# 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–1 and 23 × 10–6 °C–1 respectively.

Given:
Coefficient of Linear Expansion of iron rod : αFe = 12 x 10-6 °C-1
Coefficient of Linear Expansion of Aluminium Rod : α Al = 20 x 10-6 °C-1
Formula used:

The formula for Linear Expansion is:

Let, LFe and LAl be the original lengths of the iron and aluminium rods respectively.
Let, L’Fe and L’Al be the changed lengths (ΔL) of iron and aluminium rods respectively when temperature is changed by ΔT.
Now ,

Since , the difference in lengths is independent of temperature it shows that their difference is constant. That is, L’Fe – L’Al= LFe - LAl

By arranging the terms we get,

Hence, the ratio of the lengths of an iron rod and an aluminium rod for which the difference in the lengths is independent of temperature is 23:12.

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