If you think Maxwell’s equations describe all of electromagnetism, then you’re missing half the picture. This episode was made possible by generous supporters on Patreon. Hey Crazies. We spent a better part of the 1800s trying to figure out how electromagnetism worked. To the timeline! In 1820, we got Ampere’s law. In 1831, we got Faraday’s law. In 1835, we got Gauss’s law. In 1861, Maxwell added some stuff. And finally, in 1885, Heaviside wrote out all four in the form we know today. They’re called the Maxwell-Heaviside equations and they govern the behavior and appearance of the electric and magnetic fields. But I spent the entire last video visualizing those four laws. This video is about the fifth law. Wait, what fifth law?! The Lorentz force law! It tells us what those fields do. They exert forces. See, things like magnets and charges can exert forces on each other over a distance. That kind of seems like magic, which is a problem. These two fields are a way for us to avoid that problem. They act like a kind of middle man or intermediary. One charge creates a field and it’s that field that actually exerts a force on the other charge. The whole purpose of those fields is to explain force at a distance. But the Maxwell-Heaviside equations only tell us how fields are made. The Lorentz force law shows us how those fields affect things. They each only tell half of the story. You can’t understand electromagnetism without all five of them. And, as it turns out, those don’t even make complete sense without special relativity. Just like the field comes in two parts: electric and magnetic. The force comes in two parts: an electric force caused by the electric field and a magnetic force caused by the magnetic field. This velocity is going to be the source of our troubles. We’ve known since the time of Galileo that motion is relative. I demonstrated this a long time ago with a
tennis ball and a car. Actually, that reminds me. What ever happened to Chauffeur Clone? [Heavy Metal Music] Oh well. He’ll turn up eventually. Anyway, if this force has a velocity in it, that means it’s also relative to the observer. Which is fine most of the time, but, in certain circumstances, it could become an issue. Let’s take a look at one of those circumstances. Full-disclosure: I didn’t come up with this example myself. It’s from the Feynman lectures. Link in the doobly-doo. I also did a video on it a long time ago, which was funny, but not very informative. I can do better! Say we’ve got a conducting wire made of loosely-bound electrons held in place by some positive bits. Yes, it’s technically more complicated than
that, but this is good enough for today. The wire is electrically neutral. There are just as many positive charges as there are negative charges. So Gauss’s law tells us there’s no electric field. But let’s say, somewhere down the line, we hook up a battery. That gets the electrons moving. The wire is still electrically neutral, so no electric field. However, we know from Ampere’s law that, around any moving charge we’ll find a magnetic field. This wire has a magnetic field that wraps around it. Now that we’ve got a field, we should be able to exert a force on something. In this case, it’s a magnetic field, so we should get a magnetic force. It’ll be exerted on anything that has charge and is moving. Shrink Ray Time!! [Shrink Ray Noises] Let’s say this squirrel is positively charged and moving to the right at the same pace as the electrons. We’ve got a charge in motion inside a magnetic field. That means there’s a magnetic force. The squirrel is repelled from the wire. No problems so far. Everything’s fine and dandy until we switch points of view. This is what we might call the Lab Frame. It’s the point of view of everyone in my lab who has their feet firmly on the ground. But, what if I have a clone on a treadmill moving along with the squirrel? Instead of this, he’ll see something more like this, which we’ll call the Clone Frame. There’s a stationary squirrel, a bunch of stationary electrons, and some positive bits moving to the left. Those moving positive bits will create the same magnetic field, but the squirrel isn’t moving, which means there’s no magnetic force. Um, without a force, how does the squirrel
get repelled from the wire? It can’t. Isn’t that a problem? Oh yes, most definitely. This example breaks electromagnetism! Forces may be relative, but events are not. These are both equally valid points of view. If the squirrel is repelled in one of them, it must be repelled in the other. That can happen a slightly different way or even for a different reason, but it must happen from all points of view. Thankfully, Einstein’s special relativity can fix this problem. It’s really good at handling shifts in point of view like this. According to the model, there are only a few things that stay the same under these shifts. Charge is one of them. Everything else is relative. That means they change under these shifts. The one we’ll be taking advantage of today is length. As you shift from one point of view to another, the length of objects can change and so can the distances between them. They don’t just appear to. They actually do, like, for real. How much they change depends on how much your point of view has shifted. The shift here isn’t very big, but it’s still enough to fix our problem. I’ll exaggerate everything for the rest of the video so you can see what’s happening. In the lab frame, the squirrel is moving, so it’s contracted along that direction. Everything in the wire is fine. It happens to be neutral in this frame. Just as before, the magnetic field around
the wire exerts a magnetic force on the squirrel, repelling it away. In the clone frame, things look a little different. The squirrel and electrons are stationary, so they’re not contracted anymore. Now that the positive bits are moving, they are contracted. That means this part of the wire has a net
charge density. Charge may be invariant, but space is not. Charge density is the amount of charge divided by the space that charge occupies. In our case, we’re measuring that space as a distance. The electrons have expanded and the positive bits have contracted, which means they occupy a different amount of space. Their density has changed. Each spot along the wire has a net positive charge. According to Gauss’s law, that means there’s an electric field. If there’s an electric field, the Lorentz law tells us there’s an electric force. Bingo Bango! We’ve got a force! In the clone frame, the squirrel still repels from the wire. It’s just due to an electric force instead of a magnetic force. Woo! Crisis averted! Thanks Einstein! [DJ Air Horn] [Laughing] Like I said though, the total amount of charge is still invariant. If you look at the whole circuit you can see what’s really happened. While we have a net positive charge near the squirrel, the segment farther away has a net negative charge. The charge has just redistributed. The point is special relativity fixes our problem by making the following statement: The magnetic force in one frame of reference could easily be an electric force in another or even some combination of them in a third frame. We can show this in the Lorentz force law by pulling out the charge. You might even call this thing here the electromagnetic field. Actually, that’s got me thinking about something. If you can say this about electric and magnetic force, you should also be able to say it about electric and magnetic fields. Rather than treating these as two separate things, we should be able to treat them as one thing. But, if we want to do that, vectors aren’t going to cut it. We need tensors. The electromagnetic field tensor to be specific. It’s straight out of special relativity, four rows and four columns because there are four dimensions in spacetime. It represents the electromagnetic field as a field of tensors, one tensor at every point in space and every moment in time. It groups the electric and magnetic field together in just the right way, so the Lorentz force law still works out the same. We even get electrical power as a bonus! So, how does special relativity fix electromagnetism? By allowing measurements to change between points of view. In the lab frame, the moving electrons use a magnetic field to exert a magnetic force on the squirrel. In the clone frame, the electrons expanded and the positive bits contracted. This left the wire charged, allowing it to exert an electric force instead. The squirrel is repelled by the wire in both frames. Sometimes magnetism is just electricity in a different frame of reference. So hopefully this version of the video makes a lot more sense. Let me know in the comments if it helped. Thanks for liking and sharing this video. Don’t forget to subscribe if you’d like to keep up with us. And until next time, remember, it’s OK to be a little crazy. I asked you whether or not I should do more videos visualizing equations and the vote was a landslide. Most of you seemed to really like it. I guess I have a new type of video I can make sometimes. I’m thinking maybe Einstein’s field equations. Yeah, that’ll be fun. Anyway, thanks for watching!
