14. Areal (Superficial)l expansion

The amount by which the area of a material increases when the temperature is raised one degree is called the coefficient of superficial expansion. The unit of coefficient of superficial expansion is $K^{-1}$.

The coefficient of superficial expansion can be designated by the symbol $\mathrm{\beta }$ (beta).

Areal expansion in solid

Consider a material with the area of $A_1$ at $t_1^o$ \(c\). And, the material is heated upto $t_2^o$ \(c\). Now, the length of the material is $A_2$.

Let $\mathrm{\beta }$ be the coefficient of linear expansion.

We know that,

The change in area \(ΔA\) is proportional to the original area $A_1$, rise in temperature \(ΔT\), and material type.

$\begin{array}{l}\mathrm{\Delta }A=A_2-A_1\mathit{and}\\ \\ \mathrm{\Delta }T=t_2-t_1\\ \\ \mathrm{\Delta }A\propto A_1\mathrm{\Delta }T\\ \\ \mathrm{\Delta }A=\mathrm{\beta }{A}_1\mathrm{\Delta }T\end{array}$

Rearranging the above equation,

$\begin{array}{l}\mathrm{\beta }\hspace{0.17em}=\frac{A_2-A_1}{A_1\left(t_2-t_1\right)}\\ \mathit{Where},\\ A_1\mathit{and}{A}_2\mathit{are}\mathit{the}\mathit{areas}\mathit{of}\mathit{the}\mathit{material}\mathit{before}\mathit{and}\mathit{after}\mathit{heating}\mathit{respectively}.\\ t_1\mathit{and}{t}_2\mathit{are}\mathit{the}\mathit{temperatures}\mathit{of}\mathit{the}\mathit{material}\mathit{before}\mathit{and}\mathit{after}\mathit{heating}\mathit{respectively}.\\ \mathrm{\beta }\mathit{is}\mathit{the}\mathit{Areal}\mathit{coefficient}\mathit{of}\mathit{expansion}\end{array}$

The $\mathrm{\beta }$ will vary for different materials.