Chapter 32: Electromagnetic Waves

Welcome to Last Minute Lecture.

This free chapter overview is designed to help students review and understand key concepts.

These summaries supplement not replaced the original textbook and may not be redistributed or resold.

For complete coverage, always consult the official text.

Welcome back everyone to the Deep Dive.

Today we're going to be diving deep into electromagnetic waves.

It's going to be electrifying.

Yeah, exactly.

We've got this really cool chapter that we've been going over and you know it's kind of amazing when you think about it that light from the sun and the signals that our phones use and the the warmth that we feel from like a microwave oven are all basically the same thing.

Yeah, it really is mind -blowing that they're all unified by this one single theory.

It is.

It's pretty incredible.

Our mission I guess in this Deep Dive is to really unpack what this chapter is talking about with electromagnetic waves and hopefully make it really clear and understandable for everyone.

Yeah, I'm excited to dive in.

So the chapter starts off by talking about Maxwell's equations.

Right.

Which can sound a little intimidating.

Yeah.

They're really the foundation for understanding how this all works.

So what is it about these equations that makes them so important?

Well, I think the main thing about Maxwell's equations is that they really showed how electricity and magnetism, they're not two separate things.

They're actually two sides of the same coin, electromagnetism, and they showed how a changing magnetic field can create an electric field and a changing electric field can create a magnetic field.

So they kind of they work together and that's how electromagnetic waves can exist.

So it's like a constant interplay between those two fields that keeps the wave going.

Yeah, exactly.

They sort of feed off of each other and propagate through space.

Interesting.

So how did Faraday's law and Ampere's law play into all of this?

Well, Faraday's law, that was already known, that showed how a changing magnetic field could make an electric field.

Right.

But Maxwell, he was the one who added something called displacement current to Ampere's law.

Okay.

And that basically said that a changing electric field, it acts kind of like a current and it can also create a magnetic field and that was like the final puzzle piece that brought everything together.

I think one of the most mind -blowing things about this whole thing is that Maxwell's equations led to this prediction about how fast these waves should travel.

Yeah, that's right.

And what was that prediction?

Well, he basically took all of these equations and some fundamental constants about electricity and magnetism and he was able to calculate how fast these waves should move through space.

Okay.

And it turned out that that speed that he calculated, it was exactly the same as the speed of light, which was a huge deal because it suggested that light is just a type of electromagnetic wave.

So it connected something that people had been wondering about forever light with this new understanding of electromagnetism.

Yeah, exactly.

That's pretty incredible.

It was a huge breakthrough.

And I think that really shows the power of these equations, right?

Yeah, definitely.

So the chapter then goes on to list out Maxwell's equations in a vacuum.

We don't need to get too deep into the math here, but what do these equations really tell us about electromagnetic waves?

Well, in a nutshell, they lay down all the basic rules for how these fields behave.

Okay.

And they show us how these fields can actually propagate as waves.

So you have Gauss's law for electricity and magnetism, Faraday's law, and Ampere's law with Maxwell's addition.

And together, they basically give us the full picture of how electromagnetic waves work.

So it's really a complete set of instructions for how these fields and waves operate.

Pretty much.

Okay.

And the chapter also talks about the fact that you need accelerating charges to actually create these waves.

Why is it that a static charge or a charge that's moving at a constant speed doesn't create these waves?

Well, think about it like this.

For a wave to kind of radiate outwards, you need the electric and magnetic fields to be constantly changing because they feed off each other.

Got it.

A stationary charge just has this static electric field around it that doesn't really do anything.

It just sits there.

Exactly.

And a charge moving at a constant speed.

It creates a constant magnetic field and constant electric field in its own frame of reference.

There's no change, so there's no wave.

So nothing to really kick off that back and forth process.

Exactly.

It's only when you accelerate a charge that you get these changing fields that can actually break away from the charge and create a self -sustaining wave.

Got it.

And I think the example the chapter uses is a radio antenna.

Yes.

With the charges moving back and forth, creating those radio waves.

Exactly.

It's a perfect example.

Okay.

So we've got these electromagnetic waves being generated.

They travel at the speed of light.

Right.

The chapter then goes on to talk about this simplified model called a plane electromagnetic wave.

Okay.

So why is this model helpful?

Well, it's a really useful tool because it lets us look at the basic features of these waves without having to deal with complicated shapes.

So imagine a wave where the electric and magnetic fields are basically the same across any plane that's perpendicular to the direction the wave's moving.

Okay.

And that makes it much simpler to understand how the electric and magnetic fields relate to each other.

So it's basically a way to make the math and the concepts a little bit easier to handle.

Exactly.

And one of the things this model shows us is that both the electric field E and the magnetic field B, they're both perpendicular to the direction the wave is traveling.

That's right.

Which means it's what we call a transverse wave.

Yeah.

So what does that actually look like if we could see these fields?

So it's kind of like if you shake a rope up and down.

Yeah.

You get a wave traveling along the rope.

Right.

But the motion of your hand is perpendicular to the direction the wave's traveling.

Okay.

And it's the same with electromagnetic waves.

The electric and magnetic fields, they oscillate at right angles to the direction the wave's moving.

So they're moving side to side as the wave moves forward.

Exactly.

And they're also perpendicular to each other.

Okay.

Okay.

The chapter then goes on to apply Faraday's and Ampere's laws to this plane wave model.

Yes.

And that gives us this important relationship, which is E equals CB.

Right.

So what's the significance of that equation?

Well, it shows this really beautiful connection between the electric and magnetic parts of the wave.

Okay.

It tells us that the strength of the electric field is directly proportional to the strength of the magnetic field.

And the constant of proportionality there is the speed of light in a vacuum.

Interesting.

So they're basically tied together.

Yeah.

They're inseparable.

You can't have one without the other.

And they always have that specific ratio between their strengths.

Exactly.

Okay.

And then we can go even further and manipulate these equations and bring in those fundamental constants, epsilon zero and mu zero.

Yes.

And that leads us to this famous formula for the speed of light, which is C equals one over the square root of epsilon zero times mu zero.

That's right.

And I think it's pretty amazing how that pops out of these fundamental electrical and magnetic properties.

It really is.

It's like the universe is built on these fundamental constants.

Yeah.

And the chapter mentions that the value of CM is now actually defined.

Yeah.

So why did they switch from measuring it to just defining it?

Yeah.

So that's actually really interesting.

Yeah.

It's now fixed at this very precise number, which is like 299 ,792 ,458 meters per second.

Wow.

And the reason they did that is because now

the meter itself is defined in terms of the speed of light and the second.

So instead of trying to measure C super accurately all the time, they just decided to make it a fixed value and then use that to define the meter.

So it's kind of like they flip the script a bit.

Exactly.

It's all about having a really solid and accurate system of units.

So it's kind of a neat little detail about how we define these fundamental quantities.

Yeah, it is.

Okay.

So we've got the speed figured out, but the direction the wave travels is also important.

And the chapter says that's given by the cross product of E and B.

Yeah, the cross products.

So what's that actually telling us physically?

So that's where you have to remember the right -hand rule.

Right.

So you point your fingers in the direction of the electric field, E,

curl them towards the magnetic field.

And your thumb will point in the direction the wave's going.

That's a good way to visualize it.

Yeah, it is.

And to make the wave picture even more solid, we can actually derive the electromagnetic wave equation from Maxwell's equations.

Yes, that's right.

So what does that equation show us?

Well, it basically describes how these fields spread out as waves through space or through a material.

And because we can get this equation directly from Maxwell's laws, it just confirms that these electromagnetic disturbances really are waves and they move at the speed of light.

So it's like the final piece of evidence that ties everything together.

Pretty much.

Okay.

So to sum up what we've talked about so far,

electromagnetic waves are transverse.

Right.

The electric and magnetic fields are related by E equals C B.

Yes.

They travel at the speed of light in a vacuum and they don't need a medium to travel through.

That's right.

Which is how light from the sun can reach us through space.

Exactly.

They can move through empty space, which is pretty amazing.

It is.

So the chapter thing goes on to talk about a specific kind of electromagnetic wave, which is a sinusoidal wave.

Right.

So what's special about those?

Well, sinusoidal waves are really important because any wave, no matter how complicated, you can break it down into a sum of these simple sine waves.

And that's something called Fourier analysis.

So if we understand how sine waves work, then we can understand pretty much any kind of wave.

Got it.

So they're kind of like the building blocks of more complex waves.

Exactly.

And they also have these really well -defined properties, like frequency and wavelength, which makes them easier to work with.

And the chapter gives us these equations for a plane wave moving along the x -axis.

Yeah.

So we have E of x and t equals E max cosine kx minus omega t.

Right.

And then Bs of x and t equals B max cosine kx minus omega t.

Right.

And we also have that familiar relationship E max equals C B max.

So what are k and omegas in those equations?

So k is the wave number.

Okay.

And that tells you basically how bunched up the oscillations are, like how many waves you have in a certain distance.

And it's related to the wavelength.

It's 2 pi divided by the wavelength.

Ah, man.

And omega is the angular frequency, which is how fast the oscillations are happening in time.

Okay.

And it's related to the regular frequency, f by 2 pi.

And then that whole part in the cosine, the kx minus omega t, that's called the phase of the waves.

Yeah.

And it tells us where the wave is in its cycle at a certain point in time.

Exactly.

And there's also this important relationship between speed, wavelength, and frequency.

Right.

Which is C equals lambda f.

Yeah.

And that applies to all electromagnetic waves in a vacuum.

Yes.

That's a fundamental relationship.

So no matter what kind of wave you have, as long as it's in a vacuum, that equation holds true.

Exactly.

Okay.

And the chapter also mentions that the oscillations of the electric and magnetic fields are in phase in these sinusoidal waves.

Yeah.

So what does that mean visually?

What means that they reach their peaks and their valleys at the same time and at the same place.

So they're perfectly synced up.

Yeah, exactly.

They're moving together.

And then we get to polarization.

Okay.

Which is basically the direction that the electric field is oscillating.

Right.

So in linear polarization,

the electric field is always oscillating in the same direction.

Yeah, like up and down or side to side.

So it's like if you were shaking that rope up and down, the electric field would be going up and down.

Okay.

So what happens when these waves leave the vacuum and enter a material?

Well, that's when things get a little bit more complicated, because now the speed of the wave changes.

Right.

It's no longer C, it's some other speed, V.

Okay.

And that's because the electric and magnetic fields, they start interacting with the atoms and molecules in the material.

And those interactions, they kind of slow the wave down.

So the material is basically resisting the wave's progress in a way.

Yeah, kind of.

It's like it's having to push its way through the material.

Okay.

And this new speed V, it depends on the material's properties.

Right, yeah.

Permittivity and permeability.

Exactly.

And the chapter gives us this formula for V.

Yeah.

Which is V equals one over the square root of epsilon times mu.

Yeah.

And those are often expressed relative to their values in a vacuum.

Right.

Using what we call the dielectric constant, K, and the relative permeability, Km.

Okay.

You get V equals C over the square root of K times Km.

So those constants basically tell us how the material responds to electric and magnetic fields.

Yeah, pretty much.

And then from there, we get to this concept of the index of refraction.

Right, the index of refraction.

Which is N.

Yes.

So how's that related to this change in speed?

Well, it's basically a way to measure how much the material slows down the light.

Okay.

So it's the speed of light in vacuum divided by the speed of light in the material.

So N equals C over V.

Exactly.

And we can also express it in terms of those K and Km constants.

Yeah.

So N equals the square root of K times Km.

And for a lot of materials, Km is about one.

So we can just say N is approximately the square root of K.

Yeah, that's a good approximation in most cases.

So the index of refraction is a really useful way to compare how different materials affect the speed of light.

Yeah, it is.

And it helps us understand things like refraction, where light bends when it goes from one material to another.

Right.

So moving on, we know that light carries energy.

Yeah.

We can feel that from the sun.

Definitely.

But how do we actually quantify that energy in an electromagnetic wave?

So the energy is actually stored in the electric and magnetic fields themselves.

And there's this quantity called energy density, which is the amount of energy per unit volume.

And the formula for that in a vacuum is U equals one half epsilon zero E square plus one half zero B squared.

Okay.

But we can use that relationship E equals CB to simplify that.

Yeah.

And it turns out that the energy density is actually just epsilon zero E squared, which means the energy is equally distributed between the electric and magnetic fields.

So they both contribute equally to the wave's energy.

Okay.

The chapter then introduces something called the pointing vector.

The pointing vector.

Which is S.

Yes.

So what is that telling us about the energy?

So the pointing vector,

it basically tells us how the energy is flowing.

Okay.

Its magnitude is the power per unit area.

Right.

And its direction is the direction the energy is moving.

So it tells us both how much energy is flowing and where it's going.

Exactly.

It's a very useful tool for understanding energy flow.

Okay.

And then we have the intensity, which is the average magnitude of the pointing vector.

Right.

And the chapter gives us a few different formulas for intensity.

Yeah.

For a sinusoidal wave in a vacuum.

So you have I equals S average.

Right.

Which is also equal to E max B max over two mu zero.

Okay.

Which we can also write as E max squared over two mu zero C.

Right.

Or one half epsilon zero C E max squared.

So all of those are equivalent ways of expressing the intensity.

Yeah, exactly.

They all give you the same answer.

And that intensity basically corresponds to how bright the wave is.

Yeah, that's a good way to think about it.

All right.

No, this one might be a little less intuitive.

Okay.

But electromagnetic waves also carry momentum.

That's right.

It might seem strange because they don't have mass.

Right.

But they do have momentum.

So how is that momentum connected to the energy?

Well, that comes from Einstein's theory of relativity.

Okay.

Which says that energy and momentum, they're kind of two sides in the same coin.

So the momentum density, which is the momentum per unit volume.

Yeah.

Is given by dp over dv equals Eb over mu zero C squared.

Okay.

Which is also equal to S over C squared, where S is the magnitude of the pointing vector.

So basically, the more energy the wave has, the more momentum it carries.

Exactly.

They're directly proportional.

And this momentum can actually exert a force on a surface.

Yeah, that's called radiation pressure.

Okay.

And it depends on whether the surface absorbs or reflects the wave.

So how does that work?

So if a surface absorbs the wave completely,

then all of the wave's momentum gets transferred to the surface and the radiation pressure is I over C.

Got it.

But if the surface reflects the wave, then the momentum gets reversed.

Right.

So you get twice as much momentum transfer.

Okay.

And the pressure becomes two I over C.

Interesting.

So reflection basically doubles the pressure.

Exactly.

And even though this radiation pressure is usually pretty small,

it can be important in some situations.

Yeah, like in astrophysics, like how the tales of comets get pushed away from the sun.

Okay.

So the last part of the chapter talks about standing electromagnetic waves.

Standing waves.

So how do those form?

So imagine you have two waves with the same frequency and amplitude.

But they're traveling in opposite directions.

And those waves meet, they interfere with each other, and they create a standing wave.

And this usually happens when a wave gets reflected back on itself, like when it hits a conductor.

Got it.

And the chapter mentions this important condition for a perfect conductor, which is that the electric field parallel to the surface has to be zero.

That's right.

Why is that?

Well, in a perfect conductor, the charges can move around really easily.

Okay.

So if there was an electric field parallel to the surface, those charges would just move until they canceled out the field.

So the charges basically rearrange themselves to eliminate that field.

Exactly.

They always want to be in equilibrium.

Okay.

So the chapter gives us an example of a wave traveling in the negative x direction that gets reflected at x equals zero.

Right.

And we've got these incident and reflected electric fields.

Yeah.

So what happens to the electric and magnetic fields in the standing wave?

So the incidence electric field, which is E equals E max cosine kx plus omega t.

Okay.

When it gets reflected, it undergoes a phase change because of that boundary condition of the conductor.

So the reflected wave becomes E equals minus E max cosine kx minus omega t.

And when you add those two waves together, you get a standing wave for the electric field, which is E up x and t equals minus two E max sine kx sine omega t.

Got it.

And what about the magnetic field?

So the magnetic field is Bs of x and t equals two B max cosine kx cosine omega t.

So the spatial and temporal parts are separated now.

Yeah, exactly.

It's different from a traveling wave where they're all mixed together.

And the interesting thing here is that the points where the electric field is zero.

Yeah, the nodes.

Those are the points where the magnetic field is maximal.

Right.

And vice versa.

So the peaks of the two fields are kind of shifted in space.

Exactly.

They're out of sync spatially.

And they're also out of sync in time too.

Yeah, that's right.

At any given point, the electric and magnetic fields are 90 degrees at a phase.

So when one is at its peak, the other is at zero.

Exactly.

They take turns being strong.

And the chapter also points out that the average pointing vector in a standing wave is zero.

Yes.

So what does that mean?

So that means there's no net flow of energy in a standing wave.

The energy is still there.

It's oscillating back and forth between the electric and magnetic fields, but it's not actually moving anywhere.

So it's kind of trapped in the standing wave.

Yeah, that's a good way to think about it.

And the chapter wraps up by talking about how these standing waves can be set up in a cavity.

Right.

Like a box with conducting walls.

Yeah.

And that leads to these normal modes.

The normal modes.

So what are those?

So when you have a wave trapped in a cavity, it can only exist at certain wavelengths.

Okay.

Because the electric field has to be zero at the walls.

Right, because of that boundary cannot.

Exactly.

So only certain waves can fit in the cavity.

And those are the normal mode.

Yeah, exactly.

And each normal mode has a specific frequency, and the chapter gives us formulas for the wavelengths and frequencies for a cavity with two parallel walls.

Which is lambda n equals 2l over n.

Yeah.

And Fn equals nc over 2l.

Where n is an integer.

Yes.

And those are similar to the formulas we see for other kinds of waves, like sound waves in a pipe.

Yeah, it's the same idea.

The waves have to fit in the space.

Yeah.

So the frequencies are quantized.

Exactly.

Only certain frequencies are allowed.

And this is really important for a lot of technologies.

Yeah, like microwave ovens.

Right.

And lasers.

Hmm.

Okay, well I think that covers pretty much everything from the chapter.

Yeah, I think we hit all the major points.

So we started with Maxwell's equations, which unified electricity and magnetism.

Right.

And showed that light is an electromagnetic wave.

Yeah.

We talked about how these waves are generated and how they propagate.

And the different properties there like polarization and intensity.

Exactly.

And then we looked at how they behave in different materials.

Right, right.

And how they can form standing waves.

And how those standing waves can be used in technologies.

Yeah.

So it's really a pretty comprehensive look at electromagnetic waves.

Definitely.

We covered a lot of ground.

And I think it's pretty incredible how much we've learned about these waves since Maxwell's time.

It really is amazing how much progress we've made.

So as you go about your day,

think about all the different ways electromagnetic waves are at work.

Yeah, they're everywhere.

From the sunlight that warms us to the radio waves that let us communicate.

It's a pretty amazing universe we live in.

And with that, I think we can safely say we've covered everything in the chapter.

Yes, I think we're good.

Thanks for joining us on this deep dive into electromagnetic waves.

Yeah, it was fun.

We'll see you next time.