A parallel plate condenser has a unifrom electric field `E (V//m)` in the space between the plates. If the distance between the plates is `d(m)` and area of each plate is `A(m^(2))` the energy (joule) stored in the condenser is

Options

A. `(1)/(2) epsilon_(0)E^(2)`
B. `epsilon_(0)EAd`
C. `(1)/(2)epsilon_(0)E^(2)Ad`
D. `E^(2)Ad//epsilon_(0)`

Answer

To find the energy stored in a parallel plate condenser (capacitor), we can use two different methods. Here, we will go through the calculations step by step.

### Step 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 formula:
\[ V = E \cdot d \]

### Step 2: Determine the capacitance of the parallel plate capacitor
The capacitance \( C \) of a parallel plate capacitor is given by the formula:
\[ C = \frac{A \epsilon_0}{d} \]
where \( A \) is the area of the plates and \( \epsilon_0 \) is the permittivity of free space.

### Step 3: Calculate the energy stored in the capacitor
The energy \( U \) stored in a capacitor can be calculated using the formula:
\[ U = \frac{1}{2} C V^2 \]
Substituting the expression for \( V \) from Step 1 into this equation gives:
\[ U = \frac{1}{2} C (E \cdot d)^2 \]

### Step 4: Substitute the expression for capacitance
Now, substitute the expression for \( C \) from Step 2 into the energy formula:
\[ U = \frac{1}{2} \left(\frac{A \epsilon_0}{d}\right) (E \cdot d)^2 \]

### Step 5: Simplify the expression
Now simplify the equation:
\[ U = \frac{1}{2} \cdot \frac{A \epsilon_0}{d} \cdot (E^2 \cdot d^2) \]
\[ U = \frac{1}{2} A \epsilon_0 E^2 \cdot d \]

### Final Result
Thus, the energy stored in the parallel plate capacitor is:
\[ U = \frac{1}{2} A \epsilon_0 E^2 \cdot d \quad \text{(in joules)} \]