Ampère's mercury experiment and Aharonov-Bohm: same message?

[Roberto Pavani]

TL;DR

Ampère's mercury experiment already showed that A is physical, 140 years before Aharonov-Bohm.

I'd like to share an observation and ask if others find it compelling.

Consider a current-carrying loop that tends to expand. This is usually explained by the Lorentz force (IL×B) on each segment due to the field of the other segments.
Equivalently, one can describe it in terms of the gradient of the vector potential A generated by the rest of the circuit.

For a large loop, both descriptions give the same result.

Now imagine continuously deforming the loop, shrinking it until it becomes Ampère's mercury trough experiment: a short bar floating on mercury, with the return current flowing immediately beneath it in the liquid. The bar moves, it "wants to expand the loop."

But in this limit, by symmetry, B at the bar's location is essentially zero. The Lorentz force description becomes problematic, while the gradient of A remains well-defined and still explains the force.

This seems to me analogous to the Aharonov-Bohm effect: a physical consequence of A in a region where B vanishes. The difference is that AB is quantum (phase shift) while this is classical (mechanical force), but the underlying message appears the same, A is physically meaningful, not merely a mathematical convenience.

I find it curious that this classical "AB-like" evidence was available since 1820, long before the vector potential was even formalized.

Am I overlooking something, or is this connection well-known and I've simply missed it in the literature?

 

[Sagittarius A-Star]

But in this limit, by symmetry, B at the bar's location is essentially zero.

Why is there a symmetry?

 

[Roberto Pavani]

The symmetry I refer to: current approaches the bar through one trough and leaves through the other, in opposite directions. These two symmetric return currents produce B fields that largely cancel at the bar's location.

I agree it's not exactly zero in practice, but it becomes negligible while the force persists.

 

[Dale]

I must be misunderstanding the geometry. The B field from the return currents seems to add, not cancel.

Can you post an image or sketch of the geometry?

 

[Sagittarius A-Star]

Is this the scenario?

https://kirkmcd.princeton.edu/examples/hairpin.pdf

 

[Roberto Pavani]

@Dale I've found an interesting link: http://www.ampere.cnrs.fr/histoire/parcours-historique/lois-courants/force-obsolete/eng

@Sagittarius A-Star you beat me in speed

 

[Roberto Pavani]

I must be misunderstanding the geometry. The B field from the return currents seems to add, not cancel.

Can you post an image or sketch of the geometry?

You're right, I overstated the symmetry, B is not zero at the bar. My mistake.

Still, I notice that explaining the force on the bar using Lorentz force has historically been non-trivial (as the McDonald paper linked by @Sagittarius A-Star shows, it requires careful analysis). Whereas describing it in terms of A seems more direct: the bar moves to increase the enclosed flux.

Maybe my analogy with AB is too strong, but I still find it interesting that A gives a simpler picture here. Am I wrong about that?

 

[Sagittarius A-Star]

@Dale I've found an interesting link: http://www.ampere.cnrs.fr/histoire/parcours-historique/lois-courants/force-obsolete/eng

This is an interesting link.

I think a convincing explanation is the following, because a magnetic field only from the moving part of the circuit does not contribute to Newton's 3^rd law:

Peter Graneau at MIT observed in 1981, with currents of several hundred amperes, strong turbulence in the mercury when the movement of the hairpin was blocked. This turbulence seemed to manifest the repulsion undergone by the portion of mercury carrying the current.

A paper from Peter Graneau from 1987:
https://isidore.co/misc/Physics pap...ian recoil and the efficiency of railguns.pdf

 

[Roberto Pavani]

It seems to me that the deformation forces are all caused by $\nabla A$ ?

 

[Sagittarius A-Star]

It seems to me that the deformation forces are all caused by $\nabla A$ ?

They are caused by electromagnetic forces, not by $A$ itself.

In 4D-Spacetime Algebra, the electromagnetic field bivector is
$F=\nabla A$

Source (equation 6.6):
https://davidhestenes.net/geocalc/pdf/SpaceTimeCalc.pdf

 

[Roberto Pavani]

Yes, I think I was expressing roughly the same thing in a less rigorous way. Thank you for the precise formulation.
What I was (clumsily) trying to say is: $F = I \, \partial(\oint \mathbf{A} \cdot d\mathbf{l})/\partial x$. That's all.

 

[Sagittarius A-Star]

Yes, I think I was expressing roughly the same thing in a less rigorous way. Thank you for the precise formulation.

If you meant the (magnetic) 3D-vector potential $\vec A$, yes, I think in the lab frame it comes mainly via the magnetic field
$\vec B = \vec \nabla \times \vec A$

 

[Dale]

Whereas describing it in terms of A seems more direct

Certainly. The potentials are often more direct, convenient, or easier.

 

[Sagittarius A-Star]

This article confirms, that the force in the restframe of the railgun is mainly a magnetic force.

A railgun consists of two parallel metal rails (hence the name). At one end, these rails are connected to an electrical power supply, to form the breech end of the gun. Then, if a conductive projectile is inserted between the rails (e.g., by insertion into the breech), it completes the circuit. Electrons flow from the negative terminal of the power supply up the negative rail, across the projectile, and down the positive rail, back to the power supply.[37]

... Since the current is in the opposite direction along each rail, the net magnetic field between the rails (B) is directed at right angles to the plane formed by the central axes of the rails and the armature. In combination with the current (I) in the armature, this produces a Lorentz force which accelerates the projectile along the rails, always out of the loop (regardless of supply polarity) and away from the power supply, toward the muzzle end of the rails. There are also Lorentz forces acting on the rails and attempting to push them apart, but since the rails are mounted firmly, they cannot move.

Source:
https://en.wikipedia.org/wiki/Railgun#Theory