The temperature of a wire is doubled. The Young's modulus of elasticity

Options

A. will also double
B. will become four times
C. will remain same
D. will decrease

Explanation

To solve the problem regarding the effect of temperature on the Young's modulus of elasticity of a wire, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Young's Modulus**:
Young's modulus (Y) is defined as the ratio of normal stress to longitudinal strain. Mathematically, it can be expressed as:
\[ Y = \frac{\text{Normal Stress}}{\text{Longitudinal Strain}} = \frac{F/A}{\Delta L/L} \]
where:
- \( F \) is the force applied,
- \( A \) is the cross-sectional area,
- \( \Delta L \) is the change in length,
- \( L \) is the original length.

2. **Relating Strain to Temperature Change**:
The longitudinal strain (\(\Delta L/L\)) due to a change in temperature can be expressed using the coefficient of linear expansion (\(\alpha\)):
\[ \frac{\Delta L}{L} = \alpha \Delta T \]
where:
- \(\Delta T\) is the change in temperature.

3. **Applying the Given Condition**:
In this problem, the temperature of the wire is doubled. If we assume the initial temperature change is \(\Delta T\), then doubling the temperature means:
\[ \Delta T = 2T \]
Therefore, substituting this into the strain equation gives:
\[ \frac{\Delta L}{L} = \alpha (2T) = 2\alpha T \]

4. **Substituting into Young's Modulus**:
Now, substituting the expression for strain back into the equation for Young's modulus:
\[ Y = \frac{F/A}{\Delta L/L} = \frac{F/A}{2\alpha T} \]
This shows that Young's modulus is inversely proportional to the temperature change.

5. **Conclusion**:
Since Young's modulus is inversely proportional to the temperature change, if the temperature is doubled, the Young's modulus will be halved:
\[ Y' = \frac{Y}{2} \]
where \(Y'\) is the new Young's modulus after the temperature change.

### Final Answer:
When the temperature of the wire is doubled, the Young's modulus of elasticity becomes half of its original value.

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