Question

A charge Q is to be distributed on two conducting spheres. What should be the value of charges on the spheres so that the repulsive force between them is maximum when they are placed at a fixed distance from each other in the air.

Answer 1

Step 1, Given data

Charge q is distributed on two conducting spheres

This can be represented as

Step 2, Finding the value of the charge

We can write from the figure

$Repulsiveforce=\frac{K\times Q_1\times (Q-Q_1)}{d^2}$

$Or,F=\frac{K}{d^2}(Q{Q}_1-{Q_1}^2)$

For the force to be maximum

$\frac{dF}{d{Q}_1}=0$

Or we can write

$\frac{K}{d^2}(Q-2Q_1)=0$

$Or,Q_1=\frac{Q}{2}$

So we can say that

Force is maximum when $Q_1=\frac{Q}{2}$

Hence the charge Q should be divided in two equal halves.