To increase the length of brass rod by `2%` its temperature should increase by `(alpha = 0.00002 "^(@)C^(-1))`

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 follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between change in length and temperature**:
The change in length of a material due to temperature change can be expressed using the formula:
\[ \frac{\Delta L}{L} = \alpha \Delta T \]
where:
- \(\Delta L\) = change in length
- \(L\) = original length
- \(\alpha\) = coefficient of linear expansion
- \(\Delta T\) = change in temperature

2. **Convert the percentage increase in length to a fraction**:
We are given that the length of the brass rod should increase by 2%. This can be expressed as:
\[ \frac{\Delta L}{L} = \frac{2}{100} = 0.02 \]

3. **Substitute the values into the formula**:
We know that \(\alpha = 0.00002 \, ^\circ C^{-1}\). We can rearrange the formula to find \(\Delta T\):
\[ \Delta T = \frac{\Delta L}{L} \cdot \frac{1}{\alpha} \]

4. **Calculate \(\Delta T\)**:
Substitute the known values into the equation:
\[ \Delta T = 0.02 \cdot \frac{1}{0.00002} \]
\[ \Delta T = 0.02 \cdot 50000 \]
\[ \Delta T = 1000 \, ^\circ C \]

5. **Conclusion**:
Therefore, to increase the length of the brass rod by 2%, the temperature should increase by \(1000 \, ^\circ C\).

### Final Answer:
The temperature should increase by \(1000 \, ^\circ C\).

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