question_answer

The conducting spheres of radii r and 3r initially have charges 3q & q respectively. Their separation is much larger than their radii. If they are joined by a conductor of high resistance, the force between them will -

A) Increase continuously

B) Decrease continuously

C) First increase, then decrease

D) First decrease, then increase

Correct Answer: C

Solution :

Let charge on smaller sphere be x and on larger sphere be \[4q-x\]

Force between them is given by

\[F=\frac{kx\,\,(4q-x)}{{{d}^{2}}}\]

\[\frac{dF}{dx}=0\Rightarrow 4q-2x=0\Rightarrow x=2q\]

\[\frac{{{d}^{2}}F}{d{{x}^{2}}}=\frac{K}{{{d}^{2}}}(-\,2)<0\]

\[\therefore \]It represents a maximum.

Final charges on the smaller sphere and the larger sphere are q & 3q respectively as required by the equality of potentials

\[\therefore \]Force will increase until the charges becomes equal and after that force will decreases.