A copper rod and a steel rod are to have lengths L_C and L_S, such that the difference between their lengths is the same at all ambient temperatures. If the coefficients of linear expansion of copper and steels are α_C and α_S respectively. The lengths are related to the coefficient of linear expansion as :

Concept:

The coefficient of the linear expansion is defined as the rate of change of unit length per unit degree of change in the temperature and it is written as;

⇒ ΔL = LαΔT      

Here, ΔL = change in length, α = coefficient of linear expansion and ΔT = change in temperature.

Calculation:

Given:

Length of copper = LC, Length of steel = LS.

The coefficients of linear expansion of copper and steels are αC and αS.

Change in length of Copper:

ΔL_c = L_cα_cΔT

Change in length of Steel:

ΔLs = LsαsΔT

Given that difference between their lengths is the same at all temperatures i.e. ΔLc = ΔLs.

ΔLc = ΔLs 

LcαcΔT = LsαsΔT 

\(\frac{L_C}{L_S}=\frac{\alpha_S}{\alpha_C}\)