A parallel plate air capacitor consists of two circular plates of diameter `8 cm` . At what distance should the plates be held so as to have the same capacitance as that of a sphere of a diameter `20 cm` ?

Explanation

To solve the problem, we need to find the distance between the plates of a parallel plate capacitor such that its capacitance is equal to that of a sphere with a given diameter. Here are the steps to derive the solution:

### Step 1: Find the Capacitance of the Sphere
The formula for the capacitance \( C \) of a sphere is given by:
\[ C_{\text{sphere}} = 4 \pi \epsilon_0 r \]
where \( r \) is the radius of the sphere and \( \epsilon_0 \) is the permittivity of free space.

Given the diameter of the sphere is \( 20 \, \text{cm} \), the radius \( r \) can be calculated as:
\[ r = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} = 0.1 \, \text{m} \]

Now substituting the value of \( r \):
\[ C_{\text{sphere}} = 4 \pi \epsilon_0 (0.1) \]

### Step 2: Find the Capacitance of the Parallel Plate Capacitor
The capacitance \( C \) of a parallel plate capacitor is given by:
\[ C_{\text{plates}} = \frac{\epsilon_0 A}{d} \]
where \( A \) is the area of one of the plates and \( d \) is the distance between the plates.

The area \( A \) of a circular plate can be calculated using the formula:
\[ A = \pi r^2 \]
Given the diameter of the plates is \( 8 \, \text{cm} \), the radius \( r \) is:
\[ r = \frac{8 \, \text{cm}}{2} = 4 \, \text{cm} = 0.04 \, \text{m} \]

Now substituting the value of \( r \) into the area formula:
\[ A = \pi (0.04)^2 = \pi (0.0016) \approx 0.0050265 \, \text{m}^2 \]

### Step 3: Set the Capacitances Equal
To find the distance \( d \) such that the capacitance of the parallel plate capacitor equals that of the sphere:
\[ 4 \pi \epsilon_0 (0.1) = \frac{\epsilon_0 A}{d} \]

### Step 4: Cancel \( \epsilon_0 \) and Rearrange
Cancelling \( \epsilon_0 \) from both sides:
\[ 4 \pi (0.1) = \frac{A}{d} \]
Rearranging gives:
\[ d = \frac{A}{4 \pi (0.1)} \]

### Step 5: Substitute the Area
Substituting the area \( A \):
\[ d = \frac{0.0050265}{4 \pi (0.1)} \]

### Step 6: Calculate \( d \)
Calculating \( d \):
\[ d = \frac{0.0050265}{4 \times 3.14159 \times 0.1} \approx \frac{0.0050265}{1.25664} \approx 0.004 \, \text{m} = 4 \, \text{mm} \]

### Final Answer
The distance between the plates should be \( 4 \, \text{mm} \).

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