Illustrated Physics
physics, drawn
Concept

Continuity — where it appears

The rule that a conserved quantity can change in a region only by flowing across its boundary. For an incompressible fluid it means the same flow passes every cross-section of a channel, so water speeds up where the channel narrows; for charge it ties current to changing density.

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

Charge density and current, mixing like time and space. The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 0 and J = 2. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism behind the familiar story of two contracted lattices of charge. Here it is a coordinate change.

Charge and current are one thing

Ask what a boost leaves alone and the answer is charge. That answer forces the next one: a fixed charge in a contracting volume gives a density that transforms like a time component, and a current that transforms like a space one. So charge density and current density are the four parts of one object — and conservation of charge stops being an extra law and becomes the condition that makes the object exist.

relativity · Field transformation A bridge that never feels the boat crossing it. The weight of water an aqueduct 300 m long, 6.5 m wide and 2.7 m deep carries, against the position of an 80-tonne boat crossing it, with ten kilometres of canal joined to it. The boat pushes its own weight of water out of the aqueduct, and the water spreads along the whole reach, raising it by 1.2 millimetres. The load stays at 5265 tonnes whether the boat is on the aqueduct or not; a road bridge would gain the vehicle's whole weight. The aqueduct is designed for its water alone, and a boat crossing it is invisible to the structure.

The bridge that does not feel the boat

An aqueduct carries a canal across a valley, and a barge of eighty tonnes crosses it. The aqueduct carries no more weight with the barge on it than without, because a floating barge pushes out exactly its own weight of water, and the water goes elsewhere. Boat lifts are built on the same fact: a caisson of water weighs the same whatever floats in it, so two caissons balance whatever boats they carry, and the only thing that needs measuring is the water level.

fluids · Buoyancy

Named alongside it

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

Archimedes principleBernoulli equationBoat liftBuoyancyCharge conservationCharge densityCharge invarianceConservation lawsField transformationFour-vectorHydrostatic pressureInvariance

All concepts