Illustrated Physics
physics, drawn
Concept

Error function — where it appears

The integral of a Gaussian, scaled to run from zero to one, which describes how a sudden step spreads out under diffusion. It gives the temperature in a body cooled from its surface and the concentration in a diffusion couple, with a width that grows as the square root of time.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Where skin and the thing it touches meet. Temperature across the contact between skin at 33 °C (left) and a block at 20 °C (right), against distance from the contact in millimetres, at 0.1, 1 and 10 s after touching, for steel (blue) and pine wood (red). Each pair of curves is two error functions meeting at the contact, and the meeting temperature does not change with time: 21.0 °C against steel and 30.3 °C against wood. The nerve endings a fraction of a millimetre under the skin report something close to that interface temperature, so the steel feels like an object at 21 °C and the wood like one at 30 — although both are at 20. The profiles spread as the square root of time, much further into the steel than into the wood.

The metal that feels colder than the wood

A steel railing and a wooden bench in the same shade are at the same temperature, and the railing feels much colder. The usual explanation, that metal conducts heat better, is half the answer, and it gets water wrong. When two bodies touch, the surface between them jumps at once to a temperature that then stays fixed, a weighted average of the two in which each body's weight is the square root of its conductivity times its heat capacity per volume. Skin at 33 °C touching steel at 20 °C meets it at 21; touching wood, at 30. The nerves report the meeting temperature, not the object's.

thermodynamics · Diffusion A cooling planet, read from the top. Temperature against depth in a body cooling by conduction from a uniform start at 3870 °C, with a surface held at nought, at four ages: the exact error-function solution for a half-space. Cooling reaches downward as the square root of time, so after a hundred million years the upper few hundred kilometres have cooled and the interior knows nothing about it. The gradient at the surface is what a mine measures, and it falls as the inverse root of time. A thermal gradient therefore gives an age, and this is what Kelvin saw in 1862. He assumed the Earth was a solid conductor from the start, on a good basis, as everyone believed then, and these curves are the mathematical consequence of that belief.

The age a cooling Earth gives

In 1862 William Thomson, later Lord Kelvin, took a thermometer reading from the bottom of a mine and a piece of mathematics from Fourier and computed the age of the Earth: about a hundred million years. Geologists, needing far longer, had no physics with which to reply. The arithmetic was right. The usual story blames radioactivity for overturning it, but half the heat is not radioactive, and the error lay in assuming the inside of the planet stays still.

thermodynamics · Diffusion

Named alongside it

The objects these essays reach for when they reach for this one.

Heat equationAge of the earthDiffusionGeothermal gradientHalf-space modelHeat capacityHeat conductionMantle convectionPlate coolingRadiogenic heatThermal diffusivityThermal effusivity

All concepts