A `5.80 muF` parallel-plate air capacitor has a plate separation of `5.00 mm` and is charged to a potential difference of `400 V`. Calculate the energy density in the region between the plates, in `J/m^3`

Options

A. `0.113`
B. `0.117`
C. `0.152`
D. none of these

Explanation

To calculate the energy density in the region between the plates of the capacitor, we can follow these steps:

### Step 1: Understand the formula for energy density
The energy density (u) in the electric field between the plates of a capacitor is given by the formula:
\[ u = \frac{1}{2} \epsilon_0 E^2 \]
where:
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)
- \( E \) is the electric field between the plates.

### Step 2: Calculate the electric field (E)
The electric field (E) between the plates of a parallel-plate capacitor can be calculated using the formula:
\[ E = \frac{V}{d} \]
where:
- \( V \) is the potential difference (400 V)
- \( d \) is the separation between the plates (5 mm = \( 5 \times 10^{-3} \) m)

Substituting the values:
\[ E = \frac{400 \, \text{V}}{5 \times 10^{-3} \, \text{m}} = \frac{400}{0.005} = 80000 \, \text{V/m} \]

### Step 3: Substitute E into the energy density formula
Now, we substitute the value of \( E \) into the energy density formula:
\[ u = \frac{1}{2} \epsilon_0 E^2 \]
Substituting the values:
\[ u = \frac{1}{2} \times 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \times (80000 \, \text{V/m})^2 \]

### Step 4: Calculate \( E^2 \)
First, calculate \( E^2 \):
\[ E^2 = (80000)^2 = 6400000000 \, \text{V}^2/\text{m}^2 \]

### Step 5: Calculate the energy density (u)
Now substitute \( E^2 \) back into the energy density formula:
\[ u = \frac{1}{2} \times 8.85 \times 10^{-12} \times 6400000000 \]
Calculating this gives:
\[ u = \frac{1}{2} \times 8.85 \times 10^{-12} \times 6.4 \times 10^9 \]
\[ u = 0.02832 \, \text{J/m}^3 \]

### Step 6: Round off the answer
Rounding off to three significant figures, we get:
\[ u \approx 0.028 \, \text{J/m}^3 \]

### Final Answer
The energy density in the region between the plates is approximately:
\[ \boxed{0.028 \, \text{J/m}^3} \]

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