To solve the problem step by step, we will follow the given data and apply the relevant formulas.
### Step 1: Identify the given values
- Capacitance, \( C = 9 \, \text{nF} = 9 \times 10^{-9} \, \text{F} \)
- Dielectric constant, \( \varepsilon_r = 2.4 \)
- Dielectric strength, \( E = 20 \, \text{MV/m} = 20 \times 10^{6} \, \text{V/m} \)
- Potential difference, \( V = 20 \, \text{V} \)
### Step 2: Calculate the distance between the plates (D)
The dielectric strength \( E \) is defined as the electric field strength, which can be expressed as:
\[
E = \frac{V}{D}
\]
Rearranging this formula gives:
\[
D = \frac{V}{E}
\]
Substituting the known values:
\[
D = \frac{20 \, \text{V}}{20 \times 10^{6} \, \text{V/m}} = \frac{20}{20 \times 10^{6}} = 10^{-6} \, \text{m}
\]
### Step 3: Use the capacitance formula for a capacitor with a dielectric
The capacitance \( C \) of a capacitor with a dielectric is given by:
\[
C = \frac{\varepsilon_0 \cdot A \cdot \varepsilon_r}{D}
\]
Where:
- \( \varepsilon_0 \) (the permittivity of free space) is approximately \( 8.85 \times 10^{-12} \, \text{F/m} \)
- \( A \) is the area of the plates
- \( D \) is the distance between the plates (calculated in Step 2)
### Step 4: Rearranging the capacitance formula to find the area \( A \)
Rearranging the formula to solve for \( A \):
\[
A = \frac{C \cdot D}{\varepsilon_0 \cdot \varepsilon_r}
\]
Substituting the known values:
\[
A = \frac{9 \times 10^{-9} \, \text{F} \cdot 10^{-6} \, \text{m}}{8.85 \times 10^{-12} \, \text{F/m} \cdot 2.4}
\]
### Step 5: Calculate the area \( A \)
Calculating the denominator:
\[
8.85 \times 10^{-12} \cdot 2.4 = 2.124 \times 10^{-11} \, \text{F/m}
\]
Now substituting back into the area formula:
\[
A = \frac{9 \times 10^{-9} \cdot 10^{-6}}{2.124 \times 10^{-11}} = \frac{9 \times 10^{-15}}{2.124 \times 10^{-11}} \approx 0.423 \times 10^{-3} \, \text{m}^2
\]
This can be expressed as:
\[
A \approx 4.23 \times 10^{-4} \, \text{m}^2
\]
### Final Answer
The area of the plates is approximately:
\[
A \approx 4.23 \times 10^{-4} \, \text{m}^2
\]
