How Special Relativity Fixed Electromagnetism

If you think that Maxwell's equations describe all of

electromagnetism

, then you are missing half the picture. This episode was made possible by generous Patreon supporters. Hello Crazy. We spent much of the 19th century trying to figure out how

electromagnetism

worked. To the timeline! In 1820 Ampère's law emerged. In 1831 Faraday's law emerged. In 1835 Gauss's law emerged. In 1861, Maxwell added a few things. And finally, in 1885, Heaviside wrote all four in the form we know today. They are called Maxwell-Heaviside equations and they govern the behavior and appearance of 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 at a distance. That looks like magic, which is a problem. These two fields are one way to avoid that problem. They act as a kind of intermediary or intermediary. One charge creates a field and it is that field that actually exerts a force on the other charge. The only purpose of those fields is to explain force at a distance. But the Maxwell-Heaviside equations only tell us how fields are formed.

Lorentz's law of forces shows us how those fields affect things. Each of them only tells half the story. You cannot understand electromagnetism without all five. And it turns out that they don't even make much sense without

special

relativity

. Like the field, it comes in two parts: electrical and magnetic. The force is divided into two parts: an electric force caused by the electric field and a magnetic force caused by the magnetic field. This speed will be the source of our problems. We have known since the time of Galileo that movement is relative. I demonstrated it a long time ago with a tennis ball and a car.

Actually, that reminds me. What happened to Chauffeur Clone? Oh good. It will eventually appear. Anyway, if this force has a velocity, that means it is also relative to the observer. Which is fine most of the time, but in certain circumstances it could become a problem. Let's take a look at one of those circumstances. Full disclosure: I didn't come up with this example myself. It's from Feynman's lectures. Link in the doobly-doo. I also made a video about it a long time ago, which was fun, but not very informative. I can do better! Let's say we have a conducting wire made of loosely bound electrons and held in place by some positive bits.

Yes, it's technically more complicated than that, but that's enough for today. The cable is electrically neutral. There are as many positive as negative charges. So Gauss's law tells us that there is no electric field. But let's say that, at some point, we connect a battery. That makes the electrons move. The wire is still electrically neutral, so there is no electric field. However, we know from Ampere's law that around any moving charge we will find a magnetic field. This wire has a magnetic field surrounding it. Now that we have a field, we should be able to exert a force on something.

In this case, it is a magnetic field, so we should get a magnetic force. It will be exercised on anything that has a charge and is in motion. Reduce lightning time!! Let's say this squirrel has a positive charge and moves to the right at the same rate as the electrons. We have a charge moving within a magnetic field. That means there is a magnetic force. The squirrel is repelled from the cable. There are no problems so far. Everything is fine until we change our point of view. This is what we could call Lab Frame. It is the point of view of everyone in my laboratory who has their feet firmly on the ground.

But what if I have a clone on a treadmill moving along with the squirrel? Instead, you'll see something more like this, which we'll call the Clone Frame. There is 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 is not moving, meaning there is no magnetic force. Um, without force, how does the squirrel get repelled from the wire? Can not. Isn't that a problem? Oh yes, definitely. This example breaks electromagnetism! Forces may be relative, but events are not.

Both are equally valid points of view. If the squirrel is repelled in one of them, it must be repelled in the other. That may happen in a slightly different way or even for a different reason, but it must happen from all points of view. Fortunately, Einstein's

special

relativity

can solve this problem. It's really good at handling point of view changes like this. Depending on the model, there are only a few things that remain the same during these changes. The position is one of them. Everything else is relative. That means they change under these changes. The one we will take advantage of today is the length.

When moving from one point of view to another, the length of objects can change and so can the distances between them. They don't just look like it. In fact, they really do. How much they change depends on how much your point of view has changed. The change here is not very big, but it is still enough to solve our problem. I'll exaggerate everything for the rest of the video so you can see what's going on. In the lab frame, the squirrel moves, so it contracts in that direction. Everything on the cable is fine. It turns out to be neutral in this framework.

As before, the magnetic field around the cable exerts a magnetic force on the squirrel, repelling it. In the clone framework, things look a little different. The squirrel and the electrons are stationary, so they are no longer contracted. Now that the positive bits are moving, they contract. That means this part of the cable has a net charge density. The charge may be invariant, but the space is not. Charge density is the amount of charge divided by the space the charge occupies. In our case, we measure that space as a distance. The electrons have expanded and the positive bits have contracted, meaning they take up a different amount of space.

Its density has changed. Each point along the cable has a net positive charge. According to Gauss's law, that means there is an electric field. If there is an electric field, Lorentz's law tells us that there is an electric force. Bingo Bango! We have a force! In the clone frame, the squirrel is still moving away from the wire. It is simply due to an electrical force rather than a magnetic force. Court! Crisis averted! Thanks Einstein! However, as I said, the total amount of charge remains unchanged. If you look at the whole circuit you can see what really happened.

While we have a net positive charge near the squirrel, the segment further away has a net negative charge. The position has just been redistributed. The point is that special relativity solves our problem by making the following statement: the magnetic force in one reference frame could easily be an electric force in another or even a combination of them in a third frame. We can prove this in Lorentz's force law by removing the charge. You could even call this an electromagnetic field. Actually, that made me think of something. If you can say this about electric and magnetic force, you should also be able to say it about electric and magnetic fields.

Instead of treating them as two separate things, we should be able to treat them as one thing. But, if we want to do that, vectors won't be enough. We need tensioners. 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 space-time. It represents the electromagnetic field as a field of tensors, a tensor at each point in space and at each moment in time. It groups the electric and magnetic field together in the correct way, so Lorentz's force law still works the same. We even get electric power as a bonus!

So how does special relativity solve electromagnetism? By allowing measurements to switch between points of view. In the lab setting, 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 electrical force. The squirrel is repelled by the cable in both paintings. Sometimes magnetism is simply electricity in a different frame of reference. Let's hope this version of the video makes a lot more sense. Let me know in the comments if it helped you.

Thank you for liking and sharing this video. Don't forget to subscribe if you want to keep up to date with us. And until next time, remember, it's okay to be a little crazy. I asked you whether or not I should make more videos visualizing equations and the vote was overwhelming. Most of you seemed to really like it. I guess sometimes I have a new type of video I can make. I'm thinking perhaps of Einstein's field equations. Yes, it will be fun. Anyway, thanks for watching!