As far as I understand it, Maxwell's equations unify the theories of electricity and magnetism, however, I don't see how they show that the electric field, $\mathbf{E}$ and the magnetic field, $\mathbf{B}$ are manifestations of a single entity, the electromagnetic field?!

At least at face value, Maxwell's theory shows a mutual dependence between electric and magnetic fields, but doesn't explicitly show that they should be unified. (Adding to this point, I get that Maxwell's equations predict self propagating electromagnetic waves, but in this context they are still treated as transverse oscillations of the electric and magnetic fields).

Would it be correct to say that one cannot definitively conclude that the electric and magnetic fields should be considered as a single, unified entity, the electromagnetic field, until one takes into account special relativity?

To me this seems to be the case, since the Lorentz transformations "mix-up" the two fields, transforming one into the other, and so the notion of whether what is observed is an electric field or a magnetic field is observer dependent. The quantity that does maintain an objective reality, however, is the electromagnetic field strength, $F_{\mu\nu}$ which is Lorentz covariant, and indeed the quantity $F_{\mu\nu}F^{\mu\nu}\propto\mathbf{B}^{2}-\frac{1}{c^{2}}\mathbf{E}^{2}$ is Lorentz invariant. Since the components of $F_{\mu\nu}$ can be represented in terms of $\mathbf{E}$ and $\mathbf{B}$, one considers the electric field and the magnetic field to be different manifestations of a single unified field - the electromagnetic field.

In essence, my question is can one consider electricity and magnetism unified into a single concept of electromagnetism at the level of Maxwell's equations, or is it not until special relativity is taken into account that it becomes necessary to consider them as a single entity - the electromagnetic field?

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    $\begingroup$ Your last paragraph is correct. What is your question? $\endgroup$ – Andrey Feldman Feb 2 '17 at 14:05
  • $\begingroup$ What I want to know really is, at what stage can one consider electricity and magnetism unified? At the level of Maxwell's equations or only once special relativity is taken into account? I think it's the latter, but I want to make sure I've understood things correctly. $\endgroup$ – Feyn_example Feb 2 '17 at 14:15
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    $\begingroup$ I think you are making a false distinction. Maxwell's equations are Lorentz covariant so you cannot have the Maxwell equations without automatically taking special relativity into account. That is, you cannot separate the two. $\endgroup$ – John Rennie Feb 2 '17 at 14:26
  • $\begingroup$ @JohnRennie Ah ok, good point. Maxwell's equations are already Lorentz covariant (although he didn't notice this at the point of there formulation). However, I feel that it doesn't become obvious that the two fields become unified until one studies the Lorentz transformations. $\endgroup$ – Feyn_example Feb 2 '17 at 14:30
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    $\begingroup$ I guess we'd say they aren't manifestly covariant until we write them in relativistic notation (using the four-potential or field strength tensor) but I still think you are making a distinction that doesn't really exist. $\endgroup$ – John Rennie Feb 2 '17 at 14:37

In electromagnetism there are three stories.

The first story is about the charged subatomic particles. So electron possesses an electric monopol and a magnetic dipole moment and by this intrinsic properties the electron is surrounded by both an electric field and a magnetic field. The electric field is symmetrical in all directions, the magnetic field has always an alignement in a direction.

The second story is about the macroscopic manifestations of this fields. Separating electrons from protons gives as electric fields which are easily observable (by the force with which the separated charges will attract each other or by a current in the case the charges will be connected by a wire). With the magnetic field it is not that easy. The magnetism of permanent magnets was not explainable as long as it was not clear that charged particles have intrinsic magnetic dipole moments and in permanent magnets this moment of (from?) the involved electrons is aligned and "frozen". An other way to to get a macroscopic magnetic field was the invention of a coil of a current carrying wire. Moving in circles (on a solenoidal path) the electrons are under permanent acceleration and by this their magnetic dipole moments get aligned all in the same direction and a macroscopic magnetic field arise.

The third story is about the (later so called) electromagnetic radiation of energy by the acceleration of charged particles. The flow of electrons inside a coil is accompanied with energy losses which are higher as the losses of a straight wire of the same length (due to Ohmic resistance). The radiated from the acccelerated electrons energy doesn't need a medium for movement away. Later instead of a coil electrons were accelerated back and forth in an (antenna) rod and this invention produces an oscillating radiation. The study of such radiation led to the conclusion that (strictly exact only in vacuum) this radiation consists an electric field component and a magnetic field component, both perpendicular to the direction of propagation and perpendicular to each other. So both field components of the electron over going to the EM radiation from this charged particles.

As far as I understand it, Maxwell's equations unify the theories of electricity and magnetism, however, I don't see how they show that the electric field, E and the magnetic field, B are manifestations of a single entity, the electromagnetic field?!

Maybe this is not a direct answer to your questionnaire, but it gives explanations without any special relativity.

Let me add a post-sentence. An interesting area of scientific research would be the investigation About the inner structure of the electron which is unknown until now. But since the electron posseses permanently two fields to build a model of the constituents of this inner structure and by this of the fields should be good scientific practice. This model has to explain not only both field components of the electron but also both components of the EM radiation.

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  • $\begingroup$ Interesting answer! The problem I find is that I can see how the electric and magnetic fields are intrinsically related through Maxwell's equations, and that these equations provide a unified description of the two phenomena (electricity and magnetism), as in it unifies the two theories, however, I fail to see directly from Maxwell's equations why one should combine them into a single field - the electromagnetic field?! $\endgroup$ – Feyn_example Feb 3 '17 at 10:21

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