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 of a stretched wire, we can follow these steps:

### Step 1: Understand the formula for elastic potential energy
The elastic potential energy (U) stored in a stretched wire can be expressed in terms of stress and strain. The general formula is given by:
\[ U = \frac{1}{2} \times \text{Stress} \times \text{Strain} \times \text{Volume} \]

### 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 (L):
\[ \epsilon = \frac{\Delta L}{L} \]

### Step 3: Substitute stress and strain into the energy formula
Substituting the definitions of stress and strain into the energy formula, we get:
\[ U = \frac{1}{2} \times \left(\frac{F}{A}\right) \times \left(\frac{\Delta L}{L}\right) \times V \]
where \( V \) is the volume of the wire.

### Step 4: Express volume in terms of length and cross-sectional area
The volume (V) of the wire can be expressed as:
\[ V = A \times L \]
where \( A \) is the cross-sectional area and \( L \) is the original length of the wire.

### Step 5: Substitute volume into the energy formula
Now substituting the expression for volume into the energy formula:
\[ U = \frac{1}{2} \times \left(\frac{F}{A}\right) \times \left(\frac{\Delta L}{L}\right) \times (A \times L) \]

### Step 6: Simplify the equation
When we simplify this equation, we notice that the area \( A \) cancels out:
\[ U = \frac{1}{2} \times F \times \frac{\Delta L}{L} \]

### Step 7: Relate force to Young's modulus
Using Young's modulus (Y), which is defined as:
\[ Y = \frac{\sigma}{\epsilon} = \frac{F/A}{\Delta L/L} \]
we can express force in terms of Young's modulus:
\[ F = Y \times \frac{\Delta L}{L} \times A \]

### Step 8: Substitute force back into the energy formula
Substituting \( F \) back into the energy formula gives:
\[ U = \frac{1}{2} \times \left(Y \times \frac{\Delta L}{L} \times A\right) \times \frac{\Delta L}{L} \]
\[ U = \frac{1}{2} \times Y \times A \times \left(\frac{\Delta L}{L}\right)^2 \]

### Step 9: Final expression for elastic potential energy
This leads us to the final expression for the elastic potential energy stored in a stretched wire:
\[ U = \frac{1}{2} \times Y \times A \times \left(\frac{\Delta L}{L}\right)^2 \]

### Conclusion
Thus, the elastic potential energy of a stretched wire is given by the formula:
\[ U = \frac{1}{2} \times Y \times A \times \left(\frac{\Delta L}{L}\right)^2 \]