Origin of elecromagnetic field

[sergiokapone]

Interesting thing, the undergraduate courses of electromagnetism states the electromagnetic field caused by electric charge:

d∗F=4π/c∗J,

and students, in my opinion, mistakenly imagine the electromagnetic field as a product of charged particle.

In my opinion, it is more correct to say that the electromagnetic field is an entity that exists (in a fundamental sense, because of property of nature). And the role of charge can be viewed as something that can affect the degrees of freedom of the electromagnetic field, taking or transferring energy to the latter or creates tension in it. In addition, not so long ago, the existence of the Higgs field was established, which also simply exists by itself.

If I am wrong, please, correct me. The question here is probably the following, is it possible to initially adhere to such a concept, build a course on electrodynamics?

 

[Delta2]

What you saying might be true in the context of treating the electromagnetic field via a quantum field theory (the experts like @vanhees71 or @PeterDonis may correct me here)
However in classical electrodynamics the sources of electric field are charges and time varying magnetic fields.

 

[sergiokapone]

What you saying might be true in the context of treating the electromagnetic field via a quantum field theory (the experts...
However in classical electrodynamics the sources of electric field are charges and time varying magnetic fields...

Yes, I understand that in the classics we can assume that the electromagnetic field is generated by a charge. From the QED point, the field already exists. But the classical point of view is contained as a special case of a quantum one, therefore, for a correct physical worldview, it is probably not entirely correct to say even in the classics that an electromagnetic field is generated ... Besides, even in the classics, free oscillating fields can exist without charge, and the reason that created them is not visible (just apply $t \to -t$).

 

[Delta2]

Yes, I understand that in the classics we can assume that the electromagnetic field is generated by a charge. From the QED point, the field already exists. But the classical point of view is contained as a special case of a quantum one, therefore, for a correct physical worldview, it is probably not entirely correct to say even in the classics that an electromagnetic field is generated ... Besides, even in the classics, free oscillating fields can exist without charge, and the reason that created them is not visible (just apply $t \to -t$).

In the case of free oscillating E and B fields (charge and current densities equal to 0 everywhere), it is the fields themselves that generate each other according to maxwell-faraday's law and maxwell-ampere's law.

If you want to say classically that an E-field is not generated by a charge density or a time varying magnetic field, then you have to provide other equations than Maxwell's equations or provide some exotic interpretation of gauss's law and maxwell-faraday's law because the straightforward interpretations of those laws are that charge densities and time varying magnetic fields are the sources of E-field.

P.S my comments are based on the vector calculus form of Maxwell's equations. I am not familiar in their tensor calculus form.

 

[Gaussian97]

In the case of free oscillating E and B fields (charge and current densities equal to 0 everywhere), it is the fields themselves that generate each other according to maxwell-faraday's law and maxwell-ampere's law.

If you want to say classically that an E-field is not generated by a charge density or a time varying magnetic field, then you have to provide other equations than Maxwell's equations or provide some exotic interpretation of gauss's law and maxwell-faraday's law because the straightforward interpretations of those laws are that charge densities and time varying magnetic fields are the sources of E-field.

P.S my comments are based on the vector calculus form of Maxwell's equations. I am not familiar in their tensor calculus form.

I don't completely agree with that. First of all, we all agree that in the end physics is about doing measurable predictions, and those predictions are done using mathematics, not "intuitions" or "interpretations", so as long as your mathematical models are the same saying that the EM is created by the charges or not is not really relevant.
With this clear, Maxwell equations are the same in classical ED and QED. So as long as we all agree that in QED the field is a fundamental property and is not created by charges, then this assertion cannot contradict Maxwell equations.
Therefore, I don't see why Maxwell Equations can only be interpreted as the EM field being created by the charges.
Actually, I wouldn't like to say that the EM field is created by charges because, as you also pointed, you can have an EM field without charges. In that case, you said that then this EM field is generated by the field itself... So you need to assume that an EM already exists to be able to create an EM field... Isn't this exactly the same as assuming that the EM is not created but is a fundamental property of spacetime that is always there?

 

[sergiokapone]

Fields are fundamental entities in theory. We can "measure" them when they interact. If, for example, a charge, interacting with an electromagnetic field, can pump energy into it, which we interpret as the appearance of a photon (radiation). Photon is an excited state of the field.

In modern physics we have established the existence of the Higgs field, for example, and no one asks the question, and what gave rise to it. Fields are what our reality is woven from. From the QED point of view, as long as the fields carry out zero oscillations, they are unobservable.

 

[Delta2]

Fields are fundamental entities in theory. We can "measure" them when they interact. If, for example, a charge, interacting with an electromagnetic field, can pump energy into it, which we interpret as the appearance of a photon (radiation). Photon is an excited state of the field.

In modern physics we have established the existence of the Higgs field, for example, and no one asks the question, and what gave rise to it. Fields are what our reality is woven from. From the QED point of view, as long as the fields carry out zero oscillations, they are unobservable.

I think you are mixing classical and quantum electrodynamics.

Classically I will insist that the sources of E-field (at least in the electrostatic case) are the charges.

According to QED or QFT I don't know that if instead of "the field is created by charges" we can say "the field takes or gives energy from/to charges". I don't know much about QFT or QED so if you keep mixing ideas I ll just stop commenting.

 

[sergiokapone]

I think you are mixing classical and quantum electrodynamics.

Reality is one

Classically I will insist that the sources of E-field (at least in the electrostatic case) are the charges.

What does "source" mean, what is the meaning of this?

Maxwell's equations without sources also have a nontrivial solution. If we assume that the "source" is what generates, then why are there solutions to equations without sources?

I would interpret the "source" of the field as something that "perturbs" the field.

 

[Delta2]

I would interpret the "source" of the field as something that "perturbs" the field

Ok that does it for me. I think you can have your own non standard custom interpretations of Maxwell's equations and charges and the fields. Maybe they are correct but it is not what is mentioned at least in well known books about classical EM like Griffith's and Jackson's. As long as you keep Maxwell's equations the same you don't change anything in classical EM.

 

[sergiokapone]

Well, following Ginsburg (in Russian http://www.akzh.ru/pdf/2005_1_24-36.pdf), let's consider the approach of the Hamiltonian method in classical electrodynamics:

The potentials of the EM field can be expanded as:

\begin{equation}
\vec{A}(\vec{r},t) = \sum\limits_{\lambda,i = 1,2} q_{\lambda,i}(t) \vec{A}_{\lambda,i} (\vec{r}),
\end{equation}

where

\begin{equation}
\vec{A}(\vec{r})_{\lambda\, 1,2} = \vec{e}_{\lambda} \sqrt{8\pi}c \cos \vec{k}_{\lambda} \vec{к}.
\end{equation}

Maxwell's equations without sources lead us to the oscillatory equations

\begin{equation}
\frac{d^2 q_{\lambda,i}}{dt^2} + \omega^2_{\lambda}q_{\lambda, i} = 0.
\end{equation}

There are no sources here yet, the fields are free and behave like freely swinging oscillators. And this is classical physics.

How does the source (moving charge with $\vec{v}$) manifest itself?

The same Maxwell equations with sources give the following equations:

\begin{equation}
\frac{d^2 q_{\lambda,i}}{dt^2} + \omega^2_{\lambda}q_{\lambda, i} = (\vec{e}_{\lambda}\cdot e\vec{v}) \sqrt{8\pi}c \underset{\cos}{\overset{\sin}{}} \vec{k}_{\lambda} \vec{r}_q.
\end{equation}

We see all the same equations for the oscillators, but now these oscillators are under the influence of a "disturbing" force. We get what I already sai:

I would interpret the "source" of the field as something that "perturbs" the field.

 

[Delta2]

Sorry I am not familiar with the Hamiltonian method in classical electrodynamics. Do Griffith's or Jackson's contain a treatment of it?

What is the supposed meaning of $q_{\lambda,i}$ don't tell me they are just coefficients of the expansion of the vector potential.

 

[sergiokapone]

Sorry I am not familiar with the Hamiltonian method in classical electrodynamics. Do Griffith's or Jackson's contain a treatment of it?

You can read more about all this here.

It is curious that in classical electrodynamics the Hamiltonian method was almost never applied in the past; this method became popular only with the transition to quantum electrodynamics. However, as often happens, then “feedback” began to act. Specifically, it turned out that the Hamiltonian method is also very convenient for solving a number of classical problems, especially in the presence of a medium (see Chapters 6 and 7 below). In recent years, when many problems have already been solved, new more complex problems have arisen and, in addition, a number of powerful mathematical methods have been developed and began to be widely used (diagram technique, the method of Green's functions, etc.), the Hamiltonian method has receded into the shadows in both quantum and classical theories of radiation. We are convinced that the Hamiltonian method nevertheless retains the advantage of clarity, simplicity, and rather great versatility, which makes its presentation and use quite expedient, at least for pedagogical purposes.

What is the supposed meaning of qλ,i don't tell me they are just coefficients of the expansion of the vector potential.

In the Hamiltonian approach, these are new dynamic field variables, an analogue of generalized coordinates.

 

[vanhees71]

What you saying might be true in the context of treating the electromagnetic field via a quantum field theory (the experts like @vanhees71 or @PeterDonis may correct me here)
However in classical electrodynamics the sources of electric field are charges and time varying magnetic fields.

That's not entirely correct also in the realm of classical electrodynamics, at least not from a modern relativistic point of view, and since classsical electrodynamics is the paradigmatic classical relativistic field theory, it's the most consistent way to interpret it as such.

The quantities called "sources" of the field should be causally connected with the fields, and that are the charge and current distributions (the latter also including magnetization distributions), because the observable electromagnetic fields are functionals given by the retarded Green's functions of these quantities and their derivatives (->the Jefimenko equations, as nowadays these formulae are called):

https://en.wikipedia.org/wiki/Jefimenko's_equations

The split of the electromagnetic field in electric and magnetic components is frame dependent. So it has no direct physical meaning to say one set of components wrt. a given frame is the source of another set of such components.

 

[Delta2]

Sorry @vanhees71 I didn't quite understand, what exactly changes in relativistic classical electrodynamics such that the statement "the sources of E-field are charges and time varying magnetic fields" is not entirely correct?

 

[vanhees71]

It's a tricky business, and unfortunately usually electrodynamics is taught as if we still were in the 19th century, i.e., in terms of 3D vector calculus for the electric and magnetic components. Then you sometimes have statements that interpret Faraday's Law
$$\vec{\nabla} \times \vec{E}=-\frac{1}{c} \partial_t \vec{B}$$
as that (parts of) the electric field has a time-varying magnetic field as a source, and the Helmholtz theorem of 3D vector calculus is used to kind of "solve" Maxwell's equations. You can force this program through to a certain extent and get very complicated non-local looking expressions. All the beautiful physics is lost though.

It's much more "natural" (in the literal sense!) to treat Maxwell's equations as relativistic field equations. Then you have clear local solutions for the fields with the charge and current densities as sources. With local I mean that the fields are retarded solutions with these quantities as sources.

For a detailed analysis along these lines of argument, see, e.g.,

https://arxiv.org/abs/1609.08149v1
https://doi.org/10.1088/0143-0807/37/6/065204

 

[Dale]

in my opinion, mistakenly imagine the electromagnetic field as a product of charged particle.

In my opinion, it is more correct to say that the electromagnetic field is an entity that exists (in a fundamental sense, because of property of nature).

In my opinion this is a pointless distinction either way. Mathematically, if you have a conserved charge then that mathematically implies a potential field. Conversely, a potential field implies a conserved charge.

The logical implication works either direction, so trying to claim one or the other as primary is simply a beauty pageant where you crown the most aesthetically pleasing one as your own “Miss Fundamental”

 

[vanhees71]

I'd say in classical electrodynamics there are two fundamental dynamical entities, "matter" and "electromagnetic field". The only consistent way to describe matter within classical physics is in terms of "continuum mechanics" (fluids or solids), i.e., in a field description, and the electromagnetic field is a field anyway (despite Newton's overly strong influence with his particle theory of light in his "Opticks"), because there's no "point-particle phenomenology" of the electromagnetic field that could have been observed, while the electron was (mis)taken to be a point particle since its discovery by J. J. Thomson. The real unsolved problem in classical electromagnetism is the idea of a point particle in the literal sense, and I'm pretty sure that problem cannot and doesn't need to be solved, because at the really fundamental level you need quantum (field) theory to describe everything consistently.

 

[sergiokapone]

In my opinion this is a pointless distinction either way. Mathematically, if you have a conserved charge then that mathematically implies a potential field. Conversely, a potential field implies a conserved charge.

The logical implication works either direction, so trying to claim one or the other as primary is simply a beauty pageant where you crown the most aesthetically pleasing one as your own “Miss Fundamental”

In classical physics yes, but quantum field theory(wich is more fundamental), we should conclude EM field is independent objective entity.

 

[vanhees71]

I think, also within classical electrodynamics the "electromagnetic field" is just a fundamental dynamical entity as is "matter".

 

[Dale]

In classical physics yes, but quantum field theory(wich is more fundamental), we should conclude EM field is independent objective entity.

I don’t think that there is more than an aesthetic personal preference judgement for doing that. There are e.g. the photon field and the electron field and both are treated similarly in the math.

FYI, I share that preference, but I don’t think that it is a preference imposed by QFT.

 

[PeroK]

Interesting thing, the undergraduate courses of electromagnetism states the electromagnetic field caused by electric charge.

Are you proposing to begin an undergraduate course in EM with QED/QFT and derive classical EM from that?

 

[vanhees71]

I'd begin classical EM with an introduction to special relativity and then introduce the electromagnetic field as a relativistic field theory from the beginning , starting with the Maxwell equations as the fundamental equations, which cannot be derived (of course in an undergrad. course it depends whether they already had an experimental-physics lecture about the subject before, as it's usually the case at German universities of not, whether you have a chance to do that; if not, you need a quite detailed heuristic introduction before that). Then I'd treat the usual examples but with much less "boring" statics and more time-dependent problems, i.e., mainly waves in the vacuum as well as in simple matter in the usual linear-response approximation.

I'd finish with classical electron theory and end the course with the demonstration that classical electrodynamics is incomplete hinting at the necessity of QT and QED/QFT.