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Do electrons move rapidly through direct-current circuits?

Electrons actually move very slowly through direct current (DC) electric circuits. Remember that DC is the simple circuit you get when you connect something like a battery to a lightbulb to make a flashlight: the transfer of energy between the battery and the bulb is due to the kinetic energy of the electrons that move through the wires of the circuit.
But for something as simple as a flashlight, the electrons don’t actually need to move that fast to carry enough energy to light up the bulb. When you turn on a flashlight, it can take the electrons in the switch up to a full minute to travel through the light bulb! The reason that the light comes on instantly is because there are already many free electrons in the bulb’s filament (as well as all of the other parts of the circuit), and so there’s a net current from electrons moving throughout the entire circuit even though the electrons at any given location are slowpokes. When you switch your flashlight on, an electric field appears almost instantaneously in all parts of the circuit, but it actually pushes individual electrons along pretty slowly. The reason that the slow-moving electrons are able to get any work done at all is because there’s just so many of them!
AC current diagram
Think about it this way: when you’re at the beach surfing, you might notice that waves actually move pretty slowly compared to a swimmer or a boat---at best, waves travel a couple of meters each second. Yet a tremendous wave like a tsunami is capable of sinking ships and destroying docks purely due the combined force exerted by the tremendous mass of water that it carries. Electric circuits are very much the same way---the individual electrons travel remarkably slowly through the circuit, yet there are so many of them that they can do all sorts of useful things.

How do electrons in alternating current circuits carry energy?

Alternating current (AC) circuits carry energy due to the coordinated vibrations of neighboring electrons. While DC circuits require single electrons to (slowly!) move through the circuit and carry energy thanks to the kinetic energy carried by electrons as they drift through the wire, AC manages to carry energy without any overall motion of the electrons through the circuit.
The mechanism for this is pretty clever: when an AC circuit is activated, the power source starts shoving on electrons at one end of a wire. This shoving is periodic: the closest electron to the source gets pushed forward a tiny amount, but then it gets pulled back. Overall, the electron doesn’t go anywhere. But remember that electrons can’t stand to get too close to each other---as soon as the electron in the back of the line gets pushed forward by the power source, the electron right in front of him in the line gets pushed forward a little bit, too. There’s also a nearly negligible time delay between when the guy in the back moves forward and the guy in front of front of him moves forward. This delayed secondary “push” in turn causes the second-to-last guy to shove the third-to-last guy forward a little bit, and so on and so on throughout the entire wire. When the power source pulls the backmost electron back to his original position, the guy in front of him is then able to scoot back a little bit as well, and so on and so forth until the electrons throughout the wire are back in their original spots.
So you can visualize an AC circuit as a bunch of electrons spaced evenly apart, where the guys in the back periodically shove the guys in front of them, creating ripples that travel through the entire line until reaching the device that is connected to the power source. AC circuits use these ripples to transfer electrical energy and do work without actually requiring the electrons to travel very far. This makes AC circuits a very simple example of how waves can be used to carry energy.
DC current diagram
Remembering our water analogy, AC circuits move energy around in the same way that ripples in water carry energy. When you throw a rock into a pond, the ensuing ripples are able to travel throughout the pond and cause leaves and other floating debris to oscillate on the water’s surface. This means that energy has been transferred from the rock to the floating leaves, even though no single water molecule has actually travelled all the way from the rock’s impact point to the floating debris. The energy is carried by the waves formed on the water’s surface, in which chains of water molecules push and pull on each other in succession, transferring energy without actually moving anyone around.

Consider the following… the nervous system

Slow-moving charges are the reason that you don’t have superpowers. Your nerves act as electric circuits that carry messages from your brain to your muscles, but these circuits use salty water instead of metal wires to conduct electrical current. Because direct current moves pretty slowly (especially in salt water!), most of the information is carried in the form of alternating current. Electrical signals cause various ions dissolved in the salt water to rapidly migrate into different parts of each nerve cell, which in turn creates new electric fields that move around the ions in the next nerve cell down the line. But the actual ions and electrons conducting this current mostly stay near their particular nerve cell, and so the signals carried by the nervous system look a lot like the waves in alternating current circuits. The biggest limitation in how fast electrical signals can travel through the nervous system involves the relatively slow rate at which charged ions and electrons can migrate back to their original positions in the cell---were it not for this physical limitation, your nerves could fire arbitrarily quickly and you’d have super-fast reflexes!