Two rods of length `l_(1)` and `l_(2)` are made of material whose coefficient of linear expansion are `alpha_(1)` and `alpha_(2)` , respectively. The difference between their lengths will be independent of temperatiure if `l_(1)//l_(2)` is to

Options

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

Explanation

To solve the problem, we need to analyze the linear expansion of the two rods and find the condition under which the difference in their lengths remains independent of temperature.

### Step-by-Step Solution:

1. **Understanding Linear Expansion**:
The change in length (\( \Delta L \)) of a rod due to temperature change can be expressed as:
\[ \Delta L = L \cdot \alpha \cdot \Delta T \]
where \( L \) is the original length of the rod, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the change in temperature.

2. **Setting Up the Equations**:
For the first rod with length \( L_1 \) and coefficient of linear expansion \( \alpha_1 \):
\[ \Delta L_1 = L_1 \cdot \alpha_1 \cdot \Delta T \]
For the second rod with length \( L_2 \) and coefficient of linear expansion \( \alpha_2 \):
\[ \Delta L_2 = L_2 \cdot \alpha_2 \cdot \Delta T \]

3. **Finding the Difference in Lengths**:
The difference in lengths after expansion is given by:
\[ \Delta L = \Delta L_1 - \Delta L_2 \]
Substituting the expressions for \( \Delta L_1 \) and \( \Delta L_2 \):
\[ \Delta L = (L_1 \cdot \alpha_1 \cdot \Delta T) - (L_2 \cdot \alpha_2 \cdot \Delta T) \]
Factoring out \( \Delta T \):
\[ \Delta L = \Delta T \cdot (L_1 \cdot \alpha_1 - L_2 \cdot \alpha_2) \]

4. **Condition for Independence from Temperature**:
For the difference in lengths to be independent of temperature, the term in parentheses must equal zero:
\[ L_1 \cdot \alpha_1 - L_2 \cdot \alpha_2 = 0 \]
Rearranging gives:
\[ L_1 \cdot \alpha_1 = L_2 \cdot \alpha_2 \]

5. **Finding the Ratio**:
Dividing both sides by \( L_2 \cdot \alpha_1 \):
\[ \frac{L_1}{L_2} = \frac{\alpha_2}{\alpha_1} \]

### Conclusion:
Thus, the ratio \( \frac{L_1}{L_2} \) must be equal to \( \frac{\alpha_2}{\alpha_1} \) for the difference in lengths to be independent of temperature.

### Final Answer:
\[ \frac{L_1}{L_2} = \frac{\alpha_2}{\alpha_1} \]