The workdone in increasing the length of a one metre long wire of cross `-` sectional area `1 m m^(2)` through `1mm` will be `(Y=2xx10^(11)Nm^(-2)) :`

Options

A. `0.1J`
B. `5J`
C. `10J`
D. `250J`

Answer

To solve the problem of calculating the work done in increasing the length of a one-meter long wire of cross-sectional area \(1 \, \text{mm}^2\) through \(1 \, \text{mm}\), we will use the formula for work done based on Young's modulus.

### Step-by-Step Solution:

1. **Understand the Given Values:**
- Length of the wire, \(L = 1 \, \text{m}\)
- Cross-sectional area, \(A = 1 \, \text{mm}^2 = 1 \times 10^{-6} \, \text{m}^2\)
- Change in length, \(\Delta L = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m}\)
- Young's modulus, \(Y = 2 \times 10^{11} \, \text{N/m}^2\)

2. **Calculate Strain:**
- Strain (\(e\)) is defined as the change in length divided by the original length:
\[ e = \frac{\Delta L}{L} = \frac{1 \times 10^{-3}}{1} = 1 \times 10^{-3} \]

3. **Calculate the Volume of the Wire:**
- The volume (\(V\)) of the wire can be calculated using the formula:
\[ V = A \times L = (1 \times 10^{-6} \, \text{m}^2) \times (1 \, \text{m}) = 1 \times 10^{-6} \, \text{m}^3 \]

4. **Calculate Work Done:**
- The work done (\(W\)) in stretching the wire can be calculated using the formula:
\[ W = \frac{1}{2} \times Y \times e^2 \times V \]
- Substitute the known values into the formula:
\[ W = \frac{1}{2} \times (2 \times 10^{11} \, \text{N/m}^2) \times (1 \times 10^{-3})^2 \times (1 \times 10^{-6} \, \text{m}^3) \]

5. **Simplify the Expression:**
- Calculate \(e^2\):
\[ e^2 = (1 \times 10^{-3})^2 = 1 \times 10^{-6} \]
- Now substitute \(e^2\) back into the work done formula:
\[ W = \frac{1}{2} \times (2 \times 10^{11}) \times (1 \times 10^{-6}) \times (1 \times 10^{-6}) \]
- This simplifies to:
\[ W = \frac{1}{2} \times 2 \times 10^{11} \times 1 \times 10^{-12} \]
- Further simplifying gives:
\[ W = 1 \times 10^{-1} = 0.1 \, \text{J} \]

### Final Answer:
The work done in increasing the length of the wire is \(0.1 \, \text{J}\).

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