Illustrated Physics
physics, drawn
Concept

Vector field — where it appears

An assignment of a vector, a size and a direction, to every point of space, such as the electric field or the velocity of a flowing fluid. Its lines, divergence, curl and the points where it vanishes each carry part of what it describes.

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

The same field, sampled as arrows. The field at a grid of points, each arrow pointing the way a positive test charge would be pushed and scaled by the strength there.

The field before the lines were drawn on it

A field is a vector attached to every point of space. Drawing it as arrows on a grid is honest and ugly; drawing it as lines is beautiful and throws information away.

electromagnetism · The field concept Four places where three charges' fields cancel. Field lines in the plane of three equal positive point charges at the corners of an equilateral triangle, with every point where the field is zero marked. There are four: one at the centre, and three more on the lines from the centre to the middle of each side, 0.164 of a side from the centre. Four is (n − 1)² for three charges, the largest number Maxwell conjectured any three charges could make. Each zero is a saddle: along some directions the field pushes a test charge back and along others away, so none of them can hold anything still.

The zeros a handful of charges can make

Between two equal charges there is a point where their fields cancel. With three there can be two such points, or one, or four. Every such point is a saddle, never a place where a charge could rest. In two dimensions the count is fixed by algebra: n equal line charges make exactly n − 1 zeros, the roots of a polynomial's derivative. In three dimensions nobody knows the answer. In 1873 Maxwell conjectured that n point charges can make at most (n − 1)² zeros. Three charges on a triangle make exactly four, and the conjecture has not been proved even for three charges.

electromagnetism · The field concept

Named alongside it

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

Electric fieldField linesEarnshaw theoremField energyLaplace equationNull pointSaddle pointSuperpositionTest chargeTopology

All concepts