What will be the stress required to double the length of a wire of Young's modulus Y?

Options

A. Y
B. 2Y
C. 3Y
D. 4Y

Explanation

To find the stress required to double the length of a wire given its Young's modulus \( Y \), we can follow these steps:

### Step 1: Understand the Definitions
- **Stress** (\( \sigma \)): It is defined as the force applied per unit area.
- **Strain** (\( \epsilon \)): It is defined as the change in length divided by the original length.
- **Young's Modulus** (\( Y \)): It is defined as the ratio of stress to strain, given by the formula:
\[ Y = \frac{\sigma}{\epsilon} \]

### Step 2: Define the Initial and Final Length
- Let the initial length of the wire be \( L_0 \).
- According to the problem, the final length \( L_f \) will be double the initial length:
\[ L_f = 2L_0 \]

### Step 3: Calculate the Change in Length
- The change in length \( \Delta L \) is given by:
\[ \Delta L = L_f - L_0 = 2L_0 - L_0 = L_0 \]

### Step 4: Calculate the Strain
- Strain \( \epsilon \) can be calculated as:
\[ \epsilon = \frac{\Delta L}{L_0} = \frac{L_0}{L_0} = 1 \]

### Step 5: Relate Stress and Young's Modulus
- From the definition of Young's modulus, we can rearrange the formula to find stress:
\[ Y = \frac{\sigma}{\epsilon} \implies \sigma = Y \cdot \epsilon \]

### Step 6: Substitute the Value of Strain
- Since we found that \( \epsilon = 1 \):
\[ \sigma = Y \cdot 1 = Y \]

### Conclusion
- Therefore, the stress required to double the length of the wire is equal to the Young's modulus \( Y \).

### Final Answer
The stress required to double the length of a wire of Young's modulus \( Y \) is \( Y \).

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