Comment on `Is a magnetic field due to an electric current a relativistic effect?'

The issue of the equivalence of the ways in which electric and magnetic effects can each be seen as relativistic consequences of the other is discussed. It is shown that, in contrast to the electric case, no satisfactory non-relativistic theory of magnetism alone can be given. From this it is argued that the usual view that magnetic forces can be seen as a relativistic consequence of electric ones is sound whereas its converse is not. Résumé. La question si on peut voir les champs électrique et magnétique également comme effets relativiste est débattue. Il est demontré que, au contaire du cas électrique, une théorie non-relativiste n'exist pas pour le champ magnétique seul et donc on ne peut pas voir le champ électrique comme un effet relativiste. In a recent paper in this journal, Jefimenko (1996) has argued that the commonly held view that magnetic forces are relativistic consequences of electric ones is not unique. This relationship, it is argued, can be reversed, so that electric forces are seen as a relativistic consequence of magnetic ones. In this argument, Ampere's law for the magnetic field is taken as a starting point to `deduce' the existence of the electric field in a very similar way to the more familiar `deductions' of the magnetic field that start from Coulomb's law. For any claim of equivalence between these different deductions to be justified requires first of all that their starting points be equally well founded in some non-relativistic limits of Maxwell's equations where, in each case, one type of force appears alone. Such a limit does exist for the electric case but not it would seem for the magnetic one. The electric limit is described by the in vacuo field equations and the Lorentz force, , acting on a test charge, q, caused by the local electric field, E : The electric field given by (1) is a manifestly Galilean invariant since the electric charge density and the operator Del are both invariants. Equations (1) simply describe a field that is due to the instantaneous action at a distance of other charges and is dependent only on the positions of these charges relative to the test charge. Consequently, the force in (2) is also a Galilean invariant as it must be for consistency with the principles of Newtonian mechanics. Equations (1) and (2) are therefore a self-consistent, non-relativistic, description of a pure electric force. They form a quasistatic approximation to electromagnetism that is in fact much used in situations where electric charges are slowly moving and there are no conduction currents (e.g. atomic physics). Because of this, relativistic generalizations of Coulomb's law that lead to magnetic forces are well founded, even though some extra assumptions are needed to make the magnetic generalization rigorous (see Jackson 1975). Although the electric limit exists and is well known, it does not seem possible to give a corresponding magnetic limit that describes self-consistently a non-relativistic pure magnetic force. The use of Ampere's law, as in equation (1) of Jefimenko (1996), amounts to adopting the equations of magnetostatics which are given in vacuo by However, unlike equations (1), equations (3) are not acceptable as general field equations since they are not Galilean invariant (nor do they ensure the conservation of electric charge). The lack of Galilean invariance can be seen by considering a Galilean transformation to a different inertial reference frame when the electric current density in (3) becomes where u is the relative velocity of the primed reference frame. It can be seen that equations (3) are only invariant under such a transformation if they are augmented to include a displacement current term. Augmenting (3) in this way introduces reference to the electric field and so destroys the desired pure magnetic character of the equations. It might be thought that the above difficulty can be evaded by taking equations (3) to apply only when there is no electric charge density, so that J is then due only to conduction currents and satisfies the condition Div . Then J and B both become Galilean invariants. However, even allowing such an artifice still leaves problems with the transformation properties of the Lorentz force, , given by The velocity of the test charge, v , is defined with respect to some inertial reference frame and is not therefore Galilean invariant. Equation (5) cannot therefore be a general expression for the magnetic Lorentz force if this force is to be Galilean invariant. At best, equation (5) could apply only in some special preferred reference frame. In some other reference frame, where the test charge has velocity , it assumes the form Clearly the force in the form given by (6) cannot be interpreted as a purely magnetic one. The second term in (6) is independent of the test particle velocity and so it has the character of an electric force. These observations lead to the conclusion that there is no non-relativistic theory of purely magnetic forces from which the existence of electric forces could be inferred by a relativistic generalization. In fact the inability to give such a non-relativistic theory means that, starting from magnetic phenomena, it could be argued that considerations of Galilean invariance alone are sufficient to infer the existence of complementary electric phenomena. References Jackson J D 1975 Classical Electrodynamics 2nd edn (New York: Wiley) pp 578 - 81 Jefimenko O D 1996 Eur. J. Phys. 17 180 - 2