Electrical displacement: doubts

[quiet]

Hi. Maybe you can help me. I am reading history of physics, especially the moment when Maxwell completes Ampere's law with the displacement current. My doubts are about propagation in a vacuum.

1. Can I say that in an electromagnetic wave there are several undulating fields, one of them the electric displacement?

2. If the displacement fulfills a wave function, can the displacement wave be formulated in the context of classical electrodynamics?

3. Since the displacement vector have the followinf form
$$\vec{D} = \vec{P} + \varepsilon_o \ \vec{E}$$
Is the polarization of the vacuum implicit in Maxwell's equations?

The question does not refer to what Maxwell and other physicists of the time thought about the vacuum. It refers to what mathematically and physically involve Maxwell's equations. I think my doubts outweigh my possibilities of personally finding the answers.

 

[Dale]

I am reading history of physics,

You have to be careful with this approach. If you already understand the physics, then learning the history can give you a better understanding of the scientific method and how these great ideas came to be. But if you do not already understand the physics, then the history of physics will not be a good way to learn. This will particularly be problematic if you read a smattering of new material and try to incorporate that into a deep dive through a previous period.

1. Can I say that in an electromagnetic wave there are several undulating fields, one of them the electric displacement?

You can say it, but it is often preferable to group the fields together into groups that are linked relativistically.

2. If the displacement fulfills a wave function, can the displacement wave be formulated in the context of classical electrodynamics?

Yes. I think this is a place where the historical approach is confusing you.

Is the polarization of the vacuum implicit in Maxwell's equations?

No. Same as above.

 

[vanhees71]

Hi. Maybe you can help me. I am reading history of physics, especially the moment when Maxwell completes Ampere's law with the displacement current. My doubts are about propagation in a vacuum.

1. Can I say that in an electromagnetic wave there are several undulating fields, one of them the electric displacement?

2. If the displacement fulfills a wave function, can the displacement wave be formulated in the context of classical electrodynamics?

3. Since the displacement vector have the followinf form
$$\vec{D} = \vec{P} + \varepsilon_o \ \vec{E}$$
Is the polarization of the vacuum implicit in Maxwell's equations?

The question does not refer to what Maxwell and other physicists of the time thought about the vacuum. It refers to what mathematically and physically involve Maxwell's equations. I think my doubts outweigh my possibilities of personally finding the answers.

First things first! Start to understand "vacuum electrodynamics" first and don't be confused by the SI units and idiosyncratic factors like $\epsilon_0$ and $\mu_0$ which are simply there to convert unnatural SI units to natural units.

Also the historic approach is highly inefficient for learning physics. Particularly to study Maxwell's treatise is very difficult because since then we understand electrodynamics much better as a classical relativistic field theory. In fact, historically, the relativistic spacetime model has been discovered through understaning electromagnetism better.

From this modern point of view there's just one fundamental electromagnetic field whose sources are electric charges and currents as well as elementary magnetic moments of the elementary particles, forming the matter around us (BTW also mostly governed by the electromagnetic interaction on the scales relevant in our usual environment).

In classical physics the vacuum is just empty spacetime without any further structure. There's no vacuum polarization, which is a quantum effect and has a very different meaning from what's usually suggested in pupular-science writing. See the corresponding Insights articles

https://www.physicsforums.com/insights/vacuum-fluctuations-experimental-practice/
https://www.physicsforums.com/insights/vacuum-fluctuation-myth/

 

[quiet]

Dale, vanhess71, thank you very much for guiding me. It is always important and pleasant to read your advice. I'll look at the links.

 

[quiet]

Forgive me for loving the history of physics. I guess it attracts me because it must meet two premises. One is not to lie scientifically and another is to present a kind of verbal theorems that, read carefully, raise concerns.

What we have discussed before has joined in my head with some simple concepts of classical electromagnetism. The following emerged.

1. When in the classroom one asks how it is done so as not to confuse a thing with another, they tell us to look at the size of the analyzed system. When there is polarization, you can always find a Gaussian surface, smaller that the system, within which there is a non-zero net charge. The other condition is that there is always a Gaussian surface, which does not exceed the size of the system, within which there are pairs of symmetric collinear vectors, that give a contribution equal to zero in terms of net charge. I guess the the same conditions must be met when a wave propagates in a vacuum.

2. In the classrooms one also learns that if in $\vec {D} = \vec {P} + \varepsilon_o \ \vec {E}$ we suppress $\vec {P}$ we ran out of $\vec {D}$ and the only thing it exists, physically and mathematically, it is $\vec {E}$. So, if $\vec {D}$ is essential for the existence of the electromagnetic wave, $\vec {P}$ is also essential. This detail does not raise any doubts in the case of a wave that propagates in a material medium, because they have explained to me how it works $\vec {P}$ in that medium. In the case of the vacuum it causes me concern, because the obligation to have $\vec {P}$ in order to have $\vec {D}$ and, in consequence, to have a displacement wave in a vacuum, has never been analyzed in the classrooms where I have been.

3. When I try to analyze it, I understand that my analysis must take into account that a wave in a vacuum can be constituted by a single cycle. In this case, the measurement of the entire system in the spread direction is equal to wavelength. And within a Gaussian surface that in the spread direction does not exceed one wavelength, I must have pairs of vectors $\vec {P}$ collinear and symmetric. If I take that seriously, vectors $\vec {P}$ what I'm looking for can not be transverse, because vectorially the only thing on cross direction is $\vec {E}$, which corresponds to the part $\varepsilon_o \ \vec {E}$ and not to $\vec {P}$. That forces me to look for a longitudinal $\vec {P}$ that fulfills a wave function. All of that brings me to a rotary $\vec {D}$ vector which, in the simplest case, has the following form.
$$\vec{D} = \hat{D} \left[\ \vec{x} \ \cos{\left(\omega t-kx \right)} +\vec{y} \ \sin{\left(\omega t-kx \right)} \ \right]$$ that by the identity of De Moivre corresponds to the following.
$$\displaystyle \vec{D} = \hat{D} \ e^{i \left(\omega t -kx \right)}$$ I have symbolized $\vec {x}$ and $\vec {y}$ to the unit vectors of the respective axes, in the orthogonal Cartesian coordinate system. And I have symbolized $\hat {D}$ to the peak value of the displacement.

4. This rotative displacement has a non-zero divergence, that is, implies a $finite \ and \ continuous$ charge density where the wave is present. Suppose that the implication of a finite charge density in vacuum did not surprise me, because to be surprised one must to know more than what I learned in the classroom. But all the people who have gone through a classroom knows what continuity means. There must be charge in the environment of each point, that is, in each infinitesimal volume. That directly means that I can't think of charged particles, neither real nor virtual, because they do not work to give a continuous charge density. I have to think directly about a state of the vacuum that occurs when a wave propagates and that, acquiring such state, the vacuum physically participates in the propagation.

5. I know I can cheat the story and add knowledge that the physics acquired much after Maxwell, saying that classical functions describe averages in probabilistic terms. But that does not convince me, for the following reason. Developing in classical terms the consequences of rotating displacement, I get a finite wavefront, which is the cap of a cylinder whose length is equal to one wavelength. The Maxwellian fields they exist only within that finite cylinder. Outside the cylinder there is no fields, nor is there any Poynting vector, nor energy, nor anything that we can specify. Honestly, all that worries me, because I do not see that it leads to an alley without departure. As far as I can see, development can continue. But if it continues, the only way to accept it is to be consistent with today's knowledge. To demand that coherence is not to cheat the story, for the following. If the consequences are really consistent with today's knowledge, then we will have something else to try to understand the mental landscape of scientists from that time, because we know that nobody publishes everything they think or all calculations made to analyze an issue. In case of inconsistency with today's knowledge we all could, including high school students, understand very simply the need to replace classical electrodynamics with a quantum field theory. When something is didactically useful, it interests me a lot, provided it is lawful. In the case of displacementt wave, I do not know what one should think.

 

[Dale]

You really need to pick up a textbook instead of a history book.

⃗D=⃗P+εo ⃗ED→=P→+εo E→ \vec {D} = \vec {P} + \varepsilon_o \ \vec {E} we suppress ⃗PP→ \vec {P} we ran out of

What? I have no idea what you mean by suppressing P or running out of D. It is very non standard at best.

in that medium. In the case of the vacuum it causes me concern, because the obligation to have ⃗PP→ \vec {P} in

There is no such obligation

a wave in a vacuum can be constituted by a single cycle. In this case, the measurement of the entire system in the spread direction is equal to wavelength

I think you need to learn a bit of Fourier analysis (preferably from a textbook rather than a biography on Baron Fourier)

4. This rotative displacement has a non-zero divergence, that is, implies a finite and continuousfinite and continuousfinite \ and \ continuous charge density where the wave is present.

This is not correct. There are many vacuum solutions with no charge density.

Developing in classical terms the consequences of rotating displacement, I get a finite wavefront, which is the cap of a cylinder whose length is equal to one wavelength.

You should learn about circularly polarized waves from a textbook. You are far off base with your own musings.

 

[quiet]

Dale, thank you very much for guiding me. It is always important and pleasant to read your advice.