# The electromagnetic nature of light

Jeff Root
How was it discovered that electromagnetic radiation is electromagnetic?

-- Jeff, in Minneapolis

Mentor
How was it discovered that electromagnetic radiation is electromagnetic?
James Maxwell discovered the laws governing the interaction between electrical and magnetic fields in the middle of the 19th century. One of the consequences of these laws is electromagnetic waves propagating at the speed ##c##. Googling for “Maxwell laws radiation” will find many explanations, although you will need a bit of experience with partial differential equations and vector calculus to follow along in his footsteps.

Jeff Root
So it appears that the answer is that Maxwell calculated that
something "electromagnetic" (since it consisted of intertwined
electric and magnetic characteristics) should have a speed very
close to the previously-measured speed of light, and therefore
guessed that this predicted thing might actually BE light.

Did he also base that guess on any other known properties of
light besides its speed? Was there anything about light that
suggested to him that it might be electric and/or magnetic in
nature? Sure, electric sparks produce light, but so do flames,
which are not obviously electric or magnetic. Did he have any
reason to think that infrared or ultraviolet light might have
electric and/or magnetic characteristics before he worked out
the laws of electromagnetism?

-- Jeff, in Minneapolis

Jeff Root
I thought I set the "level" for this thread to "Intermediate" as
the first step in posting, but I see that it is now set to "Basic".

-- Jeff, in Minneapolis

2022 Award
Was there anything about light that suggested to him that it might be electric and/or magnetic in nature?
It's a "quacks like a duck" thing, isn't it? You have a theoretical prediction that there are waves of a certain speed and a physical phenomenon with about that speed. We will certainly consider the possibility that these facts are unrelated, but the idea that they are the same is a sensible starting point. In this case it turned out to be right as well. If it hadn't been (e.g. if Maxwell had been studying gravity and derived gravitational waves, which also move at ##c##), eliminating the possibility would probably have turned up evidence towards what was really going on.
I thought I set the "level" for this thread to "Intermediate" as
the first step in posting, but I see that it is now set to "Basic".
Mentors sometimes edit the level. I suspect they don't think there's anything in the basic facts of the history of electromagnetism that's likely to need undergraduate knowledge of physics.

WWGD, davenn and russ_watters
Mentor
I thought I set the "level" for this thread to "Intermediate" as
the first step in posting, but I see that it is now set to "Basic".
Mentors sometimes edit the level. I suspect they don't think there's anything in the basic facts of the history of electromagnetism that's likely to need undergraduate knowledge of physics..
That's what happened here. We'll often adjust the thread level after the fact so that it tells future visitors (who are the real audience for many threads - just compare the view counts to the post counts) what to expect when Google or other investigation brings them to the thread.

If the discussion had moved in the direction of deriving the speed of light from Maxwell's equations, then following it would require a fair amount of math (vector calculus, multi-variable differential equations) that one doesn't meet until the second year of a serious STEM program and the I-level tag would stay. Instead, the discussion has accepted that result as a given and is considering the implications for the history of science. That's going to be accesssible and interesting to a much broader audience; the B-level tag tells people that they don't need a few years of college-level math to approach and participate in the thread.

(Further discussion of thread levels probably belongs in a thread in the forum feedback section).

Last edited:
Ibix
So it appears that the answer is that Maxwell calculated that
something "electromagnetic" (since it consisted of intertwined
electric and magnetic characteristics) should have a speed very
close to the previously-measured speed of light, and therefore
guessed that this predicted thing might actually BE light.

Did he also base that guess on any other known properties of
light besides its speed? Was there anything about light that
suggested to him that it might be electric and/or magnetic in
nature? Sure, electric sparks produce light, but so do flames,
which are not obviously electric or magnetic. Did he have any
reason to think that infrared or ultraviolet light might have
electric and/or magnetic characteristics before he worked out
the laws of electromagnetism?

-- Jeff, in Minneapolis
I think Maxwell based it solely on the speed calculation. Read what he wrote in his paper “On physical lines of force”. Imagine how he must have felt when he had this flash of insight.

I like Feynman's quote on this topic,
"From a long view of the history of the world—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electromagnetism. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade."

sophiecentaur and anorlunda
Gold Member
Prior to Maxwell, Michael Faraday found that light is affected by a powerful magnet, and he says that he suspects that light is magnetic. I believe he also suggested an electromagnetic nature for light. So Clerk Maxwell had the idea in his head already.

Jeff Root

> Did he also base that guess on any other known properties of
> light besides its speed?

A possibility just occurred to me: Maxwell must have known very
well that light behaves like waves. If he calculated that the
predicted electromagnetic radiation has the form of waves, not
just energy, then that would have been a very big clue-- a much
better clue than the radiation's speed. The speed would then
be strong confirmation instead of a tenuous connection. If he
calculated the speed first and guessed that the radiation is
light before calculating that it would have the form of waves,
then the latter discovery must have been an absolutely mind-
blowing confirmation of his guess.

Which did Maxwell derive first: the waveform or the speed?
At what point did he guess that the radiation is light?

-- Jeff, in Minneapolis

Gold Member
2022 Award
It's also historically interesting that the speed of light could in this way derived from only electro- and magnetostatic measurements in the then famous experiment by Weber and Kohlrausch of 1855:

https://doi.org/10.1119/1.1934570
That's how Maxwell came to the conclusion that light is in fact an electromagnetic wave, because the value of the "electric-to-magnetic-charge ratio" as measured by Kohlrausch was in good agreement with the then accepted value for the speed of light.

Last edited:
anorlunda
Mentor
Which did Maxwell derive first: the waveform or the speed?
They’re inseparable. You write down the differential equation and solve it; the solution is waves traveling at speed ##c##.

Jeff Root
Nugatory replied:

> They’re inseparable. You write down the differential equation
> and solve it; the solution is waves traveling at speed c.

Inseparable but probably not simultaneous. Could Maxwell tell
from the form of the equation alone that the solution is a wave?
Putting in constant and variable values would then give a speed.

He might have guessed that he was working out an equation to
describe light even before he saw that it would result in a wave
or knew what speed it gave. He might have guessed after he saw
that it would result in a wave but before calculating the speed.
He might have guessed after noticing that the speed was the speed
of light but before he noticed that it was a wave. Or he might
not have guessed until he saw both, in which case I wonder which
he saw first, and how much time elapsed between the three events.

Did he say?

-- Jeff, in Minneapolis

Staff Emeritus
He might have guessed that he was working out an equation to
describe light even before he saw that it would result in a wave
or knew what speed it gave.
Are you familiar with solutions of differential equations? Guessing the form of the solution from the structure of the equations is almost universal.

See this article and it's general description of the wave equation strictly in terms of the form of the terms.

https://en.wikipedia.org/wiki/Wave_equation

The wave equation is a partial differential equation that may constrain some scalar function u = u (x1, x2, …, xn; t) of a time variable t and one or more spatial variables x1, x2, … xn. The quantity u may be, for example, the pressure in a liquid or gas, or the displacement, along some specific direction, of the particles of a vibrating solid away from their resting positions. The equation is

where c is a fixed non-negative real coefficient.

Using the notations of Newtonian mechanics and vector calculus, the wave equation can be written more compactly as

Mentor
Inseparable but probably not simultaneous. Could Maxwell tell
from the form of the equation alone that the solution is a wave?
Yes. It’s just the well-known wave equation ##\frac{\partial^2E}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 E}{\partial t^2}## for a wave propagating in the ##x## direction with speed ##c## (although I have simplified it to avoid using the notation of vector calculus).

The solutions are sums of functions of the form ##\sin(\omega t - kx)## where ##k=\omega/c## which are immediately recognizable as waves with frequency ##\omega##, wavelength ##1/k##, and speed ##c##.

Last edited:
Jeff Root
So Maxwell probably knew that his equation describes a wave
before he used it to find the value of c. In which case it seems
likely that he suspected that electromagnetic radiation is light
before he knew that the speeds match.

-- Jeff, in Minneapolis

Mentor
So Maxwell probably knew that his equation describes a wave
before he used it to find the value of c.
No, the two are inseparable. Look at the solution again: ##\sin(\omega t - kx)##.

The part outside the parentheses says “wave”, the part inside says “with speed ##c##”.

(And even that understates the inseparability. Put something else inside the parentheses and the outside might not say “wave”)

Last edited:
Jeff Root
So the moment he saw that the equation predicts a wave,
he also saw that it predicts a speed of 300,000 km/s.

Not hours apart, not minutes apart, not even seconds apart.
Simultaneously.

Can you explain how that works? It makes no sense to me.

-- Jeff, in Minneapolis

Last edited:
Homework Helper
Gold Member
So the moment he saw that the equation predicts a wave,
he also saw that it predicts a speed of 300,000 km/s.

Not hours apart, not minutes apart, not even seconds apart.
Simultaneously.

Can you explain how that works? It makes no sense to me.

-- Jeff, in Minneapolis
Well maybe a few seconds or minutes apart. Maxwell first saw that his 4 equations combined lead to the wave equation (which is known to mathematicians since Leonard Euler discover it at 1750s) and then its a matter of doing some simple secondary calculation to determine the speed of the wave.

Staff Emeritus
It is possible that someone noticed that permitivity and permeability of free space (which can be measured independently with no reference to light) can be combined to make the speed of light, before Maxwell worked out his equations. Ideas and information circulate among people in many ways simultaneously. It is not always true that significant factors come in a logical order. Nor can we infer that discovery always happens in linear fashion; A leads to B which leads to C.

From https://en.wikipedia.org/wiki/Maxwell's_equations

Staff Emeritus
It is possible that someone noticed that permitivity and permeability of free space (which can be measured independently with no reference to light) can be combined to make the speed of light, before Maxwell worked out his equations

That someone was Wilhelm Weber in 1846.

Nugatory, davenn and anorlunda
Jeff Root
Maxwell's equation appears to predict radiation of
electromagnetic energy in the form of sine waves.
It seems natural that the waves would be sine waves,
since harmonic oscillations are common in nature, and
it is easily understood how such waves arise in many
different ways.

But I want to be sure: Maxwell's equation explicitly
predicts sine waves? And light has been observed to
have the form of sine waves, not some other kind of
waves or pulses? Not square waves or sawtooth waves
or whatever else might be imagined as an alternative?

So that when electromagnetic radiation is depicted in
a diagram as a sine wave, the sine wave shape has not
been chosen just because it looks pretty or is easy
to draw, but because it accurately represents some

Not that such a diagram is accurate in every way-- it
is only a diagram, a graph of the electric and magnetic
field strengths. But the sine wave shape correctly
graphs how the fields vary in strength as the radiation
moves through space?

-- Jeff, in Minneapolis

Staff Emeritus
But I want to be sure: Maxwell's equation explicitly
predicts sine waves? And light has been observed to
have the form of sine waves, not some other kind of
waves or pulses?
Yes. Yes.

Haven't you beaten this horse to death yet.

There is no wiggle room. Sin waves.

How do we know? From the energy of photons. A square wave of some color would have different energy than a sin wave of the same color. We can measure the energy and see that the results agree with sin waves.

davenn
Mentor
But I want to be sure: Maxwell's equation explicitly
predicts sine waves?
Yes. The wave equation is ##\frac{\partial^2E}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 E}{\partial t^2}## and it is trivial to verify that the solutions are sinusoidal waves. (But note that "trivial" does not mean that you shouldn't try it for yourself! I suggested one form of the sinusoidal solution in a previous posts).
And light has been observed to
have the form of sine waves, not some other kind of
waves or pulses? Not square waves or sawtooth waves
or whatever else might be imagined as an alternative?
The wave equation is linear, meaning that if ##A## and ##B## are both sinusoidal wave solutions with different frequencies and wavelengths, then their linear combinations ##\alpha A+\beta B## where ##\alpha## and ##\beta## are arbitrary constants will also be a solution (but don't take my word for it! Try it for yourself!). It also turns out that any waveform traveling at speed ##c## can be written as a sum of these sinusoids at various frequencies (google for "Fourier analysis") so when we've successfully predicted sinusoidal solutions we've predicted all the waveforms.

These properties of the wave equation were well known long before Maxwell, so much so that when Maxwell manipulated his equations into ##\frac{\partial^2E}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 E}{\partial t^2}## the reaction would not have been "What does that mean?" It was more "Hello, old friend, I know you, you are sinusoidal waves traveling at speed ##c## and all the linear combinations of these".

They are also so well understood that most of the derivations you'll find online and in textbooks stop as soon they've gotten to the wave equation, say something along the lines of "of course the solution is sinuoidal waves traveling at speed ##c##". Once you have that, everything after that is just commentary.
So that when electromagnetic radiation is depicted in a diagram as a sine wave, the sine wave shape has not
been chosen just because it looks pretty or is easy to draw, but because it accurately represents some features of the radiation...the sine wave shape correctly graphs how the fields vary in strength as the radiation moves through space?
As @anorlunda says above, monochromatic light has been shown to be sinusoidal and there's not a lot of room for doubt here. At lower frequencies, we can observe the waveforms with an oscilloscope and see the sine wave directly on the screen.

Last edited:
Jeff Root
Nugatory replied:

> As @anorlunda says above, monochromatic light has been shown
> to be sinusoidal and there's not a lot of room for doubt here.

I was hoping it would be that clear-cut. However...

> It also turns out that any waveform traveling at speed c can be
> written as a sum of these sinusoids at various frequencies (google
> for "Fourier analysis") so when we've successfully predicted
> sinusoidal solutions we've predicted all the waveforms.

That makes it sound less clear-cut. Are light waves sine waves, or
are you just saying that their waveforms can be mathematically
represented by combinations of sine waves?

> At lower frequencies, we can observe the waveforms with an
> oscilloscope and see the sine wave directly on the screen.

Lower frequencies would be radio. Aren't the waveforms of
an AM broadcast signal? If it is sinusoidal, isn't that purely because
it was designed and made to be sinusoidal by the physical size of
the resonating chamber or electrical circuit? As far as I know, radio
waves can only be detected in very large numbers. A single photon
of visible light has enough energy to be detected by a CCD, but
doesn't it take a huge number of radio photons to be detected by
even the most sensitive detector? If so, can we say for sure that
the sinusoidal waveform is a property of the radiation itself rather
than a property of how the radio photons are packed together?
Can the waveforms of natural radio sources be observed?

-- Jeff, in Minneapolis

weirdoguy
Gold Member

yes, but it is still EM radiation and obeys the same laws

If it is sinusoidal, isn't that purely because
it was designed and made to be sinusoidal by the physical size of
the resonating chamber or electrical circuit?

it is sinusoidal and no to the second part unless a square wave ... digital data stream ...
was specifically required

As far as I know, radio waves can only be detected in very large numbers

What do you mean by that ?

Can the waveforms of natural radio sources be observed?

Of course they can. Have you not heard of radio astronomy ?
There's countless sources of natural radio sources in space
and they are just like man-made ones ... widely varying frequencies and energy
levels

D

Jeff Root
davenn,

When I say that radio waves can only be detected in very
large numbers, I mean that unlike visible light, it is not
possible to detect a single quantum of radio energy-- a
single photon, or "a single wave". It requires the energy
of a large number of radio-wavelength photons working
together, in unison, to be detected by any kind of detector
I know of, anywhere in the radio/microwave part of the
spectrum. By "detected" I mean that one can say with a
degree of certainty that there was actually something
there, that it wasn't just noise in the detector. A single
photon of visible light can be detected by a CCD or
photomultiplier tube.

sources can be observed, I am not asking whether the
radio emissions can be detected and observed-- of course
they are all the time. I am asking whether the shape of
the individual waveforms can be observed. That seems
unlikely to me since natural radio sources are generally
a mixture of different wavelengths, polarizations, and
phases, unlike the carefully-formed waves of artificial
radio signals. I do not know how the sinusoidal shape
of the waveform of a random jumble of incoherent waves
could be observed on an oscilloscope.

-- Jeff, in Minneapolis

weirdoguy
Gold Member
sources can be observed, I am not asking whether the
radio emissions can be detected and observed-- of course
they are all the time. I am asking whether the shape of
the individual waveforms can be observed.

again … of course they can, just like any other RF source …. I really don't see what your problem is ??
An RF signal is an RF signal is an RF signal …. regardless of it's origin
There is nothing special about the natural ones from space or the natural ones from the Earth
or the ones man generates with an oscillator.

Haven't you beaten this horse to death yet.

...This horse died from flogging many posts ago,

D

Last edited:
Gold Member
@Jeff Root
The main reason your thread was lowered from an I to a B tag was probably because the mentors
could see you didn't have a grasp of the basics, let alone Maxwell's equations. You need to do
some serious personal study of EM basics before delving into the deeper stuff and trying to understand
that.

To be honest, I an not an expert in the maths side of EM and I wouldn't even try to explain
Maxwell's equations to anyone, but I have a long history of practical experience working with
RF equipment and test gear, when doing my amateur radio, commercial radio ( for work) and yes
even a bit of radio astronomy in the mix. A lot of fun receiving signals generated in Earth's upper
atmosphere, from the Sun, Jupiter and other sources in the Milky Way.

There is so much that you can learn just by dipping your "feet into the pond" and experiencing
some practical stuff. And doing that would have answered so many of your prior questions

Dave

PeroK and weirdoguy
2022 Award
@Jeff Root - my interpretation of your #27 is that you seem to be asking two different questions. One is, can you see the EM field of a single RF photon, and is it sinusoidal? The other is, if you have a broad-spectrum signal, can you see the sine waves making it up?

The first you'd have to ask of someone who understands quantum field theory. Maxwell's equations are classical and only describe classical waves. My guess would be that the question doesn't make sense - you'd have to be able to detect parts of a photon and the point of quanta is that you can't do that.

The second is a yes, to arbitrary precision given a narrow enough bandpass filter and a bright enough source. You simply filter out everything except an arbitrarily narrow frequency band and, the narrower the bandpass the nearer a pure sine wave you'll pass through.

Jeff Root
Dave,
@Jeff Root - my interpretation of your #27 is that
you seem to be asking two different questions. One is, can you see the
EM field of a single RF photon, and is it sinusoidal? The other is, if you
have a broad-spectrum signal, can you see the sine waves making it up?
Yes. Both questions were raised by Nugatory in post #24, where he said:

> At lower frequencies, we can observe the waveforms with an oscilloscope
> and see the sine wave directly on the screen.

This was surprising to me, so I thought I may have misunderstood what he
meant. He may have been referring to observation of the waveforms of very
large numbers of coherent radio waves, rather than of a "single radio wave",
how looking at many examples of a thing can lead to understanding what the
thing itself looks like in general. The more samples that are examined, the
clearer and more detailed the picture one can get of the thing. But in this
case it appeared that Nugatory may have saying that the shape of a crowd
can be observed rather than saying that the shape of humans can be deduced
from looking at many individuals, each of whom is indistinct.

Since it has been my understanding that radio-frequency electromagnetic
radiation is too weak to detect individual photons, and Nugatory explicitly
says that we can see the waveforms with an oscilloscope, I suspect that he
was referring to the shape of the crowd, not the shape of the individuals in
the crowd. If it is possible to infer the shape of the individuals from the
shape of the crowd, I would very much like to know how that can be done.

I have seen the sinusoidal waveforms of artificially-produced signals on
oscilloscopes. I do not know that I have ever seen what I would call a
"sinusoidal" waveform or anything close to sinusoidal in a natural signal.
The "waveform" of natural signals, in my limited experience, is random,
looking more like grass than like a sine wave.

the answer to my first question of whether we can observe the waveform
of individual RF photons, and are they sinusoidal.

The fundamental question I have is whether electromagnetic radiation in
general actually has sinusoidal waveform. I believe it does, but that belief
may be based on stories to children. There is certainly some truth to it.
I'm trying to find out how much, from people who know the adult story.

The first you'd have to ask of someone who understands quantum field
theory. Maxwell's equations are classical and only describe classical waves.
The role of theory here is complex and interesting, but I want to cut
through those complexities as much as possible and say that I am asking
about what is observed rather than what theory says. Maxwell's theory
was put forward at the start of the thread as the connection between light
and electromagnetism. Observations of electric and magnetic phenomenon
lead to a theory which connected them with each other and with light. One
of the predictions of the theory is that light has a sinusoidal waveform. I'm
trying to understand if that waveform is what Nugatory observed on an
oscilloscope, or if the sinusoidal shapes he saw were artificial, created as
an intentional result of the design of a circuit or RF source. Since it is my
belief that RF radiation can only be detected with very large numbers of RF
photons, it is not apparent to me how the waveform on the oscilloscope
screen can represent the waveforms of individual RF photons. I am hopeful
that you or Nugatory or someone here can explain to me how the shape of
the trace on an oscilloscope screen can accurately depict the waveform of
individual photons.

My guess would be that the question doesn't make sense - you'd have to be
able to detect parts of a photon and the point of quanta is that you can't do that.
I agree that "parts of a photon" cannot be detected. That is why I question
whether the shape of a trace on an oscilloscope screen can actually represent
the waveform of the individual photons which compose the radiation being
observed. It seems more plausible that it only depicts the waveform of the
crowd of photons, and that would not necessarily be the wave predicted by
Maxwell's equation. It might instead be an artificial waveform of arbitrary
shape. My expectation is that the sinusoidal shape of light waves can be
predicted and inferred from observations, but cannot itself be observed.
So what is Nugatory's oscilloscope actually depicting? My guess is that it
is the cumulative interference of huge numbers of individual RF photons.
The people who designed the experiment know for sure. I expect that
Nugatory knows for sure.

What makes that really interesting, of course, is that interference has been
shown to occur even when only one photon reaches the detector at a time.

The second is a yes, to arbitrary precision given a narrow enough bandpass
filter and a bright enough source. You simply filter out everything except an
arbitrarily narrow frequency band and, the narrower the bandpass the nearer
a pure sine wave you'll pass through.
Can you explain how that is possible? With an incoherent light source, even
if it is filtered to allow only a single wavelength and polarization, the waves
would be out of phase. I'd expect the waveform to be completely random.
Tall grass.

-- Jeff, in Minneapolis

Last edited:
weirdoguy
Mentor
This was surprising to me, so I thought I may have misunderstood what he
meant. He may have been referring to observation of the waveforms of very
large numbers of coherent radio waves, rather than of a "single radio wave",
No, I was referring to the single waveform that is all we're ever working with, whether it's electromagnetic radiation or any of the many other phenomena desribed by the wave equation. I'm not sure where you're getting the idea that the signal contains a large number of waves, whether coherent or not, or that a single quantum is a single wave.
Indeed, any attempt to introduce quantization into a discussion to Maxwell's equations, the speed of light, and the wave nature of electromagnetic radiation suggests a misunderstanding of the relationship between photons and electromagnetic radiation. Single-photon states of the electromagnetic field are not electromagnetic waves (don't be misled by the fact that photons have a frequency - that doesn't mean what it sounds like) and electromagnetic waves cannot be analyzed as a bunch of photons the way a beach can be analyzed as a bunch of grains of sand.
There's a lot to be said for trying to forget that you ever heard the word "photon" until after you have a solid grasp of Maxwell's classical electrodnamics.
(But if you can't bring yourself to do that and also don't want t to take on a graduate-level quantum electrodynamics textbook you might take a look at http://www.physics.usu.edu/torre/3700_Spring_2015/What_is_a_photon.pdf).
Can you explain how that is possible? ... I'd expect the waveform...
Your expectation is wrong. Many posts back I suggested that you google for "Fourier analysis"; this will explain how it doesn't work the way you're expecting.

Last edited:
davenn and PeroK
2022 Award
The fundamental question I have is whether electromagnetic radiation in
general actually has sinusoidal waveform. I believe it does, but that belief
may be based on stories to children. There is certainly some truth to it.
I'm trying to find out how much, from people who know the adult story.
You can write any wave as a sum of sine waves (decomposing an arbitrary wave into a sum of sine waves is what the Fourier transform does). Asking whether classical electromagnetic radiation is "actually" sine waves is a bit like asking whether three is "actually" two plus one. You can certainly represent it that way if you wish.

Regarding the bandpass filter, you would need an infinitely narrow one to pull a pure sine wave out of true white noise. However, real sources typically have some non-whiteness that means you could pull out something very close to a sine wave with a sufficiently narrow filter. Possibly an implausibly narrow one for a real source.

Finally, note Maxwell knew nothing of quantum theory. His equations are purely classical. The sine waves talked about here are not quanta, and you cannot understand quanta of light in terms of Maxwell's electromagnetic field.

Staff Emeritus
How was it discovered that electromagnetic radiation is electromagnetic?

-- Jeff, in Minneapolis

I'm going back to your VERY FIRST post here, because this question that I have has not been clearly answered or clarified based on your posts so far.

Question: Do you know that the EM wave function can be derived DIRECTLY by using Maxwell equations, or more specifically, by using two out of the 4 Maxwell equations?

Question: Do you know that once you derived the wave function, you IMMEDIATELY get that the speed of that wave is equal to 1/√(ε0μ0), where ε0 and μ0 are the permittivity and permeability of free space, respectively, and that is equal to the speed of light in vacuum?

If you are not aware of these, then did what I wrote above answered your question? If you are aware of these, then I do not understand what the issue is here because the derivation drops the answer to your question right onto your lap!

Zz.

davenn
Gold Member
2022 Award
You need all four Maxwell equations. I use SI units here, because they are in this case more to the point answering the historical question. Usually I prefer Heaviside-Lorentz units, but here the SI units serve our purpose better.

The microscopic Maxwell equations are
$$\label{1} \vec{\nabla} \cdot \vec{B}=0,$$
$$\label{2} \vec{\nabla} \times \vec{E}+\partial_t \vec{B}=0,$$
$$\label{3} \vec{\nabla} \times \vec{B} -\mu_0 \epsilon_0 \partial_t \vec{E}=\mu_0 \vec{J},$$
$$\label{4} \vec{\nabla} \cdot \vec{E}=\frac{1}{\epsilon_0} \rho.$$
Note the arbitrary constants ##\epsilon_0## and ##\mu_0## which come from the arbitrary choice of the units of charge in the SI being Coulombs or Ampere seconds.

Now to see that there are wave solutions we consider a region in space, where no charges and currents are present, i.e., we set the right-hand sides of the inhomogeneous equations ##(\ref{3})## and ##(\ref{4})## to 0. To get rid of ##\vec{B}## we take the time derivative of Eq. ##(\ref{3})## (with ##\vec{J}=0##) and use ##(\ref{2})##:
$$\label{5} \vec{\nabla} \times \partial_t \vec{B} -\mu_0 \epsilon_0 \partial_t^2 \vec{E}=-\vec{\nabla} \times (\vec{\nabla} \times \vec{E}) -\mu_0 \epsilon_0 \partial_t^2 \vec{E}=0.$$
For the "double curl" we have, using Eq. ##(\ref{4})## with ##\rho=0##
$$\label{6} \vec{\nabla} \times (\vec{\nabla} \times \vec{E})=\vec{\nabla}(\vec{\nabla} \cdot \vec{E})-\Delta \vec{E} = -\Delta \vec{E}.$$
Plugging this into ##(\ref{5})## we finally get the wave equation
$$\epsilon_0 \mu_0 \partial_t^2 \vec{E}-\Delta \vec{E}=0,$$
from which we get the phase velocity
$$c=\frac{1}{\sqrt{\epsilon_0 \mu_0}}.$$
Now historically (though using still different units, namely electrostatic units for elecrostatics and magnetostatic units for magnetostatics) Maxwell used the known values of ##\epsilon_0## and ##\mu_0##, determind from measurements, comparing the electrostatic and the magnetostatic measures of charge, to determine the speed of the waves, getting a value very close to the value known from meausurements of the speed of light.

You can also derive the same wave equation for ##\vec{B}## and then look for the plane-wave solutions, from which you can find all wave solutions via Fourier transformation. This more detailed analysis also leads to the polarization properties of the electric and magnetic field. The plane waves are all strictly transverse, and that leads to a description of all the polarization properties of light.

Finally after some time H. Hertz for the first time could make and detect electromagnetic waves (in the radio-frequency domain though, i.e., with wavelength in the meter scale) and demonstrate all the properties predicted by Maxwell.