The Poynting Vector in a DC Circuit

Energy in a circuit flows in the electric and magnetic fields around the wires. Here's a fully-worked example of how.

Veritasium posted a followup video to the one mentioned here, which I'd highly recommend. It looks more into the dynamics whereas this video focuses on the statics.
   • How Electricity Actually Works  

Mathematica notebook for calculating the Poynting vector in a circuit: https://drive.google.com/file/d/1yf_R...
PDF version:
https://drive.google.com/file/d/14yTF...
Veritasium's video:    • The Big Misconception About Electricity  
ElectroBOOM's video:    • How Wrong Is VERITASIUM? A Lamp and P...  
Animations were made with Manim: https://docs.manim.community/en/stabl...

I tried to keep this video reasonably short but I know lots of people are going to have lots of questions, so here's a long FAQ to cover some things that I missed in the video:

Q: In the wire between the capacitor plates, why does the energy come from far away?
A: This is a quirk of using infinitely big plates, as we just assume the plates have a voltage difference without making it clear where the source is. We could picture the plates as very big disks, and at their edges are a ring of batteries connecting them providing the voltage difference.

Q: Doesn't that formula for the magnetic field only work for an infinitely long wire?
A: Yes, I had to sacrifice exactness for simplicity. The calculations for the square-shaped circuit make no such approximation, although they do assume infinitely thin wires, and this assumption needs to be treated as an approximation when calculating the total power. Give me a break, I'm trying my hardest!

Q: If electric and magnetic fields are produced by charges and currents, then surely it is correct to say that energy is carried in the wires?
A: This is really a question of semantics and there are many arguments to be made either way. Fields can exist without charges and currents in the form of electromagnetic radiation, although you could argue that the radiation had to come from particles originally. Special relativity and quantum mechanics makes these philosophical questions very difficult.

Q: I heard that charges move through a circuit very slowly. If that's true, then how do charges build up on the surfaces of wires?
A: Individual charges don't have to move very far because there are many more positive and negative charges filling the wires. The battery pushes in charges at one end and everything shifts over very slightly, resulting in more charges coming out at the other end and on the surface. Think of it like squeezing water into a teabag; the water doesn't flow through very quickly, but a decent amount still oozes out the sides. Man, that's a disgusting analogy.

Q: Why do the wires in the two examples in the video behave in such different ways?
A: In the parallel plates example, the wire is resisting and taking up the entire voltage drop across the circuit. That means energy is flowing into the wire and being turned into heat. For the battery and lightbulb, the wires are ideal and so no energy flows into them. Real wires with low resistance are somewhere between these, with the Poynting vector directed mostly parallel to the wires but also slightly into them.

Q: So... ideal wires are kind of weird?
A: Yes. If there is a voltage source across an ideal wire, the current is theoretically infinite, which doesn't make sense. The calculations only make sense if at some point in the circuit there is something to take up the drop in voltage between the terminals of the battery, e.g. a light bulb. But this isn't hard to see for yourself: if you attach a conducting wire to both ends of a battery, you'll find that it heats up very quickly. A lot of current is flowing through because of the low resistance. Be careful if you try this at home, because it's a very fast way to burn yourself.

Q: Circuits use measured quantities like resistance, but electrodynamics also has measured quantities like resistivity and permittivity. Where do these quantities come from?
A: The electric properties of a material are determined by the material's atomic structure, and studying this requires a looooot of quantum mechanics. This area of physics is called 'solid state' or 'condensed matter' physics, and it provides quantum-mechanical explanations for things like electrical properties, colour, transparency, elasticity and brittleness.

00:00 Introduction
01:20 A wire between plates
05:58 A simple circuit
09:41 Electrodynamics versus circuits
13:23 Conclusion