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Maxwell's equations reveal an interdependency between electric and magnetic fields, inasmuch as a time varying magnetic field generates a rotating electric field and vice versa. Furthermore, the equations predict that even in the absence of any sources one can have self propagating electric and magnetic fields, so called electromagnetic waves.

However, is it correct to say that although Maxwell's equations show that electric and magnetic fields are interdependent, they do not imply that the two are different aspects of the same underlying physical phenomenon?!

Given this, is it then correct to say that it is not until one takes into account special relativity that it becomes clear that electricity and magnetism are different manifestations of the same underlying phenomenon?

Indeed, if one considers a frame of reference in which only an electric (or magnetic) field is observed, then, upon a Lorentz transformation to another frame of reference, it is found that one will observe a combination of electric and magnetic fields. This implies that the two are not independent of one another, since there is no observer independent manner in which one can separate electric and magnetic fields, hence implying that they are manifestations of the same underlying field - the electromagnetic field?!


Edit

In essence, my question is, can one deduce purely from Maxwell's equations that the electric and magnetic fields are actually "the same" field, or is this (necessary) unification not explicitly confirmed until one takes into account special relativity?

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closed as unclear what you're asking by ACuriousMind, sammy gerbil, user36790, Wolpertinger, knzhou Aug 23 '16 at 22:10

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    $\begingroup$ @OrangeDog Relativity is not derived from Maxwell's equations. $\endgroup$ – lemon Aug 23 '16 at 10:19
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    $\begingroup$ they are the same object ( and not manifestations of another object ) , the magnetic field being the electric field as seen in a moving frame $\endgroup$ – user46925 Aug 23 '16 at 10:20
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    $\begingroup$ @igael Yes, you're right, I was just trying to emphasise the fact that they are unified into a single object, but that historically they were thought of as two distinct objects. Is it correct though, to say that it is through application of special relativity that such a unification is explicitly shown, in the sense that there is no observer independent way to separate the two and as such they the same object? $\endgroup$ – user35305 Aug 23 '16 at 10:38
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    $\begingroup$ not really, it was already the case with Maxwell equations, which are , on another hand compatible with special relativity rather than with classical mechanics $\endgroup$ – user46925 Aug 23 '16 at 10:46
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    $\begingroup$ perhaps, but as today, the Maxwell synthesis is sufficient to identify them, letting to other theories deeper interpretations. Agreeing on the main ... ( surely, the difference comes from the objects : previously forces, now fields ) $\endgroup$ – user46925 Aug 23 '16 at 11:02
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Electric and magnetic fields are field strengths of a gauge field. If you consider any matter field/particle Lagrangian, its coupling constant is the same for Electric and magnetic fields.

For eg. classically in the Lorentz force equation $\vec{F}=q(\vec{E}+\vec{v}\times\vec{B})$. Notice that $q$ is the parameter that gives the coupling of the particle with Electric as well as magnetic field.

This is what we mean when we say the two forces are unified.

Edit: Reply to whether historical approach is the one to use relativity

I have to disagree. Electric and magnetic field were realized to be the same once Maxwell wrote his equations and light was found out to be an electromagnetic wave by Hertz. It is true the Maxwell equations are Lorentz covariant and when you do that you have a single four potential that gives electric and magnetic fields. And then you add coupling with relativistic matter Lagrangians. Then it is natural to have a single coupling constant. So it was rather me who was taking the historical approach. Anyhow it is also modern. When you consider the standard model, weak interaction and electromagnetic interaction are given by the same $SU(2)_L$ field multiplet. But there are still two coupling constants which is why it is not a unification

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  • $\begingroup$ From an historical perspective though (and also from a pedagogical approach), before introducing the notion of gauge fields and the like, is it correct to say that it is not one takes into account special relativity that it becomes abundantly clear that electricity and magnetism are aspects of the same field - the electromagnetic field, since there is no observer independent manner in which one can separate the two?! $\endgroup$ – user35305 Aug 23 '16 at 10:45
  • $\begingroup$ I disagree. See my edit. $\endgroup$ – BoundaryGraviton Aug 23 '16 at 13:58
  • $\begingroup$ Thanks for the updated answer. The issue I find is that, although Maxwell's equations predict "self reinforcing" propagating electromagnetic waves, it is really a statement that time varying electric fields and magnetic fields can generate one another such that they create an electromagnetic wave propagating through space. However, I don't see how one can conclude that they are the same object from this, it simply implies that they are interdependent... $\endgroup$ – user35305 Aug 23 '16 at 15:42
  • $\begingroup$ ... Surely it only explicitly becomes evident that they are "the same" once one studies how they transform under Lorentz transformations, in which it is found that the two "rotate" into one another and hence are really "the same" object?! $\endgroup$ – user35305 Aug 23 '16 at 15:43
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    $\begingroup$ So is the point that the if one takes into account the form of the Lorentz force, it is evident that the coupling to the electric and magnetic fields is identical, implying that they are in fact "the same", a single entity. If they were distinct fields then they would have distinct coupling constants?! This, along with the fact that Maxwell's equations describe the dependence between electric and magnetic fields (as well as predicting the existence of electromagnetic waves), reveals the unification of electric and magnetic fields?! $\endgroup$ – user35305 Aug 23 '16 at 18:22

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