Question

a charge q is placed at the center of the line joining two equal charges Q. shoW that the system of three charge will be in equilibrium if q= -Q/4

Answer 1

assume that q is positive
Get the total force on Q and make it equal to zero as it is in equilibrium

Answer 2

Let us treat left Q charge as the test charge and other two charges q and Q as source charges.
For equilibrium force acting on the test charge should be zero.
\[Force = k\frac{q_1 q_2}{r^2}\]

Here, \[Force \ between \ q \ and \ Q = k\frac{qQ}{(r/2)^2}\]
Here, \[Force \ between \ Q \ and \ Q = k\frac{Q^2}{(r)^2}\]
Total force = 0
So
\[k\frac{qQ}{(r/2)^2} +k\frac{Q^2}{(r)^2} = 0 \]
Solving this equation gives q = -Q/4