MHT-CET Physics Electrostatics PYQ - Charge Sharing Between Spheres

21) A large insulated sphere of radius \(r\), charged with \(Q\) units of electricity, is placed in contact with a small insulated uncharged sphere of radius \(R\) and is then separated. The charge on the smaller sphere will now be:

A. \( Q(r + R) \)

B. \( Q(r - R) \)

C. \( \frac{Q}{r + R} \)

D. \( \frac{QR}{r + R} \)

Solution:

When two conductive spheres are brought into contact, charge distributes between them in proportion to their capacitances. For an isolated sphere, the capacitance is proportional to its radius. Thus, for spheres with radii \(r\) and \(R\):

The capacitance of the large sphere is proportional to \(r\) and that of the small sphere is proportional to \(R\). When they are connected, the charge divides in the ratio of their radii.

Let the charge on the smaller sphere after contact be \(Q_s\) and on the larger sphere be \(Q_L\). Since charge is conserved: \[ Q_s + Q_L = Q. \]

And because the charge divides in the ratio of \(R : r\): \[ \frac{Q_s}{Q_L} = \frac{R}{r}. \]

From this, we can write: \[ Q_s = \frac{R}{R + r} Q. \]

Therefore, the charge on the smaller sphere is: \[ \frac{QR}{r + R}. \]

Hence, the correct answer is option D.