>This is quite discouraging to read...

It's quite discouraging to see criticism from someone who didn't read what they're attacking. The article plainly states that electrons in metals are hopping all the time, even when the current is zero. Go read it. That's the whole point of that section. A bit hard to miss, unless you leapt to a wrong conclusion after only reading the title.

> because electrons "fill in" the gaps

The article specifically describes this effect. It's probably not wise to criticize things you haven't bothered to read.

That section is a simplified introduction to the concept of "electron sea" and "drift velocity." It debunks the common grade-school textbook error that electrons in metal wires only start hopping between atoms if someone connects that wire in a battery circuit (where the electrons also are supposed to remain stuck to individual metal atoms when current is zero.) This is a serious textbook misconception because it teaches kids that the electron sea doesn't exist, and it prevents them from grasping the "full pipes" fluid analogy of electric circuits as well as later concepts of metallic bond, metal reflectivity, lack of brittleness, etc.

I would argue that correct intuition about anything can be had without math. Using math may be one way of gaining that intuition, but intuition, by its very nature, is not mathematical.

I'm not sure it's useful to argue whether an intuitive picture of electrons hopping or not is more correct. (Electrons are in any case not really little balls that are in one place.) These are all models of the world made with specific goals, where the most "correct" model would be solving the schrodinger equation for a macroscopic system -- something that would not give you any particular insight as to how things behave.

I think the usefulness of an intuitive layman's understanding should be derived from whether it makes you conclude the correct things about the macroscopic behavior of electricity.

Are the models of metals /hopping you are referring to involving free outer electrons in a metal crystal lattice - as he was referring to?

Yeah - especially metals like Fe, Co, and Ni where the conduction electrons are derived from the atom's d-orbitals. The hopping is one of the key elements of the "hubbard" model. This viewpoint doesn't add very much if you want to calculate the conductivity or optical properties, but it is essential for understanding the magnetism of these materials.

What bugged me about this article was that without doing a quantitative analysis of a model and comparing it's predictions to experiments - physical interpretations like whether the electrons are hopping or not are so ambiguous that they are effectively meaningless.

Why without math?

Correctly describing electromagnetics involves invoking vector calculus, which is a tall order for 10 year old children even in education systems more rigorous than America's.

You don't need vector calculus. My demonstration of the power factor is a children's swing: if you push against your friend as they travel out you do work on them: if you make contact with your friend aas they're coming back, they do work on you.

Kids have a pretty good mechanical model of swing sets.

Do kids actually learn about power factor?

That point aside, while an intuitive understanding is nice, power factor is really a mathematical construct arising from the multiplication of two out-of-phase sinusoids (i.e. voltage and current).

I think to really understand active / reactive power and pf, you need to look at the maths. It's really not that difficult. And you certainly don't need to know vector calculus, merely high school trig.

Fully describing it, yes, but a 6th grader should at least be able to handle Ohm's law >_>

What about pictures and animations of those vectors? They don't have to perform operations on those vectors to start understanding, they just have to see the vectors. In other words, they have to see pictures of arrows.