The work per unit volume to stretch the length by `1%` of a wire with cross sectional area of `1mm^2` will be. `[Y=9xx10^(11)N//m^2]`

Options

A. `9xx10^11J`
B. `4.5xx10^7J`
C. `9xx10^7J`
D. `4.5xx10^11J`

Explanation

To find the work done per unit volume to stretch the length of a wire by 1%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- **Stress**: It is defined as the force applied per unit area.
- **Strain**: It is defined as the change in length divided by the original length.
- **Young's Modulus (Y)**: It relates stress and strain in a material and is given by the formula:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]

2. **Given Values**:
- Young's Modulus, \( Y = 9 \times 10^{11} \, \text{N/m}^2 \)
- Cross-sectional area, \( A = 1 \, \text{mm}^2 = 1 \times 10^{-6} \, \text{m}^2 \) (conversion from mm² to m²)
- Change in length (strain) = 1% = 0.01 (as a decimal)

3. **Calculate Stress**:
- Stress can be expressed in terms of Young's Modulus and strain:
\[ \text{Stress} = Y \times \text{Strain} \]
- Substitute the values:
\[ \text{Stress} = 9 \times 10^{11} \, \text{N/m}^2 \times 0.01 = 9 \times 10^{11} \times 10^{-2} = 9 \times 10^{9} \, \text{N/m}^2 \]

4. **Calculate Work Done per Unit Volume**:
- The work done per unit volume (W) is given by:
\[ W = \frac{1}{2} \times \text{Stress} \times \text{Strain} \]
- Substitute the values:
\[ W = \frac{1}{2} \times (9 \times 10^{9}) \times 0.01 \]
\[ W = \frac{1}{2} \times 9 \times 10^{9} \times 10^{-2} = \frac{9}{2} \times 10^{7} = 4.5 \times 10^{7} \, \text{J/m}^3 \]

5. **Final Answer**:
- The work done per unit volume to stretch the length of the wire by 1% is:
\[ \boxed{4.5 \times 10^{7} \, \text{J/m}^3} \]