INSTITUTE OF PHYSICS PUBLISHING

EUROPEAN JOURNAL OF PHYSICS

Eur. J. Phys. 25 (2004) 249–256

PII: S0143-0807(04)68734-4

An introduction to the vector potential D Iencinella1,3 and G Matteucci2 1

Department of Physics, University of Bologna, V/le B Pichat, 6/2, I-40127 Bologna, Italy Department of Physics and Istituto Nazionale per la Fisica della Materia, University of Bologna, V/le B Pichat, 6/2, I-40127 Bologna, Italy

2

Received 10 September 2003 Published 6 January 2004 Online at stacks.iop.org/EJP/25/249 (DOI: 10.1088/0143-0807/25/2/011) Abstract

We present a review and discussion of the physical meaning of the vector potential in electromagnetism by means of a classical experiment in which a long solenoid, connected to a sinusoidal voltage supply, produces an electric field and a vector potential in a circuit exterior to the solenoid itself. Some basic considerations are recalled concerning the meaning of action-at-a-distance and local description, in terms of ‘real fields’ of the induced electromotive force on the circuit outside the solenoid. From this viewpoint the roles played by the magnetic field, the electric field and the vector potential to account for the physical effects are reviewed. Finally, the connections between these considerations and the magnetic Aharonov–Bohm effect are recalled in the light of recent theoretical work aimed at explaining the Aharonov–Bohm phase shift in terms of the magnetic field without resorting to the vector potential.

1. Introduction

In various introductory texts to electromagnetism, electric and magnetic potentials are presented as useful mathematical tools to calculate fields4 . While an intuitive meaning is attached to the electrostatic potential as the work done in moving a unit charge in an electrostatic field, the significance of the vector potential is left as rather obscure. It is introduced as a function where the magnetic field can be derived as B = curl A, together with the vector operator identities it must satisfy to fit Maxwell’s equations. From this point of view the vector potential is considered as a mathematical device, without a precise physical meaning. Other authors discuss the role of the vector potential to calculate an induced electromotive force in a circuit [2]. Let us consider an experiment in which a time-dependent magnetic field is generated by an infinitely long solenoid or, alternatively, by a toroidal current distribution. The magnetic field is confined inside the solenoid or the torus and is zero outside them. Moreover a metallic ring is placed outside the solenoid and concentric to its axis; therefore the ring is in a region where the magnetic field is always zero. At a first non-critical approach it is not obvious how a current can be generated in the external circuit just by considering the flux enclosed by the wire itself. From an anthropomorphic point of view one would be more 3 Present address: CNR-IMM Sezione Bologna, Via Gobetti 101, I-40129 Bologna, Italy. 4 Our aim here is not to review and discuss the approaches used in the literature to introduce the vector potential. For this purpose, see, for example, [1].

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inclined to give a description of the induced electric current as dependent on the ‘state’ of the conducting ring rather than arising from the ‘magnetic configuration’ existing in the space enclosed by the conductor itself. It was just this kind of motivation that led Maxwell to use the electronic state, introduced by Faraday, as a mental picture to explain the generation of induced current without resorting to an obscure metaphysical description as something which acts at a distance. Maxwell introduced a new vectorial function A, our vector potential, which had ‘at least a mathematical significance’ [3]. With A he was able to account for the electromotive force in any circuit. Outside the solenoid there is a non-zero vector potential whose lines are concentric to the solenoid [4]. The production of the electromotive force (emf) on the ring is understandable if we consider A as having a ‘direct physical significance while B is, more or less, a mathematical aid’, see Shadowitz [2]. In other words, the vector potential acts directly on the charges of the wire, thus producing a current. Here, A is considered clearly as a real field. A rather subtle interpretation of the role of A to explain the outcome of the same experiment is given by Scott [2]: ‘. . . a detailed mental picture of the production of this emf is easier to obtain if the producing current is considered to generate a vector potential A which acts directly on the charges in the wire through the field of equation (7.14-8)’ (equation (7.14-8) is E = −∂A/∂t). A straightforward discussion about the ‘physical meaning of a mathematical quantity’ is given by Feynman [5]. In his lectures he stresses that the vector potential ‘does have an important physical significance’ although emphasis is given, in particular, to the unavoidable role of A in quantum theory. In his discussion, in which he compares the electric and magnetic fields with the corresponding electric and vector potentials, Feynman recalls that a ‘real field’ is a mathematical function used to avoid the idea of action-at-distance. In his own words: ‘If we have a charged particle at the position P, it is affected by other charges located at some distance from P. One way to describe the interaction is to say that the other charges make some “condition” whatever it may be in the environment at P. If we know that condition, which we describe by giving the electric and magnetic fields, then we can determine completely the behaviour of the particle with no further reference to how those conditions came about . . .. A ‘real’ field is then a set of numbers we specify in such a way that what happens at a point depends only on the numbers at that point’. Emphasis is given to the electric and magnetic fields: the role of vector potential is emphasized in quantum mechanics but is not mentioned as a ‘real’ field in the context of classical physics. In the present paper, the induction experiment described above is presented with the aim of addressing our attention to two alternative approaches for the evaluation of the electromotive force induced in a circuit. In the first, the current through the solenoid and the magnetic field that arises are taken into account together with the electric field produced outside the coil. With the second approach, we will show, by using the vector potential, how a more concise and straightforward interpretation of the observed physical effects can be obtained. The pedagogical interest of these results is that an introductory discussion of the role of the vector potential in electromagnetism can be delivered to students in addition to drawing their attention to the more general considerations which such a role plays in quantum physics. The Aharonov–Bohm (AB) effect is the typical quantum experiment in which the vector potential seems to be, in its original interpretation, the unavoidable physical quantity to give a local description. However, recent theoretical work based on semiclassical considerations explains the Aharonov–Bohm phase shift in terms of action and reaction magnetic forces between the solenoid and passing electrons and not directly with the vector potential. Therefore, short review considerations of this subject are given to underline the need to proceed with caution when introducing the magnetic Aharonov–Bohm phase shift as a typical case study regarding the importance of the vector potential in quantum mechanics. 2. Faraday law revisited

Our experimental arrangement is analogous to that described in an extremely stimulating paper [6] entitled: ‘What do “voltmeters” measure?: Faraday law in a multiply connected

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Figure 1. The time-dependent magnetic field confined in the solenoid induces an electromotive

force on the metallic ring.

region’, and in another article [7] which expands on some aspects of the previous one. We have adopted the same experimental configuration so that the content of the present paper should be better discussed after a careful presentation of Romer’s consideration. Figure 1 shows the experimental arrangement. A metallic ring of uniform resistance is concentric to the axis of a long solenoid. A sinusoidal alternating current through the cylindrical coil produces a slowly varying time-dependent magnetic field B(t) that, in the approximation of an infinite solenoid, is constant in space within the solenoid and negligible outside, at the ring position. According to the Faraday law, the electromotive force (emf) induced on the ring is d emf induced = − (B) (1) dt where (B) is the magnetic field flux confined inside the coil. The magnetic field B(t) can be calculated by using the circuital law [8], i.e. the emf is deduced as a function of the timedependent current flowing through the solenoid. In this way two integral relations to calculate the emf at the ring position have been used, i.e. the circuital law and the Faraday law. It is worthwhile emphasizing that the electromotive force is generated by a magnetic flux confined inside the solenoid. No field at the ring position is described. The magnetic field acts at a distance on the ring and therefore a non-local description of the physical effect is given. From this point of view the magnetic field cannot be considered as a ‘real’ field according to the definition given in the introductory considerations. If we want to supply a local account for the electric current through the ring, a field at the ring must be defined. Let us rewrite the Faraday law:
emf induced = E · dl (2) where E is the electric field at the ring. By exploiting the particular symmetry of the problem, the electric field both inside and outside the solenoid can be calculated. It turns out that the electric field lines are circles concentric to the coil axis either inside or outside the solenoid. In particular, the electric field outside the solenoid is evaluated by equations (1) and (2) and

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by the knowledge of the magnetic field by the circuital law. Finally, the electromotive force along our metallic ring can be easily calculated. The steps involved in this process can be summarized as follows. From the current through the coil, the magnetic field is first calculated. From the magnetic field a function for the electric field at the ring is derived and the induced electromotive force is finally obtained. The electric field can be specified directly in the region of interest: therefore it is a ‘real’ field, i.e. the required function to describe the induced emf locally. Let us consider now the alternative approach according to which electromagnetism is described through potentials. From Maxwell’s equations the general relation for the electric field is given by ∂A (3) ∂t where φ is the electric scalar potential and A is the vector potential [9]. Since the metallic ring is in a region free of electric charges, the equation above becomes E = −∇φ −

∂A . (4) ∂t At any point outside the solenoid, the electric field is the partial temporal derivative of the vector potential at that point. Therefore, a local description of the electromotive force requires knowledge of the electric field, as reviewed above, or equivalently of the vector potential A(t) in the region outside the solenoid, where the magnetic field B vanishes. E=−

Vector potential outside the solenoid

In our circuit, the quasi-static conditions hold true. Therefore, using the fourth Maxwell equation and the Coulomb gauge, the vector potential is related to the current density as ∇ 2 A = −µ0 j. A general solution of equation (5) can be written as  I · dl µ0 A(r) = 4π l  |r − r |

(5)

(6)

where dl is an infinitesimal element of a circuit through which current I flows, r is the position vector of the circuit element and the integral is along circuit l  . As reported in figure 2, for an infinitesimal turn of thickness dz of our solenoid of radius a, the vector potential can be written as µ0 a 2 n I dz (−y, x, 0) (7) 4r 3 where I is the current and n is the number of windings per unit length. With integration along the z axis we obtain the vector potential outside an infinitely long solenoid: µ0 a 2 n I (−y, x, 0). (8) A(r) = 2r 2 The vector potential function is represented by a number of lines, parallel to the current of the coil, circulating around the solenoid. Using equation (4) we can derive the electric field, and then by equation (2) the electromotive force can be calculated: emf = µ0 Sn I0 ω cos(ωt). With this last procedure the vector potential is determined at every point around the solenoid only by considering the current circulating through the coil. A is a vector arising from the electric current which, in turn, also produces a magnetic field confined inside the solenoid. With this approach, the induced emf can be calculated without resorting to knowledge of the magnetic field. A(r) =

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Figure 2. Geometry to calculate the vector potential at point P due to the current I through the

winding l  .

Due to the formal equivalence of the electric field and the time derivative of A,equation (4), the vector potential can be considered, according to the definition proposed in the introductory section, a ‘real’ field, as ‘real’ as the electric one. These considerations are supported by the fact that the above equations relating E, the time derivative of A and the induced electromotive force, show that an operative definition of the vector potential can be given through the measurement of the solenoid current.

Experimental apparatus and results

In our experiment we have realized a solenoid similar to that described in [7]. Quasi-static conditions are here completely satisfied. Other arrangements can be conveniently realized such as a toroidal configuration in which the magnetic field is more accurately confined inside the coil by adding a magnetic material core. A cylindrical solenoid 85 cm in length and a diameter of 10 cm was used. The coil of 570 turns was connected to a supply which provided a sinusoidal voltage with amplitude, V0 = 8.0 ± 0.3 V and current I0 = 300 ± 20 mA. The theoretical value of the induced electromotive force, derived with the vector potential, results as f = 6 ± 1 mV. The induced emf measured with an oscilloscope along the whole length of the copper ring was f = 6.5 ± 0.3 mV, which is consistent with the theoretical value. Following Romer’s considerations [6] regarding what a voltmeter measures, we can say that the meter reading is related to the line integral of −dA/dt through the meter itself once the particular line integral is duly considered according to the physical problem. Another point which is worthwhile mentioning is related to the possibility of evaluating the charge displaced in the ring as a direct measurement of the line integral of A that, in the  present case, is straightforward, Q(t) = −1/R A · dl. With this approach the calculation of the flux changes by means of the magnetic field is avoided.

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3. Discussion

We have presented an experimental approach to the introduction of the physical meaning of the vector potential in electromagnetism. This experiment can be successfully performed with simple instrumentation and allows a direct presentation of action-at-a-distance of fields and compare it with a local description of physical effects in terms of ‘real’ fields. At this introductory level, emphasis can be given to the simplified procedure required for the calculation of emf by the vector potential. Actually, as we have shown, the knowledge of the magnetic field is not needed so that A can ‘at least’ be regarded as fundamental as the magnetic field. Here B can be considered, more or less, as a mathematical aid. Further strong evidence supporting the importance of the vector potential is due to the fact that, in systems without the convenient symmetry of the present one, the vector potential values in the field are determined as the only way of calculating the magnetic field. From this point of view therefore, B is obtained as a derived quantity from A. In our example, however, apart from the simplified approach to calculate emf by the solenoid current and then by the vector potential, there are no other reasons to prefer the ‘route’ which foresees the use of the vector potential instead of the magnetic field. We obtain, in fact, equivalent results, although the physical description can be local or not. Other more advanced considerations regarding the role and the physical interpretation of the vector potential as field momentum per unit charge have already been discussed [1, 4, 10]. 4. The vector potential in quantum physics: the Aharonov–Bohm effect

The results of the present experiment constitute a good starting point to introduce the peculiarity of the so-called magnetic Aharonov–Bohm (AB) effect, although the experimental conditions are different [11]. The magnetic AB effect makes use of an electron interferometer in which a coherent electron beam is first split into two coherent parts that are subsequently recombined to form an interference pattern. An infinitely long solenoid or a toroidal magnet is placed between the two beams. These magnetic configurations are devised to allow electrons to move in field-free regions. Electron waves that pass on both sides of this rigorously localized magnetic flux suffer a phase difference that can be observed in the interference pattern. It must be emphasized that the behaviour of electron waves propagating through magnetic fields was already mentioned by Franz in 1939 [12] and subsequently clarified by Ehrenberg and Siday [13]. These last authors developed a detailed discussion of the concept of refractive index in electron optics and, in the last 20 lines of their article, suggested the electron interference experiment described above. Probably because electron interferometry had not yet been developed, the Ehrenberg and Siday suggestion remained frozen for ten years. The full significance of the problem was recognized in 1959 by Aharonov and Bohm in an article concerning the importance of electromagnetic potentials in quantum mechanics. At that time, different electron inteferometry experiments had already been performed. For a short history of these original experiments see, for example, [14] by Gottfried Moellenstedt, a pioneer in this field. The most widespread point of view regarding the explanation of the AB effect is based on the fact that, since electrons do not interact with any magnetic field, the phase shift observed in the interference region is due to the local action of the non-vanishing vector potential outside the solenoid. For a review of theoretical considerations and experimental results see [15]. A different approach to explain the phase shift has been developed by Boyer [16]. Accordingly, the solenoid can be represented as a stack of current loops, i.e. a pile of magnetic dipoles. The magnetic field arising from an electron travelling near the solenoid produces a net force on the solenoid itself. It is assumed that the action of the charge on the solenoid and the force of the solenoid on the electron satisfy Newton’s third law. This force produces a change of electron velocity along the direction of motion. A relative displacement of the

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two wavepackets passing on both sides of the solenoid is generated and a phase difference is revealed in the interference pattern. With this approach, therefore, the AB phase shift is explained without resorting to the vector potential. This force-based approach has also been used to explain the phase shift in an electron interference pattern caused by a line of electric dipoles placed between two interfering beams [16]. In this latter case, it is clear that the electrostatic forces between the passing electrons and the dipole satisfy Newton’s third law and are responsible for the observed phase shift. Therefore, a parallel between magnetic and electric experiments has been sketched to interpret the AB quantum phase shifts in terms of electromagnetic forces. 5. Conclusion

The use of electromagnetic fields and the vector potential to explain the emf induced in a resistive line surrounding a confined time-varying magnetic field has been reviewed. These considerations, in a different context, are then used as a starting point for an introductory discussion of the AB phase shift revealed in an electron interference pattern. According to Boyer, such a phase shift can be explained in terms of magnetic field without resorting directly, as widely accepted, to the vector potential. In the light of these considerations an ‘experimentum crucis’, which demonstrates unambiguously the existence of the AB effect for electrons, has not yet been realized. Therefore, in this case, an unbiased presentation of the role of the magnetic field and vector potential remains the only choice to be taken. It is curious to note that, 44 years after Aharonov and Bohm stressed the importance of electromagnetic potentials in quantum physics, theoretical considerations are still developing to restore a semiclassical description of the observed effects using electromagnetic fields. Acknowledgments

We wish to thank Professor C Moroni for useful discussions. One of the authors, D Iencinella, wishes to thank Dr Marco Beleggia for continued help in the thesis work. References [1] Semon M D and Taylor J R 1996 Am. J. Phys. 64 1361–9 [2] We quote only two books in which comments concerning the vector potential are of particular interest for the present paper. Shadowitz A 1988 The Electromagnetic Field (New York: Dover) pp 383–92 Scott W T 1959 The Physics of Electricity and Magnetism (New York: Wiley) pp 313–47 [3] Doncel M G and De Lorenzo J A 1996 Eur. J. Phys. 17 6–10 [4] For further historical notes see [1] and references therein. For more extensive considerations regarding possible vector potential configurations see Carron N J 1995 Am. J. Phys. 63 717–29 [5] Feynman R P, Leighton R B and Sands M 1965 The Feynman Lectures in Physics vol 2 (Palo Alto, CA: Addison-Wesley) para 15-4 [6] Romer R H 1982 Am. J. Phys. 50 1089–93 [7] Lanzara E and Zangara R 1995 Phys. Educ. 30 85–9 [8] Erlichson H 1999 Am. J. Phys. 67 448–50 [9] For an interesting review of the basic properties of electromagnetic fields and their interaction with particles, see Felsager B 1998 Geometry, Particles and Fields (New York: Springer) pp 3–39 [10] Konopinski E J 1978 Am. J. Phys. 46 499–502 [11] Aharonov Y and Bohm D 1959 Phys. Rev. 115 485–91 [12] Franz W 1939 Verh. Deutsch. Phys. Ges. 2 65 [13] Ehrenberg W and Siday R E 1949 Proc. Phys. Soc. 62 8–21 [14] Moellenstedt G 1999 Introduction to Electron Holography ed E Voelkl, L F Allard and D C Joy (New York: Kluwer) pp 1–15 [15] For a review of the experiments realized with the aim of revealing the existence of the magnetic Aharonov–Bohm effect and of related considerations see Peshkin M and Tonomura A 1989 Lecture Notes in Physics vol 340 (New York: Springer)

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Matteucci G, Iencinella D and Beeli C 2003 Found. Phys. 33 577–90 [16] See the following recent articles and references therein. Boyer T H 2000 Found. Phys. 30 893–905 Boyer T H 2000 Found. Phys. 30 907–32 Boyer T H 2002 Found. Phys. 32 1–39 Boyer T H 2002 Found. Phys. 32 41–9

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