The energy required to charge a parallel plate condenser of plate separtion `d` and plate area of cross-section `A` such that the unifom field between the plates is `E` is

Options

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

Explanation

To find the energy required to charge a parallel plate capacitor with plate separation \( d \) and plate area \( A \), such that the uniform electric field between the plates is \( E \), we can follow these steps:

### Step 1: Understand the relationship between electric field, voltage, and plate separation
The electric field \( E \) between the plates of a parallel plate capacitor is related to the voltage \( V \) across the plates and the separation \( d \) by the formula:
\[ E = \frac{V}{d} \]
From this, we can express the voltage as:
\[ V = E \cdot d \]

### Step 2: Calculate the capacitance of the 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.

### Step 3: Calculate the energy supplied by the cell to charge the capacitor
The energy \( U \) supplied by the cell to charge the capacitor can be calculated using the formula:
\[ U = C \cdot V^2 \]
Substituting the expressions for \( C \) and \( V \) into this equation gives:
\[ U = \left(\frac{\varepsilon_0 A}{d}\right) \cdot (E \cdot d)^2 \]

### Step 4: Simplify the expression for energy
Now, simplify the expression:
\[ U = \frac{\varepsilon_0 A}{d} \cdot (E^2 \cdot d^2) \]
\[ U = \varepsilon_0 A E^2 \cdot \frac{d^2}{d} \]
\[ U = \varepsilon_0 A E^2 \cdot d \]

### Final Result
Thus, the energy required to charge the parallel plate capacitor is:
\[ U = \varepsilon_0 A E^2 d \]