Animated Solution for Physics - Properties of Solids and Liquids: Two different wires having lengths $L_1$ and $L_2$ and respective temperature coefficients of linear expansion ${\alpha }_1$ and ${\alpha }_2$, are joined end-to-end. Then the effective temperature coefficient of linear expansion is

Imagine you have two different metal wires, one of length $L_1$ and another of length $L_2$. Because they are made of different materials, they respond differently to heat. This unique response is quantified by their respective temperature coefficients of linear expansion, ${\alpha }_1$ and ${\alpha }_2$.

When these two wires are joined end-to-end to form a single composite wire, a natural question arises: If we were to replace this composite wire with a single, uniform wire of the same total length, what would its effective temperature coefficient $\alpha$ need to be so that it expands by the exact same amount when heated?

Let's break down the physics. When the temperature of the system increases by an amount $\Delta T$, each wire expands independently according to the fundamental law of linear thermal expansion:

For the first wire, the expansion is $\Delta L_1=L_1{\alpha }_1\Delta T$.

For the second wire, the expansion is $\Delta L_2=L_2{\alpha }_2\Delta T$.

Because the wires are connected in series (end-to-end), the total change in length of the composite system is simply the sum of their individual expansions:

Now, let's introduce our hypothetical "equivalent" wire. This wire has a total length of $L=L_1+L_2$ and an unknown effective coefficient $\alpha$. If it is truly equivalent, its total expansion must match the composite system's expansion for the same temperature rise $\Delta T$.

So, the expansion of the equivalent wire is:

Equating this to the total expansion of the composite system, we get our master equation:

Notice how the temperature change $\Delta T$ appears in every term. Since the entire system experiences the same uniform heating, we can safely divide both sides by $\Delta T$, eliminating it completely from the equation:

We know that the total length $L$ is just the sum of the individual lengths, $L_1+L_2$. Substituting this into our equation gives:

Finally, to isolate the effective coefficient $\alpha$, we divide both sides by the total length $(L_1+L_2)$:

This elegant result shows that the effective temperature coefficient of a series combination is simply the weighted average of the individual coefficients, where the "weights" are the original lengths of the wires. The longer a particular wire is, the more its specific $\alpha$ dominates the overall behavior of the composite system.