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

Answer

To find the energy stored per unit volume between the plates of a parallel plate capacitor, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Energy Stored**:
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.

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

3. **Substituting Capacitance into the Energy Formula**:
Substitute the expression for \( C \) into the energy formula:
\[ U = \frac{1}{2} \left(\frac{\epsilon_0 A}{d}\right) V^2 \]

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

5. **Energy per Unit Volume**:
The energy per unit volume \( u \) is given by:
\[ u = \frac{U}{V_{\text{vol}}} \]
Substituting the expressions for \( U \) and \( V_{\text{vol}} \):
\[ u = \frac{\frac{1}{2} \left(\frac{\epsilon_0 A}{d}\right) V^2}{A \cdot d} \]

6. **Simplifying the Expression**:
Cancel \( A \) in the numerator and denominator:
\[ u = \frac{1}{2} \cdot \frac{\epsilon_0 V^2}{d^2} \]

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

### Conclusion:
The energy stored per unit volume between the plates of a parallel plate capacitor is given by:
\[ u = \frac{1}{2} \frac{\epsilon_0 V^2}{d^2} \]