The Physical Entity of Vector Potential in Electromagnetism
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Applied Physics Research; Vol. 5, No. 4; 2013 ISSN 1916-9639 E-ISSN 1916-9647 Published by Canadian Center of Science and Education The Physical Entity of Vector Potential in Electromagnetism Vladimir A. Leus1, Ray T. Smith2 & Simon Maher3 1 Sobolev Institute of Mathematics, Novosibirsk, Russia 2 Wirral Metropolitan College, Wirral, UK 3 Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool, UK Correspondence: Vladimir A. Leus, Sobolev Institute of Mathematics, 630090, Novosibirsk, Russia. E-mail: [email protected], [email protected] Received: April 18, 2013 Accepted: June 4, 2013 Online Published: July 10, 2013 doi:10.5539/apr.v5n4p56 URL: http://dx.doi.org/10.5539/apr.v5n4p56 Abstract The scalar and vector potentials were introduced into electromagnetic physics in the second half of the nineteenth century. The chief aim was to use them as auxiliary mathematical quantities in order to solve certain practical problems. Nevertheless the discovery of the Aharonov-Bohm effect (1959) in quantum mechanics has suggested that vector potential rather than magnetic field is the causal agent in such an effect. Recent research on the Maxwell-Lodge paradox--induction of voltage in the loop circling a long solenoid carrying alternating current--has confirmed that induction occurs in a region of space effectively free from magnetic field. This again reinforces the idea of vector potential as a physical entity rather than the auxiliary artificial quantity of classical electrodynamics. The present investigation is intended to provide some degree of corroboration of the previous result. The experimental arrangement consists of a ‘special’ transformer containing movable, single turn coils wound onto rectangular frames. The primary coil is powered from a signal generator providing alternating current over a variable frequency range while the secondary output voltage is connected across a C.R.O./precision voltmeter. Measurements of transformer e.m.f. were carried out at several frequencies in the range 100 Hz–20 kHz and with various conditions of shielding around the primary and secondary coils. Certain additional experiments were carried out with a long solenoid and torus solenoid supplied with different core materials. Experimental results for induced e.m.f’s are presented and in special cases correlated with the calculated values of mutual inductance. Overall the results tend to confirm the primacy of vector potential over magnetic field as an explanation of the phenomenon. Keywords: electromagnetic induction, transformer e.m.f., vector potential, Maxwell-Lodge paradox, Aharonov-Bohm effect PACS: 03.50.De Classical electromagnetism, Maxwell equations. 1. Introduction The notion of potentials, both scalar and vector , was introduced into electromagnetic physics as auxiliary quantities to facilitate calculation. In particular the potentials allowed Maxwell’s system of differential equations of the first order to be transformed into a simpler form so as to solve second order equations, e.g. Poisson’sequation, the wave equation and d’Alambert’s equation (Grant & Phillips, 1998, pp. 30-32, 149-150, 229, 433-434). Once potentials are obtained, the vectors of magnetic field strength and of electric intensity may be calculated by simple differentiation. R. A. Mould states: “Prior to the rise of gauge theory and the discovery of the Aharonov-Bohm effect, physicists tended to regard these potentials as fictitious. Much was made of the fact that electromagnetic measurement always involve forces that manifest themselves in the Lorentz force equation and, hence, that the and fields appearing in that equation are the “real ”fields. Because the potentials cannot be uniquely determined from and , they were said not to share in this reality” (Mould, p. 167). However after discovery of the Aharonov-Bohm effect (Aharonov & Bohm, 1959), quantum mechanics tended to regard vector potential rather than the magnetic field as the casual agent in any explanation of this effect. The electromagnetic potentials that are components of the four-vector potential ,/ now become the fundamental variables of the theory of quantum mechanics. Classical electromagnetism proves to be more conservative regarding the acceptance of this point of view, but recent investigations have contributed to the scientific revival of the problem which has led to a renewed interest in this topic. 56 www.ccsenet.org/apr Applied Physics Research Vol. 5, No. 4; 2013 The transformer is a simple device to set up. It could hardly be less sophisticated, without the need of moving parts, permanent magnets, sliding contacts, vacuum tubes etc. Despite the simplicity of such a device, the theory of its operation has raised fundamental questions regarding the very foundation of electromagnetism. This was first pointed out and experimentally researched more than a century ago by Oliver Lodge (Lodge, 1889). Rousseaux et al. (2008) emphasise that the Maxwell-Lodge effect still presents a fundamental problem. A very long solenoid is circled in its central plane by a conducting loop (Figure 1). When a sinusoidal current is supplied to the solenoid, there is a corresponding voltage induced in the loop, despite the fact that no sensible magnetic field exists in the vicinity of the loop. So the question is posed: “How do the charge carriers in the external coil know that the current is varying in the solenoid?” (Rousseaux et al., 2008, p. 251). It is of interest, from an historical point of view, that Sir Oliver Lodge over 120 years ago had pondered a similar question. In 1889, at University of Liverpool, Lodge carried out an experiment using a torus solenoid wound onto a ring shaped iron core. In his own words: “So I led a short wire round the outside of the ring and brought its ends into the box where the needle was, one on each side of the needle. A distinct and proper deflexion was now observed. A wire was then taken 10 times around the ring, and connected to a common quadrant electrometer. The deflexion was easy to see. It could also be just seen with only one turn of wire” (Lodge, 1889, pp. 476-77). The phenomenon associated with induction of e.m.f. in a magnetic field free region might then be appropriately termed the ‘Maxwell-Lodge Effect’. Furthermore, in his concluding remarks, Lodge makes a reference to what effect some form of electromagnetic shielding might have between two circuits: “I would not wish the Society to suppose that I regard the research as finished and complete. I should like to clear up distinctly the effect of screens” (Lodge, 1889, p. 479). So we might regard the present investigation as a continuation of Lodge’s work, using shielded and unshielded rectangular circuits in addition to work with solenoidal and toroidal coils. It should be noted that Lodge was faced with considerable experimental difficulties by contrast with today. For example he could only detect (stray) magnetic fields by the use of iron filings; alternating voltage was simulated by including a reversing key in a direct current circuit; induced voltage was detected using a quadrant electrometer plus lamp and scale. Such an instrument was very sensitive to extraneous vibrations: “There remained a few irregular disturbances, some of which could be traced to convection-currents, others to the ordinary movements about a building, and others again to the passage of London and North-Western trains in their tunnel, some 150 yards away and 60 feet down, in the sandstone rock” (Lodge, 1889, p. 475). Experimentation was certainly a much more painstaking ordeal in those days. Figure 1. Circuits for the Maxwell-Lodge effect representation Rousseaux et al. (2008) used for their experimentation a diversity of Lodge’s apparatus with voltmeter instead of a movable needle (Figure 1). They attempt to answer above question in the following way. Each vector field may be decomposed into three additive parts: ‖ ┴ , where the third one (so called harmonic part) meets the conditions 0, 0. If is vector potential, then its harmonic part is responsible for the presence in the loop of an electric field ∂. Since the harmonic part is a gradient, such an inference contradicts the conventionally adopted “gauge condition principle”, which states that it may be 57 www.ccsenet.org/apr Applied Physics Research Vol. 5, No. 4; 2013 removed by subtracting the gradient of an appropriate scalar gauge function. The authors of (Rousseaux et al., 2008) regard the “gauge principle” as incorrect and refer to the Aharonov-Bohm effect, which is explained in terms of the harmonic part of the vector potential. In (Rousseaux et al., 2008) an “experimentum crucis” corroborating this point is presented, and the authors propose to replace the generally adopted term “gauge condition” by “gauge constraint”. The significant feature of the experimental results presented in (Rousseaux et al) is that the exterior of the solenoid coil provides a region in which the magnetic field is close to zero whereas the vector potential assumes a definite value. In the present paper a different geometrical form of the same experiment is investigated which tends to validate the conclusion of (Rousseaux et al., 2008) but allows a further insight into this fundamental problem. Here the apparatus used for showing the Maxwell-Lodge effect includes a special transformer having two identical loops for both the primary and secondary circuits. This configuration allows us to operate under two different sets of experimental conditions, (1) with free magnetic field access (i.e. using ‘unshielded’ loops) and (2) without magnetic field access (i.e. using ‘shielded’ loops). As regards multi-turn coils, we used a long solenoid (similar to one in Rousseaux et al., 2008); in addition we carried out experiments with two types of fully wound torus solenoid. It is appropriate to mention that we worked in the quasi-static limit where retardation effects are negligible. 2. Experimental Arrangement The basic arrangement consists of single rectangular loops of stranded wire taped onto rectangular wooden frames of the same dimensions, 1.20×0.60 m.