Ages 8–12 · about 6 minutes
The Field Hiding Behind the Fields
How can an electron feel a magnet it never comes near?
Do this first · Try this at home
Two paths on paper
Start with your hands. The reading makes far more sense afterwards.
You need: Paper and a pencil
0 of 3 steps done
What should I notice?
Notice: You chose the phases in a drawing; you did not measure electrons. Matching and opposite phases change where the model's bright and dark bands appear. No magnets, wires or laboratory equipment are needed.
Now read on to find out why it happens.
The big idea
The magnetic flux enclosed by two electron paths can change how their waves meet.
- phase
- A wave's position in its repeating cycle, like the direction of a clock hand.
- interference
- Waves combining to make an electron more likely to be detected in some places and less likely in others.
- magnetic flux
- A measure of the magnetic field passing through a surface.
- vector potential
- A mathematical quantity used to calculate magnetic fields and electron phases.
- gauge
- A choice of mathematical description that leaves the observable result unchanged.
Electron waves can travel along two paths and meet at a detector. Their interference makes an electron more likely to be detected in some places and less likely in others. The resulting pattern has bright and dark bands. What matters is the difference between the waves' phases: where each wave is in its repeating cycle when they meet.
In 1959, Yakir Aharonov and David Bohm predicted a surprising result. The two paths can enclose magnetic flux while the magnetic field is zero along the paths themselves. The enclosed flux still changes their relative phase and shifts the interference pattern. This is the magnetic Aharonov–Bohm effect.
Akira Tonomura's team tested it in 1986 using a tiny ring-shaped magnet covered with a superconducting shield. One electron beam passed through the hole and another outside the ring. They measured a shift between the interference fringes. Shielding mattered: stray magnetic fields along the electron paths could otherwise affect the result.
A description and a measurement
Physicists use the vector potential to calculate the phase difference. But there is more than one mathematical description of the same magnetic field. Changing that description is called changing the gauge. The potential can change while the enclosed flux and measured interference stay the same. A detector does not select one uniquely correct potential value at a point.
Two controls
In this ideal model, the central circle holds confined magnetic flux and the electron paths have zero magnetic field. Compare flux settings 0, 0.5 and 1. Half a single-electron flux quantum swaps the bright and dark bands; a whole quantum brings the pattern back to its starting position. The quantum here is h/e.
The interferometer with the field switched off
Split an electron beam, send the two halves either side of a magnetic flux that is completely confined and completely shielded, and bring them back together. On the paths themselves the magnetic field is exactly zero — the shaded corridor below is field-free the whole way. The vector potential there is not zero, and the fringes move. Dial the enclosed flux and watch the pattern slide sideways by exactly one fringe for every flux quantum. Then try the second control. It changes the gauge — the potential itself, at every point on both paths, as the arrows show — and the fringes do not move at all. Both controls are live; only one of them changes anything a detector can see.
- Upper path
- Lower path
- Confined flux
- Vector potential on the paths
What is happening: The enclosed flux is near a whole number of quanta, so the pattern is near its starting position. Adding one whole quantum repeats the pattern.
- Magnetic field on the electron paths
- zero, at every setting
- Vector potential on the upper path, in this gauge
- 0.00 nWb/m
- Vector potential on the lower path, in this gauge
- 0.00 nWb/m
- Phase the gauge adds to each half
- 0.00 radians — the same for both halves, so it cancels
- Phase difference between the two halves
- 0.00 radians
- Fringe shift
- 0.00 fringe widths
- What no choice of gauge can change
- the field on the paths, the phase difference, the fringe shift
The enclosed flux controls phase difference. Changing gauge changes the potential assigned to each point, not the observable phase difference or fringe pattern. This instrument calculates phase and interference, not stored energy or power output.
Nothing happening is the result here, not a broken control. The potential is not unique: for any smooth function χ you may replace A by A + ∇χ, which changes the potential at every point and changes the magnetic field nowhere, because the curl of a gradient is zero. This instrument performs that substitution for real and then integrates the full potential, gauge term included, around the closed loop the two halves of the beam make between them. Each half does pick up a phase from the gauge term — exactly 2λ radians in this family — and both halves pick up the same one, so it cancels out of their difference. What survives is the loop integral, which is the enclosed flux, and that is the observable.
Tonomura's 1986 experiment answered the obvious objection by construction: the flux sat inside a toroidal magnet wrapped in a superconducting shield, so stray field was not merely small on the electron wave but excluded from it. The shift was still there, and it was still one fringe per flux quantum.
An ideal two-path diagram comparing steady flux settings, not an exact reconstruction of the apparatus or a simulation of changing flux over time. A real changing magnetic flux can induce an electric field. Beam energy, path length, magnet size and absolute intensity are not modelled. Φ₀ = h/e ≈ 4.136e-15 Wb.
The source sheet: Tonomura et al. 1986, Evidence for Aharonov–Bohm effect with magnetic field completely shielded from electron wave
Now leave the flux fixed and move the gauge control. The potential arrows change, but the interference pattern stays put. The arrows describe the potential; they are not force arrows pushing the electrons.
These are comparisons between steady settings. Changing flux in real equipment can also create an electric field; this diagram does not model that changeover. It is an ideal interference calculation, not a laboratory measurement or a picture of the garden coil.
Sources: Aharonov and Bohm (1959), 10.1103/PhysRev.115.485; Tonomura and colleagues (1986), 10.1103/PhysRevLett.56.792; Hitachi's account of the shielded experiment.