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General Physics I:

Subsection 10.2.1 Linear Expansion

Suppose a rod of original length \(l_{o}\) is heated for a temperature difference of \(\Delta T\text{,}\) then we can find that the change in its length is proportional to the change in temperature, i.e.,
\begin{equation*} \Delta l \,\propto \,\Delta T \text{.} \end{equation*}
If \(\Delta T \) increases \(\Delta l \) also increases by itself. We can also see that the change in length is also proportional to its original length [Figure 10.2.1.(a)], i.e.,
\begin{equation*} \Delta l \propto l_{o} \end{equation*}
Hence by combining these properties, we can have
\begin{equation*} \Delta l \propto l_{o}\times \Delta T \end{equation*}
\begin{equation*} \text{or,}\quad \Delta l =\alpha l_{o}\times \Delta T \end{equation*}
where \(\alpha\) is a proportionality constant called coefficient of linear expansion or linear expansivity. It is a material property. This distinguishes the expansion properties of one material to another.
\begin{equation*} \text{or,}\qquad \alpha =\frac{\Delta l }{l_{o}\Delta T}= \frac{l-l_{o}}{l_{o}\left(T-T_{o}\right)} \end{equation*}
\begin{equation*} \therefore \quad l= l_{o}\left(1+\alpha\Delta T\right) \end{equation*}
Hence, linear expansivity is defined as the change in length of a rod per unit original length per unit rise of temperature.