A copper rod and a steel rod are to have lengths $L_C$ and $L_S$, such that the difference between their lengths is the same at all ambient temperature. If the coefficients of linear expansion of copper and steels are $α_C$ and $α_S$ respectively. The lengths are related to the coefficient of linear expansion as :

Question

Accepted Answer

To understand why Option 1 is the correct answer, we need to consider the concept of linear thermal expansion. The linear expansion of a material is given by the formula:

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

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

For the copper rod and the steel rod to maintain the same difference in length at all ambient temperatures, the change in their lengths due to temperature changes must be proportional to their original lengths. This means that the ratio of their lengths must be inversely proportional to the ratio of their coefficients of linear expansion.

### Explanation for Option 1:

1. **Option 1: $\frac{L_C}{L_S} = \frac{\alpha_S}{\alpha_C}$**

This option states that the ratio of the lengths of the copper rod ($L_C$) to the steel rod ($L_S$) is equal to the inverse ratio of their coefficients of linear expansion. This is correct because for the difference in length to remain constant, the expansion of each rod must compensate for the difference in their expansion rates. Therefore, the rod with the higher coefficient of expansion (copper) must be shorter relative to the rod with the lower coefficient (steel) to maintain the same difference in length across all temperatures.

### Explanation for why other options are incorrect:

2. **Option 2: $L_C - L_S = \alpha_C - \alpha_S$**

This option incorrectly suggests that the difference in lengths is directly equal to the difference in their coefficients of expansion. This is not correct because the difference in lengths is not a simple subtraction of coefficients; it involves the proportional relationship between the lengths and the coefficients.

3. **Option 3: $\frac{L_C}{L_S} = \left( \frac{\alpha_C}{\alpha_S} \right)^{1/2}$**

This option suggests a square root relationship between the ratios of lengths and coefficients, which is not supported by the principles of linear thermal expansion. The relationship is linear, not quadratic or involving square roots.

4. **Option 4: $\frac{L_C}{L_S} = \frac{\alpha_C}{\alpha_S}$**

This option suggests a direct proportionality between the ratios of lengths and coefficients, which would imply that the rod with the higher coefficient should also be longer, contradicting the requirement for the difference in length to remain constant. The correct relationship is an inverse proportionality, as stated in Option 1.

In conclusion, Option 1 correctly describes the relationship needed to maintain a constant difference in length between the copper and steel rods at all temperatures, making it the correct answer.