Are all EM waves produced by accelerating charges?

[rugerts]

So, a static charge at rest produces an electric field, but no magnetic field.
A charge moving with constant velocity produces both electric and magnetic fields.

Why is it that accelerating charges are the source of all electromagnetic radiation?
How would one go about showing this using Maxwell's equations?

 

[Dale]

Why is it that accelerating charges are the source of all electromagnetic radiation?
How would one go about showing this using Maxwell's equations?

Maxwell’s equations will not show this. There are perfectly valid vacuum solutions, meaning solutions with no charges or currents.

 

[Delta2]

Maxwell’s equations will not show this. There are perfectly valid vacuum solutions, meaning solutions with no charges or currents.

It is a bit odd the way you state it. According to Maxwell's equations, EM waves are either due to time varying charge and current densities (about same thing as accelerating charges) OR we have the vacuum solutions as you say.

 

[lightarrow]

Why is it that accelerating charges are the source of all electromagnetic radiation?
How would one go about showing this using Maxwell's equations?

Using Maxwell's equations is not easy. The simplest way is to use the integrals which give the scalar and vector potentials as functions of (retarded) charge density and (retarded) current density:

Φ = 1/(4πε₀) ∫ρ(r', t-|r-r'|/c)/|r-r'| dV'

A = (μ₀/4π) ∫j(r', t-|r-r'|/c)/|r-r'|dV'

Φ = scalar potential
A = vector potential
r = position vector where you compute the potential or the field
r' = position vector where the sources (charges or currents) are located
dV' = volume element of the sources.

Then:

E = -∇Φ -∂A/∂t
B = ∇∧A = rot A

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lightarrow

 

[tech99]

Using Maxwell's equations is not easy.

As a matter of interest, although Maxwell described an EM wave, did he know how to create radiation? There were no antennas at that time. I had the impression that it was not until Larmor that the "accelerating charge" idea came about. I don't think Hertz mentioned accelerating charges.

 

[Delta2]

As a matter of interest, although Maxwell described an EM wave, did he know how to create radiation? There were no antennas at that time. I had the impression that it was not until Larmor that the "accelerating charge" idea came about. I don't think Hertz mentioned accelerating charges.

I think Hertz experiments with the "spark gap oscillator" show that he knew that time varying currents (which are being constituted by accelerating charges) of big enough frequency and big enough amplitude give rise to EM-waves. But EM waves were of pure theoretical interest at Hertz's time i believe for one main reason:
The transmission and reception technology were not yet discovered. I mean mainly the amplifier technology (with tubes which were discovered later at 1905) was not yet available, so without amplifying technology they could neither produce a strong enough EM wave (that would carry a useful signal) , neither amplify the signal from an antenna that receive a weak EM wave.

 

[ZapperZ]

So, a static charge at rest produces an electric field, but no magnetic field.
A charge moving with constant velocity produces both electric and magnetic fields.

Why is it that accelerating charges are the source of all electromagnetic radiation?
How would one go about showing this using Maxwell's equations?

This is the opposite of what we normally get in this forum. Usually we have people claiming that ALL EM radiation is due to atomic transition, not knowing about accelerating charges. Now, comes the claim going the other way.

Zz.

 

[rugerts]

Maxwell’s equations will not show this. There are perfectly valid vacuum solutions, meaning solutions with no charges or currents.

This confuses me. Why do you mention the vacuum? I wasn't limiting myself to the vacuum. Quoting from my textbook, University Physics 14th Edition (Young and Freedman), "Maxwell's equations can also be used to show that in order for a point charge to produce electromagnetic waves, the charge must accelerate. In fact, it's a general result of Maxwell's equations that every accelerated charge radiates electromagnetic energy."

 

[rugerts]

This is the opposite of what we normally get in this forum. Usually we have people claiming that ALL EM radiation is due to atomic transition, not knowing about accelerating charges. Now, comes the claim going the other way.

Zz.

It's funny that you mention this. I asked my TA about this very question and he was sort of puzzled and said that the textbook is wrong and vaguely mentioned something about energy levels. The only knowledge I have of this is the really crude, intro chemistry type stuff (so, really none). Could you expand a bit more or would it require too much? If the textbook is wrong, I don't understand why they can't take the time to add small caveats and say something along the lines of "not entirely true... keep reading".

 

[Delta2]

Your book is correct as far as we are concerned with classical electrodynamics (or classical electromagnetism) and i believe your course's name is classical electromagnetism or something like that.
But if we go into Quantum Electrodynamics your book is partially correct. Except accelerating charges there is also the transition of electrons inside the atom that can produce EM waves.

 

[rugerts]

Your book is correct as far as we are concerned with classical electrodynamics (or classical electromagnetism) and i believe your course's name is classical electromagnetism or something like that.
But if we go into Quantum Electrodynamics your book is partially correct. Except accelerating charges there is also the transition of electrons inside the atom that can produce EM waves.

Oh. So, it's not that the quantum electrodynamics completely dismissed what my book is saying, it's just that it added another layer to it (i.e. another way that we see EM waves can be produced)?

 

[rugerts]

Using Maxwell's equations is not easy. The simplest way is to use the integrals which give the scalar and vector potentials as functions of (retarded) charge density and (retarded) current density:

Φ = 1/(4πε₀) ∫ρ(r', t-|r-r'|/c)/|r-r'| dV'

A = (μ₀/4π) ∫j(r', t-|r-r'|/c)/|r-r'|dV'

Φ = scalar potential
A = vector potential
r = position vector where you compute the potential or the field
r' = position vector where the sources (charges or currents) are located
dV' = volume element of the sources.

Then:

E = -∇Φ -∂A/∂t
B = ∇∧A = rot A

--
lightarrow

Thank you for the time and in depth reply! I don't know if this would be asking too much, but do you mind expanding a bit on the thought process behind this derivation? Is there somewhere in there describing acceleration? I'm currently learning multivariable calculus alongside E&M, so I'm quite new to both.

 

[Delta2]

Oh. So, it's not that the quantum electrodynamics completely dismissed what my book is saying, it's just that it added another layer to it (i.e. another way that we see EM waves can be produced)?

Essentially yes. To be more precise, QED says that when electrons or other charged particles change states then a photon (which is the same as an EM wave for the sake of simplicity) could be emitted.
Now when an electron is accelerated , it is obviously changing states cause his energy (energy is one of the fundamental quantity that characterizes a state in quantum physics) is changing.

 

[lightarrow]

Thank you for the time and in depth reply! I don't know if this would be asking too much, but do you mind expanding a bit on the thought process behind this derivation? Is there somewhere in there describing acceleration? I'm currently learning multivariable calculus alongside E&M, so I'm quite new to both.

Deriving it it's too long to write here. In books like this:
Classical Electricity and Magnetism - Panofsky - Phillips - second edition
there is the complete derivation at chapter 14, at least for monochromatic waves. Here is the conclusion:

https://i.postimg.cc/05YyktpT/IMG-20181130-124007.jpg

The currents density j is proportional to the charges velocity, so its time derivative is of course proportional to charges acceleration.
This as answer to your question of relating radiation fields to the sources.
Concerning how to derive the equations I wrote in my previous post, again it's too long and in the Book cited here is made at the first two paragraphs of the same chapter 14. This is the conclusion:
https://i.postimg.cc/VNVtvc3B/IMG-20181130-125618.jpg

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lightarrow

 

[Dale]

Why do you mention the vacuum?

Because you asked if Maxwell’s equations can be used to show that all EM waves are produced by accelerating charges. The existence of vacuum solutions contradicts that since vacuum solutions have waves without any charges at all.

In fact, it's a general result of Maxwell's equations that every accelerated charge radiates electromagnetic energy

This statement has to be handled with caution since it can lead to wrong conclusions in certain cases. However, even accepting this, the converse is not true. All accelerated charges radiate does not imply all radiation is from accelerated charges. Vacuum solutions are a counterexample.

The existence of vacuum solutions is central to your question

 

[fluidistic]

First comment: If I have a magnet in my hand and I move it back and forth (it accelerates, yes, but there is no charge), wouldn't it create EM waves? Wouldn't that be a case of classical electrodynamics causing EM waves without any charged particle?

Second comment: I've read that there's still a debate on whether a linearly constant accelerated particle would display radiation. See https://physics.stackexchange.com/q...ing-charged-particle-emit-em-radiation-or-not.

 

[rugerts]

Because you asked if Maxwell’s equations can be used to show that all EM waves are produced by accelerating charges. The existence of vacuum solutions contradicts that since vacuum solutions have waves without any charges at all.

This statement has to be handled with caution since it can lead to wrong conclusions in certain cases. However, even accepting this, the converse is not true. All accelerated charges radiate does not imply all radiation is from accelerated charges. Vacuum solutions are a counterexample.

The existence of vacuum solutions is central to your question

How do the EM waves exist and have no charges? A quick google search led me to the following post: https://physics.stackexchange.com/questions/156707/electromagnetic-waves-in-vacuum
which, if I'm not wrong in my interpretation but very well could be, looks like there's still going to be charges associated with the EM waves in the vacuum.

 

[rugerts]

Deriving it it's too long to write here. In books like this:
Classical Electricity and Magnetism - Panofsky - Phillips - second edition
there is the complete derivation at chapter 14, at least for monochromatic waves. Here is the conclusion:

https://i.postimg.cc/05YyktpT/IMG-20181130-124007.jpg

The currents density j is proportional to the charges velocity, so its time derivative is of course proportional to charges acceleration.
This as answer to your question of relating radiation fields to the sources.
Concerning how to derive the equations I wrote in my previous post, again it's too long and in the Book cited here is made at the first two paragraphs of the same chapter 14. This is the conclusion:
https://i.postimg.cc/VNVtvc3B/IMG-20181130-125618.jpg

--
lightarrow

Thanks!

 

[tech99]

First comment: If I have a magnet in my hand and I move it back and forth (it accelerates, yes, but there is no charge), wouldn't it create EM waves? Wouldn't that be a case of classical electrodynamics causing EM waves without any charged particle?
.

How do you make a magnet which has no charged particles?

 

[Dale]

How do the EM waves exist and have no charges?

The existence of vacuum solutions implies that the existence of EM waves does not logically require the existence of charges. That said, I think there are few if any such EM waves now.

 

[Delta2]

First comment: If I have a magnet in my hand and I move it back and forth (it accelerates, yes, but there is no charge), wouldn't it create EM waves? Wouldn't that be a case of classical electrodynamics causing EM waves without any charged particle?

According to classical physics the magnetic field of permanent magnets is due to bound current density $J_b$ (though we know from quantum physics this is not the real cause of magnetization but let's stay in classical physics here). When we move a magnet back and forth is like we moving the current density $J_b$ back and forth, that is we create a spatially and temporally varying current density. It is about as moving a DC current carrying wire back and forth.

 

[fluidistic]

According to classical physics the magnetic field of permanent magnets is due to bound current density $J_b$ (though we know from quantum physics this is not the real cause of magnetization but let's stay in classical physics here). When we move a magnet back and forth is like we moving the current density $J_b$ back and forth, that is we create a spatially and temporally varying current density. It is about as moving a DC current carrying wire back and forth.

Touché! I did not know that. Impressive!
I was wrong then, sorry.

 

[lightarrow]

How do you make a magnet which has no charged particles?

This is true but in conventional magnets it's actually the electrons' magnetic moment which gives rise to the B field, not the movement of charges, so when you move the magnet (e.g.) back and forth it's the B field time variation which generates the em wave, isn't it?
I know you talked about "classical" electrodynamics and that the B field generated from permanent magnets can be formally described as due to magnetization currents, but I think the OP is probably looking for a bit deeper explanation.

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lightarrow

 

[hutchphd]

The existence of vacuum solutions implies that the existence of EM waves does not logically require the existence of charges. That said, I think there are few if any such EM waves now.

Any particular exact solution to Maxwell's equations requires specification of boundary conditions. I believe that these in turn require some charges to be (or have been) shaken...perhaps "offstage"... for an interesting result. Am I wrong here?

 

[Dale]

Am I wrong here?

Yes. All you need is the fields.

 

[Delta2]

Yes. All you need is the fields.

Isn't the vacuum solution just a theoretical solution? In the physical reality, all fields have some sources, fields without sources simply don't exist, at least that's the case for the EM-field , isn't it?

 

[Dale]

In the physical reality, all fields have some sources, fields without sources simply don't exist, at least that's the case for the EM-field , isn't it?

The oldest extant photons have their source on the surface of last scattering, so they do have material sources. However, shortly after the Big Bang you certainly had many “primordial” photons that did not originate from matter. The fields do not logically require material sources, and although currently we do not see EM fields without matter sources that was not always the case.

 

[hutchphd]

shortly after the Big Bang you certainly had many “primordial” photons that did not originate from matter

Certainly I am unconvinced. How does one know there were not accelerated electrical charges involved? Please elaborate.

 

[Dale]

Certainly I am unconvinced.

Ok, I am not going to argue about speculative conditions at the Big Bang. I will just assert that Maxwell’s equations have vacuum solutions. Therefore the presence of charges is not logically necessary regardless of how ubiquitous it is now.

 

[hutchphd]

Ok, I am not going to argue about speculative conditions at the Big Bang. I will just assert that Maxwell’s equations have vacuum solutions. Therefore the presence of charges is not logically necessary regardless of how ubiquitous it is now.

Fair enough. I think we are at a "chicken vs. egg" point! Thanks.