Illustrated Physics
physics, drawn
Concept

Displacement current — where it appears

The term ε₀ dΦ_E/dt that Maxwell added to Ampère's law, which is not a current and completes the circulation wherever charge is accumulating. For an unclosed segment of wire it supplies exactly the deficit the conduction term leaves — 0.55 of μ₀I for a wire as long as the distance — so the law is exactly true rather than approximately.

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

The same loop, spanned two ways. A 100 cm² capacitor with a 2 mm gap, charged at 1.0 million volts per second. A loop drawn round the wire can be spanned by a flat surface, which the current of 44.271 µA passes through, or by a bag-shaped surface that passes between the plates, which no charge crosses at all. Ampère's law as it stood gave two different answers for one circulation. The rate of change of electric flux between the plates, multiplied by ε₀, is 44.271 µA — the same number to every figure, because C = ε₀A/d is the same ε₀A/d either way. That is the term, and it is not an approximation or a correction: it is what makes the law consistent at all.

The term that made light

Ampère's law contradicts itself the moment a current stops being steady, and the contradiction is visible in one picture: two surfaces on the same loop, with a current through one of them and nothing through the other. The term that repairs it turns four equations into a wave, whose speed is two constants measured in a room with the lamps off.

electromagnetism · Maxwell equations 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 Jefimenko's two sources of a magnetic field. The magnetic field one metre from a long wire whose current is switched on, against time in light-metres, split into the two terms of Jefimenko's equation: the current at the retarded time divided by the square of the distance (solid blue), which is Biot and Savart's law with each piece of the wire seen as it was when its light left; and the rate of change of the current at the retarded time divided by c times the distance (solid green). Their sum (red) is the field. Nothing arrives before one metre of light-time. At 1.5 light-metres the two terms are 0.68 and 0.78; at 4, 0.97 and 0.07; the second term dies away as the switch-on recedes along the wire, and the first grows to the steady field. Neither term contains a changing electric field.

The field a changing field does not make

Faraday's law says a changing magnetic field comes with a curling electric field, and Maxwell's correction to Ampère's law says the reverse, and the two are almost always read as a chain of causes: each field makes the other, and that is how light carries itself across empty space. Solved for what the fields actually are, Maxwell's equations say something different. Every field at every place is an integral over the charges and currents on the observer's past light cone, and over nothing else. The changing fields are not sources; they are two descriptions of the same retarded currents, and the laws that relate them hold because both are computed from one history.

electromagnetism · Maxwell equations

Named alongside it

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

Electromagnetic waveMaxwell equationsAmpere lawAmperes lawBiot–SavartBiot savart lawCausalityCirculationElectric fluxFaradays lawLight coneMagnetic field

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