Find the capacitance of a system of two identical metal balls of radius . a . if the distance between their centres in equal to .b., with to `b gt gta` The system is located in a uniform dielectric with relative permittivity `epsilon`

Options

A. `C=piepsilon_(0)epsilona`
B. `C=5piepsilon_(0)epsilona`
C. `C=2piepsilon_(0)epsilona`
D. `C=2piepsilon_(0)epsilon//a`

Answer

To find the capacitance of a system of two identical metal balls of radius \( a \) with the distance between their centers equal to \( b \) (where \( b \gg a \)), located in a uniform dielectric with relative permittivity \( \epsilon \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Configuration**:
- We have two identical metal spheres (balls) each with radius \( a \).
- The distance between their centers is \( b \), and it is given that \( b \gg a \).
- The system is placed in a dielectric medium characterized by its relative permittivity \( \epsilon \).

2. **Capacitance of a Single Sphere**:
- The capacitance \( C \) of a single isolated sphere in a medium with permittivity \( \epsilon \) is given by:
\[ C = 4 \pi \epsilon_0 \epsilon a \]
- Here, \( \epsilon_0 \) is the permittivity of free space.

3. **Capacitance of Each Sphere**:
- For both spheres, since they are identical, the capacitance for each sphere is:
\[ C_1 = C_2 = 4 \pi \epsilon_0 \epsilon a \]

4. **Combine the Capacitances**:
- When two capacitors are in series, the total capacitance \( C_{\text{net}} \) can be calculated using the formula:
\[ \frac{1}{C_{\text{net}}} = \frac{1}{C_1} + \frac{1}{C_2} \]
- Substituting the values of \( C_1 \) and \( C_2 \):
\[ \frac{1}{C_{\text{net}}} = \frac{1}{4 \pi \epsilon_0 \epsilon a} + \frac{1}{4 \pi \epsilon_0 \epsilon a} \]
- This simplifies to:
\[ \frac{1}{C_{\text{net}}} = \frac{2}{4 \pi \epsilon_0 \epsilon a} \]

5. **Calculate the Net Capacitance**:
- Inverting the equation gives:
\[ C_{\text{net}} = \frac{4 \pi \epsilon_0 \epsilon a}{2} = 2 \pi \epsilon_0 \epsilon a \]

6. **Final Result**:
- Thus, the capacitance of the system of two identical metal balls is:
\[ C_{\text{net}} = 2 \pi \epsilon_0 \epsilon a \]

### Conclusion:
The capacitance of the system of two identical metal balls is given by:
\[ C = 2 \pi \epsilon_0 \epsilon a \]