Hey Crazies. Sometimes math is unavoidable. Without it, you miss out on important insights, so what if, instead of avoiding math, we try to visualize it? I want to give you a glimpse inside my head
for a few minutes. This episode was made possible by generous supporters on Patreon. Ok, so the math I want to help you visualize today is divergence and curl, but this isn’t a math channel. It’s a science channel. We need a context! Fluid flow seems like an obvious choice, but those aren’t always simple like this. Sometimes they’re rather complicated and, if we’re being honest, I don’t really understand fluid mechanics as much as I’d like to. I’m trying to teach myself, but that takes time, so we’ll reserve fluid animations like this for analogies only. Let’s focus on something I happen to know a lot about: the electromagnetic field, which is actually a combination of two different fields: An electric field shown in blue and a magnetic
field shown in orange. We did a whole series on how we discovered those fields and what they do, but I didn’t talk about divergence or curl at all and I think that was a mistake. So it’s time to fix it! We have two behaviors: divergence and curl acting on two different fields: electric and magnetic. That means we need four equations. They look something like this: the Maxwell-Heaviside equations, which most people just call Maxwell’s equations. Don’t get my started. But these constants are just there to make the units work out. They’re not actually important for understanding the equations, so I’ll just set them all equal to one. I do what I want! Alright, as we go through these one at a time, we’ll look at the equation first, and then we’ll see what we can do about making it more visual. Let’s start with this one, also known as Gauss’s law. It says the divergence of the electric field at some point equals the charge density at that same point. In other words, the reason the electric field isn’t just zero everywhere is because electric charge exists. But this law tells us something about the shape of the field too. Say we draw an imaginary sphere around some region of space. The outer surface of that sphere has electric field arrows sticking in or out of it. Since the sphere is imaginary, we can make
it any size we’d like. But, to find the divergence, we need to make that sphere really small, like, just a bit larger the size of a single point kind of small. You’re such a little adorable sphere, yes you are. Yes you are. (cough) Anyway. The divergence of a field tells us how much that field points away from that tiny sphere. If the field points into the sphere as much as it points out, then the divergence is zero. Gauss’s law tells us there’s no charge at that point. If the field points more out of that sphere than in, the divergence is a positive number. Gauss’s law tells us there’s positive charge at that point. If the field points more into the sphere than out, the divergence is a negative number. Gauss’s law tells us there’s negative
charge at that point. Electric charge is the reason the field looks like that. Of course, that’s all very static and a little boring. If it helps you can imagine this electric field is the velocity field for some fluid instead. That way it’s easier to see the inward and outward directions at any particular place, but it’s important to remember this is just a visual tool. Nothing is actually flowing in this picture. There’s no electromagnetic fluid. It’s not a thing! We can interpret the other divergence equation the same way. It says the divergence of the magnetic field at some point is always equal to zero. So, unlike with electricity, there’s no such thing as magnetic charge. In other words, there are no magnetic monopoles. Magnetic poles, like north and south, always come in pairs. Consider the magnetic field around this bar magnet. Using that same imaginary sphere, the divergence is how much the field points away. Since the divergence of the magnetic field is always zero, there is always just as much field pointed into the sphere as there is pointed out of it, no matter where we put the sphere. This becomes pretty obvious using the fluid analogy. Every streamline that goes into the sphere comes back out. That’s all there is to divergence. The curl, on the other hand. Why does this keep happening?!? Curl is a different story. Your first instinct might be to imagine those closed fluid loops again and that’s certainly an example of a curl. But this flow has a curl too without loops of any kind. That’s because curl isn’t really about the field itself. It’s about what that field might cause. In this fluid example, say we draw a tiny
imaginary circle somewhere on the screen. (baby noises) Oh come on, man! If we imagine some kind of paddle attached to that circle, the difference in flow between the top and bottom will cause it to spin. So, even though the field itself doesn’t circulate, the things inside it will. Now let’s get back to electromagnetism. Faraday’s law says the curl of the electric
field at some point equals the negative change in the magnetic
field at that same point. Say we draw a tiny imaginary circle again and we notice a magnetic field passing through that circle. If that field is changing over time, Faraday’s law tells us there must be an electric field that curls around the imaginary circle. The curl of that electric field will be opposite the change in the magnetic field. But remember, the curl isn’t about the shape of the field. It’s about what that field might cause. If our imaginary circle happens to be a metal ring, we’ll get a circulating current. That result is what matters. So where was I again? Divergence of electric field, divergence of magnetic field, curl of electric field. Oh, right! There’s one equation left: the curl of the magnetic field. It’s pretty obvious from the bar magnet example that the magnetic field curls. But let’s take a look at Ampere’s law and see what it actually says. A curl in the magnetic field at some point can happen for one of two reasons: An electric current passing through that point or a changing electric field at that point. It’s the longest of the four equations, which is why I saved it for last. Ahhhh! Don’t panic! We can handle this! Say we’ve got an electric current in a wire. If we draw our imaginary circle so electric current passes through it, Ampere’s law says the magnetic field should curl around that circle. Now let’s say, somewhere along this wire, we’ve got a capacitor. It’ll act like a gap in the circuit, which means no steady current, which means no field. Unless that current is alternating! (dramatic music) (laugh) Alternating current, or AC, opens up a lot of possibilities for us. Since the current in the wires is changing on either side of the capacitor, the electric field inside the capacitor is also changing. That changing electric field creates a curl in the magnetic field around our circle, just like the current does in the wire. This keeps the behavior of the magnetic field smooth across the entire picture. There’s a curl in the magnetic field all
along the circuit chain. So what’s the best way to visualize Maxwell’s equations? Using tiny imaginary shapes and a little imaginary fluid, divergence tells us how much a field points toward or away from a point in space. For electric fields, that’s caused by charges. For magnetic fields, it’s always zero, so we know magnetic poles always come in pairs. Curl tells us which way something inside the field might spin or circulate. In electrodynamics, that means we’ll probably get some kind of electric current. Just remember, those shapes are imaginary and that fluid isn’t really there. They’re just tools to help us visualize the math. So what do you think? Should I do more videos like this? Please share your thoughts in the comments. Also, maybe, answer the poll that pops up over here somewhere. If you want to see the actual math of divergence and curl, there’s a whole chapter in my eBook on vector calculus. Big news though! It’s a physical book now! What?! I know, right? It’s crazy. It makes it feel so much more real to me. It’s available through Lulu dot com, which helped me keep the price reasonable. Links for both the eBook version and the physical version can be found in the doobly-doo. 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. Chaos Fever mentioned that, in Greece, they call the nabla symbol “upside down delta” That’s kind of cool actually because that’s exactly what it is, but when I type that Greek word into Google Translate, it says it means lazy. I wonder if they’re trying to say an upside
down delta is lazy. Anyway, thanks for watching!
