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As far as I can check, the Aharonov-Bohm effect is not -- contrary to what is claimed in the historical paper -- a demonstration that the vector potential $A$ has an intrinsic existence in quantum mechanics, neither that the magnetic field $B$ has an intrinsic existence in classical mechanics, but rather that the magnetic flux is the relevant, measurable quantity. Indeed, the phase difference is either $\int A \cdot d\ell$ or $\int B \cdot dS$ (in obvious notations with surface $dS$ and line $d\ell$ infinitesimal elements), and these two quantities are always equal, thanks to the theorem by Stokes.

Then, one could argue that classically one can measure the flux as defined above, whereas in QM one has only the Wilson loop $\exp \left[\mathbf{i} \int A \cdot d\ell / \Phi_{0}\right]$, with $\Phi_{0}$ the flux quantum, as a relevant, measurable quantity.

I just elaborated above about the Aharonov-Bohm effect since it was the first time I realised the possible experimental difference between measuring a flux and a field. Let me now ask my question.

My question is pretty simple I believe: Does anyone know a classical experiment measuring the magnetic field $B$ and not the flux $\int B \cdot dS$ ? As long as we use circuits the answer is obvious, since one can only have an access to the flux and the voltage drop $\int E \cdot d\ell$ ($E$ the electric field), but do you know other experiment(s)?

As an extra, I believe the same problem arises with force: *Does anyone know a classical experiment probing the force field $F$, and not the work $\int F \cdot d\ell$ ? *

NB: The question does not rely on how smart you can be to find a protocol that gives you an approximate local value of the field, say $\int F \left(x \right) \cdot d\ell \rightarrow F\left(x_{0} \right)$ with $x_{0}$ a (space-time) position. Instead, one wants to know whether or not a classical measurement can a priori gives a local value of the field, without a posteriori protocol. Say differently, one wants to know how to interpret a classical measure: is it (always ?) an integral or can it -- under specific conditions -- give a local value ?

Post-Scriptum: As an aside, I would like to know if there is some people already pointing this question out (wether it's the flux or the local magnetic field or vector potential that one can measure), and if it exists some reference about that in the literature.

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I must be missing the point, because this seems obvious to me. A bathroom scale is a classical device that measures force, not work. A Hall effect sensor is a classical device that measures B, not flux. – Ben Crowell Apr 26 '13 at 3:28
Re classical force: even if I measure it by measuring the acceleration of a test particle, I still have to make a real measurement over a distance. Since force is a dynamical quantity, something has to change. The scale, where you read the pointer when equilibrium is reached, is just saying that two forces are equal. There still has to be some dynamical measurement somewhere along the chain of design and production when the balance was callibrated. – twistor59 Apr 26 '13 at 8:16
I guess that we measure a force field at a point by calculating particle accelerations from measurements over finite distances and interpret the value of the field at the point as a limit as the distances tend to zero. In a similar way, in reality we can't grab an "infinitesimally thin" single B field line, we have to grab a bunch of them to make measurements, and interpret the B field as a limit, hence flux sort of creeps into the picture for practical reasons. – twistor59 Apr 26 '13 at 9:49
All I'm really saying is that classical things like $\bf{B}$ fields are elements of a model, hence are idealizations, and can only be approximated by measurements. I'm not familiar with the arguments for and against, but I have the impression that the AB effect is about a much deeper nonlocality than that caused by the fact that practical experiments inevitably involve some spatial smearing. – twistor59 Apr 26 '13 at 10:38
There is still the question of how the electrons feel a magnetic flux when there is no magnetic flux intersecting their trajectory. This is what makes the AB effect "non-local", whether you think that the potential, field or flux is the fundamental quantity. In my view, neither of these quantities can be considered fundamental in the AB setup, since they appear as classical constraints rather than dynamical degrees of freedom. Yet the EM field is quantum, not classical. I believe that a first step towards a resolution of this problem was taken in this paper. – Mark Mitchison Apr 27 '13 at 13:19

The simplest device to measure magnetic fields (and not integrated magnetic flux) is an electron. The spin-up/-down energy splitting in a magnetic field is $g_S\mu_B B$ where $g_S \mu_B \sim 2.8~\mbox{MHz/G}$. This can be measured in a beam machine. The next simplest is the Zeeman splitting of atomic energy levels. The latter forms the basis of some of the most sensitive magnetometers, for example John Kitching's microfabricated gas cell magnetometers (external link to a pdf describing the device) at NIST.

Building upon Ben Crowell's comment, a balance scale will measure the gravitational force on an object while keeping $\Delta l$ arbitrarily small (the more accurate your measurement of the scale's tilt, the less the masses move during the measurement).

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Thanks a lot. Nevertheless, a spin is not a classical variable. I was explicitly asking for a classical measure. The Zeeman splitting is not eligible as such :-). As for the balance, I'm always wondering if equilibrium is a measure of force. I have to think about that. Thanks again. – FraSchelle Apr 26 '13 at 6:28
I don't agree that the electron measurement is not classical. The measurement relies on the magnetic moment of the electron which is a classical quantity in the same way the charge of the electron is. The quantum mechanical effects are in the quantization of the projection which produces the discrete energy splitting in the magnetic field. Even if the projection were not quantized, a Stern-Gerlach apparatus would show a spread of electron positions due to the interaction of the magnetic moment with the B field gradient (instead of just two distinct locations). – Jason A Apr 27 '13 at 1:59
I understand your frustration, but a Stern-Gerlach apparatus integrate the magnetic field gradient along a trajectory. It is definitely not a measure of $B$. So, measuring the charge or the associated magnetic momentum is a classical measure which indeed records a flux or a work. – FraSchelle Apr 27 '13 at 7:42
Not frustrated, just trying to understand :-). On a pedantic side note, all measurements require some indicator. This implies a change between two or more discrete states with potential barriers separating the states. Always some work involved in making the change. – Jason A Apr 27 '13 at 15:53

Measure the Lorentz force acting on a moving point charge, e.g. electron in a Penning trap or cyclotron resonance.

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Thanks for your answer. Would you please consider expanding a bit. It's true that a measure of $F=q\left(E+v\times B\right)$ is a measure of a force (and at the same time of the field, not of the flux), but I do not understand how you measure it. Could you please give us a setup ? Or a reference of course. Thanks in advance. – FraSchelle Jun 7 '13 at 18:07
The Lorentz credited for the force law (among other things) has a "t". You may be confusing him with the Lorenz who is best known for his gauge condition. Or maybe that was just a typo. :) – Mike Jun 12 '13 at 19:33

If the magnetic field from the solenoid is strong enough you might be able to place a physical magnet on a sensitive force meter to provide a reading from which the strength of the magnetic field could be calculated. I have never heard of it being done except in relatively simple situations. The force between bar magnets is the calculation that would be necessary to carry out. The setup would entail treating the solenoid as one of the two magnets. This may or may not be possible. I have no experience doing this. I am merely pontificating like a dilettante.

This was meant more as a comment than an answer, but considering the large number of comments I desired to keep it from getting lost in the discussion.

Say differently, one wants to know how to interpret a classical measure: is it (always ?) an integral or can it -- under specific conditions -- give a local value ?

Unless you are willing to infer a local value from the trajectory of an electron as Jason mentioned, or from the same bending of electrons in the hall effect sensors, then there is always going to be some area involved. However if area/flux is the issue to avoid then you could always try the Gilbert model, which Griffiths recommends against using for any quantitative work.

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@JoelHobbit Thanks a lot for your answer. Would you please elaborate a little bit more it ? As far as I understand, you answer saying "Measuring a force can be done with a force meter..." Can you give us a setup, or a reference about what you call a "sensitive force meter" ? Also, I believe the force on a magnet is related to the flux on its surface, isn't it ? Thank you in advance. – FraSchelle Jun 12 '13 at 6:30
@JoelHobbit Thanks a lot for your update. I nevertheless believe this is still not a measure of a force (or the magnetic field), since you need to provide a way to measure it. I continue to believe you need to measure the work of the Lorentz force. Note that the Lorentz force is recovered from the Faraday law, which is a law describing electromagnetic fluxes, not field. See this answer for more details . Thanks a lot again. – FraSchelle Jun 12 '13 at 10:46
Have you seen TH Boyer's interpretation of the Aharonov-Bohm effect? – Dale Jun 12 '13 at 17:11
@JoelHobbit I think I've read them long ago, but I never really understood what he tried to do. There is also this discussion associated with some paper by Vaidman I still do not understand. By the way, I think I have to change the title of this question, because it is not really related to the AB effect. It's more about wether or not can we measure a field, or are we stuck with flux . Thanks for your comment. – FraSchelle Jun 12 '13 at 17:49

A compass is a classical magnetic field measurement. It's nontrivial to get the magnitude from a compass but the direction is certainly easy. But if you read Feyman you'll see that it is impossible to explain magnetism classically. Our models of electrons in teeny Amperian loops is bogus. In fact the famous Bohr–van Leeuwen theorem shows that classical magnetic fields are impossible. So magnetism is fundamentally a quantum mechanical phenomenon.

And here's a nit to pick with some of the answers. Almost no measurement is direct. A bathroom scale does NOT measure force. If it has a spring, it measures displacement and converts that to force via Hooke's law. Thus you're assuming you know the spring constant and that the spring is kept in it's linear regime. If it's a strain gage device it doesn't even directly measure strain directly. It measure tiny changes in resistance using a Wheatstone bridge and converts these to strain and thence to "weight." Almost every measurement you can think of is actually an inference of what you want from what you can really measure.

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