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. `(L_(1))/(L_(2))=(alpha_(1))/(alpha_(2))`
B. `(L_(1))/(L_(2))=(alpha_(2))/(alpha_(1))`
C. `L_(2)^(2)alpha_(1)=L_(1)^(2)alpha`
D. `(alpha_(1)^(2))/(L_(1))=(alpha_(2)^(2))/(L_(2))`

Explanation

To solve the problem, we need to analyze the change in lengths of the two rods due to temperature variation. The change in length of a rod due to thermal expansion can be expressed using the formula:

\[ \Delta L = L_0 \cdot \alpha \cdot \Delta T \]

where:
- \(\Delta L\) is the change in length,
- \(L_0\) is the original length of the rod,
- \(\alpha\) is the coefficient of linear expansion, and
- \(\Delta T\) is the change in temperature.

For the two rods, we can denote the changes in length as follows:

1. For the first rod of length \(l_1\) with coefficient of linear expansion \(\alpha_1\):
\[ \Delta L_1 = l_1 \cdot \alpha_1 \cdot \Delta T \]

2. For the second rod of length \(l_2\) with coefficient of linear expansion \(\alpha_2\):
\[ \Delta L_2 = l_2 \cdot \alpha_2 \cdot \Delta T \]

We are interested in the difference in lengths of the two rods after the temperature change, which 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 \(\Delta T\):

\[ \Delta L = \Delta T \cdot (l_1 \cdot \alpha_1 - l_2 \cdot \alpha_2) \]

For the difference in lengths to be independent of temperature, \(\Delta L\) must be zero. This leads to the condition:

\[ l_1 \cdot \alpha_1 - l_2 \cdot \alpha_2 = 0 \]

Rearranging this gives:

\[ l_1 \cdot \alpha_1 = l_2 \cdot \alpha_2 \]

Dividing both sides by \(l_2 \cdot \alpha_1\):

\[ \frac{l_1}{l_2} = \frac{\alpha_2}{\alpha_1} \]

Thus, we find that the ratio of the lengths \(l_1\) and \(l_2\) is given by:

\[ \frac{l_1}{l_2} = \frac{\alpha_2}{\alpha_1} \]

This means that for the difference in lengths to be independent of temperature, the ratio of the lengths of the two rods must be equal to the ratio of their coefficients of linear expansion.

**Final Answer:**
\[ \frac{l_1}{l_2} = \frac{\alpha_2}{\alpha_1} \]

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