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

Options

A. will also double
B. will becomes four times
C. will remain same
D. will decreases

Explanation

To solve the problem of how the Young's modulus of elasticity changes when the temperature of a wire is doubled, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Young's Modulus**:
Young's modulus (Y) is defined as the ratio of stress (force per unit area) to strain (relative change in length). Mathematically, it is given by:
\[ Y = \frac{F/A}{\Delta L/L} \]
where \( F \) is the force applied, \( A \) is the cross-sectional area, \( \Delta L \) is the change in length, and \( L \) is the original length.

2. **Linear Expansion**:
When the temperature of a wire changes, it expands. The change in length (\( \Delta L \)) due to a change in temperature (\( \Delta T \)) is given by:
\[ \Delta L = L \alpha \Delta T \]
where \( \alpha \) is the coefficient of linear expansion.

3. **Substituting into Young's Modulus**:
We can express strain (\( \Delta L/L \)) in terms of temperature change:
\[ \frac{\Delta L}{L} = \alpha \Delta T \]
Substituting this into the equation for Young's modulus gives:
\[ Y = \frac{F/A}{\alpha \Delta T} \]

4. **Effect of Doubling the Temperature**:
If the initial temperature is \( T \) and it is doubled, the new temperature becomes \( 2T \). The change in temperature (\( \Delta T \)) can be considered as \( T \) (from initial to doubled temperature).

5. **Comparing Young's Modulus**:
Let \( Y' \) be the Young's modulus at the new temperature. The relationship can be established as:
\[ Y' = \frac{F/A}{\alpha (2T)} = \frac{Y}{2} \]
Thus, the Young's modulus at the doubled temperature is half of the original Young's modulus.

6. **Final Result**:
Therefore, when the temperature of the wire is doubled, the Young's modulus of elasticity becomes:
\[ Y' = \frac{Y}{2} \]

### Conclusion:
The Young's modulus of elasticity is halved when the temperature of the wire is doubled.

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