# Fourier's law

**Fourier's law** relates thermal (heat) conduction to the temperature gradient. Thermal conduction is the transfer of internal energy by microscopic collisions of atoms or molecules and the movement of electrons within a body. The rate at which energy is conducted as heat between two bodies is a function of the temperature gradient between the two bodies. Heat spontaneously flows from a hotter to a colder source.

Fourier's law in a single dimension is

(1) |

where

- is the
**heat flux**, which is expressed as energy per unit area per unit time. - is the
**thermal conductivity**. The dimension is area per unit time, so typical units for expressing it would be m^{2}/s. - is the temperature.
- is position, the dimension of which is length.

In two or more dimensions we use , or the gradient operator. Independent of the coordinate system, Fourier's law is expressed as

(2) |

where denotes the heat flux vector.

**Integral Form of Fourier's Law**

If you integrate the single dimension form, shown above, over the materials surface area **S** you get the integral form:

(3) |

where

- is the heat transferred per unit time in
**W** - is the oriented surface area element in

## References

^{[1]}

- ↑ “Thermal Conduction.” Wikipedia, Wikimedia Foundation, 2 Dec. 2020, en.wikipedia.org/wiki/Thermal_conduction.