A parallel plate capacitor has a uniform electric field E in the space between the the plates. If the distance between the plates is d and area of each plate is A, the energy stored in the capacitor is

Options

A. `(1)/(2)epsi_(0)E^(2)`
B. `(E^(2)Ad)/(epsi_(0))`
C. `1/2epsi_(0)E^(2)Ad`
D. `eisi_(0)E^(2)Ad`

Explanation

To find the energy stored in a parallel plate capacitor with a uniform electric field \( E \), distance between the plates \( d \), and area of each plate \( A \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between electric field, voltage, and distance:**
The electric field \( E \) between the plates of a capacitor is related to the voltage \( V \) and the distance \( d \) between the plates by the equation:
\[ V = E \cdot d \]

2. **Determine the capacitance \( C \) of the parallel plate capacitor:**
The capacitance \( C \) of a parallel plate capacitor is given by:
\[ C = \frac{\varepsilon_0 A}{d} \]
where \( \varepsilon_0 \) is the permittivity of free space.

3. **Substitute the expression for voltage into the energy formula:**
The energy \( U \) stored in a capacitor is given by the formula:
\[ U = \frac{1}{2} C V^2 \]
Substituting \( V = E \cdot d \) into the energy equation gives:
\[ U = \frac{1}{2} C (E \cdot d)^2 \]

4. **Substitute the expression for capacitance \( C \):**
Now, substituting \( C = \frac{\varepsilon_0 A}{d} \) into the energy equation:
\[ U = \frac{1}{2} \left(\frac{\varepsilon_0 A}{d}\right) (E \cdot d)^2 \]

5. **Simplify the expression:**
Simplifying the equation:
\[ U = \frac{1}{2} \cdot \frac{\varepsilon_0 A}{d} \cdot E^2 \cdot d^2 \]
\[ U = \frac{1}{2} \varepsilon_0 A E^2 d \]

### Final Answer:
The energy stored in the parallel plate capacitor is:
\[ U = \frac{1}{2} \varepsilon_0 A E^2 d \]