How much should the temperature of a brass rod be increased so as to increase its length 3% (`alpha` for brass is `2xx10^(-5)//""^(@)C^(-1)`)?

Options

A. `1500^(@)C`
B. `150^(@)C`
C. `2000^(@)C`
D. `2500^(@)C`

Explanation

To solve the problem of how much the temperature of a brass rod should be increased to achieve a 3% increase in length, we can use the formula for linear thermal expansion. The formula relates the change in length of a material to its original length, the coefficient of linear expansion, and the change in temperature.

### Step-by-step Solution:

1. **Understand the Problem**: We need to find the change in temperature (ΔT) required to increase the length of a brass rod by 3%. The coefficient of linear expansion (α) for brass is given as \(2 \times 10^{-5} \, \text{°C}^{-1}\).

2. **Set Up the Equation**: The formula for linear expansion is given by:
\[ \frac{\Delta L}{L} = \alpha \Delta T \]
where:
- \(\Delta L\) = change in length
- \(L\) = original length
- \(\alpha\) = coefficient of linear expansion
- \(\Delta T\) = change in temperature

3. **Express the Change in Length**: Since we want a 3% increase in length, we can express this as:
\[ \frac{\Delta L}{L} = \frac{3}{100} = 0.03 \]

4. **Substitute Values into the Equation**: Now we can substitute the known values into the equation:
\[ 0.03 = (2 \times 10^{-5}) \Delta T \]

5. **Solve for ΔT**: Rearranging the equation to solve for ΔT gives:
\[ \Delta T = \frac{0.03}{2 \times 10^{-5}} \]

6. **Calculate ΔT**: Performing the calculation:
\[ \Delta T = \frac{0.03}{2 \times 10^{-5}} = \frac{0.03}{0.00002} = 1500 \, \text{°C} \]

7. **Final Answer**: The temperature of the brass rod should be increased by **1500 °C** to achieve a 3% increase in its length.