Weber-Fechner law posits that the perceived intensity of a stimulus is proportional to the logarithm of its actual intensity.In the 19th century, psychophysics was just emerging. Ernst Heinrich Weber, a German anatomist and physiologist, conducted experiments with weights: blindfolded subjects compared heaviness. He derived a rule: the ratio of the just-noticeable increment in weight to the original weight is a constant. Later, Gustav Theodor Fechner, a philosopher and physicist, gave it mathematical form: S = k log I, where S is the sensation magnitude, I is the stimulus intensity, and k is a constant. Thus the Weber–Fechner law was born. It explained why we can distinguish 1 candle against a background of 10, but not against 1000. The law works for vision, hearing, and taste, but with caveats: at very low and very high stimulus intensities, the proportionality breaks down.
Ernst Heinrich Weber, a German physiologist, published the work 'De tactu' in 1834, describing experiments on weight discrimination. He found that the minimum noticeable difference in weight is about 1/30 of the original. Later, he extended the principle to vision and hearing. Gustav Fechner, Weber's student, mathematically formalized the law in his 1860 book 'Elements of Psychophysics', introducing a logarithmic scale of sensations. Fechner is considered the founder of psychophysics. The law spurred the development of experimental psychology and sensory physiology.
The law is applied in designing audio equipment (decibel scale), in ergonomics (pedal force), and in the food industry (perception of sweetness). Limits of applicability: Weber's law breaks down at very low intensities (near-threshold) and very high intensities (pain threshold); for many modalities, Stevens' power function S = k Iⁿ fits better.
In audio engineering: volume is adjusted on a logarithmic scale because the ear perceives changes in loudness in relative units, not absolute ones.
Fechner was a mystic who believed that everything in the Universe has a soul. He considered his psychophysical law a bridge between matter and spirit — a mathematical proof of the connection between body and mind.
| I | variable | Transmitted intensity — intensity watt per square metre |
| I_0 | variable | Incident intensity — intensity watt per square metre |
| σ | variable | Absorption cross-section per absorber — cross section square metre |
| n | variable | Number density of absorbers — number density per cubic metre |
| x | variable | Path length through the medium — length metre |
| \Gamma | variable | Reaction rate — reaction rate per second |
| E_a | variable | Activation energy — energy joule |
| T | variable | Absolute temperature — temperature kelvin |
| A | constant | Frequency factor, related to collision frequency and steric effects — pre exponential factor = material-dependent per second |
| k_B | constant | Relates temperature to energy — boltzmann constant = 1.380649e-23 joule per kelvin |
| e^{-E_a/k_B T} | operator | Exponential decay with inverse temperature — exponential |
| J | variable | Diffusive flux (amount per unit area per unit time) — diffusive flux mole per square metre per second |
| c | variable | Concentration of the diffusing species — concentration mole per cubic metre |
| x | variable | Spatial coordinate along the diffusion direction — position metre |
| D | constant | Material-specific proportionality constant relating flux to concentration gradient — diffusion coefficient = varies square metre per second |
| ∂/∂x | operator | Partial derivative with respect to position — partial derivative |
| Bi | variable | Bingham number — bingham number |
| σ_y | variable | Yield stress of the material — yield stress pascal |
| σ_prop | variable | Characteristic propulsion stress — propulsion stress pascal |