The elastic potential energy of a stretched wire is given by

Options

A. `U=(AL)/(2Y)l^(2)`
B. `U=(AY)/(2L)l^(2)`
C. `U=(1)/(2)((All)/(Y))l`
D. `U=(1)/(2)*(YL)/(2A)*l`

Explanation

To find the elastic potential energy (E) of a stretched wire, we can follow these steps:

### Step 1: Understand the formula for elastic potential energy
The elastic potential energy stored in a stretched wire is given by the formula:
\[ E = \frac{1}{2} \times \text{Stress} \times \text{Strain} \times \text{Volume} \]

### Step 2: Define stress and strain
- **Stress** is defined as the force (F) applied per unit area (A):
\[ \text{Stress} = \frac{F}{A} \]

- **Strain** is the ratio of the change in length (ΔL) to the original length (L):
\[ \text{Strain} = \frac{\Delta L}{L} \]

### Step 3: Substitute stress and strain into the energy formula
Substituting the definitions of stress and strain into the energy formula, we have:
\[ E = \frac{1}{2} \times \left(\frac{F}{A}\right) \times \left(\frac{\Delta L}{L}\right) \times V \]
Where \( V \) (Volume) is given by \( A \times L \).

### Step 4: Substitute volume into the equation
Now substituting \( V = A \times L \) into the equation:
\[ E = \frac{1}{2} \times \left(\frac{F}{A}\right) \times \left(\frac{\Delta L}{L}\right) \times (A \times L) \]

### Step 5: Simplify the equation
Now simplifying the equation:
\[ E = \frac{1}{2} \times F \times \frac{\Delta L}{L} \]
Since \( \text{Strain} = \frac{\Delta L}{L} \), we can rewrite it as:
\[ E = \frac{1}{2} \times F \times \text{Strain} \]

### Step 6: Relate force to stress and strain
Using the relationship between force, stress, and strain, we can express the force as:
\[ F = \text{Stress} \times A = \frac{F}{A} \times A \]

### Step 7: Final expression for elastic potential energy
Finally, we can express the elastic potential energy as:
\[ E = \frac{1}{2} \times \text{Stress} \times \text{Strain} \times A \times L \]
This can be simplified to:
\[ E = \frac{1}{2} \times \text{Stress} \times \text{Strain} \times V \]

### Conclusion
The final expression for the elastic potential energy of a stretched wire is:
\[ E = \frac{1}{2} \times \text{Stress} \times \text{Strain} \times V \]

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