Thermal Expansion

$\Delta L=\alpha L\Delta T$

• $\Delta L$ is the change in length
• $\alpha$ is the coefficient of linear expansion
• $L$ is the original length of the object
• $\Delta T$ is the change in temperature

$\Delta V=\beta V\Delta T$

• $\Delta V$ is the change in volume
• $\beta$ is the coefficient of volumetric expansion
• $V$ is the original volume of the object
• $\Delta T$ is the change in temperature of the object

$\beta =3\alpha$

• Derivation:
  Approximate as a cube

$$V=L^3$$ $$\begin{array}{rl}\Delta V& \approx 3(\Delta L{L}^2)\text{ for }\Delta L\ll L\\ & =3([\alpha L\Delta T]L^2)\\ & =(3\alpha )(L^3)\Delta T\\ & =\beta V\Delta T\end{array}$$

$\alpha$ values

Desmos Demo