Asymmetry in Electromagnetism and the Theory of Relativity

Electrodynamic asymmetry and Principle of relativity

Einstein in the beginning of his 1905 paper, On the Electrodynamics of moving bodies, says “It is known that Maxwell’s electrodynamics- as usually understood at the present time- when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena.” His whole dealing in this paper, including the introduction of the Theory of special relativity seems to rest on this inherent problem that Einstein has. By the end of section 6, after having established the kinematic ground of relativity through Lorentz transformation and having applied it to Maxwell- Hertz equations, he concludes “.. it is clear that the asymmetry mentioned in the introduction as arising when we consider the currents produced by relative motion of a magnet and a conductor, now disappears.” What is this inherent asymmetry that Einstein sees in Maxwell equations. How, using relativity, does Einstein believes those to have disappeared by the end of chapter 6?

First, it is necessary to ascertain what exactly Einstein thinks is asymmetric in Maxwell’s electrodynamics. In taking the example of the electrodynamic action of a magnet and a conductor, he says:

The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighborhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighborhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise – assuming equality of relative motion in the two cases discussed – to electric currents of the same path and intensity as those produced by the electric forces in the former case. (On the electrodynamics of Moving bodies, pg. 37)

Let us consider the two cases that Einstein presents us. In the first case, the magnet is in motion and the conductor is at rest. Here, since the magnet is in motion, the magnetic field is moving as well. Hence from the frame of reference of the conductor, there is a change in magnetic flux. This change in magnetic flux according to Faraday’s law, produces an electric field. (∇ E= -dBdt) Here ∇ is an operator, with which the electric flux is being derived for each spatial dimensions. There is a change in magnetic flux (B) with respect to time (t). Hence the more the rate of change (i.e. the faster the change in terms of time), the greater the electric field produced. This electric field in turn also creates a current in the conducting wire.

Now let us consider the second case that Einstein proposes. The magnet is now at rest but the conductor is in motion (towards the magnet with the same velocity as the previous case). In this case, if we were to use Faraday’s law, although magnetic field is present, there is no change in the magnetic field. Hence the right hand term -dBdt = 0 and therefore, as Einstein mentions, “no electric force arises.” However, there acts upon it another force, an electromotive force, because a (non-current carrying) conductor is moving in the magnetic field created by the magnet. This force, also called the Lorentz force is equal to q/c(vB). This force generates an electric current in the conductor the same size as the previous case. Clearly, two different explanations are given for these two cases. In the first case it is direct, that change in magnetic field creates an electric field which in turn induces current in the conductor. In the second case, it is indirect - that there is no electric field created, but rather an electromotive force, which generates the electric current in the conductor. However, if we were to plug in a galvanometer to the ends of the conductor, (and also considering the identical set-up and same relative velocities in both cases) we would get the same observable current. We get this same observable phenomena in terms of the deflection of the galvanometer, whether the magnet is moving or the conductor with a certain given velocity but we are given two different physical and mathematical explanations. Einstein seems to imply that it is asymmetric because it is “the same observable phenomenon” which does not need different explanations other than “the relative motion of the conductor and the magnet”.

Another way to see this asymmetry is considering the charged particle moving with a constant velocity (v) on an electric field ( E ) as well as magnetic field ( B ). This charged particle with unit charge q, experiences the Lorentz force, F=qE + q/c(v B). Now the first term of the right hand side of this equation (qE) indicates that the particle experiences an electric force irrespective of the particle’s motion, just because of its virtue of being in an electric field. The second term q/c(v B)indicates when the particle is moving and there is a magnetic field, it experiences a force perpendicular both to the direction of its velocity and the direction of the magnetic field. In a given frame of reference where the particle is moving with a uniform translational motion through an electric and a magnetic field, the particle experiences the force with two components - one associated with the electric field and the other with the magnetic field. If we were to consider the particle, from a different frame of reference which is moving with the same velocity and the same direction as the particle, (i.e. the electric field and magnetic at rest relative to the particle), the particle would experience the force only due to an electric field. Again we can see how using Lorentz force, whether charge particle is at rest relative to the electric field or not, we get different explanations. These differing explanations perhaps could be more simple if only the relative motions of the particle and the field it is moving in is considered.

Of these above examples, classical electrodynamics tells two states of affairs apart, because of the existence of an absolute state of rest. Observationally, however, the two cases concerning the magnet and conductor in relative motion are indistinguishable. The state of affairs which are not observationally distinct should not be distinguished by the theory. Thus, absolute velocities (and rest) should be eliminated from the theoretical account of electromagnetic induction. Up until this point, however, we are not justified to remove the absolute state of rest from all electromagnetic phenomena. If there is a medium, like