To solve the problem step by step, we will follow the outlined procedure to find the minimum area of the plates required for the capacitor.
### Step 1: Identify the given values
- Potential \( V = 1 \, \text{kV} = 1000 \, \text{V} \)
- Dielectric constant \( K = 6\pi \)
- Dielectric strength \( = 10^7 \, \text{V/m} \)
- Maximum electric field \( E_{max} = 10\% \) of dielectric strength \( = 0.1 \times 10^7 \, \text{V/m} = 10^6 \, \text{V/m} \)
- Capacitance \( C = 50 \, \text{pF} = 50 \times 10^{-12} \, \text{F} \)
- Permittivity of free space \( \epsilon_0 = \frac{1}{36\pi} \times 10^{-9} \, \text{F/m} \)
### Step 2: Calculate the distance \( D \) between the plates
Using the relationship between potential \( V \), electric field \( E \), and distance \( D \):
\[
V = E \cdot D
\]
Substituting the known values:
\[
1000 = 10^6 \cdot D
\]
Solving for \( D \):
\[
D = \frac{1000}{10^6} = 10^{-3} \, \text{m} = 1 \, \text{mm}
\]
### Step 3: Use the capacitance formula for a parallel plate capacitor
The capacitance \( C \) of a parallel plate capacitor with a dielectric is given by:
\[
C = \frac{K \cdot \epsilon_0 \cdot A}{D}
\]
Rearranging to find the area \( A \):
\[
A = \frac{C \cdot D}{K \cdot \epsilon_0}
\]
### Step 4: Substitute the known values into the area formula
Substituting \( C = 50 \times 10^{-12} \, \text{F} \), \( D = 10^{-3} \, \text{m} \), \( K = 6\pi \), and \( \epsilon_0 = \frac{1}{36\pi} \times 10^{-9} \):
\[
A = \frac{(50 \times 10^{-12}) \cdot (10^{-3})}{(6\pi) \cdot \left(\frac{1}{36\pi} \times 10^{-9}\right)}
\]
### Step 5: Simplify the expression
Calculating the denominator:
\[
(6\pi) \cdot \left(\frac{1}{36\pi} \times 10^{-9}\right) = \frac{6 \cdot 10^{-9}}{36} = \frac{10^{-9}}{6}
\]
Now substituting back into the area formula:
\[
A = \frac{(50 \times 10^{-12}) \cdot (10^{-3})}{\frac{10^{-9}}{6}} = (50 \times 10^{-12}) \cdot (10^{-3}) \cdot \left(\frac{6}{10^{-9}}\right)
\]
\[
A = 50 \times 10^{-12} \cdot 10^{-3} \cdot 6 \cdot 10^{9} = 300 \times 10^{-6} \, \text{m}^2
\]
### Step 6: Convert the area to mm²
\[
A = 300 \times 10^{-6} \, \text{m}^2 = 300 \, \text{mm}^2
\]
### Final Answer
The minimum area of the plates required for safe working is \( 300 \, \text{mm}^2 \).
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