Maxwell's Equations Explained Intuitively

So, electrostatics versus magnetostatics versus electrodynamics versus electromagnetics. What do these things even mean? And the only way to fully understand what they mean is to understand Maxwell's equations. Maxwell's equations are essentially four equations that describe the behavior of electricity and magnetism, at least the way we see it. And the four equations are essentially written like this, where the first one tells you that electric field will diverge based on some electric charge with respect to the permittivity of space. That charge is going to exert an electric field essentially like all directions. And this is very similar to gravitational fields. For example, if I have this eraser, this eraser has mass. And this mass is going to exert a gravitational field. Although it's going to be pulling, force will be in the same direction as the field. And this will attract anything that has mass to it. Just like why we are attracted to the Earth. Because the Earth has such a massive gravitational field similar to this. So you can think of this as Earth. And instead of these arrows going outwards, you'll have these arrows going inward, such that if like there's like you, you'll be pulled inward due to this very strong gravitational field of the Earth. But this explains to you the first law of Maxwell's equations. And this is Gauss's law. Now, this is assuming that the charge is not changing. The charge density is there and static. The charges are static. So this is called electrostatics. So whenever you're dealing with a field. So again, like let's go back to the electron. And the field lines in this case are not changing because the charge itself has not changed and it has not moved. This stays the same essentially forever or for a very long time. And this is electrostatics. Hence the electric field is static because the charge is static. Okay, now the second part of Maxwell's equation, and this is equation number one. This is called Gauss's law for electric fields. Now, if we want to think of the second equation in Maxwell's equations, it's very similar, but it's actually way simpler. So again, we are looking for some sort of divergence behavior. But in this case, instead of just thinking of the electric field, we want to think of the magnetic field. Is there a magnetic charge? Is there a magnetic monopole that would exert something? So where it was charged density for electric fields, for magnetic fields, it's actually nothing, zero. And that is for the simple reason that basically magnets and magnetism does not have this divergent behavior, right? And if you look at a magnet, it will always have a north pole and a south pole. And the behavior will always have these curved lines. You'll never have a magnet that has lines that go like this, that just does not exist. It's always going to be this north pole, south pole polarity. And because there's no monopole, essentially there's no divergent behavior for the magnetic field. This is Gauss's law for magnetic fields. And essentially this tells you that in order to create a magnetic field, you can't just have some type of magnetic stuff. Like the magnetic field has to come from somewhere, right? It has to come from charges. And we're going to see that later on in the third and fourth law. But so this is also Gauss's law for magnetic fields. And all you really need to know from this is there is no such thing as a magnetic monopole. Magnetic fields don't exist because of some type of magnetic stuff. They exist as a by-product of electric stuff, most likely. So this brings us to equation number three, which is, this is based on the famous experiment of Michael Faraday, or I guess array of experiments. And he was the person that discovered that if you pass magnet through a coil, that coil starts inducing current. In other words, if you have a changing magnetic field dv, again vector, it has direction, over dt. So the magnetic field is changing with time. And I believe there's a minus. This has to do with the direction. And this changing magnetic field will essentially be equivalent to some change in the electric field based on that rate of change. And it's called Faraday's law. And the other side of this coin is the Ampere-Maxwell's law, which says the reverse. Now, if you have a changing electric field, or in this case, like we again use the flux because we don't want to deal with permittivity. So we are going to have a changing electric flux, which you can think of it as changing electric field, plus some current density. And that's going to, again, equal the cross del operator with the cross of the changing magnetic field. And this is going to be an h. So if you have a changing electric field, that's going to induce a magnetic field that's going to be changing. If you have a constant current, j, that's also going to induce a magnetic field. But the magnetic field is not going to be changing. It's going to be static. So this right here, this j, if this exists on its own and this were not a factor, this would be a case of magnetostatics, where you have a steady current, non-changing current, and that steady current will induce a static magnetic field. And this is true, for example, if you were to run DC current through a wire, there will be a static magnetic field around that wire. And it's not going to be changing. It's going to be static, hence magnetostatics. Electrostatics is when the electric charges are static. Magnetostatics is when the current is static and not changing. So you'll have a magnetic field. And then in electrodynamics, you have changing electric fields, which cause changing magnetic fields and vice versa. So if you have a basic understanding of these four equations, again, I'm going to summarize them, again, very quickly. You essentially understand what these things mean. I'm going to write them in a different color. So again, we said the first one was the charge density is equal. If you have a charge that's existing in space, like any charge that's just chilling there, that charge will have an electric field. And that electric field will basically diverge in space, in free space, or in whatever material you have it in. E-naughts, in this case, would be free space. Or it would be, if it's a different material, this value would change. And then this essentially would be the electric field, which is a vector, because the electric field has to have a direction, hence there's electric field lines. And this is going to be... Oh, it's not a cross, it's a dot, because it's divergence. Okay. So this is the law of electrostatics. And then we have a similar case for... This is the second part of Gauss's law that tells you that there is no such thing as divergent magnetic behavior. There's no magnetic monopoles. This is equation one for Gauss's law. This is equation two for Gauss's law. And the third, in this case, is going to be a curl, it's going to be a cross product. So if we are interested in observing a changing magnetic field, so if we observe a changing magnetic field, then what we're going to have is a curling electric field or an electric field that's changing as well. And likewise, if we want to observe, in this case, I'm going to use d over dt plus j. If we have a changing electric field, that's going to induce a magnetic field. And likewise, if we have current or a bunch of charges going, basically, going steadily through a wire, in this case, like a static current, that's also going to induce a magnetic field. So this is, again, to the best of my ability, as of now, the way to explain these things. It is four in the morning and I can't sleep. So this is the best explanation I could come up with at four in the morning. If you have questions, let me know in the comments. And generally, I'm hesitant to post technical content because there's tons of amazing people teaching this kind of stuff. But some of you guys requested that I do it. So this is my take on it. And with that, I'll see you in the next video. Peace, love.