A spherical capacitor has an inner sphere of radius 10 cm and outer sphere of radius 12 cm . The outer sphere is earthed and the inner sphere is given a charge of `5 mu C` . The capacitance of the capacitor is

Options

A. `2.40 ( pi epsilon_(0))`
B. `4.80 ( pi epsilon_(0))`
C. `2.00 ( pi epsilon_(0))`
D. ` 3.00 ( pi epsilon_(0))`

Answer

To find the capacitance of a spherical capacitor, we can use the formula for the capacitance \( C \) of a spherical capacitor, which is given by:

\[ C = \frac{4 \pi \epsilon_0 R_1 R_2}{R_2 - R_1} \]

where:
- \( R_1 \) is the radius of the inner sphere,
- \( R_2 \) is the radius of the outer sphere,
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{F/m} \).

### Step 1: Identify the radii
From the problem, we have:
- Inner sphere radius \( R_1 = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Outer sphere radius \( R_2 = 12 \, \text{cm} = 0.12 \, \text{m} \)

### Step 2: Substitute the values into the capacitance formula
Now, substituting the values into the capacitance formula:

\[ C = \frac{4 \pi (8.85 \times 10^{-12}) (0.1)(0.12)}{0.12 - 0.1} \]

### Step 3: Calculate the denominator
Calculate the difference in the radii:

\[ R_2 - R_1 = 0.12 - 0.1 = 0.02 \, \text{m} \]

### Step 4: Calculate the numerator
Now calculate the numerator:

\[ 4 \pi (8.85 \times 10^{-12}) (0.1)(0.12) = 4 \pi (8.85 \times 10^{-12}) (0.012) \]

Calculating this gives:

\[ = 4 \times 3.14 \times 8.85 \times 10^{-12} \times 0.012 \approx 1.675 \times 10^{-13} \, \text{F} \]

### Step 5: Calculate the capacitance
Now, substituting the values back into the capacitance formula:

\[ C = \frac{1.675 \times 10^{-13}}{0.02} \approx 8.375 \times 10^{-12} \, \text{F} \]

### Step 6: Convert to microfarads
To express the capacitance in microfarads:

\[ C \approx 8.375 \times 10^{-12} \, \text{F} = 8.375 \, \text{pF} \]

### Final Answer
The capacitance of the spherical capacitor is approximately \( 8.375 \, \text{pF} \).

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