In a parallel plate capacitor, the distance between the plates is `d` and potential difference across the plate is `V`. Energy stored per unit volume between the plates of capacitor is

Options

A. `(Q^(2))/(2V^(2))`
B. `(1)/(2) epsilon_(0)(V^(2))/(d^(2))`
C. `(1)/(2)(V^(2))/(epsilon_(0)d^(2))`
D. `(1)/(2) epsilon_(0)(V^(2))/(d)`

Explanation

To solve the problem of finding the energy stored per unit volume in a parallel plate capacitor, we can follow these steps:

### Step 1: Understand the Energy Stored in a Capacitor
The energy (U) stored in a capacitor is given by the formula:
\[ U = \frac{1}{2} C V^2 \]
where \( C \) is the capacitance and \( V \) is the potential difference across the plates.

### Step 2: Determine the Capacitance of a Parallel Plate Capacitor
The capacitance \( C \) of a parallel plate capacitor is given by:
\[ C = \frac{A \epsilon_0}{d} \]
where:
- \( A \) is the area of one of the plates,
- \( \epsilon_0 \) is the permittivity of free space,
- \( d \) is the distance between the plates.

### Step 3: Substitute Capacitance into the Energy Formula
Substituting the expression for capacitance into the energy formula, we get:
\[ U = \frac{1}{2} \left( \frac{A \epsilon_0}{d} \right) V^2 \]

### Step 4: Calculate the Volume Between the Plates
The volume \( V_{volume} \) between the plates of the capacitor is given by:
\[ V_{volume} = A \cdot d \]

### Step 5: Find the Energy per Unit Volume
To find the energy stored per unit volume, we divide the total energy stored \( U \) by the volume \( V_{volume} \):
\[ \text{Energy per unit volume} = \frac{U}{V_{volume}} = \frac{\frac{1}{2} \left( \frac{A \epsilon_0}{d} \right) V^2}{A \cdot d} \]

### Step 6: Simplify the Expression
Now, simplifying the expression:
\[ \text{Energy per unit volume} = \frac{\frac{1}{2} \epsilon_0 V^2}{d^2} \]

### Final Result
Thus, the energy stored per unit volume between the plates of the capacitor is:
\[ \text{Energy per unit volume} = \frac{1}{2} \epsilon_0 \frac{V^2}{d^2} \]