA = AB = x/2
OB = x
Force on the charge at B due to the charge at O is,
F1 = kQ2/x2
Force on the charge at B due to the charge at A is,
F2 = kQq/(x/2)2
For the system to be in equilibrium,
F1 + F2 = 0
=> kQ2/x2 + kQq/(x/2)2 = 0
=> q = -Q/4