Question

Two rods of different materials have lengths '\(l_1\)' and '\(l_2\)' whose coefficient of linear expansions are '\(\alpha_1\)' and '\(\alpha_2\)' respectively. If the difference between the two lengths is independent of temperature then

• A. \(\alpha_1^2 l_1 = \alpha_2^2 l_2\)
• B. \(\frac{l_1}{l_2} = \frac{\alpha_2}{\alpha_1}\)
• C. \(\frac{l_1}{l_2} = \frac{\alpha_1}{\alpha_2}\)
• D. \(l_1^2 \alpha_2 = l_2^2 \alpha_1\)

Hint:

If expression independent of temperature → coefficient of \(\Delta T\) must be zero.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
When an object is heated, its length increases due to thermal expansion.
The condition that the difference in lengths remains independent of temperature means that both rods must expand by the exact same amount for any given temperature change.
Step 2: Key Formula or Approach:
The change in length \( \Delta l \) of a rod of initial length \( l \) with a linear expansion coefficient \( \alpha \) subjected to a temperature change \( \Delta T \) is given by \( \Delta l = l \alpha \Delta T \).
Step 3: Detailed Explanation:
Let the initial lengths at some reference temperature be \( l_1 \) and \( l_2 \).
Let the temperature change be \( \Delta T \).
The new lengths will be \( l'_1 = l_1 + \Delta l_1 = l_1 + l_1 \alpha_1 \Delta T \) and \( l'_2 = l_2 + \Delta l_2 = l_2 + l_2 \alpha_2 \Delta T \).
The difference in lengths at the new temperature is:
\[ l'_1 - l'_2 = (l_1 + l_1 \alpha_1 \Delta T) - (l_2 + l_2 \alpha_2 \Delta T) \] Grouping terms, we get:
\[ l'_1 - l'_2 = (l_1 - l_2) + (l_1 \alpha_1 - l_2 \alpha_2) \Delta T \] The term \( (l_1 - l_2) \) is the initial difference in lengths.
For the difference to be independent of the temperature change \( \Delta T \), the coefficient of \( \Delta T \) in the expression must be zero.
Therefore, we set the coefficient to zero:
\[ l_1 \alpha_1 - l_2 \alpha_2 = 0 \] Rearranging the equation yields:
\[ l_1 \alpha_1 = l_2 \alpha_2 \] This means the absolute expansions must be equal: \( \Delta l_1 = \Delta l_2 \).
From \( l_1 \alpha_1 = l_2 \alpha_2 \), we can write the ratio of lengths as:
\[ \frac{l_1}{l_2} = \frac{\alpha_2}{\alpha_1} \] Step 4: Final Answer:
The required condition is \( \frac{l_1}{l_2} = \frac{\alpha_2}{\alpha_1} \).