What is the Capacitance of a spherical capacitor when either Inner/outer shell is grounded (Earthed) The capacitance of a spherical capacitor is given by $$C=\frac{4\pi \varepsilon_{0}}{\left ( \frac{1}{a}-\frac{1}{b} \right )}$$
 
How can I find out the capacitance of the capacitor when


*

*The inner sphere is earthed.

*The outer hollow sphere is earthed.

 A: Consider the following cases in relation to your question:


*

*Inner sphere is grounded.


a) grounding the outer surface of the inner sphere
If you ground the outer surface of the inner sphere, the inner sphere becomes irrelevant and you get single spherical capacitor (the other one at infinity) of radius b. The capacitance is now $4\pi\epsilon_{o}b$.
b) grounding the inner surface of the inner sphere
Now charges can be stored on the outer surface of the inner sphere, inner surface of the outer sphere and outer surface of the outer sphere.
So you have a spherical capacitor system as usual along with a single spherical capacitor of radius b
Total capacitance is now: $4\pi\epsilon_{o}\dfrac{ab}{b-a}+4\pi\epsilon_{o}b$.


*Outer sphere is grounded.


a) grounding the outer surface of the outer sphere
Capacitance due to usual system, given by: $4\pi\epsilon_{o}\dfrac{ab}{b-a}$
b) grounding the inner surface of the outer sphere
If this is done, then irrespective of whether we induce charge on the outer surface of the outer sphere or on the inner surface of the inner sphere, the only charge that will be relevant is what is induced on the outer surface of the outer sphere. 
This gives the capacitance of an isolated capacitor: $4\pi\epsilon_{o}b$.
