arXiv:physics/0511103v3 [physics.class-ph] 4 May 2006 IST/CFP 2.2006-M J Pinheiro Do Maxwell’s equations need revision? - A methodological note Mario J. Pinheiro Department of Physics and Center for Plasma Physics, & Instituto Superior Tecnico, Av. Rovisco Pais, & 1049-001 Lisboa, Portugal * (Dated: February 2, 2008) We propose a modification of Maxwell’s macroscopic fundamental set of equations in vacuum in order to clarify Faraday’s law of induction. Using this procedure, the Lorentz force is no longer separate from Maxwell’s equations. The Lorentz transformations are shown to be related to the convective derivative, which is introduced in the electrodynamics of moving bodies. The new for- mulation is in complete agreement with the actual set of Maxwell’s equations for bodies at rest, the only novel feature is a new kind of electromotive force. Heinrich Hertz was the first to propose a similar electrodynamic theory of moving bodies, although its interpretation was based on the existence of the so called ”aether”. Examining the problem of a moving circuit with this procedure, it is shown that a new force of induction should act on a circuit moving through an inhomogeneous vector potential field. This overlooked induction force is related to the Aharanov-Bohm effect, but can also be related to the classical electromagnetic field when an external magnetic field is acting on the system. Technological issues, such as the so called Marinov motor are also addressed. PACS numbers: 03.50.De;03.50.-z;03.30.+p;01.55.+b Keywords: Classical electromagnetism, Maxwell equations; classical field theories; Special relativity; General physics I. INTRODUCTION The theory proposed by James Clerk Maxwell successfully unified optics and the electrodynamics of moving bodies. In 1855 he tried to unify Faraday’s intuitive field lines description and Sir William Thomson’s mathematical analogies to the laws of hydrodynamics, in particular, making use of his 1842 analogy relating heat propagation to electrostatic theory. In 1861 Maxwell proposed a complete set of equations including the displacement current, from which the electromagnetic wave equation could be obtained [1]. Despite its success, however, Maxwell’s equations are still the subject of conceptual difficulties and controversy. Notwithstanding its widespread technological applications, it remains particularly painful for scientists and engineers to apply Michael Faraday’s law of induction [2, 3]. Faraday himself gave an empirical rule to determine when an induced voltage should be expected in a circuit. The explanation of this rule requires the use of Maxwell’s equations if the magnetic field changes with time, but the Lorentz force (considered necessary to define the fields E and B) if the circuit is displaced [4]. The title of this paper is intended to call attention to some of these conceptual difficulties [5, 6, 7, 8]. In our view the changes proposed in this work do not challenge Maxwell’s equations as written for bodies at rest. The novelty of our approach lies in a new form for the expected electromotive force and a more systematic presentation of the Maxwell’s fundamental set of equations. We show that the problem of electromagnetic induction can be more clearly understood using an appropriate procedure. Although it is not usually mentioned in the literature, Heinrich Hertz was the first to propose this procedure and give a systematic treatment of Maxwell’s macroscopic equations for the case of moving bodies. This inquiry has its origin in the thought that the theory’s current difficulties must be intimately connected to the transport process of physical quantities. II. ELECTRODYNAMICS OF MOVING BODIES For a better understanding of Maxwell’s fundamental set of equations we must return to its main experimental sources: Faraday’s law of induction and Amp` ere’s law. To facilitate the analysis we consider all our sources to be in a vacuum. With his creative mind, Maxwell built his theory on two basic concepts imported from fluid theory: the notions of circulation and flux. Whenever the flux of a well-behaved vector field through a given surface S is calculated, * Electronic address: mpinheiro@ist.utl.pt