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.