Beer-Lambert law

When a light beam passes through an absorbing medium, the intensity $I$ decreases exponentially with the thickness $x$:

$I(x) = I_0\,e^{-\mu x},$

where $\mu$ is the absorption coefficient. The half-value thickness is $x_{1/2}=\dfrac{\ln 2}{\mu}$. It is the same law as radioactive decay, with time replaced by thickness.

Example — Red in a swimming pool

In a swimming pool, red is attenuated to $1/2$ of the initial intensity after about $5\,\text{m}$. After $20\,\text{m}$ only $1/16$ remains: this is why objects deep down appear blue.

Links

Topics: Exponential function
Concepts: Exponential decay · Euler’s number
Functions: Exponential function
Skills: Modelling · Using formulae