A spherical condenser has innder and outer spheres of radii a and b respectively. The space between the two is filled with air. The difference between the capacities of two condensers formed when outer sphere is earthed and when inner sphere is earthed will be

Options

A. zero
B. `4 pi epsilon_(0)a`
C. `4 pi epsilon_(0)b`
D. `4pi epsilon_(0)a((b)/(b-a))`

Answer

To solve the problem, we need to calculate the capacitance of a spherical capacitor in two different scenarios: when the outer sphere is earthed and when the inner sphere is earthed. We will then find the difference between these two capacitances.

### Step-by-Step Solution:

1. **Understanding the Spherical Capacitor:**
- A spherical capacitor consists of two concentric spherical conductors: an inner sphere of radius \( a \) and an outer sphere of radius \( b \).
- The space between the two spheres is filled with air.

2. **Capacitance when Outer Sphere is Earthed (C1):**
- When the outer sphere is earthed, it is at zero potential.
- The potential \( V \) at the inner sphere (radius \( a \)) due to a charge \( Q \) on it is given by:
\[ V = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q}{a} \]
- The potential at the outer sphere (radius \( b \)) is zero because it is earthed.
- The capacitance \( C_1 \) is given by:
\[ C_1 = \frac{Q}{V} = \frac{Q}{\frac{Q}{4\pi \epsilon_0 a}} = 4\pi \epsilon_0 \frac{ab}{b-a} \]

3. **Capacitance when Inner Sphere is Earthed (C2):**
- When the inner sphere is earthed, its potential is zero.
- The potential \( V \) at the outer sphere (radius \( b \)) due to a charge \( Q \) on the outer sphere is:
\[ V = \frac{1}{4\pi \epsilon_0} \cdot \left( \frac{Q}{b} - \frac{Q}{a} \right) \]
- Setting the potential of the inner sphere to zero gives:
\[ \frac{Q}{b} - \frac{Q}{a} = 0 \implies Q = \frac{Q}{b} \cdot a \]
- The capacitance \( C_2 \) is given by:
\[ C_2 = \frac{Q}{V} = \frac{Q}{\frac{Q}{4\pi \epsilon_0 b} - \frac{Q}{4\pi \epsilon_0 a}} = 4\pi \epsilon_0 \frac{ab}{a-b} \]

4. **Finding the Difference in Capacitance:**
- We need to find the difference \( C_2 - C_1 \):
\[ C_2 - C_1 = 4\pi \epsilon_0 \left( \frac{ab}{a-b} - \frac{ab}{b-a} \right) \]
- Simplifying this expression:
\[ C_2 - C_1 = 4\pi \epsilon_0 \left( \frac{ab(b-a) - ab(a-b)}{(a-b)(b-a)} \right) \]
- This simplifies to:
\[ C_2 - C_1 = 4\pi \epsilon_0 a \]

5. **Final Result:**
- The difference between the capacitances when the outer sphere is earthed and when the inner sphere is earthed is:
\[ \boxed{4\pi \epsilon_0 a} \]