Energy per unit volume for a capacitor having area A and separation d kept at potential diffeence V is given by : -

Options

A. `1/2epsilon_0 V^2/d^2`
B. `1/(2epsilon_0) V^2/d^2`
C. `1/2CV^2`
D. `Q^2/"2C"`

Explanation

To find the energy per unit volume for a capacitor with area \( A \), separation \( d \), and potential difference \( V \), we can follow these steps:

### Step 1: Understand the capacitor's properties
A capacitor consists of two plates with an area \( A \) and separated by a distance \( d \). The potential difference between the plates is \( V \).

### Step 2: Calculate the capacitance
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 stored in the capacitor
The energy \( E \) stored in a capacitor is given by:
\[ E = \frac{1}{2} C V^2 \]
Substituting the expression for capacitance \( C \):
\[ E = \frac{1}{2} \left( \frac{\varepsilon_0 A}{d} \right) V^2 \]
This simplifies to:
\[ E = \frac{1}{2} \frac{\varepsilon_0 A V^2}{d} \]

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

### Step 5: Calculate energy per unit volume (energy density)
The energy density \( u \) (energy per unit volume) is given by:
\[ u = \frac{E}{V_{cap}} \]
Substituting the expressions for \( E \) and \( V_{cap} \):
\[ u = \frac{\frac{1}{2} \frac{\varepsilon_0 A V^2}{d}}{A \cdot d} \]
This simplifies to:
\[ u = \frac{1}{2} \frac{\varepsilon_0 V^2}{d^2} \]

### Final Answer
Thus, the energy per unit volume for the capacitor is:
\[ u = \frac{1}{2} \frac{\varepsilon_0 V^2}{d^2} \]