Illustrated Physics
physics, drawn
Concept

Biot–Savart — where it appears

The rule giving the magnetic field of a current as a sum over its elements, each falling as the inverse square of its distance. It is what computes a field where symmetry is absent and Ampère's law therefore yields only a number: a wire as long as the distance to the field point gives 45 per cent of the infinite-wire answer.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The field of a current loop, seen edge on. Magnetic field lines integrated from the Biot–Savart law for the current shown. Every line closes on itself: there is nowhere for one to start and nowhere for it to end, because no magnetic charge exists to end on.

The field with no ends, and the force that does no work

Magnetic field lines never start and never stop. That single absence is a law, it has survived every attempt to break it, and it makes the magnetic field a different kind of object from the electric one.

electromagnetism · Magnetism How much of the answer a finite wire gives back. The field beside a straight segment of wire, divided by what the infinite-wire formula would give, against the length of the segment in units of the distance to the field point, on a logarithmic horizontal axis. The field is integrated element by element along the segment rather than evaluated from a closed form. A wire ten times as long as the distance already gives 92.8 per cent of the infinite answer, and one as long as the distance gives 45 — which is the practical content, and the reason the infinite-wire result is used for laboratory wires without apology. The second column is the part that matters for the law rather than for the number. The circulation of this field round a circle of radius d is not μ₀I; it falls short by exactly the fraction the segment fails to subtend. What makes up the difference is the displacement current of the charge piling up at the segment's two ends, and the two terms, both integrated here, sum to μ₀I to within 2.2e-8 per cent at every length. So Ampère's law is not approximately true for an open circuit and exactly true for a closed one — it is exactly true always, and it is a computation only when symmetry supplies the direction and the magnitude along the loop. Symmetry is doing the work; the law is doing the bookkeeping.

The law that is always true and rarely useful

Ampère's law holds for every loop and every current. Apply it to a wire of finite length and it gives an answer that is wrong by half — until the displacement current of the charge piling up at the wire's two ends is put back in, whereupon the two terms sum to μ₀I exactly at every length. The law was never approximate. It was never a computation either.

electromagnetism · Ampere law Two loops, two calculations, one number. The coupling between a circular loop of radius 10 cm and a rectangle of 5 by 3 cm tilted at 35° and offset sideways, against how far apart they are along the axis. The two loops differ in area by a factor of 21 and in shape entirely. Two quantities are plotted and they lie on top of one another. One is the flux through the rectangle when a current runs in the circle, obtained by integrating the circle's Biot–Savart field over the rectangle's tilted surface. The other is the flux through the circle's disc when the same current runs in the rectangle, integrated over a disc a hundred times the area. Nothing is shared between the two calculations except the positions of the wires, and they agree to 0.075 per cent — a difference which is the quadrature's, not the physics's. This is reciprocity, and it is not obvious: there is no reason from the geometry why a small loop should catch as much of a big one's field as the big one catches of the small one's. It follows from the double line integral the two quantities can both be reduced to, which is symmetric in the two loops — and that reduction is the argument, not this figure, which is the check that the argument is true of the actual fields.

The coupling that is the same both ways

A small loop and a large one catch the same fraction of each other's field. Nothing in the geometry suggests it — one has twenty-one times the area of the other — and the two quantities are computed here by two integrals with nothing in common, over two surfaces of different shapes, agreeing to seven parts in ten thousand.

electromagnetism · Induction

Named alongside it

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

Magnetic fieldSymmetryAmpere lawAntennaCirculationDisplacement currentFaraday's lawField linesFluxInductanceLinearityThe Lorentz force

All concepts