A certain charge Q is divided into two parts q and Q-q. How the charge Q and q must be related so that when q and (Q-q) is placed at a certain distance apart experience maximum electrostatic repulsion?

Q = 2q

q = 3q

Q = 4q

Q= 4q + c

Solution

Q = 2q

The electrostatic force of repulsion between the charge q and (Q-q) at separation r is given by

$F = \frac{1}{4 {πε}_{o}} \frac{q (Q - q)}{r^{2}} = \frac{1}{4 {πε}_{0}} \frac{qQ - q^{2}}{r^{2}}$

If F is maximum then $\frac{\partial F}{\partial q} = 0$

i.e.

$\frac{1}{4 {πε}_{0}} . \frac{(Q - 2 q)}{r^{2}} = 0 As \frac{1}{4 {πε}_{0} r^{2}} is constant, therefore, Q - 2 q = 0 Q = 2 q$

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