The Origins Podcast with Lawrence Krauss - Physics for Everyone, Lecture 5: The Unification that Changed the World

Episode Date: July 7, 2026

I began this lecture by saying I was daunted by what I was about to talk about. There are several reasons for this. First, I will lead you through the development of what was certainly most important theoretical development in the 19th century, and what could easily be considered the most important mathematical framework in all of physics. Second, this development affected our modern picture of the universe more than almost any other. Next, this development made our modern technological society possible. And finally, it allowed a calculation that ultimately unveiled the true nature of light. A calculation we will perform together during this lectureI am going to take you through what is often taught in a whole semester course in physics. So hold onto your hats. Take it step by step. I think the reward of understanding will be worth the effort. I hope at the end of this lecture you agree with me. This month I will be traveling with the Origins Project and a group of 27 intrepid travelers to Cyprus and Greece. So the next Origins podcast will not air until August. I hope your July is enjoyable. As always, an ad-free video version of this podcast is also available to paid Critical Mass subscribers. Your subscriptions support the non-profit Origins Project Foundation, which produces the podcast. The audio version is available free on the Critical Mass site and on all podcast sites, and the video version will also be available on the Origins Project YouTube. Get full access to Critical Mass at lawrencekrauss.substack.com/subscribe

Well, hello and welcome to the Origins Podcast, our fifth lecture in our 12-part series on physics for everyone. I'm really excited but also daunted by today's lecture because I'm going to talk about basically the fundamental basis of modern physics, the origin of modern physics, and perhaps the single most important result that determined the course of the 20th and 21st centuries. That may sound like hyperbole, but I don't think it is. And, you know, I don't do things hierarchically, but I think this is perhaps one of the most important results in physics, and what I love about it,

and I loved as an undergraduate in physics, I want to walk you through, I'm going to lead you up to what is perhaps the most important calculation that was done in the 19th century and maybe ever, that changed the world. And so that's where I'm going to lead you, and we're going to talk about electricity and magnetism. Well, I want to talk about it not just in terms of the sort of bland history

and discussion of electricity and magnetism. I want to talk about it because they change the way we think about the universe in profound ways. They change the way we think about the world. They also change the way the world acts. The modern world will not be possible, clearly. without them. So those are the ideas I'm going to cover and I'm going to cover what is basically a whole term or a year of physics in this lecture and abstract out what I think is important

and create the basis for understanding the nature of space and time and a modern theory of the universe. So here we go. I'm a little as I say I'm a little daunted. We'll see how long it takes and I hope we can do it in one podcast worth. So, you know, when you first learn electricity and magnetism, and everyone's done it, I think, at some point he's ever taken electricity and magnesium, it seems like deja vu all over again, as Yogi Berra would say. Because we've learned before, you know, that the force of gravity, as we talked about, was this thing,

It was proportional to the product of the masses of the two objects, and there was a one over R squared law. That set up the basis for everything we talked about, understanding the motion of the planets, weighing the sun, weighing the universe, and doing much more. And it came, I remind you, from Newton, thank you about the nature of force, after Galileo basically set up our understanding of motion.

And the reason I'm mentioning this is because they'll both come back, certainly Galileo, but also Newton, in a profound way. The things, these bases are not just sort of something that was developed 400 years ago, but they come back and they're central to all the ideas that we think about

even today at the forefront of modern particle physics and quantum mechanics. Okay, so everyone knows, maybe not everyone knows. But as Leonard Cohen would say, everybody knows. That electromagnetism looks, electricity.

Electromagnetism, but the electric force between charges looks identical to the gravitational force in a way. I should have said there's a minus sign here But a minus sign is it's attractive The gravitational force is always attractive between two masses gravity sucks Doesn't blow but for normal matter now So the gravity the electric force Looks just the same between two charges And you know, I remember when I was a kid, and I bet a lot of you thought the same thing,

wow, isn't that amazing, these look alike? Maybe, you know, and you could picture a single atom as an electron going around a proton, just like the Earth goes around the sun, maybe we're fundamental atoms in some wonderful other beings, meta-universe. I'm sure everyone's thought of that, that they're so similar. But they're similar, but really quite different.

And one of the things that makes it quite different, and I want to stress the kind of things to make them different in a modern perspective. So this is a discussion that really is different than a standard physics class, because I want to emphasize the things that really matter for understanding the universe today, not for mastering the details of figuring out

electricity and magnetism and how to build things, although I'll talk about building a few things. But what I want to talk about are the differences, because there are profound differences, and those differences lead us to understand how electricity and magnetism change the nature of space and time in a fundamental way. Gravity, of course, will come back later

and change the nature of space and time in a very different way. But the first thing you can realize is that the two forces are very different in strength. Well, there's something else here. This is a positive sign. Two light charges repel, two like masses attract. That's a big difference. Of course, the difference between charge and masses,

charge comes in two different signs, plus and minus. Math only comes in plus. So if you have opposite charges, plus and minus, that minus sign comes into here and it becomes an attractive force. Okay, everyone knows all this stuff. Okay. But the key thing that's really important is the gravitational, just, I usually don't plug these things in, the units don't matter, but just give you a sense of the comparative size of this. In units of meters, kilograms, and seconds, which is what the civilized world uses for units everywhere outside the United States. This has units of six times 10 to the minus 11, and the units are Newton's meters over kilogram,

meter squared over kilogram squared. What are Newton's? Newton's are a unit of force, and this is a force. So clearly, this has units of meters squared over kilograms squared, So when you plug it in to two, there's a kilogram and a kilogram and an over meter, these cancel out and you end up with something that has units of Newton's. It's a unit of force. We can call it after Newton for obvious reasons. He was the one who first described that. Anyway, and that's that's that's that's G but the the constant for electromagnetism, which by the way, I can't help and I can't putting in this form We write it for reasons that I'll explain in a while as 1 over 4 pi times this quantity, epsilon or not.

It's just arbitrary what we call it, but that's the way we describe it, is 9 times 10 to the 9th Newton's, and it would be meter squared over something called coulome squared. Cool-oam is a unit of charge. A cool-oam is a unit of charge so that and it's, we now know the fundamental charges are electrons that are moving through materials and how many electrons on them? And a cool-oam, a lot of them.

Almost 10 to the 20th, well, 10 to the 19th electrons. But the way to think about it is that we measure, you're used to amps, one amp, 2 amps, 10 amps, 15 amp fuses, the kind of electric charge is moving around the currents in your house. An amp is a cool home of charge moving past any point per second.

So it's a unit of charge, it doesn't really matter. Now, of course, units are arbitrary, and I can make things, you know, when you look at this, wow, this has got a 10 of the 9th, this has got 10 minus 11th, but that could be, I could make that up. I could make a different unit where this number was smaller.

But it turns out, Let's just try and think of the relative strength, which is really what I want to talk about, of gravity and electromagnetism, just to make it clear. And I've talked about this in the first lecture, that gravity is the weakest force in nature and so much smaller.

So the charge on an electron, by the way, is 1.6 times 10 and minus 19 coulums, just so you know. It doesn't really matter. But the mass on an electron, so let's just think of the ratio. So let's just think of two electrons here. And there's a force of repulsion due to electromagnetism,

and there's a force of attraction due to gravity. And what's the ratio of those two? And I think that's instructive. So it turns out the mass of an electron in units or kilograms, is about 10 to minus 30 kilograms. And that's really all you have to know, because the mass, we know the charge now, an electron, and this charge and that charge,

we know the mass of an electron, and we know, it doesn't matter, let's put them one meter apart, it really doesn't matter, so R is one, and R squared is one here, so we can go into it. So we get the force of gravity, the ratio of the force of gravity of attraction of those electrons versus their force of repulsions, versus their force or repulsion.

Dividing these two, the R squared's cancel. And you just have the ratio of G over K and then M1, M2 times Q1, Q2, so M1 squared times Q1 squared. And let's do that. So G, let's make it 10 to minus, 6 times 10 to minus 11, let's make that 10 to minus 10. between friends, doesn't matter.

And then the mass is 10 to minus 30 squared, so that's 10 to minus 60 in those units. And then, and so you end up, that's so the result will end up in Newton's. And the charge on the electron is about 10 to minus 19 squared, right, in coulomes. And then this is nine times 10 of the ninth, so we'll make that 10 to the 10.

So this is 10 to the minus 70 over 10 minus 10 to minus 38, 10 to minus 28. 70 minus 28 when I last was able to do addition, which is now, is 10 to the minus 42. The force of gravitational attraction of one electron with another is 42 orders of magnitude smaller, than the force of electric propulsion between those two electrons. And so now these numbers have actually put, we put quantitative values to show you not only how weak gravity is for electrons,

but it explains so many things about gravity versus electromagnetism. It explains why gravity, why, for all intents and purposes, electromagnetism is the force that governs everything we do. The only is true that gravity keeps us on Earth. But as I've told you before, It's only because while the gravitational force for each Adam in the earth against each atom in our body is so small we could it's almost impossible to measure it

That gravitational force adds up over the entire earth to produce a measurable effect But while it's measurable it holds us down Note that it's just the electric forces between my feet and the ground right there just touching right there that's able to counter the gravitational force of the end entire earth pulling me down. That's because the force is 42 orders of magnitude greater locally from the electrons, the charges and the atoms in the ground on the charges and the atoms in my feet compared to the attraction of the earth. And that's why the solid earth holds me up a little bit of

Earth can hold me up. And I think the example that I don't know if I used it here, but but but I think I was first learned it from Feynman was that if you want to to understand this relative strength of gravity, push someone out of a 10-floory building. Don't do it. But it takes them 10 stories for gravity to accelerate them all the way down, but it takes a fraction of an inch.

They don't even make a dent in the concrete for electromagnetism to stop them. Electricity is much, much, much larger. So that's the first thing I wanted to point out. There's one fundamental difference. Electricity is really relevant for us today because it's much, much stronger than gravity. You know, and a little bit of charge goes a long way, therefore.

I wonder, I just mentioned something that, again, I think I learned from Feynman. And that is, you know, thunderstorms and weather and are profound right now across the country when I'm recording this while I'm in sunny California. Nevertheless, much of the country is having storms, snowstorms, and other parts of the country are having thunder and lightning. The earth is charged, but my understanding is that the total electric charge of the earth is simply a million coulomes.

And that's just, it's discharged by lightning storms and thunderstorms. This charge of the earth, by the way, I believe produces, I haven't talked about voltage, but a voltage in the atmosphere that changes, I think, by 100 volts per meter. But we'll get to what voltage is maybe. It's not that important for the moment. But if you work that out, that's 10 to 6 coolums is about 10 to the 25 electrons.

So there's an excess of 10 to the 25 electrons, which is not a lot. They're 10 to the 24 electrons in a gram of material. But there's that excess is enough to cause all the thunderstorms. But if the Earth discharges it, these that have to have a lot of the Earth. that has to get recharged and that's from cosmic rays coming down. And if you work out how do you get, you know, if you're going to in a day replenish your Earth by 2010 to the 25 electrons, that's one electron per cubic per square centimeter

coming down to the Earth every 10 seconds, which is about the rate of cosmic rays on the Earth, charged by, it doesn't have to be electrons, it would be other charged particles. So there's a, you know, if you take a square centimeter, and if you take a little Geiger counter, and I meant to bring one on Earth, you'll hear, you know, cosmic rays are coming

about once every 10 to 20 seconds, boom, for each square centimeter of the Earth. And those particles bombarding us from space are enough to keep the Earth charged to power all those thunderstorms that would otherwise discharge the Earth in no time. Okay. So that's a fun fact.

irrelevant to what I'm going on, but I think it's kind of neat when I first learned that. Okay. Now, now I want to get to the me to things, the most amazing developments into this seemingly, otherwise maybe boring thing

that sounds just like gravity. And some of you have seen my general lectures, I've seen this guy. I'm sorry for the low resolution, but I don't have a better picture of Michael Faraday. But that's Michael Faraday, one of my heroes. And I've talked about this in public lectures, but I want to emphasize it here

because Faraday is going to play an important role in everything that comes from. Michael Faraday was the most important experimental physicist of the 19th century. His story is remarkable. I'm going to sit down for it, and he inspires me for many, many reasons. He was just a normal fella and actually made a point of not wanting any honors or anything else.

But most importantly, he rose, he was a self-made person in every way. He did not come from a family that gave many education. He was a bookbinder's apprentice, and therefore he was indentured for seven years to learn how to bind books, basically and he attended the lectures of Sir Humphrey Davy who was the head of the Royal Institution at the time a chemist famous chemist and he attended those lectures and took beautiful notes and then bound those notes up into a beautiful book because after all he was a bookbinder's apprentice and one day presented them to Humphrey Davy and said it was duly impressed and said basically can I be your assistant

I've often told students this is a good example you should always suck up to your instructors. And he certainly did. And Humphrey Davy was sufficiently oppressed that he made Michael DeVaraday, his apprentice, his assistant. And he assisted with the ongoing experiments

that were ongoing at the Royal Institution at the time. And of course, Rose eventually to become the director of the Royal Institution and discover the law, the result, Faraday's law, as we call it, that changed the world,

that made the model. world possible and in his laboratory there are so many apocryphal stories about his laboratory as he developed it he also was a wonderful teacher when I when I taught it at places like Yale or at Harvard or wherever any sort of leading research institution the number of courses I would teach would be about one course per term which still felt onerous when I was doing it but people but it's amazing people like Michael Faraday would be doing all the stuff he was doing and he would prepare these beautiful public lectures at the Royal Institution because

he loved to lecture on them at Christmas lectures and others and they became famous and the Christmas lecture still happened to the Royal Institution I almost gave one one time and I didn't and I always wished I had but in any case he was he loved to teach but he was a wonderful experimentalist and he had all these things going on that looked like they would have no use whatsoever And I'll tell you the two apocryphal stories which are And I'm sure neither are true They involve the prime minister and they may have been Gladstone or the queen may have been Victoria

Coming into his laboratory and looking around and seeing all these obscure Instruments jumping frogs and you know putting electrodes to fraud you can do that and make dead frogs jump and all the rest and other just all this this obscure stuff and saying of what use is any of this and his answer I've heard two answers One is Of what use is a newborn baby Which is of course a beautiful poetic answer the other answer was what do you mean? It's going to be so useful you'll tax us for it and he was exactly right Now the thing about

Faraday was because he had no formal training in mathematics he he he He pictured things. He was an experimental physicist, but he tried to understand what was happening. Why was there an electric force between two charges? It was a question that Newton never really asked for gravity. He, Newton, said this famous Latin phrase, hypothesis non-fingo, I frame no hypotheses, just explained how gravity worked. The why wasn't there. Now, Verde wanted to sort of get a picture so he could intuitively understand

what was happening between two charges, so he could think about it his mind in a pictorial way because he couldn't do the algebra. Maybe he could, but he wasn't comfortable enough with it if he could. So he thought of an electric charge is following. Electric charge there.

And he said, okay, it's a positive charge. I can think of that positive charge. I can think of these things I'm going to call them feel lines going out from the charge. And if it's a positive charge, those feel lines all point out. And the number of feel lines, number of lines, is proportional to the charge. The bigger the charge, the more lines.

Okay? And the way you can think about this is that if I have another charge here, Q, the force on the charge Q due to this charge, we'll call Big Q is the magnitude of that charge times the strength of this thing he called a field. The field is the number of field lines per unit area. If there are more feel lines going through any given area, because there's more charge, the field is bigger.

And the force is proportion to that. So I can tell what's going to happen. If I put a charge here, it's going to be the force is going to be in that direction. If I put the charge here, the force is going to be in that direction, and so on. And he could think, therefore, that what was happening, this action at a distance, which is the same action at a distance in Newton's form of gravity,

where the Earth and the Sun suddenly attract each other at a distance in his picture instantaneously, although we now know it's not instantaneous, and this will lead us to understand why the electric force is not instantaneous. But he would say instead of this charge being repelled by this charge This charge creates this imaginary field Throughout space which you can picture he could picture this way and that field throughout space Would interact with the part so the particle isn't interacting with this the particle sitting here and it feels a field It feels something right where it is so there's no action and a distance in that way

It's interacting with this with the space at that point and space is at that point has this imaginary field that he was thinking. And now the particle knows what to do, because there's a field line, and it, and it, and it, and it, and it, and it goes out. And, and then you could see, and then you could draw pictures, right? I mean, if I drew a negative charge here, I have this, so a negative charge here, I'd have these feel lines going out, the same kind of field lines. but the lines are point inward, because if there's a negative charge in here and I put this positive charge Q here,

it'd be attractive because this is positive. But then you can think of this as these field lines, if you put these two charges together, if I have one here, the field lines will connect up. So field lines begin on a positive charge, and they can end on a negative charge. And this is a familiar picture for what the electric force looks like.

around two charged particles, a positive negative charge. So if this is a negative charge, this is a positive charge, wherever you put a particle, now Faraday could do the math, and what's amazing is this reproduces the math exactly, right? Because wherever you put a charge here, you know what the force is gonna be.

You put a charge there, the particle's gonna get pushed in that direction. You put a charge here, it's gonna be pulled in that direction. You put a positive charge here, going to be pushed in that direction. So just pictorially, it was a beautiful imaginary aid that he created that allowed him to picture the electric field and the electric forces, well, the electric forces between particles. Note also something else. It automatically tells you about the strength of electricity. Because if the number of field lines depends,

just on the charge coming out of it. Then if I draw a sphere around here, and I have all these colors, I should use them, I suppose. And I know all of you are going to say, oh, you should be doing this with fancy graphics on a computer, and you can, but the point is the fancy graphics don't improve anything except that maybe the quality of the images, perhaps.

But the ideas are what's important. important. It's the ideas that are important. And the ideas don't depend on fancy graphics. So if I draw a sphere around that charge, the total number of feel lines coming out of that sphere is always the same, right? If there's 20 of them, because a charge has some value, there's always 20 coming out of them. But the area of the sphere goes as a square of the radius of the sphere. And therefore, if the force, if the strength of the field or the force between particles depends on number of field lines per unit area, and the total number of field lines is constant, then the force is going to go down as 1 over our squared. Because the number of field lines per unit area, if the total number of field lines is constant,

the density of field lines goes down as the area of the sphere, and therefore the strength of the field goes down as one. 1 over R squared. So it reproduces exactly this 1 over R squared result of what electromagnetism is just pictorially. It's beautiful. Of not electromagnetism. I'm sorry, I keep saying electromagnetism. We haven't talked about magnetism yet. Of electricity. And by the way, the last not so pretty picture I'll draw, that may be obvious. You know, what's kind of amazing to me is if I asked people ask, maybe even a physicist, if I draw, if I draw, draw two, let's say, two, I'll use red for the moment, two negative charges. And I put a positive charge here. Where would the, what would the force be on it? Well, of course, you can do this algebraically, you can say the force here is, is in one direction from this and another

direction there, and I took the vector sum of all this. Or you could say, okay, well, there are field lines coming from that go out from that sorry go in because it's a negative charge go in and they they look radial near the charge so I'll draw these field lines going in in in in and in but the other thing you know about field lines is they don't cross so they can't keep going out like this so So they basically have to bend up or bend down so they won't cross. And let's take another field line there, it's going to bend up. Another field line there is going to bend up. And this one will bend down.

And now you can see, and remember the arrows are always pointing inward, this, once again, looking at the field, you can see this positive charge is now going to experience a force in that direction. So in any complicated configuration of charge, you can use these pictures of Faraday, and you get the exact same thing as just doing the algebra. And that was a wonderful crutch for Faraday, but it turned out to be much more than a crutch.

That's what I want to talk about. It is amazing how the imagination of people, the hallucinations we have in our blackboards or rooms late at night, can sometimes turn out to be true. And his hallucination, not only was true, but it literally helped change the world. Even though he's an experimental physicist,

and his experiment changed the way the world works in a greater way than any other experiment that's ever been performed, perhaps, except for the first person to develop the wheel, and we don't know who that is. The most important experiment for modern civilization that was ever done was done by Michael Faraday. But Michael Faraday's picture gives us a law, which we call Gauss's law, and we're going to, by the end of this lecture, we're going to have derived the four Maxwell's equations, and we'll explain how they change the world.

And we'll introduce how they change our thinking about ultimately motion. And in the next lecture, we'll talk about how they change our understanding of space. in time. That's where we're heading in case you're wondering. But Gauss of course was a famous mathematician and he, you know, we call this because it's a general property of something called a vector field, but that doesn't matter.

In electromagnism, I've already given you Grauss's law. And Gauss's law basically says that the total number of field lines going out of crossing any sphere,

any sphere of radius R, It doesn't matter what the radius is. This is a sphere. The total number of field lines, which is like the strength of the field times the area of the sphere, for a field like this, is just proportional

to the charge enclosed inside the sphere. That's all. And since this, since the strength of the field of this giving charge goes as one over a squared and the area goes as R squared, you can see that those cancel and it's always a constant.

And so, and that's why we use that 4 pi, by the way, because if we really want to keep this, the area, remember, of a sphere is 4 pi r squared. So the strength of the field times 4 pi r squared is a constant, and therefore you see the field is proportional to some constant over 4 pi r squared. And what we do, now is we call that constant 1 over epsilon not,

where epsilon not is called the permittivity of empty space. I'm not even gonna write that number down because I never, or that word down, because I never remember it. And memorizing words is not part of physics, so I didn't become a biologist because I didn't like memorizing terms. Biology is a lot more than that now,

but when I was growing up, that's all it was, and that's why I didn't become a doctor like my mother wanted. But the great thing about being a physicist is you don't have to memorize anything. If you're as good as fine, you just work it all out. But that's why we call, you know, we give the force of the electric force is Q1, Q2 over 4 pi epsilon not R squared. So we call K1 over 4 pi epsilon not just because of that area a bit.

That's just a convention, of course, but it is what it is. Anyway, that's the first of Maxwell's equation, something that was known for some time. By the way, one of the things that I don't think I talked about when I talked about energy in the last area, energy, of course, we talked about gravitational potential energy, the ability to do work. In case of electricity, there's another kind of potential we call it, we call it potential. It's called a voltage. If we apply an electric force on some charge and have it go across a certain distance D, it does work on it. Lift it up. The force repels it, lifts it up, or maybe in a battery it pushes from one end of the electrode to another.

That force pushes it a given distance D. It gives it a certain energy, a potential energy, and we call that a voltage. So a voltage is basically the energy that you've given by a battery so that when you let that charge go just like when you let a rock fall, the potential energy that you've given it by lifting it up can be released

due to its motion when it hits your head. When we have a 5-volt battery, we've given a certain potential energy to the charges. When we let them go by attaching us a wire to it, they can then do work by moving. and that's where voltage comes from, just so you know. I thought I'd just mention that.

Now, I want to, and as an aside, prove a theorem that actually, well, wasn't really proved probably accurately or in its modern form, maybe till the 20th century, I'm not sure. A version of it probably was proved by Gauss earlier, but it's true for gravity, but it's also true for electromagnetism. It's kind of neat. Here's a very modern theorem. The universe, we don't know that,

I told you, we think the universe is probably flat. Well, it is flat. We don't think it's, on the scales we can measure, there's no observed curvature to space on large scales. But I also think I said that we actually think, on the largest scales, and if I didn't now, we'll go over this in a future lecture. On the largest scales,

it seems reasonable to think of our universe is that in fact being closed, that scales larger than we can see, larger than the visible size of our universe today, that space may close in on itself. So if we're able to look far enough in that direction, farther than we can actually ever see, you could see the back of your head.

Okay. There's a theorem that says in a closed universe, the total charge in a closed universe must be zero. A universe can't be charged. And this sounds like, wow, you'd have to know a lot of physics to know this. And I remember not learning this result till I was well after graduate school. And it was my lovely late friend, Sidney Coleman,

and Professor Harvard, who I think first explained this to me. In a closed universe, you can't have any net electric charge. Now that sounds like something that's so deep that you'd have to know everything about the forces of nature and to know about it. But the reason is kind of simple. And it's related to Gauss's law what we've just shown.

So let's imagine a closed universe. I'm going to try and draw it better. You know, imagine it as like a, well, like a sphere. So I guess I should do this with a line and then dash lines behind the sphere. Okay. And let's imagine I have a charge here and I have field lines going out on that curved surface.

So eventually the field lines follow what are called great circles. the curved surface. So they go all the way around, behind. And what I'm trying to draw, if you look at it, I should draw this with a straight line when they're on front and dots when they're behind, is if all the field lines are coming out here, what happens is at an antipital point, the point the opposite part of the, if this is a sphere, and of course I can't draw a curve three-dimensional space, but this is a picture. imagine our universe is closed and I'm getting rid of two of the dimensions. Then what you see is all the feel lines are going out here will converge. Take another feel line here. They'll all converge at a point here all heading in. So there'll be a point on the opposite part of the universe where all the field lines have

to come together. And of course what does that correspond to? That corresponds to a negative charge. equal and opposite magnitude to this. So if you ever have field lines going out in a closed universe, they have to converge again somewhere else and those convergence of those points will be essentially equivalent to an electric charge. Now, in the quantum universe, we realize what happens is the field energy gets great enough there, the strength of things comes great enough that quantum mechanics allows you to actually create quantum mechanics and relatively

allow you to create a negative charge there that will appear even if it wasn't the before. The energy of space will be enough to pop a particle into existence and create a negative charge there in a way. And so anyway, this is Gauss's law in a curved universe tells us that we have to live in a neutral universe, which is pretty important because you know a lot of people who may wonder how do we know the universe is neutral and one of the ways we know is if it wasn't. If there was just a small excess charge in our solar system, as can be understood now because we've talked about the relative strength of electricity and magnetism and electricity versus gravity, a small excess charge would overwhelm

the gravitational fields of the sun on the planets. So we're darn lucky that there aren't a lot of excess charge, and you've seen that small excess charge in Earth. Just 10 of the 6 kouloms is responsible for all the thunderstorms and lightning that does all that devastation on Earth. So we're darn lucky that the Earth is almost neutral. and that the universe is,

otherwise we'd be crushed by electric forces. Okay, I think that's everything that's relevant about electricity. All the news that's fit to print. Now, the second part, magnets. Everyone is played with magnets, and when you grow up, you, I mean, people do play with magnets. I know that Einstein did, and it got him, too.

We all love magnets. Magnets are weird. You know, you see electricity, you see static electricity, your hair stands in, you get a shock, you touch the wall, you can take a birthday balloon and rub it against the ball like I used to do when my daughter was young and it will stick to the wall because you're scraping off charged particles

and then they get attracted to the wall and the balloon will stick. And that's lovely, and if you haven't done it, you should do it. But you don't really see it manifest. But magnets are manifest. Magnets are in your face. They're wonderful to play with. And we all know magnets have North and South Poles.

And we can play with them. And if you want to see what magnets do, we all do this. We put iron filings around. I'll use blue here. If we put iron filings around the magnets, you'll see the island filings line up along what we call the direction of the magnetic field. And you see the field lines go from north to south,

producing this nice pat picture and wonderful. And the neat thing is you put another magnets, North poles repel, South Poles repel, just like positive charges repel and negative charge they repel each other. But the strange thing, the first thing that puzzled people for the longest while is if I cut this magnet in two, I end up with two magnets. But when I don't get a North Pole, I end up with a magnet has a North Pole and a South Pole, and another magnet has a North Pole and a South Pole.

and that is strange. And if I cut that again, I end up with another magnet, four magnets that have a north and south pole, a north and south pole, a north and south pole, and a north and south pole. And I can keep doing this ad nauseum.

And I'm sure many of you, at least I don't know how many of you, but I certainly wondered when I was a kid if you kept cutting a magnet, and cutting a magnet, would you eventually get a north pole? And the answer is no. One of the properties of our world is there no isolated North Poles.

This leads you to wonder how come magnets even exist, what causes it? And we now sort of understand it as I'll talk about it. But that is a very important property of the universe. In fact, it's another one of Maxwell's equations in a sense. There are no isolated North Poles or South Poles, as far as we can tell. What then is the source of magnetism? That already was an interesting question. And a person who understand that, I think it was in 1820,

was a guy named, I think it's Hans Christian Orsted. I like Hans Christian. It was 1820. Orsted, let me give it. I don't like, you don't have to remember names or anything in physics, but I'm going to write it down because I won a medal once called the Orsted medal for teaching physics around the country. And so I have a soft spot in my head for Mr. Orsted.

But what he showed is that the source, well, he actually described how it could be, is that the source of magnetism, and this should get you thinking, in retrospect it should get you thinking. The source of magnetism is moving charges. Now this is profound from a philosophical perspective. Because remember that the source,

of electricity, and I want to keep throwing this out all the time because you're going to see it over and over again, the source of electricity is static charges, namely a charge just sitting there produces an electric field, but a moving charge produces a magnetic field in a really weird way. So you can see that magnetism and electricity already have a connection even though there's no obvious connection between magnets and electric charges, until we realize this, they are related, and they're related by motion.

And it's therefore not too surprising that understanding electricity and magnetism and unifying electricity and magnetism as well do in this lecture, changes our understanding of space and time, because what is motion as opposed to non-motion? It's how things move through space over time. But, you know, while the electric charge produces this imaginary field that goes out radially, it's much stranger.

It's much stranger. And a source of great confusion for physics students over generations, and sometimes they're instructors. If a charge is moving, so a charge is moving, it creates a magnetic field around it. But the magnetic field goes in a circle around the motion. Well, see, I'm not good at drawing these things. Let me. And you're going to say, therefore, you should use a computer to draw it. And I have a few computer pictures, but I like to draw it

because I think low-tech is, people sometimes get too enamored with technology, in my mind, and not enough with what the technology is also trying to explain. So the magnetic field, We can imagine a magnetic field, just we did like we did an electric field. We call it B, and I don't know why we call it B, but we do. I could have looked it up, but it doesn't matter. Names aren't important in physics. And it goes, it's sort of perpendicular to the motion. If the motion is here, the field is perpendicular to it, and therefore a moving charge will create a magnetic field around it. By the way, that means if we do the opposite, if you think about this, and this will become relevant later, let me take a charge and move it around in a circle. So it goes around in a circle, like a circuit.

So this is going around in a circle. If you think about it, what this does is it produces a magnetic field that looks like that. And what does that look like? That looks like a little magnet. And so a char, a little current loop, looks like a magnet. And now we can think about what are the fundamental sources of magnetism in the real world, okay? And we now understand that an elementary level

in atoms basically, it's not only the motion of electrons around protons that produce little magnets, fundamental magnetic dipoles, as this is called, in north-south magnet is called the dipole. But even electrons themselves or parliamentary particles act like they're spinning. And if they're not, if they're not a finite, if they're not of zero size,

classically, if they're a finite size, spinning means charge is moving around in a circle and it's not too surprising that an electron has what's called a magnetic dipole. So does a proton because it's charged. Now all of that is a classical picture of something that's actually happening quantum mechanically, but it really means that at a fundamental level

we're getting these little magnetic dipoles in atoms, in iron or whatever, and if they all line up in the same direction, then you end up with something that looks like a magnet, a macroscopic magnet. You add up these microscopic dipoles, and they're a little essentially current loops.

And by the way, this is an neat experiment you can do here that I always enjoy doing, and it surprises people. So an iron is full of these little dipoles, and they're normally moving in random directions. Remember, the Earth has a magnetic field. God, I should have done this. If I was thinking about it, I do what I normally do in physics class

and just do the demonstration. The Earth has a magnetic field. We all know. We have a compass and points north because the Earth has spinning currents, charges moving in its core that causes a magnetic field thanks to the fact that it's molten

in its core, and it's still molten and still, therefore, has it a magnetic field, unlike some planets. Magnetic fields really important. Causes cosmic rays to bend and protects us from many of them. But if you take a nail and you think and you point it in the direction

of the Earth magnetic field, that's really simple to think about this. So point it towards the north and think about what the field lines would look like if you're a certain point while you're not, you know, if you're right at the North Pole, they're pointing into the Earth. But if you're not, they're kind of,

you know, pointing at some angle towards the earth and take a hammer and just keep hitting that nail, jiggling around those fundamental little magnetic dipoles, and in the background field of the earth, once they're jiggling, you loosen them up in a little bit, they'll want to point just like the iron filings do in the direction of the magnetic field. And if you do that enough in the direction of this magnetic field, you'll magnetize the nail. And that's, by the way, not just why nails often when you pick them up, act like they magnetized, but hammers do too, because they get magnetized from that.

And so you'll often find that in your toolbox, males are magnetized or hammers are, and that's the reason. In any case, this is the key analogy. So those are some cute facts, but this is the key thing I want to emphasize. The source of magnetism is moving charges, the source of electricity is static charges. Now, let me take this further.

If static charges create an e-field, which produces a force on static charges, and this is, of course, not how things happened in the real world, but after the fact we can always look at things and try and understand it. And if moving charges produce a magnetic field, maybe magnetic field will produce a force on moving charges. after all, making it symmetric. One can suppose this, but of course the answer is it does.

It does, and in a very weird way. Just as a moving charge produces a magnetic field that goes around in a perpendicular direction, there's a really weird, and I think it's Amper's Law. Well, it's related to Amper's Law. It's this force. I forget who in the fourth. is named after, if it's named after anyone.

But again, names don't matter in physics. It's perhaps the most confusing thing, and I don't want to spend a lot of time on it, and people get hung up on this, because it is confusing in many ways, but we can sort of pictorially do it at least. Let me put it in green, since I haven't used green. If I have a charge,

and it's moving in a magnetic field, and let's say the magnetic field lines are going in the border with little arrows, so I can, this is like, the cross on the back of an arrow. And if they're moving out of the board, you just see a point. So if they're going into the board, and a charge is moving perpendicular to those magnetic fields, it will experience a force and, let's see,

it'll experience a force, and I use, it's called the right-hand rule, it doesn't matter, but it'll experience a force perpendicular to both its motion and the magnetic field. called the, and it's crazy, but the strength of the force, let's say it's always perpendicular. The strength of the force is equal basically to the strength of the charge times the velocity

of the particle times the strength of the magnetic field. The stronger the field, the bigger the force. The stronger, the faster the motion, the bigger the force. The bigger the charge, the bigger the force. And just for fun, we can put perpendicular, which means the force is always perpendicular to both the motion and the magnetic field. So if what charge is moving in the direction

of a magnetic field, there's no force at all. Only if it's perpendicular. So that'll come back to us. So now this leads us to basically two of the other equations that lead to Maxwell's equations. We wrote the one, the first one was that the total number of field lines going out of a sphere, an electric charge is proportional number,

just the charge. The other two of Maxwell's equations are pretty obvious. For a, if I have a sphere and have a magnet in it, those magnetic field lines will come out, let me draw this sphere. So we'll draw this sphere like this, and it looks like that. The magnetic field lines will come out and come back in. They never begin or end. They're always continuous. So that means for every field line going out of the sphere, another field line is going back in. So the total number of field lines going out of a sphere is always zero. That's equivalent to saying there are no magnetic charges. That's one. I don't like this green. Maybe it's because I'm colorblind.

The second one, and this is called Amper's Law, is more or less what I told you before, it codifies so that's that's that's pretty obvious let me keep that equation up there the second one we call Amper's Law just says if I have a current and a current is nothing other than a current we'll call it I a number of charges moving through some surface per second an amp is a cool oam of charge moving through a circuit surface any you know moving past a point every second so one amp means 10 to the 19th electrons are moving past that point every second okay or it doesn't have to be electrons 10 to the 19th things with the charge of the electron so current you know this is proportional to charge and its velocity if it's moving up then it will create as we told you as I told you a magnetic field around it goes around it and it's basically this just says the magnetic field times the sum times the length of the law of the loop is just proportional to some other

constant, we call mu not, times the current. The more of the current, the bigger the field. Okay? And once again, if I have a line of current and I'm a distance R away from it, then the circumference of this loop is 2 pi R, if this radius is R. and since the product of the magnetic field times the length of that loop is a constant if the length of the loop goes like 2 pi r you now see if I have a

current a wire carrying current then the magnetic field away from it goes like mu not times the current over r because and we'll just put the 2 pi in there for fun to get rid of two pies.

Because that means this times this is a constant well that's exactly a solution of this. If this is two two pi r, I divide this by this, I get mu not i over two pi r. So we see that the magnetic field away from a

line of current goes down as one over r. But the important point is this is called the permeability of space. Who the hell cares? Well, because it determines the strength of magnetism. The strength of the repulsion of two magnets goes like, you know, mu not, like the strength of magnetic field.

Because magnetic field lines also repel, just like electric field lines, they don't cross, and they produce a force. That's why two magnets, when you play all those funny games with magnets, you know, you can levitate a magnet from another magnet. That's due to the magnetic force. So the important thing I want to point out is that what we've learned, we're almost at the threshold of discovery, of power,

and with great power comes great responsibility, is the fact that we've learned that the strength of electromagnetism depends upon this one quantity, a property of nature. It's just a number. Why is that number the way it is? We don't know. It is what it is. Get over it.

That's the permativity. And now we've learned this thing. other constant, which gives us the strength of magnetism, the permeability determines the strength of magnetism. So these two fundamental constants are properties of the universe. And I think that's all I have to say about magnetism. Directly. Now time for the great discovery. But before we do the great discovery, I think I owe it to talk about two things that have changed the world based on this. I probably shouldn't have. I'll keep that here.

Remember, a charge moving in a magnetic field experiences this kind of force because we use it. Let's look, once again, at a charge moving at a velocity of VE in a background

magnetic field where once again we'll put the magnetic field into the board, pointing into the board. So all these cross- mean the background magnetic field is pointing into the board. Well, it's going to experience a force. I think I put it up before, but we'll change the magnitude of the sign of the charge. So it'll experience a force downward. But then it starts to go like that. It'll always experience a force perpendicular to its motion. And we know from Newton that if an object is always experiencing a force perpendicular to its motion, because that's what's happening for an object orbiting the earth, we know it. What happens? The charge will go in a circle. So a charge in a background field like that, going in that direction, will always go in a circle.

And you may not remember this, but we use this in Newton's law, that the force required to make something going in a circle of radius R is mv squared over R. That was, we used that for gravity to determine Newton's law. So that's the magnitude of force to make an object go in a circle of radius R. But if that force comes from electromagnetism, a background field, then this is true. And that means that a particle of mass M moving in a velocity V will go in a circle whose radius is determined by the strength of the charge and the strength of the magnetic field. This can be used, of course, to determine properties of objects, like a mass spectrometer.

If I have a charge, and it may be going so fast that its radius curvature takes you out, if I have a charge zooming through here, it'll bend down, and an object with more momentum will have a bigger radius of curvature. How can it have more momentum? Well, it can be going faster or its mass can be greater. But if I shoot things in with a known velocity, then by determining how they bend, I can determine the mass of the object if I know it's charged. But in particular, I can work out that Q over M is equal to V over BR.

So I can work out the charge-to-mass ratio of objects. And that means if there are independent ways of determining mass, I can determine charge. But this allows me, if I take an object, that contains lots of different stuff and I want to separate it out into the different particles or the different atoms or the different compounds if I put them in this Object which might I call it mass spectrometer then out in certain directions Each will become

objects of known charge to mass ratio Assuming they all come in with the same velocity and the chart and so I can determine what these objects are and that's one way to analyze materials and And that, by the way, I think I'll skip this, but it's a simple calculation. J.J. Thompson in 1897 was the person who discovered the electron, really set the stage for the modern world. The first elementary particle that was ever known was the electron. It was seen something called cathode ray tubes, which were known at the time if you took a tube and you got most of the gas out of it. But you put a battery on it, you'd see this glow. and they were called cathode rays.

It was beautiful. It was mysterious. And he hypothesized, as it was others, that they were due to motion of charges through this very faint gas, causing the gas to get excited, and that's how you get fluorescent lights and, et cetera. But he wondered what those objects were,

that current that was flowing through that gas. And he put the whole thing in a magnet, and by basically applying a known voltage, a known energy to the charges and energy being one half mv squared for particles that are moving and therefore a voltage if you drop an electric charge

through a voltage at the end it'll come out with a kinetic energy equal to its original potential energy and if you plug in it in the equation it's not very difficult he could determine the charge to the mass ratio of an electron and he showed that they were all the same

they were unique objects who's charged a mass ratio is the charge to mass ratio we know for an electron. And he discovered that that's how we discovered that they were all individual fundamental particles and called electrons. And by the way, if you do the same thing with a proton, you didn't know what proton, but a hydrogen atom, the central hydrogen atom is a proton surrounded by an electron,

get rid of the electron, so you just have a, proton, if you threw it through the same apparatus, you'd get an answer that's different by a factor of about 1,800 or so, maybe a little bit, almost that. Because the proton has the same charge as an electron, but it's 1,800 times, yeah, 1,800 times more, times heavier than the electron. And so that's one way you could see.

And that's, by the way, one of the ways, when we look for, when originally in the 1950s, when people were looking and trying to discover new elementary particles, they would have these chambers called clown chambers where they could see the tracks, because the particles, when they came through, if they were charged, would ionize the gas and produce these beautiful tracks, a bubble chamber.

I knew the guy who invented the bubble chamber. He went to the university. I used to be the chairman of the apartment of, a lovely guy, Don Glazer. But if you put in a magnetic field, there you could look for curvature and by using that you could see if you were discovering new elementary particles particles of new mass okay assuming because we now know that basically all elementary particles come in charges that are most multiples of the charge on the electron and most of them are equal in opposite so the weight at which they bent would tell us if we were discovering new particles and it's one of the ways that in accelerators we look for new particles and speaking of accelerators the last thing I want to talk about

was an accelerated developed by a guy I particularly like. And if you saw the movie Oppenheimer, you would have seen this guy, Ernst Lawrence. I like him because of the name Lawrence. I was lucky enough, on a 100th anniversary for Lawrence Livermore Labs, I guess, had me come and given a centenary lecture in honor of Ernst Lawrence.

And I found that not only a great honor, but, well, it was a great honor. Anyway, but the neat thing Lawrence did, And it's kind of a neat, it's really an amazing result. If I have a particle going in a circle in a magnetic field, okay? Remember, we got mv equals QBr. That was, that was, right?

Just because mv squared over r was QVB, okay? And so a circle of radius r, a particle will go in that circle. Now this is kind of interesting, because what happens if the velocity gets bigger? Well, it'll go on a radius that's bigger. But what's the circumference of that radius? It's 2 pi r. So the velocity is proportional to the radius.

The period, if this particle is going at a speed V, the time it takes to go around the circle is 2 pi. Pry r over v. The distance traveled over time. The distance traveled divided by velocity is the time. Okay? I'm gonna get that right? I hope so.

Velocity is, if I put the velocity up here, yes, there would be distance over time. So time is equal to pi r over v. See, I don't remember equations. I just try and figure out if it makes sense. So great. But if v is proportional to r, the period is independent. No matter how, so if you, so if I have a little accelerator here with a little battery,

and I, and I kick that particle with a little more energy, it'll now do a bit, it'll do a bigger radius, but the period will be exactly the same. So if I just, if I just give this particle kick once a second, I always kick it when it's at the same point. So I'm not kicking it when it's here in that direction, I'm kicking it here to go in that direction. So it's like a swing. you're basically kicking it the same time, and like a swing, you can resonate and get bigger. And so this particle will get accelerated. All you need is a little space with a battery

that you can turn on at the right time to give this particle a little kick in here, and it'll accelerate, and the time is independent of the energy, the particle. So you can always just kick it at the same time. And this was what Lawrence realized when he created something called the Slytotron, And what is now the Lawrence Berkeley Labs,

and it's probably in the movie Oppenheimer. And it's a beautiful, cute thing. To create a little circular accelerator, you just have to keep kicking the particle, just like you would on a swing at exactly the same time. And it doesn't matter how much energy it gets, it'll keep getting kicked. And the energy you get greater and greater and greater,

and the radius will get greater and greater and greater. And so the total energy of the particle that comes out at the end, because at the end you just let an exit, and you've got these particles coming out, the total energy at the end will just depend upon the size of the cyclotron and the magnetic field that you've got. The bigger the magnetic field

and the bigger the size, the more of the energy. And that's the way the early cyclotron created energetic particles, which were used to do both nuclear physics and particle physics. Okay, enough applications, but those are some maybe cute applications. Now we come to the denouement.

We come to the Pieste de Resistence. Michael Faraday and James Clerk Maxwell, coming together to change the world. To change the way we think about the world and to change the way the world operates. Faraday to change the way the world operates and Maxwell to change the way the world works. Okay, or the way we think about the world, sorry.

So I'm not even going to write down. I've drummed it into your heads, that electricity, static charges, produces electricity, which produce a force on static charges. Moving charges produce magnetism, which produces a force on moving charges. But, so this was a growing connection to realize that magnetism and electricity weren't different.

They're really connected. But how were they connected? And for a long time, the question was, does the magnetic field produce any kind of force on a static charge? If magnetism is really due to charges that are moving, can you somehow get a force on a static charge? And there were lots of experiments that were done, and people are out of big magnets and nothing would happen. And one way to do it is to take a current loop, but no current, just a wire.

And then, you know, if you measure the current, if there is a current, in a closed loop, and then have another wire nearby it in which there is a current, right? And if there's a current here, we know there's a little magnetic field, and the magnetic field will come here. And so the magnetic field due to this wire will affect the particles in this wire. But if nothing's happening, if the charges aren't moving... If you have a current here, of course that's another magnet, the two magnets are a rappel. But if there isn't a current, and it's just sitting there, and I have a very strong magnetic field, can I produce a current in this wire?

And the answer was no. And Faraday tried for the longest time to do that till by accident he discovered something. We now called Faraday's Law. By accident, he discovered if he had a switch here, a switch, and he could close the switch, and if he closed the switch, the current flow, a little battery, attached to a little battery,

a little battery here, if he closed the switch, a current of flow, when he turned, when he opened or closed the switch, when this current was suddenly starting or stopping, when the magnetic field was changing, therefore through this loop, then he noticed a current produced in this loop. And what he discovered is that a changing magnetic field,

the changing magnetic field due to the changing current here when you opened or closed the switch, the changing magnetic field would produce a current. And so what we learned, is a changing, this is profoundly important, a changing magnetic field produced a current. What causes a current? Well, a current is caused.

Remember, a static charge can only experience an electric force. So a changing field somehow produced an electric force on that charge. Or a changing magnetic field produces an electric field. A moving charge, which means moving E field produces a magnetic field. It's not too surprising in retrospect then

that a changing electric field, a changing magnetic field I should say, a changing magnetic field produces an electric field. And the result looks a lot like Amper's Law. It says if I have a loop, and this is length L, a radius R, and I have magnetic field going through it, and if it's L, the area is pi r squared, right,

if this is radius R, that the magnitude of the field times the length of the loop, in this case 2 pi r, is proportional to the total amount of magnetic field lines that goes through the loop. The magnetic field is the number of field lines per unit area times the area. So, sorry, that's not due to that, that's due to the change of that per time. If there's no change, there's no electric field induced around the loop.

And remember, E times L is basically the voltage. It's the electric force times the region around the loop. So what I'm telling, what this tells us is, if the field is not changing, there's no voltage induced, but if you change either the magnetic field, If you change the magnetic field over time quickly, it induces a voltage in this loop. But there's another way to do it. If I take the loop and I change its size, so the area enclosed in the loop is different, then the number of field lines going through the loop changes, you'll also produce a voltage.

And if you think about it, if I change the angle of the loop, when the loop is like this, there's no fuel lines going through it, When the loop is like this, there are a lot. So if I change the angle of the loop in a fixed magnetic field, so the magnetic field itself is a fixed magnitude, but the number of field lines going through that loop changes, I'll also produce a current. This is the third Maxwell's equation.

There are four of them. And it's incredibly important because now we've learned a complete symmetry. Static charges, electric field, moving charges, magnetic field, magnetic field produces a force on moving charges, electric fields produce a force on static charges. A changing electric field, namely a current, will produce a magnetic field. So a changing, moving charges will produce a magnetic field.

A moving magnetic field will produce an electric field. We're almost completely there. And I've even said words that led Maxwell to change things a little bit, to change Amper's law a little bit. Now, I'm always amazed how in retrospect things are obvious. And in retrospect, this could have been, this didn't require an accident. It didn't require Mr. Faraday, but it's always easy after the fact to realize how things work, and things are always easier once you understand them.

But this result could have been understandable by thinking about a different frame of reference. And thinking about frames of reference is important because it'll lead us to the next lecture. relativity. But let's think about that loop, okay? And now, yeah, let's let's let's let's let's think about that about that loop and now imagine imagine now that instead of changing the magnetic field, I'm going to twist this loop around like this. So the magnetic field lines are say going down. So they're coming from a thing here and the magnetic field lines are going through the loop like that.

Well, if I think about it, if this is a charge and I turn the loop, well this is a charge moving in a magnetic field. Before I was thinking of the magnetic field is changing, but now I'm thinking of the charge moving in the magnetic field. And what's it gonna experience? It's gonna experience a force in that direction.

So the fact that if I took a current loop like this and twisted it, well, these static electrons, now when they're moving, will all experience as charges, it'll cause them to go around here, and that will act like a voltage. So knowing that a moving charge experiences a force and magnetic field could have led you to realize that if you take a current loop and you twist it in a magnetic field, it will produce a voltage,

voltage and therefore again thinking about a different frame if I take a static electric I take a static I just I consider the frame of the loop so I'm going along with the loop as far as I'm concerned the loop is not changing in this case the number of field lines going through the loop is changing so the field is changing I'll get exactly the same effect so in one frame of reference I'm in the lab watching the loop twist and I just see you know Amper's law But in another frame, I'm on the loop, and it's the lab that's twisting. And the lab that's twisting is producing magnetic field that's changing in direction, and the number of field lines are going through here is changing.

And that frame of reference should give me exactly the same result, because the result should happen independent of which way you're, whether you're on the loop or not, a current should flow or a voltage should be produced. So thinking about frames of reference tells us that maybe Faraday's strange results, which changed the world, maybe wasn't so strange after all. But Faraday's result was that a changing, that, you know, if I change the current in one wire, a changing current in one while,

I'll produce an electric field in that wire. But that wasn't entirely consistent. And the person who showed that that wasn't consistent, that wasn't consistent, was my other favorite physicist of the 19th century, James Clerk Maxwell. James Clerk Maxwell is a brilliant Scotsman, who by the time he was a little over half my age was dead, but he did an incredible amount, and he was a brilliant theoretical physicist. He was Scottish. He originally didn't get a job in Scotland, so he had to go down to Cambridge where he got a job. They gave it to some other philosopher in, I think it was in Edinburgh.

I think, or maybe Glasgow, I forget which of the universities, maybe it was Glasgow, anyway. And James Clark Maxwell was amazing. He just revolutionized theoretical physics in so many ways. The understanding ultimately of electricity and magnetism, the understanding of color, the understanding of gases, and temperature and gases. He realized that the temperature of a gas

was portion of the speed of the molecules, moving the gas, and basically the fundamental thermodynamics. He was a towering, tongue figure and he died I think it in his early 40s I think he was 48 he might have been 38 I can't remember anymore um anyway Maxwell realized that all these results of Faraday and Maxwell was was slightly almost a generation younger than Faraday he'd list he he he knew that what was going on and uh that all these results the ones I've told you about could be codified in four equations The fact that the total number of electric field lines going out of any sphere is fixed.

It's proportional to charge going through it. The fact that the total number of magnetic field lines going throughout any sphere is zero. There are no magnetic charges. The fact that a changing magnetic field will produce a current, will produce an electric field, which will produce a current. So an electric field in a loop will be proportional to the rate of change of the magnetic field going through the loop. And then he realized that Amper's Law, remember,

which said that if I have, if I have a current, if I have a current going through this loop, that remember the strength of the field times the length of the loop was proportional to this constant times the current going through, he thought of an, he realized that that wasn't consistent Because he imagined a loop a little wire like this with a little battery or a little two plates. And imagine a current going through it.

Well, a current going through that would mean around that little loop around that wire there would mean magnetic fields. But what would happen here? If the charge were flowing here, you'd eventually produce a positive charge there and a negative charge there. So this empty space would not not have a current flowing through it, because we know there'd be empty space there, but there'd be a growing electric field in there.

And in order for things to be magnet, so I'll draw the field that way, in order for things to be mathematically consistent, if there was a magnetic field around this current loop produced by the current, there must be a magnetic field that also goes like that produced by the growing electric field. So just as a changing magnetic field, produces an electric field, Maxwell, in the fourth Maxwell's equation, showed that in fact

there also be a magnetic field produced around a loop when you just change the electric field. You don't have to actually have particles. So if I change the electric field times the area or the electric flux, same as the magnetic thing. The number, if you have the number of field lines times the area of the loop, number of field lines period area times the area of the loop, which is the total number of field lines going through the loop, if that changes with time, you'll produce a magnetic field. So you can have a real current or you can just have a changing electric field. Now what makes this neat, well it's profoundly important, it's mathematically consistent,

is I want to take you back to Faraday, 50 years earlier, who developed this idea of an electric field as just a hypothetical construct. a crutch because he couldn't do the math. It was an invention of his imagination. But now you see that this field seems to have a real existence on its own. You don't need the actual charged particles that create the electric field. Remember, he said a charged particle creates an electric field around there, and that's why another charged particle gets retracted or repelled.

It was a crutch. It was a way of understanding action at a distance. But now just the field itself, no particles around. If it changes, things happen. The field itself, this crutch of Faraday, has a real existence. And the existence is so real, well, there's nothing more real than it because you can see it before your very eyes.

And there's two things I want, so I want to point out how it changed our philosophical picture of the world in a way that'll be relevant, and how it also practically changed the world. And then I want to walk you through the final thing in this lecture, will be, we'll work together

to derive the most stunning result that Maxwell ever produced. And to me, the most important calculation you can do as an undergraduate in physics or as a graduate student in physics, or as a person. The first thing we've now learned, this relationship with electricity and magnetism,

is magnetism is due to moving charges, electricity is due to static charges. But what we've also seen is that one person's magnetism is another person's electricity. Remember, if we're just to be able to be able to be Remember, if we took that loop of wire with the static charge on it, and we have a magnetic field going through the wire, if I twist the wire, then we know there's a magnetic force on a moving charge. Because the charge is moving up, if we're twisting the wire this way, the charge is moving up with a velocity like that, and it will cause the charge to experience a force, that Amper's Law Force, that QVB,

force. So when it's moving, it's experienced magnetic force. But now if we go to the frame of where it's at rest, it's not moving. And a static charge can only experience electric force, but what's moving are the magnetic feel lines. Magnetic field lines are moving now. And so that charge experiences, we now know that magnetic field lines end up producing electric force. That charge sees it as electric force. So that charge is experiencing electric force in its own rest frame. In the laboratory, when we look at it, we're saying it's experiencing a force due to its motion in a magnetic field. So one person's, one person's magnetism is another person's electricity. And the difference between the two is a frame of reference. I mean, I think it should be obvious to you now where this is going to lead us eventually. But this, again, at the time, it wasn't thought of in this way. This is something, at least I don't think it was.

You don't ever hear about it, anyway. But we now, in retrospect, can picture this, is that electricity and magnetism are not just related. They are the same thing. One person's electricity is another person's magnetism. It's just a point of view. It's just your frame of reference.

They're not different. They're the same. And that is, remarkable. That is philosophically remarkable, I think. It tells us something profoundly important about nature, but also it allowed us to change the world. And so because some of you like pictures, nice pictures instead of my drawings, it changed the world in two ways. We can use this to produce the power that powers the studio and everything in our lives. This way.

If I take a loop, current loop, and I have a magnet, a fixed magnet here, and I turn that current loop in the magnet, there'll be a current that will be forced to flow, do what I showed you, this law where, where Faraday's law, where the number of field lines are going through that loop changes over time. It will cause a current to flow. It'll produce a voltage. And that's exactly what happens in Niagara Falls or anywhere else. In fact, all, you know, when we talk about nuclear power or any other kind of power, particularly nuclear power, you know, the nuclear power is all it's doing is heating up water, water's heating up steam, and steam is going into a turbine

and causing the turbine this loop to go around in a circle in a magnetic field. That's it. It's not that the nuclear power itself is producing electricity, in that case. The nuclear power is just heating up water or whatever substance you want that's producing steam and powering a turbine to go in a circle. And when it goes in a circle, instead of your hand,

this is the nuclear power, basically, you're producing, in this case, an alternating current. And that's what happens in Niagara Falls, where they have big turbines, Niagara Falls, Water falls on these turbines, causing the twist around, like big paddles, like little paddles in a paddle-wheel boat. Those paddles turn in a big magnet, that turning in the magnet turns the mechanical energy,

of the water into the electrical energy of a circuit. And that's why Niagara Falls is such a great power source. So Faraday's law, his discovery, allowed us produce electricity, and of course we get taxed for it. So he is absolutely right. But of course, we can use it another way. If instead of turning the crank, if we have a battery here and we have a wire, and if the current in the wire is alternating,

because we go back and forth, it's a different kind of battery, maybe it's not a good picture. If you have an alternating source of current, so the current is flowing one way, and then it goes in the other way. In one direction, because it's moving charges in a magnetic field, it'll cause that wire to twist,

and then if it's just the right frequency, I switch the direction of the current, I can cause it to keep I can keep going so I can cause in this case a motor and those are the motors that power most things including now more automobiles it's just you just have batteries and you and you put and you do it correctly in a magnetic field and if you time it right you'll get that you'll and you'll you'll have the the wire go around and you'll have you'll have electric motor so we have motors that produce much of what's happening in the world, like cars nowadays, among other things, and we have power sources.

All that came from those jumping frogs and those little things in Faraday's laboratory, it changed the world, it produced the modern world. But it changed the world in another way, because it changed the way we think of space and time. And that's the last thing I wanna go over. So it's taking us about 90 minutes to get here, but that's not bad, it's better than a whole term.

And now we'll produce, what I want to actually work out for you, in a way that I hope is understandable and not fancy, it doesn't require any calculus or anything else. A profound discovery that Maxwell made. Let's imagine an antenna connected to a battery. I think that's the way you draw batteries. Anyway, I don't remember anymore. Okay, and so I connect this antenna, it's just a wire.

it's this battery, and it causes charge to flow. So a current goes there, and this becomes positive and say this gets connected to the other things, so this becomes negative. So what's going to happen is I'm going to create an electric field, a dipole field, right? But when the currents, you know, once we've established that, but while the current is flowing,

because I have a charge with a current flowing, it's going to create a magnetic field too. While the charge is flowing, it's going to create a magnetic field. And down here, maybe going out of the board. The board here everywhere. Sorry. Into the board everywhere.

Okay. Now, that's a disturbance in the field, if you're thinking in terms of Star Wars. So you experience that, but the point is, what does that disturbance happen? Does that disturbance propagate? We have electric fields, we instantaneously do that. Is there an effect over here? We don't think anything happens instantaneously.

It will take time for this disturbance to reach over here. But if during that time, I suddenly flip the direction of the battery and the current. So now I make the current flow here and flow here and actually... So let's just take this picture and say now it's had a little bit of time a few seconds ago. That field... I should draw the things of the same color. That field configuration that I had a few seconds ago looks like that and magnetic field lines are here.

But now I flip everything. Now in the nearby region, things look different. Now it looks like the magnetic, the field lines go in the opposite direction. So they, now they go like this, the electric field lines go like that. And the magnetic field is coming out of the board. And then that travels along. And what you can see if you think about this, so as that travels along, you can travels along, you've got these electric field lines that are going down and then they're coming up.

Okay, traveling along and you have these magnetic field lines that are going out of the board, mean into the board, and then out of the board. And so you have these little packets of of disturbances where the direction of the electric field is changing and the direction of the magnetic field. is changing over space. And this disturbance is traveling along and what of course what does it look like it looks like a wave because you have let's just let's just draw the electric field at some point in space if we see what happens over time as this wave travels past the electric field is first pointing up and then it's pointing down and then it's pointing up and then it's pointing up and then it's pointing

down and then it's pointing up. And if you look at the magnetic field, it's pointing, I flip my colors, but it doesn't really matter, it's pointing into the board, and then it's pointing out of the board, and it's pointing into the board, and it's pointing out of the board. And we have an electromagnetic wave. So if we have this oscillating charge,

if we take a charge and go back, a single charge, you have it go back and forth, or have an antenna hooked up to an oscillator so the current goes back, and forth in the antenna, we produce an electric magnetic wave that looks like this. Okay, so what Maxwell's equations tell us, and the four Maxwell's equations, all based on what Faraday gave us, is that if you have a charge and jiggle it upside down, you'll get a wave of fields, a wave of fields, big deal. deal. But that wave of fields will move at a speed and we can calculate the speed of that wave. And that's the beauty of Maxwell's theory because it allows us to calculate things exactly.

And I want to show you how to do it and we'll calculate the number. In this next picture, and the number is remarkable. And this is what Maxwell did. And as I say, it's probably the most remarkable. calculation of physics. Let's think of what happens. If a wave is coming by, let's think of the electric field. So we, let's draw a loop. We have a wave coming through.

And the wave is sort of, you know, we can think of the way, the electric field is zero there, and it's kind of a, and it's got a maximum value there. And it's got a wavelength, which depends on its velocity. The faster, if the wave were faster, the wave would look like that. But, and so let's think about this now. We have an electric field that points up here, and there's zero electric field here. So here the field is some non-zero value, here the field is zero.

Okay? Okay, great. But now let's think of a loop in the perpendicular direction. direction. What do we have here? We have and it's area, in some time, the wave goes from here to here in that time. And so the length of this line here is the velocity of the wave times whatever time we have it to go from before the wave front goes from here to there. Okay? And we have an electric field going through this loop now, this one in this perpendicular direction. Now what we're told is that a changing electric field

produces a magnetic field around the loop, around this loop, right? And the changing, remember, we have that a magnetic field is proportional to mu not times epsilon not times the rate of change of EA over T, where E is the field and a. A is the area. So the area is fixed.

The area has length, say, Z in this direction, and length VT in this direction. So area, so the area is fixed. It's Z times VT. And E starts

out, E is zero and then becomes non-zero. So E in that time, I should say delta, the time it takes to get from there to there. So the rate of change, so this, so, so, so, so E grows as time, so E final after delta T is E, and E initial here is zero at time zero.

So, so delta, so delta, delta E over delta t is just E so we get the magnitude of the field again and it's the magnitude of the field times the length of the loop around this loop now the field is zero here so you got B along this direction and and nothing happening here and nothing happening here because there's no E in those directions so B times Z and is equal to mu-not, epsilon-not, times z-e-e-v-v, because this was delta-t, the delta-t, is canceled. And so, we get a magnetic field that's mu-not, epsilon-0, e-t times v.

Okay, you have to convince yourself of that. I didn't do it very clearly. But if the wave is going through here, there's a change in electric flux, and considering the B is, you know, the only, when B is zero B is, is, is big over here. And so only one part of this loop experiences magnetic field, and that's B times Z. And the area changes and the area doesn't change, but the area doesn't change, but the area The loop depends upon the distance that wave travels in a given time times the size of that loop. And we can do the same thing in the opposite direction. We now remember that there are magnetic field lines going that way.

And the same kind of argument will really, exactly the same argument will tell us that the changing magnetic field lines going through the loop in that direction will result in an electric field whose value is v times b. So the changing magnetic field lines going through there will produce an electric field and the changing electric field lines will produce a magnetic field and that's what's causing the changing electric field produce the magnetic field the magnetic field. The changing magnetic field produces an electric field. But this is the quantitative values. The changing magnetic field will produce an electric field, the changing electric field, the changing electric field produced a magnetic field. Now let's plug that in here and we get the result B, B, equals mu-not, epsilon-not, v times b.

v squared, sorry, times b, because there's a v there and a v there. And this is all that matters. Well, if b is mu-not, mu-not- epsilon-not times v squared b, then this equals 1. 1 equals mu-0, epsilon-not, v squared, or the square of the speed of this wave goes

like one goes like one over mu not epsilon. Maxwell's equations, and I put in all the factors there, tell us that we can calculate if an electric field, if a changing electric field produces a changing magnetic field and a changing magnetic field produces a changing electric field, and we use Maxwell's equations, we come up, we can calculate the speed of that wave, the speed of the disturbance.

And let's just plug in the next. numbers. What's the speed of that disturbance? This is the calculation. It took him a little longer than this blackboard, but he also probably described it better than I just did. But let me put the values. I never remember these numbers, but the permittivity of empty space is 8.85 times 10 to the minus 12, I think. Kulombs squared over Newton meter squared. And that's so that when you put it in the electric force law, the coulomes and the meters cancel, and you get newtons for the force of electricity.

And the permeability of space turns out, and this is again just convention four pints times 10 to the minus 7, Newton, second squared over coulome squared, because then when you multiply that by current, which is coulomes per second, you get Newton's. You know, you get two, this is the force between two magnets, two currents, so you got each current has a coolomes per second,

cooling per second that cancels that, and you get Newton. Similarly, the force between two charges, it's coulomb squared over radius squared. So that cancels that. But if we put those two numbers together, what do we get? we get v and this is all the same set of units

you get v squared goes like 1 over 9 times 10 the minus 12 and 4 pi is 12 times 12 times 10 to minus 7 meter squared per second squared so we can

using our order magnitude estimation 9 times 12 is 100 so this is 1 what's 100 100 is 10 squared This is 10 to the minus 19th. We add those. But 10 to the minus 19th times 100 is 10 to the minus 17th. This is 1 over 10 to the minus 17. But that's equal, 1 over that is equal to 10 to the 17.

What's 10 to the 17? That's equal to 10 times 10 to the 16. So let's take the square root of that to get the velocity. Take the square root of that. The square root of 10 is 3. It's error to 10 to the 16th is 8. so this becomes three times 10 to the eighth meters per second. And any of you who've taken physics knows

that the speed of light is three times 10 to the eighth meters per second. So what James Kirk Maxwell discovered is that if you shake a charge, it moves at a speed that you can calculate from two fundamental quantities that are fundamental properties of nature. You plug in those two numbers without ever measuring your thing, you'll get a disturbance, and what do you discover? the disturbance travels to the speed of light.

So what do you think light is? Light is an electromagnetic wave. These fields that Faraday invented in his mind as a crutch because he couldn't do the math are not only real, they're so real you can see them. They're the light before your very eyes. And fundamental physics at a level that we can do on a blackboard without a calculus

gives us a number that tells us that you measure those two fundamental concepts in nature. I'll tell you the speed of light, and it's exactly what you measure the speed of light to be. And it's not surprising that that would change our picture of space and time. Because remember, electricity and magnetism are related by motion. Electricity due to static charges, magnetism due to moving charges. And of course, motion is what relates space and time, the movement of objects through space over time. And it's therefore not particularly surprising that this result would lead a very

creative young man who when he was 16 or 17 started to wonder about the light waves that Maxwell had said must exist. And he started to wonder, what would it be like to sit on a light wave? And what he discovered changed our picture of space and time. And that'll be what we'll talk about in the next lecture. And I wanted to go through this because in a finite amount of time, we went through the important discoveries that not only just changed engineering and applications

and the way we live, but changed our picture of one of the most fundamental forces in nature, or two of them, electricity and magnetism, unified them to show that there were different facets of the same thing, depending upon your point of view, and then allow us to do a calculation that gives us incredible power. That this calculation is so beautiful because you don't know where you're going. And in the end, after a little bit of just analysis, you come up with the speed of light,

and you realize that that disturbance, that light is an electromagnetic wave. And that's why you see physics students going around with T-shirts that have the four Maxwell's equations written out in a little more mathematical form than I've given them here. And then at the bottom it says, let there be light. Because it's fine for the Bible to say let there be light, but it doesn't explain why except the whim of a deity. In this case, we know why. There has to be light because there's electricity and magnetism

and they obey the laws that we measure in the laboratory that take us places we didn't know where we were going. We didn't impose the existence of light. We derived it. And that is the power of physics. Next lecture, Einstein. Thanks. Hi, it's Lawrence again.

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