Coefficient of linear expansion of brass and steel rods are `alpha_(1)` and `alpha_(2)`. Length of brass and steel rods are `l_(1)` and `l_(2)` respectively. If `(l_(2) - l_(1))` is maintained same at all temperature, which one of the following relations holds good?

Options

A. `alpha_(1) l_(2) = alpha_(2) l_(1)`
B. `alpha_(1) l_(2)^(2) = alpha_(2) l_(1)^(2)`
C. `alpha_(1)^(2) l_(2) = alpha_(2) l_(1)`
D. `alpha_(1) l_(1) = alpha_(2) l_(2)`

Explanation

To solve the problem, we need to analyze the relationship between the lengths of the brass and steel rods when they are subjected to temperature changes. We are given that the difference in lengths of the two rods, \( (l_2 - l_1) \), remains constant at all temperatures.

### Step-by-Step Solution:

1. **Understanding Linear Expansion**:
The linear expansion of a material is given by the formula:
\[ \Delta L = \alpha L_0 \Delta T \]
where \( \Delta L \) is the change in length, \( \alpha \) is the coefficient of linear expansion, \( L_0 \) is the original length, and \( \Delta T \) is the change in temperature.

2. **Initial Lengths**:
Let the initial lengths of the brass and steel rods be \( l_1 \) and \( l_2 \) respectively.

3. **Final Lengths After Temperature Change**:
After a temperature change \( \Delta T \):
- The new length of the brass rod, \( l'_1 \), is:
\[ l'_1 = l_1 + \alpha_1 l_1 \Delta T = l_1 (1 + \alpha_1 \Delta T) \]
- The new length of the steel rod, \( l'_2 \), is:
\[ l'_2 = l_2 + \alpha_2 l_2 \Delta T = l_2 (1 + \alpha_2 \Delta T) \]

4. **Setting Up the Equation**:
According to the problem, the difference in lengths remains constant:
\[ l'_2 - l'_1 = l_2 - l_1 \]
Substituting the expressions for \( l'_1 \) and \( l'_2 \):
\[ l_2(1 + \alpha_2 \Delta T) - l_1(1 + \alpha_1 \Delta T) = l_2 - l_1 \]

5. **Simplifying the Equation**:
Expanding both sides:
\[ l_2 + l_2 \alpha_2 \Delta T - l_1 - l_1 \alpha_1 \Delta T = l_2 - l_1 \]
Rearranging gives:
\[ l_2 \alpha_2 \Delta T - l_1 \alpha_1 \Delta T = 0 \]
Factoring out \( \Delta T \) (assuming \( \Delta T \neq 0 \)):
\[ l_2 \alpha_2 - l_1 \alpha_1 = 0 \]

6. **Final Relation**:
This leads to the relation:
\[ l_1 \alpha_1 = l_2 \alpha_2 \]

### Conclusion:
The relation that holds good is:
\[ l_1 \alpha_1 = l_2 \alpha_2 \]