Thermal Expansion

#Physics

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

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

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

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

$\displaystyle \beta = 3\alpha$

• Derivation:
  Approximate as a cube
  $$
  \displaystyle V=L^{3}
  $$
  $$
  \begin{align}
  \Delta V & \approx 3(\Delta L L^{2}) \text{ for }\Delta L \ll L\\ & = 3([\alpha L\Delta T]L^{2}) \\ & = (3\alpha)(L^{3}) \Delta T \\ & = \beta V \Delta T
  \end{align}
  $$

$\displaystyle \alpha$ values

Desmos Demo