Problem 3: A charge q is placed at the center of a line joining two charges each of magnitude Q. Prove that the system of three charges will be in equilibrium if q = -Q/4.

Solution

Charge q is at the center of two equal charges. Hence, it experiences equal and opposite forces from the two charges which cancel each other. However, the remaining two charges each of magnitude Q experience forces from each other and charge q. Therefore, the system will be in equilibrium if the forces on the two charges also cancel each other. This is possible when the charge q at the center has a suitable magnitude.

Consider the diagram. The two charges are each of magnitude Q are at point A and B. Charge q is at the center of both of them and its distance from both charges is r. Now forces on the charge at point A due to charge q and Q (at B) are

respectively.

Similarly, forces on the charge at B due to charge q and Q at A are

So for the system to be in equilibrium,

 Cancelling the common terms k, Q and r²

Rearranging and simplifying

Hence proved.