The dielectric to be used in a parallel-plate capacitor has a dielectric constant of 3.60 and a dielectric strength of `1.60 xx 10^7 V//m`. The capacitor is to have a capacitance of `1.25 xx10^-9F` and must be able to withstand a maximum potential difference of `5500 V`. What is the minimum area the plates of the capacitor may have?

Explanation

To find the minimum area of the plates of a parallel-plate capacitor given the dielectric properties and the maximum potential difference, we can follow these steps:

### Step 1: Understand the relationship between capacitance, area, and dielectric properties
The capacitance \( C \) of a parallel-plate capacitor is given by the formula:
\[ C = \frac{A \epsilon_r \epsilon_0}{d} \]
where:
- \( C \) is the capacitance,
- \( A \) is the area of the plates,
- \( \epsilon_r \) is the dielectric constant,
- \( \epsilon_0 \) is the permittivity of free space (\( 8.85 \times 10^{-12} \, \text{F/m} \)),
- \( d \) is the separation between the plates.

### Step 2: Calculate the maximum separation \( d \)
The maximum separation \( d \) can be determined using the dielectric strength \( E_{max} \) and the maximum potential difference \( V_{max} \):
\[ d = \frac{V_{max}}{E_{max}} \]
Given:
- \( V_{max} = 5500 \, \text{V} \)
- \( E_{max} = 1.60 \times 10^7 \, \text{V/m} \)

Substituting the values:
\[ d = \frac{5500}{1.60 \times 10^7} = 3.4375 \times 10^{-4} \, \text{m} \]

### Step 3: Rearrange the capacitance formula to solve for area \( A \)
Rearranging the capacitance formula to find \( A \):
\[ A = \frac{C \cdot d}{\epsilon_r \cdot \epsilon_0} \]

### Step 4: Substitute the known values into the area formula
Given:
- \( C = 1.25 \times 10^{-9} \, \text{F} \)
- \( \epsilon_r = 3.60 \)
- \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \)

Substituting these values into the area formula:
\[ A = \frac{(1.25 \times 10^{-9}) \cdot (3.4375 \times 10^{-4})}{3.60 \cdot (8.85 \times 10^{-12})} \]

### Step 5: Calculate the area \( A \)
Calculating the numerator:
\[ 1.25 \times 10^{-9} \times 3.4375 \times 10^{-4} = 4.296875 \times 10^{-13} \]

Calculating the denominator:
\[ 3.60 \times 8.85 \times 10^{-12} = 3.186 \times 10^{-11} \]

Now, substituting these into the area equation:
\[ A = \frac{4.296875 \times 10^{-13}}{3.186 \times 10^{-11}} \approx 0.0135 \, \text{m}^2 \]

### Final Result
The minimum area of the plates of the capacitor is approximately:
\[ A \approx 0.0135 \, \text{m}^2 \]

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