A spherical capacitor consists of two conducting concentric spherical shells of radii a and b  $\left(a<b\right)$:

detailed solution

Correct option is A

If outer shell is earthed then its potential becomes equal to zero. When a battery is connected between two shells, an electric field gets established in the space between the shells only. Hence given system will act as a spherical capacitor having radii a & b. Therefore, its capacitance will be equal to 4π∈0abb−a .Hence (A) is correct.When inner shell is earthed then the potential of this shell becomes equal to zero. Potential at infinity is also equal to zero. Hence, when a battery will be connected between the shells, an electric field will be established not only in the space between the shells but outside the shells also. Therefore, the system may be considered as a parallel combination of two capacitors.A spherical capacitor having radii a and b respectively. Its capacitance is C1=4π∈0abb−a .A spherical capacitor of radii b and infinite or an isolated sphere of radius b.Its capacitance is C2=4∈0b The equivalent capacitance is given by  C=C1+C2=4π∈0abb−a+4π∈0b=4π∈0b2b−a