The capacity of a spherical conductor is

Options

A. `(R)/(4 pi epsilon_(0))`
B. `(4 pi epsilon_(0))/(R)`
C. `4 pi epsilon_(0)R`
D. `4 pi epsilon_(0)R^(2)`

Answer

To find the capacitance of a spherical conductor, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Capacitance**:
- Capacitance (C) is defined as the ability of a conductor to store electric charge per unit potential difference (V) across it. Mathematically, it is given by:
\[ C = \frac{Q}{V} \]
where \( Q \) is the charge on the conductor and \( V \) is the potential difference.

2. **Electric Field of a Spherical Conductor**:
- For a spherical conductor with charge \( Q \), the electric field \( E \) at a distance \( r \) from the center is given by:
\[ E = \frac{kQ}{r^2} \]
where \( k \) is Coulomb's constant.

3. **Potential Difference Calculation**:
- To find the potential \( V \) at the surface of the conductor, we need to calculate the potential difference from infinity to the surface (at distance \( R \)):
\[ V = -\int_{\infty}^{R} E \, dr \]
- Substituting the expression for \( E \):
\[ V = -\int_{\infty}^{R} \frac{kQ}{r^2} \, dr \]
- This integral evaluates to:
\[ V = -\left[-\frac{kQ}{r}\right]_{\infty}^{R} = \left(0 - \left(-\frac{kQ}{R}\right)\right) = \frac{kQ}{R} \]

4. **Substituting into Capacitance Formula**:
- Now, substituting \( V \) back into the capacitance formula:
\[ C = \frac{Q}{V} = \frac{Q}{\frac{kQ}{R}} = \frac{R}{k} \]

5. **Using the Value of \( k \)**:
- Recall that \( k = \frac{1}{4\pi \epsilon_0} \), where \( \epsilon_0 \) is the permittivity of free space. Therefore:
\[ C = \frac{R}{\frac{1}{4\pi \epsilon_0}} = 4\pi \epsilon_0 R \]

6. **Final Expression for Capacitance**:
- Thus, the capacitance of a spherical conductor is:
\[ C = 4\pi \epsilon_0 R \]