In optics, the formula of Hagen-Rubens expresses the coefficient of reflection of a conducting medium according to the frequency of incidental radiation and conductivity of the medium.

This relation is written:

$R=1-2\; \backslash \; left\; (\backslash \; frac\; \{2\; \backslash \; epsilon\_0\; \backslash \; Omega\}\; \{\backslash \; sigma\}\; \backslash \; right)\; ^\; \{1/2\}$

In the system of units IF.

Demonstration

One leaves the Maxwell's equations and one expresses the density of current according to the field electric by using the Law of Ohm.

One obtains:

$\backslash \; nabla\; \backslash \; times\; \backslash \; mathbf\; \{B\}\; =\; \backslash \; mu\_0\; \backslash \; sigma\; \backslash \; mathbf\; \{E\}\; +\; \backslash \; mu\_0\; \backslash \; epsilon\_0\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; mathbf\; \{E\}\}\; \{\backslash \; partial\; T\}$

and:

$\backslash \; nabla\; \backslash \; times\; \backslash \; mathbf\; \{E\}\; =\; \backslash \; frac\; \{\backslash \; partial\; B\}\; \{\backslash \; partial\; T\}$

By eliminating the magnetic field, one finds a differential equation with the derivative partial of the second order for the electric field. This equation is written:

$\backslash \; nabla^2\; \backslash \; mathbf\; \{E\}\; =\; \backslash \; mu\_0\; \backslash \; partial\; sigma\; \backslash \; frac\; \{\backslash \; \backslash \; mathbf\; \{E\}\}\; \{\backslash \; partial\; T\}\; +\; \backslash \; frac\; \{1\}\; \{c^2\}\; \backslash \; frac\; \{\backslash \; partial^2\; \backslash \; mathbf\; \{E\}\}\; \{\backslash \; partial\; t^2\}$

One seeks a solution of this equation in the form $E\; (X,\; T)\; =E\_0e^\; \{ikx-i\; \backslash \; Omega\; T\}$, what gives:

$k=\; \backslash \; frac\; \{\backslash \; Omega\}\; \{C\}\; \backslash \; left\; (1+\; \backslash \; frac\; \{I\; \backslash \; sigma\}\; \{\backslash \; Omega\; \backslash \; epsilon\_0\}\; \backslash \; right)\; ^\; \{1/2\}$

A metal thus behaves like a medium of complex index of refraction

$N\; (\backslash \; Omega)\; =\; \backslash \; left\; (1+\; \backslash \; frac\; \{I\; \backslash \; sigma\}\; \{\backslash \; Omega\; \backslash \; epsilon\_0\}\; \backslash \; right)\; ^\; \{1/2\}$

The imaginary part of $N\; (\backslash \; Omega)$ is responsible for the effect of skin of metal.

By applying the formula giving the coefficient of reflection to a medium of index $N\; (\backslash \; Omega)$ and by supposing a frequency such as $\backslash \; sigma\; \backslash \; gg\; \backslash \; epsilon\_0\; \backslash \; Omega$ the relation of Hagen-Rubens is obtained.