Aharonov–Bohm Phase Shift Calculator

What this calculator measures

The Aharonov–Bohm effect is one of the most striking reminders that quantum mechanics responds to more than the local force you would describe in a basic classical picture. A charged particle can acquire a measurable phase shift even when it travels through a region where the magnetic field on the path is effectively zero, provided the two interfering paths enclose a magnetic flux. This calculator turns that idea into numbers you can inspect immediately. From a magnetic field B and a circular loop radius r, it estimates the enclosed flux, the number of flux quanta inside the loop, and the associated phase shift Δφ in radians.

This page is intentionally focused on the simple circular-loop case that appears in many classroom derivations and first-pass lab estimates. You enter a field in tesla and a radius in meters. The calculator then uses the loop area πr², compares the resulting flux with the flux quantum Φ₀ = h/e, and reports how many phase cycles have accumulated. That makes it useful when you want to check whether an interference pattern should shift noticeably, estimate how sensitive a mesoscopic ring is to field changes, or build intuition for how geometry and magnetic flux are tied together.

The physics behind the result

In a two-path interference experiment, what matters at detection is relative phase. The magnetic Aharonov–Bohm contribution can be written several ways, but the most practical chain for this calculator is simple: calculate the flux through the enclosed area, divide by the flux quantum to get the number of flux quanta n, and multiply by 2π to get the phase shift. If n increases by 1, the phase increases by one complete cycle. Many interference observables are periodic in 2π, so a phase of Δφ and a phase of Δφ + 2π land at the same point on a repeating fringe cycle, but the unwrapped value still tells you how many full turns were accumulated along the way.

The result panel on this page reflects that interpretation. It shows Δφ in radians and also shows n, the number of enclosed flux quanta. The interface labels cases with |n| < 1 as Partial Quantum and cases with |n| ≥ 1 as Multi-Quantum. That label is not a complete physical classification by itself; it is simply a helpful summary of whether the enclosed flux has crossed one or more full flux-quantum intervals. In practice, that quickly tells you whether you are still within the first phase cycle or have already wound through several cycles.

There is one especially important scaling rule hidden in those formulas: radius enters through r². That means a small percentage change in loop radius causes a larger percentage change in area, flux, and phase. Many users expect the result to move gently with radius, then are surprised when a modest geometry change pushes the system through a full extra phase cycle. This is also why unit mistakes on radius are so destructive. Confusing micrometers with millimeters changes the area by a factor of a million, which completely changes the physics.

Inputs and units

The first input is Magnetic Field B (T). Enter the field in tesla, not millitesla or gauss unless you have already converted the number. A negative value is allowed and physically meaningful. Reversing the field direction reverses the sign of the enclosed flux, which reverses the sign of the phase shift. The second input is Loop Radius r (m). Radius must be greater than zero because the area formula uses the magnitude of the enclosed loop size. The page assumes the loop is effectively circular, so r is the effective radius of the area that matters for the interference path.

Scale matters here. In mesoscopic experiments, realistic radii are often in the micrometer or nanometer range, and useful magnetic 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 small-looking entries are not suspicious; they are often exactly the regime where the phase sits near one flux quantum and the calculator becomes most intuitive. If you instead enter centimeter-scale radii with laboratory-scale fields, the calculated flux-quanta count can become enormous. That is still mathematically correct, but it may not match the experiment or teaching example you had in mind.

A quick sanity check is to think about direction and scaling before you press the button. If you increase the magnitude of B while keeping r fixed, the magnitude of the result should grow linearly. If you increase r while holding B fixed, the magnitude should grow faster because the area scales quadratically. If the output does not follow those trends, the usual culprit is a unit mismatch rather than a problem with the formula.

Worked example

Consider a ring 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 flux quantum used by the script is 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 the regime where the numbers are easy to interpret. The phase is clearly 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 increases by a factor of 1.44. That pushes the flux-quanta count above 1 and the phase above one full cycle, which is exactly the kind of transition the result panel is designed to make easy to notice.

Scenario comparison

The table below keeps the values in the same general experimental range while changing one major quantity at a time. It is not a replacement for the live calculator, but it is a quick way to see how the model responds. The last row is the important one to notice: even though the radius change is modest, the phase crosses a full flux-quantum threshold because the area term is quadratic.

If you reproduce these cases in the form below, the panel will round to three decimal places, so tiny numerical differences are normal. What matters is the trend. Field changes act linearly. Radius changes can be more dramatic than intuition suggests. That pattern is the heart of most practical interpretation on this page.

How to interpret the result

The result panel gives you two complementary views of the same calculation. Δφ is the magnetic phase shift in radians. That is the quantity you would use if you were combining this contribution with another phase term in a larger model. n is the number of enclosed flux quanta. That is often the cleaner quantity for intuition because one additional flux quantum corresponds to one additional full 2π phase cycle. If you are thinking in terms of periodic fringe shifts, n is often the easier number to talk about.

It is also worth remembering what the calculator does and does not mean when the phase is large. A phase of 12.7 rad is not somehow more observable than a phase of 0.134 rad simply because the first number is bigger; many interference observables depend on where you land modulo 2π. What the larger value does show is that the system has wound through multiple full cycles. That matters when you sweep field values, compare nearby operating points, or reason about the periodicity of oscillations in a device.

If the sign is negative, the phase shift goes in the opposite direction, which corresponds to reversing the sense of the magnetic contribution. The periodic structure is unchanged, but the pattern moves the other way. In other words, the sign tells you direction, the magnitude tells you how far you have moved, and the flux-quanta count tells you how many whole phase turns are wrapped into that movement.

Assumptions and limits

This calculator is a first-pass model, not a full device simulation. It assumes the enclosed area can be approximated by a circle of radius r, and it assumes the relevant magnetic flux is well approximated by a uniform field times that area. Real structures can have 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 and then move to a geometry-specific model.

The page also isolates the magnetic Aharonov–Bohm contribution. It does not include electrostatic phase contributions, path-length mismatch, scattering, visibility loss from dephasing, temperature effects, or material-specific corrections. If your experiment measures a total phase difference, the output here is one ingredient, not the whole story. The script also assumes a single elementary charge because the flux quantum is hard-coded as h/e. A situation involving a different effective charge would require a different flux quantum and therefore a different result.

In practical use, that means this calculator is excellent for order-of-magnitude checks, sensitivity studies, and conceptual demonstrations. It is the right tool when you want to see how changing field or radius moves the phase, whether a design sits below or above one flux quantum, or whether a proposed measurement range is plausible. It is not the last tool you should use before publishing a device model or making a geometry-sensitive engineering choice.

General modeling note

Even though this page is about a specific quantum effect, the underlying workflow is the same as in many technical calculators: gather inputs, normalize units, apply a model, and present the output in a form that is easy to compare across scenarios. In the most abstract sense, a result is simply a function of the chosen inputs, and many scientific models can also be written as weighted sums. The preserved MathML below shows those general patterns without changing the specific Aharonov–Bohm formulas used above.

For this calculator, those generic templates collapse to one concrete chain: magnetic field and radius determine flux, flux determines the number of flux quanta, and the flux-quanta count determines phase. That is why the page is useful not only as a number generator, but as a compact explanation of how the number is produced.