How do electrons find the “path of least resistance”? [video]

> With time-varying signals, the return current follows the path of least reactance, which is also the path of least impedance. This means the return current path in your PCB is determined entirely by the impedance of the circuit that carries the return current.

I was also a bit surprised by this. But I do recall doing equations regarding capacitive and inductive reactance. In both cases there is a frequency-dependent effect which alters the shape of the waveform. It being frequency dependent, it disappears in the presence of DC (inductors becoming irrelevant and capacitors becoming nonconductive).

So I can kind of see it. But not enough to turn it into a return-path-is-suprising type example.

There's electric field created by the L-shaped conductor a millimeter away. No wonder that it affects electrons in the larger conductor.

School physics teaches us that metals are perfect conductors of electric field, so the field in the large conductor should be the same across it, and not follow the L shape.

But it's only true for a stationary situation, basically DC. At higher frequencies the fact that the conductor (every part if it!) has some capacitance and inductance starts to play a major role in how a fast-changing signal propagates over it. Both the capacitance and the inductance of the part of the large conductor under the L-shaped conductor on the other side are affected by the L-shaped conductor.

These considerations could help see the result as "more intuitive".

Add "when you have a ground plane" into the GP's fact. When you don't, it isn't true, but most circuits do have one.

Basically, capacitance and inductance are important for oscillating signals. And that plane has a huge capacitance.

All the basic electric circuit theory is just based on a bunch of simplifications that only apply under certain conditions. You get anywhere outside of assumptions for those simplifications, and all kinds of "weird" things start happening. And they're weird just because you're failing to abandon the simplifications, and start looking at Maxwell equations which are a bit more fundamental.

A popular video explaining some of the practical aspects of this phenomenon: https://www.youtube.com/live/ySuUZEjARPY?feature=share

It can be explained using the language of basic circuit theory. DC current prefers the path of the least resistance, but AC / RF current (1 MHz and above [0]) prefers the path of the least impedance, which is the combination of resistance and reactance. The main source of reactance in practical circuits is parasitic inductances - all real-world components, like wires, have unwanted parasitic inductance at AC, in the same way that all conductors have unwanted parasitic resistance at DC. The farther two conductors in a loop are located apart, the greater the parasitic inductance. Thus, when you have a big loop and a small loop, current prefers to flow via the smaller loop with lower parasitic inductance - even if the end points of the big loop have a shorter "straight line" distance with a smaller DC resistance. When you have two inductors in parallel, current prefers the smaller one - just like when you have two resistors in parallel, current prefers the smaller one.

Now going from circuit theory to physics, we would see that inductance is not the property of an individual component called the "inductor", but it's the result of the magnetic field of the current around a closed loop of a circuit. Thus, every circuit must have a parasitic inductance. To a first approximation, this inductance is proportional to the loop area enclosed by the circuit. Thus, for high-speed digital / RF signal transmission, we often want to make the return conductor be as close to the signal conductor as possible [1]. This is why ground planes are often used in circuit boards, and why twisted pairs are often used in cables.

[0] The point of transition depends on many factors, including the physical size of the circuit. Sometimes it can be as low in the 100 kHz range.

[1] I ignored the issue of characteristic impedance

Don't think of the path of least resistance (this is the limit case for f = 0 Hz), but about the path of least impedance and a lot of your intuition will translate. Also remember: 50/60Hz is almost zero on a pcb board and you can basically think of it as DC.

I design sensors, currently for medical applications where I use varying signal to measure properties of a medium. Tiny electrodes in an array that are excited with AC, through different effects (capacitive...), things like density, conductivity... can be deduced from the measure.

Parent didn't say electrons move slowly. Parent said electron flow is slow which is true. The article tries to say that in that average there are fast-moving parts and slow-moving parts. Fine.

In what way is it not?

The movement of water molecules is typically much slower than the wave speed of water.

When you turn on flow to a filled pipe, the water is reluctant to enter the pipe faster than other water is leaving the opposite end (mediated by pressure waves).

It can be modelled, but with a slight twist. If in a water analogy we prefill the maze with a water and then connect it to a source and a drain, then water level will raise near the source and fall near the drain. We'll see then how it propagates through the maze, these waves will meet somewhere, then some more hard to describe action will follow and then the system will settle to a dynamic equilibrium.

This way the water level changes probably can propagate faster then water moves. Though probably to get this result we'll need to make a "deeper" maze. I mean make walls higher so a cross section of a path would have bigger area, so a slow flow multiplied by this area would give us a big volume of a water moving. Big enough to raise level of a water fast in a whole section of the maze.

It's funny how we used to say "a picture is worth a thousand words", and now a YouTube video is worth about five words.

Nah it’s a good video. Solid change in general.

That's analogy is not really correct. Nor is the one in the video. With electrons, or light for that matter you're dealing with wave particle duality and the appearance of "shortest path" is a result of the particle's wave nature. Prof Morales did a good explanatory series in Ars https://www.google.com/amp/s/arstechnica.com/science/2021/01...

Water doesn't actually "solve" a maze either https://youtu.be/81ebWToAnvA

it doesn't know. it 'samples' or whatever inadequate word you wish to use.

> This begs the question the video is trying to answer. Imagine you have no experience or knowledge of what electricity is or how it works. Then you learn the bare basics. You might have this same question, "how does it 'know' which path has the least resistance?"

You're still saying "path of least resistance" though.

A very important part of the answer is that it doesn't find it. All paths carry electricity.

So "take all paths" isn't begging the question, it's correcting the question, and "weighted by resistance" isn't begging the question, it's giving a pretty easy answer to anyone that understands the concept of "resistance" in a non-electrical sense that some paths let things through more easily than others.

And if you have a bunch of stubs, the electricity does go onto them. It just gets stuck at the end.

> E&M (whatever that means)

electricity and magnetism

A (classical) analog computer is not a quantum computer and various conjectures on complexity say that they should all be efficiently (polynomial-time) simulated by a probabilistic Turing machine.

Or.. use non-conductive coated wire to wire up every possible path, with all the wires connected at a start and finish point, and then see which wire gets hot first.

It's not going to save you any time, since wiring up every path is basically the same as brute forcing every path anyway, but it would work.

What the electrons are doing seems closer to Djikstra than TSP