question_answer

Two conducting spheres of radii r and 3 r initially have charges 3 q & 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 \[F=\frac{k\,x\,(4q-x)}{{{d}^{2}}}\]

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

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

\[\therefore \] 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 become equal and after that force will decrease.