Force exerted by a moving electric current on a stationary or co-moving charge: Maxwell’s theory versus relativistic electrodynamics

Abstract

The force exerted by a slowly moving current-carrying loop on a stationary or co-moving charge is derived within two distinct frameworks: Maxwell’s electrodynamics classically interpreted (operating in the Galilean space and time) and relativistic electrodynamics (operating in Minkowski space-time). A comparison between the ‘classical Maxwellian’ and relativistic solutions is presented, offering some intriguing insights that have been neglected in earlier discussions of the issue.

Are Maxwell's equations Lorentz-covariant?

“…However, as Bartocci and Mamone Capria pointed out, the plain possibility of achieving Lorentzcovariance of Maxwell's equations can be regarded as nothing more than an interesting mathematical property devoid of any physical contents [17]. It is perhaps instructive to recognize that the formal covariance can be employed as a handy tool, quite outside the relativistic framework [10].…”

“…where E ′ i are given by identities (10). Employing formulae for changing the corresponding partial differential coefficients…”

“…are obviously primed velocity field components. Thus, transforming equations (1)-(4), (12)-(14) by transformation of variables (5), one obtains that, in primed variables, equations (6)-(9), (21)-(23) of the same form apply under the proviso that E ′ i and B ′ i therein be defined by equations (10). Consequently, if E i and B i satisfy unprimed equations (1)-(4), (12)- (14), one knows that E ′ i and B ′ i determined by equations (10) satisfy primed equations (6)-(9), (21)-(23).…”

“…As the last characteristic example, I quote from a recent book by Christodoulides [9]: 'It is obvious that electromagnetic theory, as expressed by Maxwell's equations, is a relativistic theory, whose equations needed no modification in order to become compatible with the Theory of Relativity, at least as these apply to the vacuum. ' Recently, I pointed out that the above statements should be taken magno cum grano salis; when Lorentz-covariance of Maxwell's equations is at stake, nothing is fulfilled automatically [10]. I noted briefly that, for example, the so-called source-free Maxwell's equations, curlE = −∂B/∂t and divB = 0, are Lorentz-covariant if one defines E ′ and B ′ via E and B as given by the well-known transformation rules (see, e.g., [2]).…”

Subtleties of the clock retardation

“…As is well known, in order to be relativistically valid, a clock must operate according to some Lorentz-covariant laws. This can be fulfilled for Jefimenko's clock, since its operation is based on Maxwell's equations (which can be made to be Lorentz-covariant (cf, e.g., [9], [10])) and the equation of motion of a charge q * in the electromagnetic field d dt…”