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
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
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If expression independent of temperature â coefficient of \(\Delta T\) must be zero.
Updated On: Apr 26, 2026
- \(\alpha_1^2 l_1 = \alpha_2^2 l_2\)
- \(\frac{l_1}{l_2} = \frac{\alpha_2}{\alpha_1}\)
- \(\frac{l_1}{l_2} = \frac{\alpha_1}{\alpha_2}\)
- \(l_1^2 \alpha_2 = l_2^2 \alpha_1\)
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The Correct Option is B
Solution and Explanation
Concept:
Length after temperature change: \[ L = l(1 + \alpha \Delta T) \] Step 1: Write lengths after expansion. \[ L_1 = l_1(1 + \alpha_1 \Delta T) \] \[ L_2 = l_2(1 + \alpha_2 \Delta T) \]
Step 2: Condition given. Difference is constant: \[ L_1 - L_2 = \text{constant} \]
Step 3: Expand. \[ l_1 + l_1\alpha_1 \Delta T - (l_2 + l_2\alpha_2 \Delta T) \] \[ = (l_1 - l_2) + (l_1\alpha_1 - l_2\alpha_2)\Delta T \]
Step 4: For independence from temperature. Coefficient of \(\Delta T = 0\): \[ l_1\alpha_1 = l_2\alpha_2 \]
Step 5: Rearrange. \[ \frac{l_1}{l_2} = \frac{\alpha_2}{\alpha_1} \]
Step 6: Conclusion. Correct option is (B).
Length after temperature change: \[ L = l(1 + \alpha \Delta T) \] Step 1: Write lengths after expansion. \[ L_1 = l_1(1 + \alpha_1 \Delta T) \] \[ L_2 = l_2(1 + \alpha_2 \Delta T) \]
Step 2: Condition given. Difference is constant: \[ L_1 - L_2 = \text{constant} \]
Step 3: Expand. \[ l_1 + l_1\alpha_1 \Delta T - (l_2 + l_2\alpha_2 \Delta T) \] \[ = (l_1 - l_2) + (l_1\alpha_1 - l_2\alpha_2)\Delta T \]
Step 4: For independence from temperature. Coefficient of \(\Delta T = 0\): \[ l_1\alpha_1 = l_2\alpha_2 \]
Step 5: Rearrange. \[ \frac{l_1}{l_2} = \frac{\alpha_2}{\alpha_1} \]
Step 6: Conclusion. Correct option is (B).
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