Verified Solution • 2025–26 Session moderate

QPages 12-14: Q18: HC Verma: Concepts of Physics - Volume 2 Heat and Temperature Pages 12-14: Q18 Solution: Find the ratio of the lengths of an iron rod and an aluminium…

⚠️ Common Exam Traps & Conceptual Pitfalls

Students frequently set up $L_{Fe}\alpha_{Fe}=L_{Al}\alpha_{Al}$ correctly but then solve for $L_{Fe}/L_{Al}$ as $\alpha_{Fe}/\alpha_{Al}$, swapping the fractions. Remember to isolate $L_{Fe}/L_{Al}$ by dividing both sides by $L_{Al}\alpha_{Fe}$, giving $L_{Fe}/L_{Al}=\alpha_{Al}/\alpha_{Fe}$.

Find the ratio of the lengths of an iron rod and an aluminium rod for which the difference in the lengths is independent of temperature. Coefficients of linear expansion of iron and aluminium are 12×10−6 �C−112 \times 10^{-6} \text{ �C}^{-1} and 23×10−6 �C−123 \times 10^{-6} \text{ �C}^{-1} respectively.

Step-by-Step Verified Solution

**Answer:** LFe/LAl=23/12L_{Fe}/L_{Al} = 23/12 **Step-by-Step Explanation:** For difference in lengths to be independent of temperature: LFeαFe=LAlαAlL_{Fe}\alpha_{Fe} = L_{Al}\alpha_{Al}.
LFeLAl=αAlαFe=23×10−612×10−6=2312\frac{L_{Fe}}{L_{Al}} = \frac{\alpha_{Al}}{\alpha_{Fe}} = \frac{23 \times 10^{-6}}{12 \times 10^{-6}} = \frac{23}{12}

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