The plates of a parallel-plate capacitor in vacuum are `5.00 mm` apart and `2.00 m^2` in area. A potential difference of `10,000 V` is applied across the capacitor. Compute
(a) the capacitance
(b) the charge on each plate, and
(c) the magnitude of the electric field in the space between them.

Explanation

To solve the problem step by step, we will compute the capacitance, charge on each plate, and the magnitude of the electric field between the plates of the parallel-plate capacitor.

### Given Data:
- Distance between plates, \( D = 5.00 \, \text{mm} = 5.00 \times 10^{-3} \, \text{m} \)
- Area of plates, \( A = 2.00 \, \text{m}^2 \)
- Potential difference, \( V = 10,000 \, \text{V} \)
- Permittivity of free space, \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \)

### (a) Calculate the Capacitance
The capacitance \( C \) of a parallel-plate capacitor is given by the formula:
\[ C = \frac{\epsilon_0 A}{D} \]

Substituting the values:
\[ C = \frac{(8.85 \times 10^{-12} \, \text{F/m}) \times (2.00 \, \text{m}^2)}{5.00 \times 10^{-3} \, \text{m}} \]

Calculating this:
\[ C = \frac{17.7 \times 10^{-12}}{5.00 \times 10^{-3}} = 3.54 \times 10^{-9} \, \text{F} \]

### (b) Calculate the Charge on Each Plate
The charge \( Q \) on each plate of the capacitor can be calculated using the formula:
\[ Q = C \times V \]

Substituting the values:
\[ Q = (3.54 \times 10^{-9} \, \text{F}) \times (10,000 \, \text{V}) \]

Calculating this:
\[ Q = 3.54 \times 10^{-5} \, \text{C} \]

### (c) Calculate the Magnitude of the Electric Field
The electric field \( E \) between the plates of a parallel-plate capacitor can be calculated using the formula:
\[ E = \frac{V}{D} \]

Substituting the values:
\[ E = \frac{10,000 \, \text{V}}{5.00 \times 10^{-3} \, \text{m}} \]

Calculating this:
\[ E = 2.00 \times 10^{6} \, \text{V/m} \]

### Summary of Results:
- (a) Capacitance, \( C = 3.54 \times 10^{-9} \, \text{F} \)
- (b) Charge on each plate, \( Q = 3.54 \times 10^{-5} \, \text{C} \)
- (c) Electric field, \( E = 2.00 \times 10^{6} \, \text{V/m} \)