### Theory:

Expansion of Solids:

Whenever we heat a solid, there is an increase in the dimensions of the body. It is known as the expansion of solids.
The solid undergoes three types of expansions.

1. Linear expansions
2. Areal expansions
3. Cubical expansions

Linear expansions:

When we heat a solid, if there is an increase in the body's length, then it is called linear expansion. It is also known as longitudinal expansion.

The increase in the length is proportional to the rise in temperature, original length, and material type. The amount by which the unit length of a material increases when the temperature is raised by one degree is called the coefficient of linear expansion. The coefficient of linear expansion is used to find out the actual increase in length. It can be represented by the symbol $\mathrm{\alpha }$ (alpha).

Consider a material with the length of ${l}_{1}$ at ${t}_{1}^{o}$ $$c$$ and the material is heated up to ${t}_{2}^{o}$ $$c$$, now the length of the material is ${l}_{2}$. Let $\mathrm{\alpha }$ be the coefficient of linear expansion, then we have:

$\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{{l}_{2}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{l}_{1}}{{l}_{2}\left({t}_{2}^{o}-{t}_{1}^{o}\right)}$ ----  eqn 1

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

Expansion in solids

In the above diagram,

${l}_{1}$ = $$1$$ $$m$$

${t}_{1}^{o}$ = ${20}^{o}$$$c$$

${t}_{2}^{o}$ = ${120}^{o}$$$c$$

${l}_{2}$ = $$1.001$$ $$m$$

then, $\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{{l}_{2}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{l}_{1}}{{l}_{2}\left({t}_{2}^{o}-{t}_{1}^{o}\right)}$

$\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{1.001\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}1}{1.001\phantom{\rule{0.147em}{0ex}}\left(120-20\right)}$ = $$0.0000099$$${}^{o}c^{-}{}^{1}$.

We shall calculate the length after the expansion using equation,

$\begin{array}{l}{l}_{2}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}{l}_{1}\left(1+\mathrm{\alpha }t\right)\\ \\ \mathit{where},\\ t=\left({t}_{2}-{t}_{1}\right)\end{array}$

We shall calculate change in length using equation,

$\begin{array}{l}{l}_{c}={l}_{1}×\mathrm{\alpha }×t\\ \mathit{where},\\ {l}_{c}=\phantom{\rule{0.147em}{0ex}}\mathit{Change}\phantom{\rule{0.147em}{0ex}}\mathit{in}\phantom{\rule{0.147em}{0ex}}\mathit{length}\\ t=\left({t}_{2}-{t}_{1}\right)\end{array}$

Areal expansions:

When we heat a solid, if there is an increase in the area of the body, then it is called areal expansion. It is also known as superficial expansion

The amount by which the area of a material increases when the temperature is raised by one degree is called the coefficient of superficial expansion.

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

$\begin{array}{l}\mathrm{\beta } =\phantom{\rule{0.147em}{0ex}}\frac{{A}_{2}-{A}_{1}}{{A}_{1}\left({t}_{2}-{t}_{1}\right)}\\ \mathit{Where},\\ {A}_{1}\mathit{and}\phantom{\rule{0.147em}{0ex}}{A}_{2}\phantom{\rule{0.147em}{0ex}}\mathit{are}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{areas}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{material}\phantom{\rule{0.147em}{0ex}}\mathit{before}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{after}\phantom{\rule{0.147em}{0ex}}\mathit{heating}\phantom{\rule{0.147em}{0ex}}\mathit{respectively}.\\ {t}_{1}\mathit{and}\phantom{\rule{0.147em}{0ex}}{t}_{2}\phantom{\rule{0.147em}{0ex}}\mathit{are}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{temperatures}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{material}\phantom{\rule{0.147em}{0ex}}\mathit{before}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{after}\phantom{\rule{0.147em}{0ex}}\mathit{heating}\phantom{\rule{0.147em}{0ex}}\mathit{respectively}.\\ \mathrm{\beta }\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{areal}\phantom{\rule{0.147em}{0ex}}\mathit{coefficient}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{expansion}\phantom{\rule{0.147em}{0ex}}\end{array}$

Cubical expansions:

When we heat a solid, if there is an increase in the volume of the body, then it is called Cubical expansion. It is also known as Volumetric expansion.

The amount by which the volume of a material increases when the temperature is raised by one degree is called the coefficient of volumetric expansion.

The coefficient of volumetric expansion can be represented by the symbol $\mathrm{\gamma }$ (gamma).

$\begin{array}{l}\mathrm{\gamma } =\phantom{\rule{0.147em}{0ex}}\frac{{V}_{2}-{V}_{1}}{{V}_{1}\left({t}_{2}-{t}_{1}\right)}\\ \mathit{Where},\\ {V}_{1}\mathit{and}\phantom{\rule{0.147em}{0ex}}{V}_{2}\phantom{\rule{0.147em}{0ex}}\mathit{are}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{volumes}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{material}\phantom{\rule{0.147em}{0ex}}\mathit{before}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{after}\phantom{\rule{0.147em}{0ex}}\mathit{heating}\phantom{\rule{0.147em}{0ex}}\mathit{respectively}.\\ {t}_{1}\mathit{and}\phantom{\rule{0.147em}{0ex}}{t}_{2}\phantom{\rule{0.147em}{0ex}}\mathit{are}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{temperatures}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{material}\phantom{\rule{0.147em}{0ex}}\mathit{before}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{after}\phantom{\rule{0.147em}{0ex}}\mathit{heating}\phantom{\rule{0.147em}{0ex}}\mathit{respectively}.\\ \mathrm{\gamma }\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{volumetric}\phantom{\rule{0.147em}{0ex}}\mathit{coefficient}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{expansion}.\end{array}$

Relation between $\mathrm{\alpha },\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathrm{\gamma }$:

$6\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}4\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}2\mathrm{\gamma }$.

Reference:
https://i.stack.imgur.com/3Qoah.png