B Interpreting light as Maxwell's EM wave

[Rev. Cheeseman]

TL;DR Summary

Curious which one is the closer representation of wavy configuration of light

Sometimes I cannot imagine light as the popular Maxwell's blue and red electric and magnetic wave https://simply.science/images/content/physics/Electromagnetism/em_waves/emv.jpg but I found the image below to be the more accurate representation of how light actually looks like as waves.

https://www.semanticscholar.org/pap...e5ea2c82ee3cde216ea5e2f970822498a7b0/figure/0
Is that accurate? I think light as a wave in reality are more chaotic in their configuration than Maxwell's blue and red EM wave propagating straight with the electric wave waving vertically and magnetic waves waving horizontally.

 

[BvU]

Imagining things gets you nowhere. Study the solutions of the maxwell equations.

The first picture is one aspect. The second is a mess.

$\$

 

[Dale]

TL;DR Summary: Curious which one is the closer representation of wavy configuration of light

Sometimes I cannot imagine light as the popular Maxwell's blue and red electric and magnetic wave

I think your best bet is to get some computer algebra software and plug in either a dipole wave or a plane wave, verify that it solves Maxwell’s equations, and plot it yourself.

That last part, plotting it yourself, should help you understand those diagrams better. The EM field is a 10 dimensional object, and the experience trying to capture salient features on a 2 dimensional figure is useful.

 

[Rev. Cheeseman]

I think your best bet is to get some computer algebra software and plug in either a dipole wave or a plane wave, verify that it solves Maxwell’s equations, and plot it yourself.

That last part, plotting it yourself, should help you understand those diagrams better. The EM field is a 10 dimensional object, and the experience trying to capture salient features on a 2 dimensional figure is useful.

Is it appropriate to use this http://www.falstad.com/wavebox/? Thoughts?

 

[Dale]

Is it appropriate to use this http://www.falstad.com/wavebox/? Thoughts?

No. You should do it yourself. You won’t get the experience of figuring out how to present the important features of 10 dimensional data in a 2 dimensional plot if you let someone else do it for you. Also, don’t use animations.

 

[Rev. Cheeseman]

No. You should do it yourself. You won’t get the experience of figuring out how to present the important features of 10 dimensional data in a 2 dimensional plot if you let someone else do it for you. Also, don’t use animations.

Sorry, but how to do that?

 

[Dale]

Sorry, but how to do that?

Get a computer algebra software. I like Mathematica. If you don’t have one through your school then SageMath is free and I have used it before

 

[vanhees71]

I think your best bet is to get some computer algebra software and plug in either a dipole wave or a plane wave, verify that it solves Maxwell’s equations, and plot it yourself.

That last part, plotting it yourself, should help you understand those diagrams better. The EM field is a 10 dimensional object, and the experience trying to capture salient features on a 2 dimensional figure is useful.

In which sense is the EM field a 10-dimensional object? That doesn't make any sense to me. You start with 6 field degrees of freedom, $(\vec{E},\vec{B})$. The free-field Maxwell equations ($\rho=0$, $\omega=0$) finally lead to 2 polarization states for each Fourier (plane-wave) mode, as it should be. Any field can be described in terms of these Fourier modes.

The most economic description is through the radiation-gauge potentials,
$$\Phi(t,\vec{x})=0, \quad \vec{A}(t,\vec{x}) = \sum_{j=1}^2 \int_{\mathbb{R}^3} \frac{\mathrm{d}^3 k}{(2 \pi)^3 \sqrt{2 \omega_k}} \vec{\epsilon}_j(\vec{k}) [A_j(\vec{k}) \exp[-\mathrm{i} (\omega_k t-\vec{k} \cdot \vec{x})] + cc.],$$
where $\vec{\epsilon}_1(\vec{k})$, $\vec{\epsilon}_2(\vec{x})$, and $\vec{k}/|\vec{k}|$ build a right-handed Cartesian basis and $\omega_k=c |\vec{k}|$. The fields are given by
$$\vec{E}=-\partial_t \vec{A}, \quad \vec{B}=\vec{\nabla} \times \vec{A}.$$
A physical em. field is given by any two square-integrable functions, $A_j(\vec{k})$.

 

[BvU]

Now, how do we make this accessible for 'gifted amateurs' to get some insight ?

$\$

 

[vanhees71]

Just tell the gifted amateur to carefully fill out the steps in the above calculation, using a good electromagnetics textbook ;-)).

 

[Dale]

In which sense is the EM field a 10-dimensional object?

3 dimensional vector for E and 3 dimensional vector for B over 3 dimensions of space and 1 dimension of time

The most economic description is through the radiation-gauge potentials

We are not looking for the most economic description. The OP has found some figures in the literature that confuse them. I am suggesting an approach for them to understand both what the figures mean and why it is so difficult.

 

[Rev. Cheeseman]

So the result from the calculations using those math software systems will give us what EM waves look like? I just want to see the closest representation of EM rays if we can somehow look at them like when we are looking at apples, houses, etc. I'm really bad at mathematics

Or, is it actually impossible to imagine what EM waves look like exactly in reality?

 

[Dale]

So the result from the calculations using those math software systems will give us what EM waves look like?

Just look around your room. What you see is what EM waves look like. There is no need for software for that.

The calculations will help you understand what EM waves are and how they work. The diagrams are intended to convey that, not merely what they look like.

 

[Rev. Cheeseman]

Just look around your room. What you see is what EM waves look like. There is no need for software for that.

The calculations will help you understand what EM waves are and how they work. The diagrams are intended to convey that, not merely what they look like.

Thank you, Dale. Anyway, is this software good? Have anyone tried it https://www-fourier.ujf-grenoble.fr/~parisse/giac.html

 

[Dale]

Thank you, Dale. Anyway, is this software good? Have anyone tried it https://www-fourier.ujf-grenoble.fr/~parisse/giac.html

I have not tried that one. I have tried this free one:
https://www.sagemath.org/

I have also used these paid ones:
https://www.wolfram.com/mathematica/
https://www.maplesoft.com/products/Maple/
https://www.mathworks.com/products/matlab.html

 

[vanhees71]

3 dimensional vector for E and 3 dimensional vector for B over 3 dimensions of space and 1 dimension of time

This is a strange way to count field-degrees of freedom, I've never heard about before, and I've no clue what it should tell me.

We are not looking for the most economic description. The OP has found some figures in the literature that confuse them. I am suggesting an approach for them to understand both what the figures mean and why it is so difficult.

The first picture shows $\vec{E}$ and $\vec{B}$ of a linearly polarized plane-wave mode.

The two pictures in the second file show the superposition of single-frequency modes, the first is a coherent superposition, and the second is a superposition with random phase shifts. The latter describes in coherent light.

 

[Dale]

This is a strange way to count field-degrees of freedom, I've never heard about before, and I've no clue what it should tell me.

I never claimed to be counting degrees of freedom.

Frankly, I don’t think that this little exchange between you and me is helpful for the OP, so I am not going to pursue it further. I think my comments to them, encouraging them to get some math software, write some known solutions to Maxwell’s equations, and plot them will be a helpful exercise for them, so I will focus on that.

 

[Rev. Cheeseman]

The first picture shows E→ and B→ of a linearly polarized plane-wave mode.

The first picture shouldn't be taken literally when it comes to imagining what EM waves look like in reality. Correct?

 

[berkeman]

The first picture shouldn't be taken literally when it comes to imagining what EM waves look like in reality. Correct?

In what ways? I mean the actual fields are not color coded like that, and it's just a snapshot in time of the traveling wave moving to the right, but otherwise it seems to be fairly accurate...

 

[Ibix]

Or, is it actually impossible to imagine what EM waves look like exactly in reality?

The point is that they don't "look like" anything at all because they're not something you can bounce something off to form an image. The visualisations you see are illustrative, but that's all.

The first image is depicting the direction and magnitude of the electric and magnetic field along a single one dimensional line parallel to the direction of a wave's propagation. But it isn't really a bunch of red and blue arrows, any more than your velocity is a little arrow following you around. The second image is showing the electric field only (probably for clarity) along several lines in order to illustrate the coherence (or lack thereof) of a wave.

Again, neither is "really" what an EM wave "looks like" and the question doesn't really make sense. Both are useful visualisations of different things

 

[Dale]

The first picture shouldn't be taken literally when it comes to imagining what EM waves look like in reality. Correct?

Again, what EM waves look like is what you see when you open your eyes. The figures show aspects of what EM waves are and how they work.

 

[Drakkith]

In what ways? I mean the actual fields are not color coded like that, and it's just a snapshot in time of the traveling wave moving to the right, but otherwise it seems to be fairly accurate...

View attachment 324850

I'd argue that this isn't even what EM waves 'look like'. Or, rather, it's only part of what an EM wave looks like. This is a diagram. A representation showing how the electric and magnetic field vectors oscillate over time and how they are oriented perpendicular to each other.

The big issue here is that defining what light 'looks like' is rather ambiguous in its meaning. If this were a diagram of a water wave (minus the red e-field wave) it would be more intuitive since we can almost immediately understand that the diagram represents the height of the water, since the height of a water wave is something we can see and understand without any knowledge of physics.

But, unlike a water wave, an EM wave has no easy to see physical effects that we are used to observing in our everyday lives. You can't see the strength and direction of a field, after all, and relating these wavy lines to how charged particles behave is much more complicated than understanding that the height of the line on a diagram represents the height of a water wave.

If we interpret 'is this what does an EM wave look like' to mean 'does this accurately represent how an EM wave behaves' we get a much clearer question, to which the answer is "yes, for a single point in space over time or a line through space at a single point in time this accurately represents the field strengths and orientation of a certain EM wave."

But this isn't the only viable diagram. See this picture:

This also represents an EM wave. But instead of showing the field vectors for a single point or line, I believe it is showing regions of similar field strength and direction. EM waves also 'look' like this.

The point is that they don't "look like" anything at all because they're not something you can bounce something off to form an image. The visualisations you see are illustrative, but that's all.

In the end that is what it boils down to. Every picture, illustration, or diagram of an EM wave is simply conveying some set of information to you. You can ask if EM waves 'look' like the illustrations only so long as you understand that you're using 'look' as shorthand for asking if the diagram is reasonably accurate in what it depicts.

 

[vanhees71]

In what ways? I mean the actual fields are not color coded like that, and it's just a snapshot in time of the traveling wave moving to the right, but otherwise it seems to be fairly accurate...

View attachment 324850

Of course such a plane wave can not be literally realized in nature, because we have only a finite amount of energy, and such a plane-wave mode has a diverging total field energy. You should take it as a field mode, i.e., you can decompose any solution of the Maxwell equations in terms of these plane-wave modes, i.e., in terms of a Fourier transformation, as explained in #8. Realistic fields are given with the functions $A_j(\vec{p})$ being square-integrable functions.

 

[Rev. Cheeseman]

If we interpret 'is this what does an EM wave look like' to mean 'does this accurately represent how an EM wave behaves' we get a much clearer question, to which the answer is "yes, for a single point in space over time or a line through space at a single point in time this accurately represents the field strengths and orientation of a certain EM wave."

I'm sorry for this terrible drawing below, I'm trying to draw a laser beam because it seems to be a fitting example for something like "a single point in space over time or a line through space at a single point in time" as laser beam is a concentrated type of light.

Not sure if the drawing is actually correct. Just curious about what others think about it.

 

[Rev. Cheeseman]

Can we actually distinguish an electric field from a magnetic field in reality? Without these red and blue colour-coding and if we include every electric and magnetic fields instead of only two such as the vertical waves and horizontal waves, this is what I thought usual light and laser will look like if they are waves.

Sorry for terrible drawings.

Light from a flashlight.

Laser beam.

 

[Ibix]

Can we actually distinguish an electric field from a magnetic field in reality?

Rub a balloon on your hair. Hold it next to a magnet. Are they attracted the way two magnets are? No? Then there's a difference between the electric field of the charged balloon and the magnetic field of a magnet.

 

[Dale]

Can we actually distinguish an electric field from a magnetic field in reality?

Yes, very much.

this is what I thought usual light and laser will look like if they are waves.

Again your eyes are already designed for looking at EM waves. You have seen what they look like many times already.

I recommend getting the math software and plotting some known solutions to get a better understanding of how they behave.

 

[Drakkith]

I'm sorry for this terrible drawing below, I'm trying to draw a laser beam because it seems to be a fitting example for something like "a single point in space over time or a line through space at a single point in time" as laser beam is a concentrated type of light.

Not sure if the drawing is actually correct. Just curious about what others think about it.

It's fine, but it's limited. It might help to find an illustration that shows the wavefronts of an EM wave instead of one that just shows the field vectors. The wavefront is an imaginary line/plane that connects parts of the wave that are at equal intensities. In other words, regions where the phase of the wave is equal. We often draw them to show the peaks and troughs of a wave, but you don't have to. If you draw four 'squiggly lines' equidistant from each other such that they form a square, you could connect each peak of the four to form a plane. Do this for each set of peaks and you have a series of wavefronts for a plane wave.

Can we actually distinguish an electric field from a magnetic field in reality? Without these red and blue colour-coding and if we include every electric and magnetic fields instead of only two such as the vertical waves and horizontal waves, this is what I thought usual light and laser will look like if they are waves.

I'm not quite sure what you're asking. Note that there are an infinite number of potential locations that our diagram could represent for every EM wave of any size, no matter how big or small. Also, remember that the two field components overlap each other. That is, even though the arrows are drawn perpendicular to each other in the diagram, any two such pairs represent the field strength and direction at the point on the axis where the two arrows touch. We could try to overlay a whole bunch of these wavy lines with arrows in our diagram, but it would just be a big mess and just isn't a good way of illustrating things. That's why other illustrations exist that show different things, like the diagrams that show wavefronts.

And to reiterate, even though I'm sure you already know this, remember that the lines and arrows don't actually exist. They're just marks on paper or a screen used to represent something. I could just as easily draw colored numbers for the two field vectors at each location instead of using a squiggly line and some arrows.

Edit: Here's an illustration of wavefronts for a plane wave. Note that the arrows and squiggly line are indicative of the phase at any location in the plane, not just one side of the plane. There are an infinite number of these squiggly lines that we could draw and overlap since there are an infinite number of points in each plane. We only draw a few because you wouldn't be able to see anything if we tried to draw an infinite number of them.

 

[Rev. Cheeseman]

Thank you so much for these responses. Much appreciated.

 

[jtbell]

A long time ago, I attempted to describe a procedure for mentally constructing a three-dimensional diagram of an electromagnetic plane wave, in a similar spirit to Drakkith's diagram and discussion:

https://www.physicsforums.com/threads/electromagnetic-waves.71621/#post-533190

In this example the wavefronts are planes perpendicular to the "paper" or x-axis. At all points on a wavefront, the $\vec E$ field has the same magnitude and direction at a given time; likewise for the $\vec B$ field.

 

[Rev. Cheeseman]

I wonder if there are actually instances or if there are possibilities for the electric fields and magnetic fields to be parallel (or non-perpendicular) to each other instead of being perpendicular.

What happens if the electric fields and magnetic fields are parallel (or non-perpendicular) to each other?

 

[Drakkith]

I wonder if there are actually instances or if there are possibilities for the electric fields and magnetic fields to be parallel (or non-perpendicular) to each other instead of being perpendicular.

I don't think they can be anything but perpendicular to each other.

 

[Dale]

I wonder if there are actually instances or if there are possibilities for the electric fields and magnetic fields to be parallel (or non-perpendicular) to each other instead of being perpendicular.

What happens if the electric fields and magnetic fields are parallel (or non-perpendicular) to each other?

Such fields are called "near fields". The near fields are called "near" because they do not propagate very far from the antenna. Away from the antenna the fields are called "far fields" and are orthogonal. That is what radiates far away.

 

[robphy]

This might help.

https://glowscript.org/#/user/Rob_Salgado/folder/My_Programs/program/robphy-maxwell-EMwave-2022

It's an animation of an electromagnetic plane wave in vacuum (no source charges or currents),
together with an animation of the Maxwell Equations
...the Gauss Laws for E and for B (using the Gaussian surface shown),
and the Faraday Law and the Ampere-Maxwell Law ( the "mechanism" for telling B and E (at each point) how to change with time, depending on the curls of E and of B ).

The result of what B and E should do is
to propagate the configuration of the entire electromagnetic field
by a little displacement in the direction of $\vec E \times \vec B$. Repeat.Tap the 'w' key to see the wavefront.

 

[Rev. Cheeseman]

Any thoughts on this https://www.researchgate.net/figure...asurements-l780-nm-of-parallel_fig1_51591625?

 

[robphy]

Can we actually distinguish an electric field from a magnetic field in reality?

Note, in a plane electromagnetic wave, the wave travels parallel to $\vec E\times \vec B$.
(Since $\vec B\times \vec E \neq \vec E\times \vec B$ in general, we have that $\vec E$ and $\vec B$ are distinguishable.)

Note:
A proton would feel an electric force parallel to the electric field.
A proton would feel a magnetic force perpendicular to the magnetic field (and perpendicular to its velocity vector).
$\vec F_{Lorentz}=q\vec E + q\vec v\times \vec B$

(This is associated with the fact that electric field is an ordinary [polar] vector
and the magnetic field is an axial vector [so, a pseudovector] .)

 

[Rev. Cheeseman]

Is this an example of a parallel electromagnetic wave? https://qph.cf2.quoracdn.net/main-qimg-d5afa8922da2e560dda3342b9c2cbda3-pjlq

 

[renormalize]

Is this an example of a parallel electromagnetic wave? https://qph.cf2.quoracdn.net/main-qimg-d5afa8922da2e560dda3342b9c2cbda3-pjlq

The electric and magnetic fields are parallel in that image so it can't be any type of far-field electromagnetic wave that I'm aware of. And the increasing-pitch helical motion of the charge suggests that those fields are steady, not oscillating like a wave.

 

[vanhees71]

No, this is the motion of a charged particle in an electromagnetic field. Of course one cannot say much from the picture alone. You need to give more context, to understand, what's shown there.

 

[Rev. Cheeseman]

The electric and magnetic fields are parallel in that image so it can't be any type of far-field electromagnetic wave that I'm aware of. And the increasing-pitch helical motion of the charge suggests that those fields are steady, not oscillating like a wave.

No, this is the motion of a charged particle in an electromagnetic field. Of course one cannot say much from the picture alone. You need to give more context, to understand, what's shown there.

So, we are still not sure? Sorry, I'm confused. The only context I can find is here https://www.embibe.com/exams/motion-in-combined-electric-and-magnetic-fields/ and I'm not sure if this is the right context.

 

[vanhees71]

That's the motion of charged particles in an em. field. It's not about electromagnetic waves, i.e., time-dependent solutions for the fields themselves.

 

[Rev. Cheeseman]

That's the motion of charged particles in an em. field. It's not about electromagnetic waves, i.e., time-dependent solutions for the fields themselves.

How about this https://cdn.comsol.com/wordpress/sites/1/2022/07/TE11-mode-10-GHz.png the electric fields and magnetic fields are both perpendicular and parallel there.

 

[Dale]

How about this https://cdn.comsol.com/wordpress/sites/1/2022/07/TE11-mode-10-GHz.png the electric fields and magnetic fields are both perpendicular and parallel there.

That looks like the field near some source in the center. Hence it is a "near field" as I described above.

So, we are still not sure?

I am sure about what I have stated. I cannot speak for you or others.

 

[vanhees71]

How about this https://cdn.comsol.com/wordpress/sites/1/2022/07/TE11-mode-10-GHz.png the electric fields and magnetic fields are both perpendicular and parallel there.

If you don't give a clear context, I can't say anything. Pictures don't speak for themselves!

 

[Rev. Cheeseman]

That looks like the field near some source in the center. Hence it is a "near field" as I described above.

Now I got it. First I was a bit confused about what is near field. Thank you.

I am sure about what I have stated. I cannot speak for you or others.

If you don't give a clear context, I can't say anything. Pictures don't speak for themselves!

Nevermind, even myself are not sure what the context is.

 

[Rev. Cheeseman]

@Dale @vanhees71 regarding unpolarized electromagnetic field, are the electric and magnetic fields still can be distinguished from each other? Are there pictures that show the orientations or angles of electric fields and magnetic fields if the electromagnetic field is unpolarized? Thank you.

 

[Dale]

are the electric and magnetic fields still can be distinguished from each other?

Yes. They exert different forces on charges.

Are there pictures that show the orientations or angles of electric fields and magnetic fields if the electromagnetic field is unpolarized?

I don’t have any.

 

[Drakkith]

@Dale @vanhees71 regarding unpolarized electromagnetic field, are the electric and magnetic fields still can be distinguished from each other? Are there pictures that show the orientations or angles of electric fields and magnetic fields if the electromagnetic field is unpolarized? Thank you.

Did you mean an unpolarized EM wave instead of field? If so, here's a few depictions of varying accuracy:

Note that these all show the electric field vectors along a single axis. The magnetic field vectors would be perpendicular to each e-field vector.

 

[Rev. Cheeseman]

Did you mean an unpolarized EM wave instead of field? If so, here's a few depictions of varying accuracy:
View attachment 325414
View attachment 325413
View attachment 325415

Note that these all show the electric field vectors along a single axis. The magnetic field vectors would be perpendicular to each e-field vector.

Because the electric fields are distributed randomly in unpolarized electromagnetic fields, between every magnetic fields will be electric fields too. I hope I can see more pictures of unpolarized EM fields.

 

[Ibix]

Because the electric fields are distributed randomly in unpolarized electromagnetic fields, between every magnetic fields will be electric fields too. I hope I can see more pictures of unpolarized EM fields.

Are you thinking that these waves actually have spatial extent? If so, that's wrong. The arrows illustrate the magnitude and direction of the electric and magnetic fields at some point (usually the base of the arrow) but say nothing about the electric or magnetic field anywhere except that single point.