# Does special relativity unify the two phenomena at the base of Faraday's flux law (was Feynman wrong in this case)?

Consider Faraday's flux law for the EMF generated in a conductor loop:

$$\varepsilon = - \frac{d \phi}{dt},$$

where $\varepsilon$ is the EMF, and $\phi$ is the magnetic flux through the loop.

There are two possible causes for the flux to variate over time: variations in the magnetic field ("transformer EMF") and variations in the area enclosed by the loop ("motional EMF").
Feynman has noted that this is a unique case where a single rule is explained by two different phenomena:

We know of no other place in physics where such a simple and accurate general principle requires for its real understanding an analysis in terms of two different phenomena. Usually such a beautiful generalization is found to stem from a single deep underlying principle. Nevertheless, in this case there does not appear to be any such profound implication.
—Richard P. Feynman, The Feynman Lectures on Physics (Volume II, 17-2).

Is it really true that there is no way to view these two phenomena as one?
For example, it says in this Wikipedia article that this apparent dichotomy was part of what led Einstein to develop special relativity.
Does special relativity give us a unifying principle to derive Faraday's law from?

## 1 Answer

The two phenomena Feynman referred to are the $q\vec v\times \vec B$ part of the force acting on an electric charge carrier; and the $\nabla\times \vec E =-\partial_t \vec B$ dynamical Maxwell's equation. In the most general situation when both the magnetic field and the shape of the wire is changing, we have to use and add both terms. That's true in each reference frame because for a complicated space-dependent, time-dependent geometry of the wires and fields, there won't be any inertial frame in which one of the phenomena would completely vanish.

I believe that it's more likely (but not certain) Feynman only meant this thing – that there's no way to eliminate or "explain" one of the terms in the general situation.

On the other hand, the fact that both terms have the same origin is a consequence of special relativity and it was indeed one of the motivations that led Einstein to his new picture of spacetime. It seems somewhat plausible to me that Feynman was ignorant about this history.

The full theory of electrodynamics is nicely Lorentz-covariant and it implies both terms. However, these terms aren't not really the same. The Lorentz force comes from the integral $q A_\mu dx^\mu$ over the world lines of charged particles while Maxwell's equations arise from the $-F_{\mu\nu}F^{\mu\nu}$ Maxwell Lagrangian. So Feynman would also be right if he said that the two terms can't be transformed to each other by any symmetry transformation.

Still, one can make physical arguments that do involve such transformations and imply that the two phenomena are inseparable. For example, if we assume the Maxwell equation, it follows from the Lorentz symmetry that $\vec E$ must transform as the remaining 3 components of the antisymmetric tensor whose purely spatial components give $\vec B$. But then it follows that the force acting on a charged particle, $q\vec E$, must also be extended by the remaining term $q\vec v\times \vec B$ for the theory to be Lorentz-invariant. Or one can run the argument backwards. Still, we are dealing with transformations of two different terms in the action that just happen to have a "unified, simply describable" impact on the EMF in wires with magnetic flux.

The simplification and unity only occurs if we assume that the two different kinds of phenomena are in the action to start with but they deal with the same fields which respect the same Lorentz symmetry; and if we study situations that are "understood" or "simplified" on both sides in which some effects, e.g. the magnetic ones, are absent.

Let me say it differently: if the area enclosed by a wire goes to zero, it's an objective thing that is clearly independent of the reference frame. So one shouldn't expect the shrinking of the area is just a matter of inertial systems; it's a frame-independent fact. On the other hand, it's not shocking that the EMF ultimately only depends on one thing, the change of the flux, which has a simple form although it may have different origin.

I would conclude that Feynman was more right than wrong. Lorentz symmetry operates in both phenomena and it's the same one which is a constraint on the most general theory; however, the fact that both possible sources of the changing flux influence the EMF in the same way is a sort of "coincidence", at least if we use the conventional variables to describe the electromagnetic phenomena.

• Thank you for the detailed answer, parts of which I'm still struggling to understand. Two questions: In the third paragraph from the end, I didn't understand what situations you refer to, in which the simplification and unity can supposedly occur. Or better yet, in what situations do they not occur? Also, I understand the fact that both phenomena influence the EMF in the same way is a "coincidence" if we use the conventional variables, but is it still a coincidence in covariant variables? Is there a covariant formulation of the flux law? – Joe Feb 1 '13 at 20:46
• Dear Joe, by the simplification, I meant the fact that the EMF only depends on the change of flux and not on some more complicated function of changes of the area, the magnetic field, or some combination of them. When one considers the induction in the case of uniform magnetic fields only etc., this simplification for this formula for EMF always occurs. But as Feynman correctly suggests, this simplification does not imply that the terms that combine to the full answer are related by symmetries or "unified" in the conventional sense. – Luboš Motl Feb 3 '13 at 7:44
• Otherwise, I was using (Lorentz) covariant variables all the time and the terms don't and didn't unify into "a term of the same origin" so yes, covariance isn't enough to explicitly explain the simplification. It was the bulk and key point of my answer, I feel that I am already answering the same question for the 4th time. So no, Feynman wasn't wrong in his statement. Maybe you don't like the answer because you were predecided that the right answer should be very different? – Luboš Motl Feb 3 '13 at 7:46
• In fact I did, but I was wrong. Sorry for being slow to understand this, and thank you for all the explanations. – Joe Feb 3 '13 at 7:52
• @LubošMotl : The Maxwell-Ampere law can likewise be generalized to a variable loop, so that the variation in displacement flux through the loop is due to two causes: variation in the D field, and movement of the loop. Tai (p.941) calls the latter effect "motional mmf". So again we have "2 phenomena". OTOH, motional emf is due to the magnetic Lorentz force, which can be derived from Faraday's law for a fixed loop; see this answer. So I think the "2 phenomena" are linked. – Gavin R. Putland Aug 14 '18 at 7:31