A wire of natural length l, young's modulus Y and ares of cross-section A is extended by x. Then, the energy stored in the wire is given by

Options

A. `1/2(YA)/(l)x^(2)`
B. `1/3(YA)/(l)x^(2)`
C. `1/2(Yl)/(A)x^(2)`
D. `1/2(YA)/(l^(2))x^(2)`

Explanation

To find the energy stored in a wire that has been extended, we can follow these steps:

### Step 1: Understand the parameters given
We have:
- Natural length of the wire, \( L \)
- Young's modulus, \( Y \)
- Area of cross-section, \( A \)
- Extension of the wire, \( x \)

### Step 2: Recall the formula for energy stored in a stretched wire
The energy stored in a wire when it is stretched is given by the formula:
\[ \text{Energy} = \frac{1}{2} \times \text{Young's Modulus} \times \text{Strain}^2 \times \text{Volume} \]

### Step 3: Calculate the strain
Strain is defined as the change in length per unit length. Here, the change in length (extension) is \( x \) and the original length is \( L \). Therefore, the strain \( \epsilon \) can be expressed as:
\[ \text{Strain} = \frac{\Delta L}{L} = \frac{x}{L} \]

### Step 4: Calculate the volume of the wire
The volume \( V \) of the wire can be calculated using the formula:
\[ \text{Volume} = \text{Area} \times \text{Length} = A \times L \]

### Step 5: Substitute the values into the energy formula
Now, substituting the values of strain and volume into the energy formula:
\[ \text{Energy} = \frac{1}{2} \times Y \times \left(\frac{x}{L}\right)^2 \times (A \times L) \]

### Step 6: Simplify the expression
Now, simplify the expression:
\[ \text{Energy} = \frac{1}{2} \times Y \times \frac{x^2}{L^2} \times (A \times L) \]
\[ = \frac{1}{2} \times Y \times A \times \frac{x^2}{L} \]

### Final Result
Thus, the energy stored in the wire is:
\[ \text{Energy} = \frac{Y \cdot A \cdot x^2}{2L} \]

### Conclusion
The energy stored in the wire when it is extended by \( x \) is given by:
\[ \text{Energy} = \frac{Y \cdot A \cdot x^2}{2L} \]