The distance between the circular plates of a parallel plate condenser 40 mm in diameter, in order to have same capacity as a sphere of radius 1 m is

Options

A. 0.01 mm
B. 0.1 mm
C. 1.0 mm
D. 10 mm

Answer

To find the distance between the circular plates of a parallel plate capacitor that has the same capacitance as a sphere of radius 1 m, we can follow these steps:

### Step 1: Understand the given data
- Diameter of the parallel plate capacitor (d) = 40 mm
- Radius of the sphere (R) = 1 m

### Step 2: Convert the diameter to radius
The radius (r) of the parallel plate capacitor is half of the diameter:
\[ r = \frac{d}{2} = \frac{40 \text{ mm}}{2} = 20 \text{ mm} = 20 \times 10^{-3} \text{ m} \]

### Step 3: Write the formula for capacitance of a parallel plate capacitor
The capacitance (C1) of a parallel plate capacitor is given by:
\[ C_1 = \frac{\epsilon_0 A}{d} \]
where \(A\) is the area of the plates and \(d\) is the distance between the plates.

The area \(A\) of the circular plates is:
\[ A = \pi r^2 \]
Substituting this into the capacitance formula gives:
\[ C_1 = \frac{\epsilon_0 \pi r^2}{d} \]

### Step 4: Write the formula for capacitance of a sphere
The capacitance (C2) of an isolated sphere is given by:
\[ C_2 = 4 \pi \epsilon_0 R \]

### Step 5: Set the capacitances equal to each other
Since we want the capacitance of the parallel plate capacitor to be equal to that of the sphere:
\[ C_1 = C_2 \]
This leads to:
\[ \frac{\epsilon_0 \pi r^2}{d} = 4 \pi \epsilon_0 R \]

### Step 6: Simplify the equation
We can cancel \(\epsilon_0\) and \(\pi\) from both sides:
\[ \frac{r^2}{d} = 4R \]

### Step 7: Solve for \(d\)
Rearranging the equation gives:
\[ d = \frac{r^2}{4R} \]

### Step 8: Substitute the values
Substituting \(r = 20 \times 10^{-3} \text{ m}\) and \(R = 1 \text{ m}\):
\[ d = \frac{(20 \times 10^{-3})^2}{4 \times 1} \]
Calculating \( (20 \times 10^{-3})^2 \):
\[ (20 \times 10^{-3})^2 = 400 \times 10^{-6} \text{ m}^2 \]
Now substituting back:
\[ d = \frac{400 \times 10^{-6}}{4} = 100 \times 10^{-6} \text{ m} \]

### Step 9: Convert to mm
To convert meters to millimeters:
\[ d = 100 \times 10^{-6} \text{ m} = 0.1 \text{ mm} \]

### Final Answer
The distance \(d\) between the circular plates of the parallel plate capacitor is:
\[ \boxed{0.1 \text{ mm}} \]