To increase the length of brass rod by `2%` its temperature should increase by `(alpha = 0.00002 "^(@)C^(-1))` a) 800 degree C b) 900 degree C c) 1000 degree C d) 1100 degree C

Options

A. `800^(@)C`
B. `900^(@)C`
C. `1000^(@)C`
D. `1100^(@)C`

Explanation

To solve the problem of how much the temperature should increase to achieve a 2% increase in the length of a brass rod, we can use the formula for linear expansion:

### Step-by-Step Solution:

1. **Understand the formula for linear expansion**:
The formula relating the change in length (\( \Delta L \)) to the original length (\( L \)), the coefficient of linear expansion (\( \alpha \)), and the change in temperature (\( \Delta T \)) is given by:
\[ \Delta L = \alpha \cdot L \cdot \Delta T \]

2. **Express the percentage increase in length**:
We want to increase the length by 2%, which can be expressed as:
\[ \frac{\Delta L}{L} = \frac{2}{100} = 0.02 \]

3. **Rearrange the formula to find \( \Delta T \)**:
Rearranging the linear expansion formula gives:
\[ \Delta T = \frac{\Delta L}{\alpha \cdot L} \]
Substituting \( \Delta L = 0.02L \) into the equation, we get:
\[ \Delta T = \frac{0.02L}{\alpha \cdot L} \]

4. **Cancel \( L \) from the equation**:
Since \( L \) is in both the numerator and the denominator, it cancels out:
\[ \Delta T = \frac{0.02}{\alpha} \]

5. **Substitute the value of \( \alpha \)**:
Given \( \alpha = 0.00002 \, ^\circ C^{-1} \):
\[ \Delta T = \frac{0.02}{0.00002} \]

6. **Calculate \( \Delta T \)**:
Performing the division:
\[ \Delta T = 1000 \, ^\circ C \]

7. **Identify the correct option**:
The calculated temperature increase is \( 1000 \, ^\circ C \), which corresponds to option (c).

### Final Answer:
The temperature should increase by **1000 degree C** (Option c).

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