A wire of length `50 cm` and cross sectional area of 1 sq. mm is extended by `1mm`. The required work will be `(Y=2xx10^(10) Nm^(-2))`

Options

A. `6xx10^(-2) J`
B. `4xx10^(-2) J`
C. `2xx10^(-2) J`
D. `1 xx10^(-2) J`

Explanation

To solve the problem, we need to calculate the work done in extending a wire using the given parameters. Here is a step-by-step solution:

### Step 1: Identify the given values
- Length of the wire, \( L = 50 \, \text{cm} = 50 \times 10^{-2} \, \text{m} \)
- Cross-sectional area, \( A = 1 \, \text{mm}^2 = 1 \times 10^{-6} \, \text{m}^2 \)
- Extension of the wire, \( \Delta L = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)
- Young's modulus, \( Y = 2 \times 10^{10} \, \text{N/m}^2 \)

### Step 2: Calculate the stress
Stress (\( \sigma \)) is defined as force (\( F \)) per unit area (\( A \)):
\[ \sigma = \frac{F}{A} \]
From Young's modulus, we have:
\[ Y = \frac{\sigma}{\epsilon} \]
where \( \epsilon \) is the strain, defined as:
\[ \epsilon = \frac{\Delta L}{L} \]
Thus, we can express stress in terms of Young's modulus and strain:
\[ \sigma = Y \cdot \epsilon \]

### Step 3: Calculate the strain
Using the values:
\[ \epsilon = \frac{\Delta L}{L} = \frac{1 \times 10^{-3}}{50 \times 10^{-2}} = \frac{1 \times 10^{-3}}{0.5} = 2 \times 10^{-3} \]

### Step 4: Calculate the stress
Now substituting the strain into the stress formula:
\[ \sigma = Y \cdot \epsilon = 2 \times 10^{10} \cdot 2 \times 10^{-3} = 4 \times 10^{7} \, \text{N/m}^2 \]

### Step 5: Calculate the work done
The work done (\( W \)) in stretching the wire can be calculated using the formula:
\[ W = \frac{1}{2} \cdot \sigma \cdot \epsilon \cdot V \]
where \( V \) is the volume of the wire given by:
\[ V = A \cdot L \]
Substituting for \( V \):
\[ V = (1 \times 10^{-6}) \cdot (50 \times 10^{-2}) = 5 \times 10^{-6} \, \text{m}^3 \]

Now substituting \( \sigma \), \( \epsilon \), and \( V \) into the work done formula:
\[ W = \frac{1}{2} \cdot (4 \times 10^{7}) \cdot (2 \times 10^{-3}) \cdot (5 \times 10^{-6}) \]

### Step 6: Calculate the final work done
Calculating this gives:
\[ W = \frac{1}{2} \cdot 4 \cdot 2 \cdot 5 \cdot 10^{7} \cdot 10^{-3} \cdot 10^{-6} = \frac{1}{2} \cdot 40 \cdot 10^{-2} = 20 \times 10^{-2} = 2 \times 10^{-2} \, \text{J} \]

### Final Answer
Thus, the required work done is:
\[ W = 2 \times 10^{-2} \, \text{J} \]

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