How the Aharonov–Bohm phase shift is estimated
This Aharonov–Bohm calculator turns the circular-loop derivation into a quick estimate you can test with your own numbers. A charged particle can pick up a measurable phase even when the magnetic field is zero along the path, as long as the interfering paths enclose magnetic flux. Enter a field B and a circular-loop radius r, and the page estimates the enclosed flux, the flux-quanta count, and the phase shift Δφ in radians.
This calculator stays with the simple circular-loop geometry that is common in textbook derivations and first-pass lab checks. It uses the area πr², compares the resulting flux with the flux quantum Φ0 = h/e, and reports how far the phase has advanced in 2π cycles. That makes it useful when you want to gauge whether a ring should show a noticeable fringe shift, how sensitive the result is to field changes, or how strongly the geometry alone can swing the phase.
Why the enclosed flux sets the Aharonov–Bohm phase
In an Aharonov–Bohm interference measurement, the detector responds to relative phase rather than to a local force along the path. For the magnetic contribution, this calculator follows the standard chain: compute the flux through the enclosed area, divide by the flux quantum to get n, and multiply by 2π to get the phase shift. If n rises by one, the phase advances by one full cycle. Many fringe observables repeat modulo 2π, so Δφ and Δφ + 2π point to the same place on the repeating pattern, but the unwrapped value still tells you how many full turns have accumulated.
The result panel is built around that interpretation. It shows Δφ in radians and n, the number of enclosed flux quanta. The Partial Quantum label appears when |n| < 1, and the Multi-Quantum label appears when |n| ≥ 1. That is not a full physical classification on its own; it is simply a compact way to say whether the enclosed flux has crossed one or more complete flux-quantum intervals.
The most important scaling rule in those formulas is the r² term. A small percentage change in radius becomes a larger percentage change in area, flux, and phase, so a loop that looks only a little larger can jump by a surprisingly large phase increment. That is also why unit mistakes are so damaging: treating micrometers as millimeters changes the area by a millionfold and completely changes the answer.
Aharonov–Bohm inputs and units
The first input in this Aharonov–Bohm page is Magnetic Field B (T). Enter the field in tesla, not millitesla or gauss unless you convert it first. Negative values are allowed and meaningful; reversing the field reverses the sign of the enclosed flux and therefore the sign of the phase shift. The second input is Loop Radius r (m). Radius must be greater than zero because the area comes from the size of the loop that the interfering paths enclose.
Scale matters in the regimes where the effect is usually discussed. Mesoscopic rings are often micrometers or nanometers across, and useful fields may be millitesla or smaller. A 1 μm ring should be entered as 0.000001 m. A field of 1 mT should be entered as 0.001 T. Those numbers may look tiny, but they are often exactly the range where the flux is near one quantum and the phase is easy to interpret. If you instead use centimeter-scale radii with ordinary laboratory fields, the flux-quanta count can become very large. That is mathematically fine, but it may not match the device or classroom example you had in mind.
A quick check before you compute is to compare directions and scaling. Increase |B| with r fixed and the output grows linearly. Increase r with B fixed and the output grows faster because the area is quadratic. If the result does not move that way, the usual issue is a unit mismatch, not the formula.
Worked example: a 1 μm loop in a 1 mT field
Take a circular Aharonov–Bohm loop with radius 1 μm in a nearly uniform magnetic field of 1 mT. In the form, enter B = 0.001 and r = 0.000001. The area is πr² ≈ 3.1416 × 10⁻¹² m². Multiplying by the field gives a flux of about 3.1416 × 10⁻¹⁵ Wb. The script uses the flux quantum h/e ≈ 4.1357 × 10⁻¹⁵ Wb, so the loop encloses n ≈ 0.760 flux quanta. Multiplying by 2π gives a phase shift of about 4.77 rad.
That example is useful because it sits in an easy-to-read region of the scale. The phase is substantial, but the flux is still below one full flux quantum, so the calculator reports the Partial Quantum label. If you keep the field fixed and increase the radius to 1.2 μm, the area rises by a factor of 1.44. That pushes the flux-quanta count above 1 and the phase beyond one complete cycle, which is exactly the transition this page is built to highlight.
Aharonov–Bohm scenario comparison
The table below stays in the same experimental neighborhood while changing one major quantity at a time. It is a quick way to see the circular-loop model respond, not a replacement for the live calculator. The row to watch is the larger loop: even though the radius increases only modestly, the quadratic area term pushes the phase past a full flux-quantum threshold.
| Scenario |
B (T) |
r (m) |
Flux Φ (Wb) |
n = Φ/Φ₀ |
Phase Δφ (rad) |
| Lower field |
0.0008 |
0.000001 |
2.513 × 10⁻¹⁵ |
0.608 |
3.82 |
| Baseline |
0.0010 |
0.000001 |
3.142 × 10⁻¹⁵ |
0.760 |
4.77 |
| Larger loop |
0.0010 |
0.0000012 |
4.524 × 10⁻¹⁵ |
1.094 |
6.87 |
If you run these same numbers in the form below, the result panel rounds to three decimal places, so tiny numerical differences are normal. The important pattern is the trend: field changes scale linearly, radius changes can be much more dramatic, and the phase responds exactly as the Bπr² model predicts.
How to interpret the Aharonov–Bohm result
The result panel gives two views of the same Aharonov–Bohm calculation. Δφ is the magnetic phase shift in radians, which is the number you would use when combining this contribution with other phase terms. n is the number of enclosed flux quanta, and it is often the easier quantity for intuition because each extra flux quantum corresponds to one more full 2π phase cycle. If you are thinking in terms of repeating fringes, n is usually the cleaner number to talk about.
A large phase does not automatically mean a more visible fringe change than a small one, because many interference observables depend on the phase modulo 2π. What the larger number shows is that the system has advanced through several complete cycles. That matters when you sweep field values, compare nearby operating points, or think about the periodicity of oscillations in a ring or loop device.
If the sign is negative, the phase shift runs in the opposite direction, which corresponds to reversing the magnetic contribution. The repeating pattern stays the same, but it shifts the other way. In practice, the sign tells you direction, the magnitude tells you how far you have moved, and the flux-quanta count tells you how many whole turns are hidden inside that movement.
Assumptions and limits of the circular-loop approximation
This calculator is a first-pass Aharonov–Bohm model, not a full device simulation. It assumes that the enclosed area can be approximated by a circle of radius r and that the magnetic flux is well described by a uniform field times that area. Real devices can involve finite-width paths, noncircular geometry, fringe fields, shielding, and effective areas that differ from a simple geometric estimate. When those details matter, use this page as a benchmark or teaching tool, then move to a geometry-specific model.
The page also isolates the magnetic Aharonov–Bohm contribution. It does not include electrostatic phase terms, path-length mismatch, scattering, visibility loss from dephasing, temperature effects, or material-specific corrections. If your experiment measures a total phase difference, the number here is only one ingredient. The script assumes a single elementary charge because the flux quantum is fixed at h/e. A different effective charge would need a different flux quantum and therefore a different result.
In practical use, that makes this calculator best for order-of-magnitude checks, sensitivity studies, and conceptual demonstrations. It is a good fit when you want to see how field or radius changes move the phase, whether a design sits below or above one flux quantum, or whether a proposed range is plausible. It is not the last word before you publish a device model or make a geometry-sensitive engineering choice.
Aharonov–Bohm modeling note
The Aharonov–Bohm workflow here is the same as in many technical calculators: gather the inputs, keep the units consistent, apply the model, and present the output in a form that is easy to compare across scenarios. For this page, the chain is concrete rather than generic: magnetic field and radius determine flux, flux divided by h/e determines the flux-quanta count, and that count determines the phase in radians. The live formulas above and the result panel below are the only relationships that matter here.
That is why the page works as both a calculator and a compact explanation of the physics. You can use it to estimate a single operating point, compare nearby radii or fields, or check how quickly the phase winds as the loop gets larger. If the numbers surprise you, the surprise usually traces back to the r² dependence rather than to anything exotic in the code.