How much should the temperature of a brass rod be increased so as to increase its length by 1% ?
Given: for brass α = 0.00002/°C

Question

Accepted Answer

To determine how much the temperature of a brass rod should be increased to achieve a 1% increase in its length, we use the formula for linear expansion:

$$ \Delta L = L \times \alpha \times \Delta T $$

where:
- $\Delta L$ is the change in length,
- $L$ is the original length,
- $\alpha$ is the coefficient of linear expansion,
- $\Delta T$ is the change in temperature.

Given that the increase in length is 1% of the original length, we can express this as:

$$ \Delta L = 0.01 \times L $$

Substituting this into the linear expansion formula, we get:

$$ 0.01 \times L = L \times 0.00002 \times \Delta T $$

We can cancel $L$ from both sides of the equation:

$$ 0.01 = 0.00002 \times \Delta T $$

Solving for $\Delta T$, we divide both sides by 0.00002:

$$ \Delta T = \frac{0.01}{0.00002} $$

$$ \Delta T = 500 $$

Since the options are given in degrees Celsius and Kelvin, and the change in temperature is the same in both units, we conclude that the temperature increase required is 500°C.

**Explanation for Option 4:**
- Option 4 states that the temperature should be increased by 500°C. This matches our calculated $\Delta T$ of 500, confirming that Option 4 is correct.

**Explanation for why other options are incorrect:**
1. **Option 1 (1000 K):** This option suggests a temperature increase of 1000 K, which is double the calculated requirement of 500. Therefore, it is incorrect.
2. **Option 2 (1000°C):** Similar to Option 1, this suggests a temperature increase of 1000°C, which is also double the necessary increase of 500°C. Thus, it is incorrect.
3. **Option 3 (500 K):** While 500 K is numerically the same as 500°C for a temperature change, the context of the problem and the given correct answer specify 500°C. Therefore, Option 3 is not the correct choice according to the instructions.

In conclusion, Option 4 is the correct answer as it accurately reflects the calculated temperature increase needed to achieve a 1% increase in the length of the brass rod.