Beer-Lambert Law

The Beer-Lambert law (also called Beer’s law) describes the attenuation of light or radiation as it passes through a material.

Statement

For monochromatic radiation passing through a homogeneous medium:

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

where:

• $I_0$ = incident intensity
• $I$ = transmitted intensity
• $\mu$ = attenuation coefficient (units: 1/length)
• $x$ = path length through the material

Differential Form

The intensity decreases proportionally to the current intensity:

$$ \frac{dI}{dx} = -\mu I $$

This is why attenuation is multiplicative (exponential) rather than additive.

For Inhomogeneous Media

When the attenuation coefficient varies with position:

$$ I = I_0 \exp\left(-\int_0^L \mu(x) \, dx\right) $$

Taking the logarithm:

$$ \ln\left(\frac{I_0}{I}\right) = \int_0^L \mu(x) \, dx $$

This line integral is exactly what CT scanners measure — the Radon transform of the attenuation coefficient.

Absorbance

In spectroscopy, the absorbance $A$ is defined as:

$$ A = \log_{10}\left(\frac{I_0}{I}\right) = \varepsilon c \ell $$

where:

• $\varepsilon$ = molar absorptivity
• $c$ = concentration
• $\ell$ = path length

Applications

• Spectrophotometry: Measuring concentrations
• Medical imaging: X-ray and CT attenuation
• Atmospheric science: Light propagation through atmosphere
• Optical fibers: Signal attenuation in telecommunications

Limitations

The law assumes:

• Monochromatic radiation
• No scattering (pure absorption)
• Dilute solutions (for spectroscopy)
• Uniform illumination

See Also

• Computed tomography
• Radon transform