An air capacitor has plates 6cm in diameter. At what distance should the plates be placed so as to have the same capacitance as a sphere of diameter 90cm.

Explanation

To solve the problem of finding the distance between the plates of a parallel plate capacitor such that it has the same capacitance as a spherical capacitor, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values:**
- Diameter of the parallel plate capacitor: \(6 \, \text{cm}\)
- Diameter of the spherical capacitor: \(90 \, \text{cm}\)

2. **Calculate the radius of the plates:**
- The radius \(r\) of the parallel plate capacitor is half of the diameter:
\[ r = \frac{6 \, \text{cm}}{2} = 3 \, \text{cm} = 0.03 \, \text{m} \]

3. **Calculate the radius of the spherical capacitor:**
- The radius \(R\) of the spherical capacitor is half of its diameter:
\[ R = \frac{90 \, \text{cm}}{2} = 45 \, \text{cm} = 0.45 \, \text{m} \]

4. **Write the formula for the capacitance of a parallel plate capacitor:**
- The capacitance \(C\) of a parallel plate capacitor is given by:
\[ C = \frac{\varepsilon_0 A}{d} \]
- Where \(A\) is the area of one plate and \(d\) is the distance between the plates.

5. **Calculate the area \(A\) of the plates:**
- The area \(A\) of a circular plate is given by:
\[ A = \pi r^2 = \pi (0.03)^2 = \pi \times 0.0009 \, \text{m}^2 = 0.002827 \, \text{m}^2 \]

6. **Write the formula for the capacitance of a spherical capacitor:**
- The capacitance \(C\) of a spherical capacitor is given by:
\[ C = 4 \pi \varepsilon_0 R \]

7. **Set the capacitances equal to each other:**
- To find the distance \(d\) such that the capacitance of the parallel plate capacitor equals that of the spherical capacitor:
\[ \frac{\varepsilon_0 A}{d} = 4 \pi \varepsilon_0 R \]

8. **Cancel \(\varepsilon_0\) from both sides:**
- This simplifies to:
\[ \frac{A}{d} = 4 \pi R \]

9. **Rearrange to find \(d\):**
- Rearranging gives:
\[ d = \frac{A}{4 \pi R} \]

10. **Substitute the values:**
- Substitute \(A = 0.002827 \, \text{m}^2\) and \(R = 0.45 \, \text{m}\):
\[ d = \frac{0.002827}{4 \pi (0.45)} \]

11. **Calculate \(d\):**
- First calculate \(4 \pi (0.45)\):
\[ 4 \pi (0.45) \approx 5.6549 \]
- Now calculate \(d\):
\[ d = \frac{0.002827}{5.6549} \approx 0.000499 \, \text{m} = 0.499 \, \text{mm} \approx 0.5 \, \text{mm} \]

### Final Answer:
The distance \(d\) between the plates should be approximately \(0.5 \, \text{mm}\).