Physics Rediscovered #22: Let there be light

The power of Maxwell's equations

by Luis Dalvan

The fact that you can see this text, gentle reader, is nothing short of a miracle.

I don’t mean the technology here, although it is incredibly impressive in its own right. No, I mean something different.

If you cover the screen, you no longer see what’s on it. Profound insight, right? But think about what it means. It seems that whatever the screen is displaying is not projected directly into your mind. Instead, something is happening in the space between the device your reading this on and your eyes.

Exactly what it is was a question that people asked since asking questions was a thing. It proved to be dauntingly elusive for most of humanity’s existence. Though we couldn’t understand it or name it properly, Nature always knew itself and evolved the eye to be the antenna for picking up whatever was happening around. Today we know that seeing is a side-effect of a fundamental force of the Universe in action.

But wait, this is Physics Rediscovered, not Physics State-The-Obvious-And-Move-On, and as often happens here, we are getting ahead of ourselves.

It’s time to laugh at some historical figures, watching as they struggled to explain what light is, while being terribly unequipped to do so. All this to make ourselves feel better about ourselves, as the rates of mathematical and scientific literacy across the world plummet.

Light be chunky

So let’s start with the Greeks. These old geezers thought that light is something that shoots out from our eyes, like with DCU’s Superman and MCU Cyclops (see, we cater to all audiences here). It then hits things, bounces back and that’s how we see said things.

A few obvious problems here, for example, why can’t we see well at night as during the day. Regardless, this type of thinking prevailed and continued to dominate until the fall of the Roman Empire and the whole messy aftermath of that called the Dark Ages. They were called like that because of the lack of light, obviously.

Once the scientific center shifted to the Middle East, Abu Ali ibn al-Haytham grew ever certain that it should be the other way around and flipped the script. It turned out that this was a solid idea and simplified thinking about optics a lot. In my opinion, a pretty cool example of how reversing a problem can help you solve it.

Anyway, time went on without significant breakthroughs, until René Descartes decided to take interest in physics. He took a look at sound waves and how they seem to bend when crossing from one medium to another. Experiments with light, prisms and crystals showed that light was behaving the same way. Therefore, Descartes thought, light must be a wave also.

That’s interesting, thought Newton. It’s also wrong, thought Newton even harder. This type of thinking seemed misguided to a man, for whom the world was all about mechanics. Newton believed everything could be explained by forces acting on pieces of matter and light was no exception. In this spirit, light must’ve consisted of particles, he thought, that interact with matter, maybe gravitationally, resulting in the various phenomenon we can observe. He had thought he was able to explain reflection, refraction (it’s when light bends as it crosses from one material to another) and even maybe polarization.

Yes, they knew about polarization back then. They realized that rotating certain crystals, as light passes through them, can make the light disappear and appear again. Newton struggled with that one. His best guess was that the particles of light were not necessarily spheres, but rather something with sides, like a rectangle maybe. Perhaps particles with sides could have a preferred directionality, so that’s why they somehow react to polarization crystals? Sound good?

Not to Christian Huygens it didn’t. None of it did. We’ve already met the Dutch astronomer (among other things) here, where he battled with Newton over the nature of gravity. There, I promised I would pay more attention to him in the future so now let’s give the man’s ideas some space.

Light is light, is light, is light…

Huygens was a man of many talents, one of which was mathematics. His interests in this domain frequently oscillated around things that rotated, revolved and vibrated. With a mind primed in such a way, thinking about waves as foundations of reality came to him effortlessly.

Especially since the discovery of water.

You see, water was a cheap and readily available source of inspiration when it came to deciphering the behaviour of waves. Throw a pebble in a pond and you get concentric circles expanding outward, reflecting from things and bending around them. Throw two pebbles and you get two sources of circles, that sometimes can get in each other’s way and create hypnotizing patterns.

Thinking about all this, Huygens came up with a principle that, by a shocking turn of events, got named the Huygen’s principle. Funny how things work out…

It said that all the time, each point on the front of incoming light is a new source of a semi-circular, longitudinal wave, spreading forward. Longitudinal means, that the waves wave in the plane, in which it moves. This is in contrast to a transverse wave, which constantly waves above and below its plane of motion.

This is an incredibly powerful approach to analyzing the behaviour of light. It perfectly explained the phenomenon of reflection. Take a look:

First, a front (blue) is formed from several sources of light (infinite in principle, but three here, coz you know…). Each point on the front is a new source of an expanding wave, which results in the front moving forward. This happens constantly, so when the front hits a reflective surface (white), each point there, again, becomes a source of light wave. When the left edge of the front hits the surface, it creates an expanding circle, which has some time to grow before the right edge hits. The resulting is a reflection of the front.

But wait! There’s more!

The same reasoning can be used when light refracts, like when you have something partially submerged in water and it looks all weird and crooked when viewed from below the surface. Refraction works like this:

Here, the beginning of the story is the same as before. However, a very important thing happens, when light crosses the white line that separates different materials. When the left edge of the front touches the line, a new wave expands below the separation, but much slower than it would above it. So does every other new source on the travelling front. The new front, that forms below, now travels at a different angle, just like we see in reality. Something really weird is happening. To Huygens, it seems like light is slowing down.

This is a biggie, because it demands that the speed of light cannot be infinite: something that wasn’t super obvious at the time. This idea didn’t come out of the blue, mind you. Giovanni Domenico Cassini and his assistant Ole Christensen Rømer, two big-time, OG astronomers, observed how Jupiter eclipses its major moons and realized that the time between the eclipses got shorter when Earth and Jupiter got closer to one another on their orbital voyages. Conversely, it got longer when the planets went their separate ways. They concluded that it must take light more time to reach their eyes, in the latter case. Thus, a finite speed of light.

As you can see, gentle reader, Huygens came up with a phenomenal idea. It even predicted that light could bend around corners, although that wasn’t observed at the time. However, he also struggled to explain polarization of light. On top of that, Newton proved to be the more influential of the two and, eventually, Isaac’s corpuscular ideas prevailed.

For a while.

Slitting the throat of light particles

Newton’s idea of a corpuscular light dominated the minds of scholars for over a century. However, not all of them were so easily swayed. Riding a wave of resistance was one Thomas Young - another one of those know-it-alls of the 19th century.

In 1801 he did the unspeakable and proved Newton insufficient. But before he did that, he had spent countless hours investigating the behaviour of water and sound in the context of wave mechanics. Witnessing how two sources of water waves collided with each other, creating even larger waves or completely calm seas, he grew ever suspicious of Newton’s ideas. This notion was further reinforced, hearing two sources of sound overlap, creating spaces of eerie silence or unsettling beats. Thus, at the turn of the century, aware of Newton’s shortcomings, he became convinced that light must behave the same way.

Seeing how water waves get all bendy with minimal effort suggests that doing the same thing to light should be equally simple, right?

Well no.

If you make a wave on water, say by throwing a rock in the pond, you can try and put stuff in the way of it, like islands. If the islands are far, far apart, you’ll likely not see much of an effect. However, if you push the islands closer together, like on the picture below, you start to see hypnotizing patterns.

The longer I stare at it, the more faces I see

The key here is that the islands don’t have to be super close together, because the waves on water are relatively large, not it terms of height, but in terms of the distance between each up and down.

This is a problem because if light is indeed a wave, then these distances must be microscopic. Why? Well, because so far no one was able to make an obstacle small enough to force light to behave like water.

Fortunately, the beginning of 19th century witnessed a leap in slit-making technology. We were making large slits, funny slits, sad slits, but most importantly, really small slits. Micron slits, to be specific. Once such slits became available, everything changed.

Young got two of those and decided to shine some light on them. Now, if light was supposed to be made of particles, he should’ve expected to see two blobs of a slit-shaped pattern on the screen placed after the slits. Instead, he saw this:

Three words: minds were blown. There are bright bars and dark bars. Just as sounds from two sources were loud in one place but silent in another. The only reason light should do something like this is because it was a wave and not particles. In your face, Newton!

But what’s doing the waving?

As we enter the 19th century, the technological progress and far reaching exploration of nature results in physics experiencing something like a golden age of discoveries. Folks over there are trying to figure out what heat is (might wanna start here) and folks over here are scratching their heads at the mysterious behaviour of compass needles.

Among other things, certain James Clerk Maxwell is well embedded in both worlds. Propelling humanity forward on brain-fueled rocket, he gave us the foundations of statistical mechanics and, by now, gave us a comprehensive description of all electromagnetic phenomena.

Ok, almost all. Arguably the greatest one is still ahead of him and that’s why we are here. Before we proceed it might be helpful, dear reader, for you to revisit this:

Otherwise, things might get somewhat confusing.

Done with that? Great! Now, charge!!!

Or maybe not just yet. We need to lay some ground first.

Maxwell and others, grew strongly convinced, especially after Young’s double-slit slap, that light is made of waves travelling through… well, something. This is problematic because waves of water are the particles of water bouncing around, waves of sound are the particles of air compressing and decompressing, so particles of what is light?

Just ordinary matter? Not really, because when you suck out all air, creating a vacuum, light still moves about and heat still heats.

There, everyone concluded, there must be something there, some invisible form of matter, different from ordinary one, that is everywhere all time and waves of that thing is what light is. Huygens called it the luminiferous ether and the name stuck.

At the same time, electricity and magnetism also seemed to move through vacuum just fine. Could it be that light and electromagnetism are related?

Unsurprisingly, Michael Faraday, again came in to save the day, as he usually did. If you ever get quizzed on electromagnetism and forget the answer, just say Faraday did it. Whatever it is, there’s a solid chance he did.

He had this experiment where he would shoot light, polarized at a know angle, through something like glass, for example. Nothing special so far, but Faraday would introduce a magnetic field to act on the glass. When he turned it on, the light coming out from the glass, had a different polarization angle. No one could properly explain why that was happening, but in Maxwell, it only reinforced the long harbored suspicion that electromagnetism might be the source of light.

Light hits like a brick wall

Having seen enough, in 1864 Maxwell sets off in his “A Dynamical Theory of the Electromagnetic Field“ with the following:

“It appears therefore that certain phenomena in electricity and magnetism lead to the same conclusion as those of optics, namely, that there is an aetherial medium pervading all bodies, and modified only in degree by their presence…“

He starts with showing the geometry of the electromagnetic wave. Maxwell’s approach is fairly mathematical and we don’t want to get bogged in that. Instead, let’s take one of his magnetic equations and play with the imperfect, but helpful intuition of closed surfaces again, like we did here. The equation is:

\(\frac{\mu}{4\pi} \left( \frac{d\alpha}{dx} + \frac{d\beta}{dy} +\frac{d\gamma}{dz} \right) = 0\)

It tells us that the amount of magnetic lines (represented by greek letters in each x, y and z direction) going in and out of an enclosed, completely arbitrary surface must be equal. It’s another way of saying that there is no such thing as a magnetic particle.

Now let’s imagine that a wave of light is coming at us, face on, like a brick wall. Actually, let the shape of that brick wall be our enclosed surface. Let’s make is simpler by making the wall coming at us in the x direction. So the wall is wide in y, tall in z and, say, one brick deep in x.

Now, as the wall is rushing at us, we want to make sure that, at any given time, the amount of magnetic lines coming in and out of that wall, in all directions, is the same. So, the lines going through the top and bottom of the wall (the z direction) seem to do that just fine. The same goes for lines going through the sides (y).

However, the lines in x have a problem.

This is the direction the wall is speeding towards us. We can imagine that lines trying to go out in this direction hit the backside of the wall sooner, as it is rushing to meet them. The lines that want to escape through the front face of the wall will have a hard time keeping up as the front is escaping them, eager to smash us in the face.

All this tortured analogy is supposed to explain, is that, at any given time, the amount of lines going in and out in x (the direction the wave is travelling) is unbalanced. But, the equation above requires them to be balanced, just like we said. What does that mean? It means, that the right hand side of the equation would not be equal to zero and that would mean the presence of magnetic particles. But there aren’t any!

It turns out that the direction of the magnetic waving must be perpendicular to the direction of the wave’s propagation. Always, when in vacuum.

The same argument can be repeated for electric waving, remembering that there can be no electric charges in a vacuum. Because it’s a vacuum.

Little more mathematics show that the electric and magnetic components of the wave must be perpendicular to each other, also. So if the wave of electromagnetism is indeed like a wall coming at us from x, then the electromagnetic components can only wave in the yz plane. Huh…

A need for speed

So this already is a spectacular result, shedding light on the nature of light.

But how are we so certain we should be talking about waves of light and not something else.

This is where Maxwell’s absolutely murders the problem, cuts it into small pieces and dumps it into the river.

He joins several of his equations together and does a lot of smart algebra on them. None of this would work, mind you, if it hadn’t been for his recent idea of the displacement current. It was this “fake” current that Maxwell invented in order to explain the behaviour of isolators. While they themselves do not conduct electricity, the do show signs of charge polarization. Maxwell imagined that there’s this additional current present in them, different from the regular current in wires, that displaces the charges in the isolators (more here), hence the name.

Since we in a vacuum, there are no isolators, however, this luminiferous ether still can produce these displacement currents. Today, we refer to this as how an electric field that’s changing over time right here can induce magnetic fields over there. Without this time dependent piece, everything falls apart.

Several mathematical transformation later Maxwell arrives at the following result:

\(V = \sqrt{\frac{k}{4\pi\mu}}\)

V is the speed of that brick wall heading towards us. μ measures how well the medium transports magnetism. k tells us what is the electric elasticity of the medium, through which the wave travels, or how susceptible it is to these displacement currents, or as we would call it today, to electrical induction.

So, the obvious question is how much is V?

Maxwell looks to experiments to figure it out. He only has access to reasonable values for the ones performed in air. By definition μ in air is 1 so that’s easy. As for k, there were several experiments, which gave him a sense for the value. Once he plugged it in, he got V to be something around 300 000 km/s.

That itself is a very interesting result, but without additional context it doesn’t mean much. Fortunately, it turned out that many people at the time tried go get a sense of how fast light really is. Here is a nice video summarizing the history of these efforts.

One of the cleanest experiments in this area, cleanest in the sense that it assumed nothing about electromagnetism, came from Léon Foucault. He used a rotating mirror to send light from a fixed source one way, there it would bounce back onto the mirror. However, during that time, the mirror would rotate a bit, sending the light back to the source but no exactly. Knowing that offset and how fast the reflecting mirror revolved allowed him to get a handle on the speed of light in air. The result he got was 298 000 km/s.

This was eerily close to what Maxwell got and he figured that it was no coincidence. It was the joining piece between Maxwell model of electromagnetism and the nature of light itself.

One of the oldest and most fundamental mystery of reality was solved.

With that you might think Maxwell was crowned king of the world. That didn’t happen, though, and just because the is no such institution. People were just like: yeah it’s fine. Fine.

I don’t know what the speed of discovery is but it took over 20 years for Maxwell to be validated. It came from the mind of a guy called Hertz, just like the unit of frequency.

But that is a story for the next episode of Physics Rediscovered.