The elastic energy stored per units volume in a streched wire is

Options

A. `(1)/(2)(("Stress"))/(Y)`
B. `(1)/(2)(("Stress")^(2))/(Y)`
C. `(1)/(2)(("Stress")^(2))/(Y^(2))`
D. `(1)/(2)(("Stress"))/(Y^(2))`

Explanation

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

### Step-by-Step Solution:

1. **Understand the Formula for Elastic Energy**:
The elastic energy (E) stored in a material can be expressed in terms of stress and strain:
\[ E = \frac{1}{2} \times \text{Stress} \times \text{Strain} \]

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

3. **Use Young's Modulus**:
Young's modulus (Y) relates stress and strain:
\[ Y = \frac{\sigma}{\epsilon} \]
From this, we can express strain in terms of stress and Young's modulus:
\[ \epsilon = \frac{\sigma}{Y} \]

4. **Substitute Strain into the Elastic Energy Formula**:
Now, substitute the expression for strain into the elastic energy formula:
\[ E = \frac{1}{2} \times \sigma \times \left(\frac{\sigma}{Y}\right) \]
Simplifying this gives:
\[ E = \frac{1}{2} \times \frac{\sigma^2}{Y} \]

5. **Final Expression for Elastic Energy per Unit Volume**:
The elastic energy stored per unit volume in a stretched wire is:
\[ E = \frac{\sigma^2}{2Y} \]

### Conclusion:
The elastic energy stored per unit volume in a stretched wire is given by the formula:
\[ E = \frac{\sigma^2}{2Y} \]