Faraday's Law of Induction and the Electromagnetic Vector Potential. UNPUBLISHED. WORK IN PROGRESS
Jeffrey F. Gold Department of Physics, Department of Mathematics University of Utah
Abstra ct Faraday's Law of Induction is characterized by two expressions- the line integral form f E 路dl, and the rate of change of flux , -~fit J B 路da. These equations describe two different manifestations of the same phenomenon but are equivalent under very specific circumstances. It is possible to hypothesize a Gedanken experiment in which the equality of the two forms of Faraday's Law does not hold. Although the physical conditions for this hypothesis do not exist, an exploration of this issue raises important points about the role of the electromagnetic vector potential and the topology of the space in which it resides. The physical situation described possibly alludes to a generalization of the Aharonov-Bohm effect [1].
Introduction In 1831, Michael Faraday presented his findings on electromagnetic induction which were published in 1839 as Experimental Researches in Electricity. According to this established phenomenon, a current is generated in a sensing wire r if the magnetic flux <I>", through a surface a bounded by the wire, changes
1
Faraday's Law of Induction and the Electromagnetic Vector Potential.
2
with time. The electromotive force, [, is given by [ = -~ d<I>u
c dt ' where <I> u is the magnetic flux given by
where B is the magnetic field and the dii are areal segments of some surface u bounded by r. Because V路 B = 0, the surface u is not unique; hence any closed surface with the boundary r satisfies this criterium. The changing magnetic field produces an accompanying electric field. 1 This allows us to formulate the secondary form of the electromagnetic force as the line integral
拢= where
E is the electric field
iE路df,
and the df are line segments along the boundary
r.
The Gedanken Experiment Because the two formulations characterize two different manifestations of the same phenomenon, we hypothesize an experiment involving a magnetic field and a sensing wire which will contrast the differences of the physical mechanisms involved. The geometrical arrangement in Fig. 1 illustrates an experiment to determine the action of a completely shielded toroidal magnetic field on a Cu sensing wire. As seen in the figure, the Cu wire is linked to the toroid in such a manner that any flat surface u, bounded by the wire, necessarily contains a cross-section of the toroid. This toroid, driven by an alternating current source, generates an alternating magnetic field which is contained entirely within the physical dimensions of the toroid; that is to say, no magnetic field is in "physical" contact with the Cu sensing wire. Because real wires have non-zero dimensions, there will also exist a component of the magnetic field which deviates from that of the confined field due to the pitch in the wrapped wire; it is parallel to the axis of the toroid and is generated by the resulting surface current . To remedy this stray field, and for reasons of this investigation, the toroidal magnet is completely shielded on all surfaces by a hypothetical enclosure which does not permit any magnetic fields to stray beyond the space within the enclosure. The magnetic enclosure is also lThis, in fact, is the formal statement of Faraday 's Law and is one of Maxwell's four equations.
Faraday's Law of Induction and the Electromagnetic Vector Potential.
3
Figure 0.1: Set-up of Gedanken Experiment . The items labeled are: (1) Fecore toroidal magnet, (2) field-shielding enclosure, (3) Cu sensing wire, and (4) alternating current source. not permitted to establish internal currents which would screen the enclosed magnetic field. The Gedanken experiment is an attempt to decouple the electric field from the magnetic field. In other words, our aim here is to shield the toroidal magnet with some "exotic" material (exotic here meaning, of course, any material that would produce the desired effect), which would topologically isolate the toroidal magnetic field from the surrounding space.
The Strong and Weak Conditions of Faraday's Law Supposing such an "exotic" material exists, and that no electric field E is present in the space surrounding the enclosure, the line integral
Faraday's Law of Induction and the Electromagnetic Vector Potential.
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implies that the current in the sensing wire is zero. However, the Electromotive Force given by
1-
1 d £=--B·da c dt u is of a finite, non-zero value. This kind of scenario stipulates that the equality
r:E . df
lr
= -
~~
r:B . da
c dt lu
does not hold in the case of this hypothetically enclosed magnetic field. Since a current is sensed in the wire, if the experiment is performed, this exploration forces us to conclude the following points: that the form £=
iE·df
is the stronger condition of Faraday's Law and that 1d £=--c dt
1-
B·da
u
is a weaker form of Faraday's Law. That is to say, since the magnetic field in this hypothetical experiment has absolutely no "physical" contact with the sensing wire, we must conclude that the electric field (generated by the magnetic field, but embedded in the electromagnetic vector potential) is the driving mechanism of the current exhibited in the sensing wire.
The Electromagnetic Vector Potential This Gedanken experiment alludes to some interesting ramifications regarding the nature of space, its topology, and the electromagnetic vector potential. What this implies, is the fact that, since no such magnetic enclosure can exist, the electromagnetic vector potential must exist in a topologically isolated space different from the space containing the magnetic and electric fields. While electric and magnetic fields may, in fact, be shielded using other electromagnetic fields, the electromagnetic vector potential seems to permeate all space and matter with impunity. Since a current is in fact generated in the sensing wire, it must be through an action similar to that of the Aharonov-Bohm effect. In 1959, Y. Aharonov and D. Bohm [1] proposed a theory outlining the effects of electromagnetic vector potentials in the quantum regime. In their thought experiment, which has since been verified at the quantum mechanical level [2] and mesoscopic scale [3], electrons are acted upon directly by the electromagnetic vector potential A in a field-free region . It seems that the same mechanism is at work here, i.e., this Gedanken experiment, which could easily be performed, is a classical analogue of the Aharonov-Bohm effect.
Faraday 's Law of Induction and the Electromagnetic Vector Potential.
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Conclusion From the Gedanken experiment we may infer that the equality of the two terms of Faraday's Law, i.e. , the line integral form f E 路 dl, and the rate of change of flux equation, - ~ J B 路da, holds because the electromagnetic vector potential seemingly permeates all space unobstructed; that is to say, there exists no way of topologically isolating one region of space from another with respect to the electromagnetic vector potential.
ft
References [I] Y. Aharonov and D. Bohm, Phys. Rev. 115(3) , 485 (I959). [2] R. G. Chambers , Phys. Rev. Let. 5(I), 3 (I960). [3] J . Imry and R. Webb , Sci. Am. 260, 56 (I989) . [4] Jorge Pullin. Personal communication.
[5] Robert H. Romer, Am. J . Phys. 50(I2), I089 (I982). [6] M.V. Berry, R. G. Chambers, M.D. Large, C. Upstill, and J . C. Walmsley, Eur. J. Phys. 1, I54 (I980).
[7] E. Merzbacher, Am. J . Phys. 30(4), 237 (Apr. I962) . [8] H. Erlichson, Am. J . Phys . 38(2), I62 (Feb. I970). [9] G. Casati and I. Guarneri, Phys. Rev. Let. 42(24), I579 (I979) . [IOJ S. M. Roy, Phys . Rev. Let . 44(3) , Ill (I980) . [11] T. T. Wu and C. N. Yang, Phys. Rev. D 12(I2) , 3845 (I975) .
jgold@physics. utah. edu