On stretching a wire, the elastic energy stored per unit volume is,

Options

A. F dl/2 Al
B. F A/2l
C. Fl/2A
D. F l/2

Explanation

To find the elastic energy stored per unit volume in a stretched wire, we can follow these steps:

### Step 1: Understand the concept of elastic energy
The elastic energy stored in a material when it is deformed (stretched or compressed) can be expressed in terms of stress and strain.

### Step 2: Define stress and strain
- **Stress (σ)** is defined as the force (F) applied per unit area (A):
\[ \sigma = \frac{F}{A} \]
- **Strain (ε)** is defined as the change in length (ΔL) divided by the original length (L₀):
\[ \epsilon = \frac{\Delta L}{L_0} \]

### Step 3: Write the formula for elastic energy
The total elastic energy (U) stored in a volume (V) of the material can be expressed as:
\[ U = \frac{1}{2} \times \text{Stress} \times \text{Strain} \times V \]

### Step 4: Substitute stress and strain into the energy formula
Substituting the definitions of stress and strain into the energy formula gives:
\[ U = \frac{1}{2} \times \left(\frac{F}{A}\right) \times \left(\frac{\Delta L}{L_0}\right) \times V \]

### Step 5: Express volume in terms of area and length
The volume (V) can be expressed as:
\[ V = A \times L_0 \]
Substituting this into the energy equation gives:
\[ U = \frac{1}{2} \times \left(\frac{F}{A}\right) \times \left(\frac{\Delta L}{L_0}\right) \times (A \times L_0) \]

### Step 6: Simplify the equation
This simplifies to:
\[ U = \frac{1}{2} \times F \times \frac{\Delta L}{L_0} \]

### Step 7: Find the elastic energy per unit volume
To find the elastic energy per unit volume (u), we divide the total energy (U) by the volume (V):
\[ u = \frac{U}{V} = \frac{\frac{1}{2} \times F \times \frac{\Delta L}{L_0}}{A \times L_0} \]

### Step 8: Final expression
This gives us:
\[ u = \frac{1}{2} \times \frac{F}{A} \times \frac{\Delta L}{L_0} \]
Thus, the elastic energy stored per unit volume in a stretched wire can be expressed as:
\[ u = \frac{1}{2} \sigma \epsilon \]

### Conclusion
The final answer for the elastic energy stored per unit volume in a stretched wire is:
\[ u = \frac{1}{2} \times \frac{F}{A} \times \frac{\Delta L}{L_0} \]