Calculate capacitance of a parallel plate capacitor with area of each plate 1 `cm^(2)` and separation 1 mm.

Answer

To calculate the capacitance of a parallel plate capacitor, we can use the formula:

\[ C = \frac{A \epsilon_0}{d} \]

Where:
- \( C \) is the capacitance,
- \( A \) is the area of one plate,
- \( \epsilon_0 \) is the permittivity of free space (approximately \( 8.85 \times 10^{-12} \, \text{F/m} \)),
- \( d \) is the separation between the plates.

### Step-by-Step Solution:

**Step 1: Convert the area from cm² to m².**
- Given area \( A = 1 \, \text{cm}^2 \).
- To convert cm² to m², we use the conversion factor \( 1 \, \text{cm}^2 = 10^{-4} \, \text{m}^2 \).

\[ A = 1 \, \text{cm}^2 = 1 \times 10^{-4} \, \text{m}^2 \]

**Step 2: Convert the separation from mm to m.**
- Given separation \( d = 1 \, \text{mm} \).
- To convert mm to m, we use the conversion factor \( 1 \, \text{mm} = 10^{-3} \, \text{m} \).

\[ d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \]

**Step 3: Substitute the values into the capacitance formula.**
- Now we can substitute \( A \), \( \epsilon_0 \), and \( d \) into the capacitance formula.

\[ C = \frac{A \epsilon_0}{d} = \frac{(1 \times 10^{-4} \, \text{m}^2)(8.85 \times 10^{-12} \, \text{F/m})}{1 \times 10^{-3} \, \text{m}} \]

**Step 4: Simplify the expression.**
- Calculate the numerator:

\[ A \epsilon_0 = 1 \times 10^{-4} \times 8.85 \times 10^{-12} = 8.85 \times 10^{-16} \, \text{F m} \]

- Now divide by \( d \):

\[ C = \frac{8.85 \times 10^{-16} \, \text{F m}}{1 \times 10^{-3} \, \text{m}} = 8.85 \times 10^{-13} \, \text{F} \]

**Step 5: Final result.**
- The capacitance of the parallel plate capacitor is:

\[ C = 8.85 \times 10^{-13} \, \text{F} \, \text{or} \, 0.885 \, \text{pF} \]