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 given the uniform electric field \( E \), distance between the plates \( d \), and area of each plate \( A \), we can follow these steps:

### Step 1: Understand the relationship between capacitance, voltage, and energy
The energy \( U \) stored in a capacitor can be expressed as:
\[ U = \frac{1}{2} C V^2 \]
where \( C \) is the capacitance and \( V \) is the voltage across the capacitor.

### Step 2: Determine the capacitance of a parallel plate capacitor
The capacitance \( C \) of a parallel plate capacitor is given by the formula:
\[ C = \frac{\varepsilon_0 A}{d} \]
where \( \varepsilon_0 \) is the permittivity of free space, \( A \) is the area of the plates, and \( d \) is the distance between the plates.

### Step 3: Relate the electric field to voltage
The electric field \( E \) between the plates is related to the voltage \( V \) and the distance \( d \) by the equation:
\[ E = \frac{V}{d} \]
From this, we can express the voltage \( V \) as:
\[ V = E \cdot d \]

### Step 4: Substitute the expressions for capacitance and voltage into the energy formula
Now, substituting \( C \) and \( V \) into the energy formula:
\[ U = \frac{1}{2} \left(\frac{\varepsilon_0 A}{d}\right) (E \cdot d)^2 \]

### Step 5: Simplify the expression
Substituting \( V \) into the energy equation gives:
\[ U = \frac{1}{2} \left(\frac{\varepsilon_0 A}{d}\right) (E^2 d^2) \]
This simplifies to:
\[ U = \frac{1}{2} \varepsilon_0 A E^2 d \]

### Step 6: Final expression for energy stored in the capacitor
Thus, the energy stored in the parallel plate capacitor is:
\[ U = \frac{1}{2} \varepsilon_0 A E^2 d \]

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

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