Question

The coefficient of linear expansion of brass and steel are ${\alpha _1}$ and ${\alpha _2}$. If we take a brass rod of length ${l_1}$ and steel rod of length ${l_2}$ at $0^\circ C$ and their difference in length $(\,{l_2}\, - \,{l_{1\,}})$ will remain the same at all temperatures if
(A) ${\alpha _1}{l_2} = {\alpha _2}{l_1}$
(B) ${\alpha _1}{l_2}^2 = {\alpha _2}{l_1}^2$
(C) ${\alpha _1}^2{l_2} = {\alpha _2}^2{l_1}$
(D) ${\alpha _1}{l_1} = {\alpha _2}{l_2}$

Answer 1

Lengths of rods made up of brass and steel are ${l_1}$ and ${l_2}$ and their coefficient of linear expansion are ${\alpha _1}$ and ${\alpha _2}$ at $0^\circ C$ temperature. The length of a metal rod will increase from ${l_1}$ to ${l_1}\,(\,1 + {\alpha _1}\,)$. Here $\alpha$ is the coefficient of linear expansion. We will then have to check each option to find out which equation is correct.

Complete step by step answer
Coefficient of linear expansion $(\alpha )$:
For $1^\circ$ rise in temperature, the ratio of the increase in length of wire to its actual length of wire is known as coefficient of linear expansion.
$\alpha = \dfrac{{\Delta l}}{{\Delta T}}$
Reason for expansion:
When heat is supplied to the object, intermolecular force of attraction between the molecules decreases and molecules start to separate. This results in increase in spacing between the molecules and therefore results in increase in size and change in shape of substance. This is the reason why length, area or volume of substance expands.
In this case we will study about length expansion only.
At temperature T,
Length of brass rod $= {l_1}\,(\,1 + {\alpha _1}\,T\,)$
Length of steel rod $= {l_2}\,(\,1 + {\alpha _2}\,T\,)$
Difference in lengths $= {l_1}\,(\,1 + {\alpha _1}\,T\,) - {l_2}\,(\,1 + {\alpha _2}\,T\,)$
$= (\,{l_2}\, - \,{l_{1\,}}) + T\,(\,{l_1}{\alpha _1} - \,{l_2}{\alpha _2}\,)$
Length will be independent of temperature only when coefficient of temperature will be equal to zero.
$(\,{l_1}{\alpha _1} - \,{l_2}{\alpha _2}\,)\, = \,0$
${l_1}{\alpha _1} = \,{l_2}{\alpha _2}\,$

Since D is a satisfying option. Hence it is the correct relation.

Note
If $(\,{l_2}\, - \,{l_{1\,}})$ equals to zero then both the lengths will become equal which cannot be possible. So, it is a wrong understanding.
According to the concept coefficient of linear expansion is multiplied by length of same material used then option A, B and C are not possible.
Hence, we are left with only one option I.e. D. so the correct solution is D.