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Canadian Journal of Philosophy 37.2 (2007) 263-282

Is Classical Electrodynamics an Inconsistent Theory?1
Gordon Belot
University of Pittsburgh
Pittsburgh, PA 15260

I Introduction

Classical Electrodynamics is a classical theory treating systems comprising an electromagnetic field and electrically charged matter. In a recent book Mathias Frisch argues that this theory is inconsistent within its intended domain of application and that a number of interesting methodological morals follow from this conclusion (2005, 33-5; all citations of Frisch refer to this work).

More precisely, Frisch argues that one can derive a contradiction from the following five premises: (1) there exists a charged particle and an electromagnetic field; (2) the field obeys Maxwell's equation (with the particle providing source terms); (3) the electromagnetic force on the particle is given by the Lorentz force law (for the given electromagnetic field); (4) the particle accelerates; (5) energy is conserved. He goes on to argue that the inconsistency of (1)-(5) has the following fairly direct consequences: (a) there exist inconsistent theories; (b) consistency is a theoretical virtue like any other; (c) we have here a victory for the models-based approach to philosophy of science far away from its home territory of application and approximation; and (d) we must reform our notion of theory acceptance. [End Page 263]

My purpose here is to discuss Frisch's argument for the inconsistency of electrodynamics and the conclusions he draws from it. I contend that although the argument is valid it relies on a contentious understanding of the Lorentz force law. So I think that there is plenty of room to be skeptical of Frisch's conclusion. I also contend that even if this conclusion is granted, the connection with the methodological consequences (a)-(d) is not nearly so tight as Frisch takes it to be.

Sections II and III address preliminary matters concerning the framework of classical electrodynamics. One of the topics addressed is the nature of particles in classical electrodynamics. Another is an important ambiguity in the construal of the Lorentz force law: one has to decide whether to employ the total field or only the field external to the particle in applying this law. This ambiguity is crucial: under one reading of the law, (1)-(5) above are consistent; under the other they are inconsistent (Frisch favours this latter reading). Sections IV and V are devoted to these points. Section VI weighs the question which version of the law we should take to be constitutive of classical electrodynamics. Here I differ from Frisch, and so I conclude that classical electrodynamics is consistent. In Section VII I turn to the question whether there is in fact a tight connection between Frisch's methodological morals and his thesis that classical electrodynamics is inconsistent.

Despite the critical tone of this paper, I want to emphasise that there is much that I admire in Frisch's book — and I encourage people to read it to see how fruitful for philosophy of physics and for philosophy of science engagement with less than fundamental theories can be.

II The Maxwell-Lorentz Equations

The fundamental equations of the classical electrodynamics are Maxwell's equation and the Lorentz force law, with the latter coming in two versions.

MAXWELL'S EQUATIONS. Choosing an inertial frame, we can decompose the electromagnetic field into the electric field E and the magnetic field B. Then we can write Maxwell's equations as:

・E = ρ
c∇ x E = -∂tB
c∇ x B=J+∂tE. [End Page 264]

Here ρ and J are the charge and current density corresponding to some given configuration of charged matter in spacetime, and c is the velocity of light.

THE LORENTZ FORCE LAW FOR THE TOTAL FIELD. This version of the Lorentz force law reads


It determines the electromagnetic force that a particle of charge q and velocity v located at spacetime point x feels in the presence of electric field E and magnetic field B.



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