A spherical capacitor consists of two concentric spherical shells of outer radius r1 and inner radius r2, held in position by suitable insulating supports. calculate the capacitance of this spherical capacitor.

Options

A. `(4piepsi_(0)r_(1)r_(2))/(r_(1)-r_(2))`
B. `(4piepsi_(0)-(r_(2)-r_(1)))/(r_(1)r_(2))`
C. `(r_(1)r_(2))/(4 pi epsi_(0)(r_(2)-r_(1)))`
D. `((r_(1)-r_(2)))/(4 pi epsi_(0)r_(1)r_(2))`

Explanation

To calculate the capacitance of a spherical capacitor consisting of two concentric spherical shells with outer radius \( r_1 \) and inner radius \( r_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two concentric spherical shells: the inner shell (radius \( r_2 \)) and the outer shell (radius \( r_1 \)).
- The inner shell is given a positive charge \( +Q \), which distributes uniformly over its surface.

2. **Induction Effect**:
- The positive charge on the inner shell induces a negative charge \( -Q \) on the inner surface of the outer shell (radius \( r_1 \)).
- Consequently, the outer surface of the outer shell will have a charge of \( +Q \) (since the outer shell is grounded, it will lose some positive charge to the ground).

3. **Electric Field Calculation**:
- The electric field \( E \) in the region between the two shells (from \( r_2 \) to \( r_1 \)) can be calculated using Gauss's law. The electric field due to a uniformly charged sphere at a distance \( r \) (where \( r_2 < r < r_1 \)) is given by:
\[ E = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q}{r^2} \]

4. **Potential Difference Calculation**:
- The potential difference \( V \) between the inner shell (at \( r_2 \)) and the outer shell (at \( r_1 \)) is given by:
\[ V = V_A - V_B = -\int_{r_2}^{r_1} E \, dr \]
- Substituting the expression for \( E \):
\[ V = -\int_{r_2}^{r_1} \frac{1}{4\pi \epsilon_0} \cdot \frac{Q}{r^2} \, dr \]
- Evaluating the integral:
\[ V = -\frac{Q}{4\pi \epsilon_0} \left[-\frac{1}{r}\right]_{r_2}^{r_1} = \frac{Q}{4\pi \epsilon_0} \left(\frac{1}{r_2} - \frac{1}{r_1}\right) \]

5. **Capacitance Calculation**:
- The capacitance \( C \) of the spherical capacitor is defined as:
\[ C = \frac{Q}{V} \]
- Substituting the expression for \( V \):
\[ C = \frac{Q}{\frac{Q}{4\pi \epsilon_0} \left(\frac{1}{r_2} - \frac{1}{r_1}\right)} = \frac{4\pi \epsilon_0}{\left(\frac{1}{r_2} - \frac{1}{r_1}\right)} \]
- Simplifying further:
\[ C = \frac{4\pi \epsilon_0 r_1 r_2}{r_1 - r_2} \]

### Final Result:
The capacitance of the spherical capacitor is given by:
\[ C = \frac{4\pi \epsilon_0 r_1 r_2}{r_1 - r_2} \]