an autodidact meets a dilettante…

easy peasy

Canto: So we were talking about eddy currents, but before we get there, I’d like to note that, according to one of the various videos I’ve viewed recently, this connection between electricity and magnetism, first observed by Faraday and Henry, and brilliantly mathematised by James Clerk Maxwell, has transformed our human world perhaps more than any other discovery in our history. I think this is why I’m really keen to comprehend it more thoroughly before I die.

Jacinta: Yeah very touching. So what about eddy currents?

Canto: Okay, back to Wikipedia:

Eddy currents (also called Foucault’s currents) are loops of electrical current induced within conductors by a changing magnetic field in the conductor according to Faraday’s law of induction or by the relative motion of a conductor in a magnetic field. Eddy currents flow in closed loops within conductors, in planes perpendicular to the magnetic field. They can be induced within nearby stationary conductors by a time-varying magnetic field created by an AC electromagnet or transformer, for example, or by relative motion between a magnet and a nearby conductor.

Jacinta: Right. All is clear. End of post?

Canto: Well, this ‘perpendicular’ thing has been often referred to. I’ll steal this Wikipedia diagram, and try to explain it in my own words.

So, the eddy currents are drawn in red. They’re induced in a metal plate (C)…

Jacinta: What does induced actually mean?

Canto: That’s actually quite a difficult one. Most of the definitions of electrical induction I’ve encountered appear to be vague if not circular. Basically, it just means ‘created’ or ‘produced’.

Jacinta: Right. So, magic?

Canto: The fact that an electric current can be produced (say in a conductive wire like copper) by the movement of a magnet suggests strongly that magnetism and electricity are counterparts. That’s the central point. That’s why we refer to electromagnetism, and electromagnetic theory, because the connections – between the conductivity and resistance of the wire and the strength and movement of the magnet (for example it can be made to spin) will determine the strength of the electric field, or the emf, and all this can be calculated precisely via an equation or set of equations, which helps us to use the emf to create useful energy.

Jacinta: Okay, so this metal plate is moving, and I’m guessing V stands for velocity. The plate is a conductor, and the nearby magnet (N – that’s the magnet’s north pole) produces, or induces, a magnetic field (B) – or it just has a magnetic field, being a magnet, and this creates a current in the plate.

Canto: Which is perpendicular to the magnetic field, because what causes the current in the plate is the movement of electrons, which can’t jump out of the plate after all, but move within the plane of the plate. And the same would go for a wire. There’s also the matter of the direction, within the plane, of the current – clockwise or anticlockwise? And many other things beyond my understanding.

Jacinta: Would it help to try for a historical account, going back to the 18th century – Franklin, Cavendish, even Newton? The beginning of the proper mathematisation of physical forces? I mean, all I wanted to know was how an induction stovetop worked.

Canto: That’s life – you wonder why x does y and you end up reflecting on the origin of the universe. I’ve looked at a couple of videos, and they explain well enough what happens when a magnet goes inside an electrified coil, but never really explain why. But let’s just start with Faraday. He was a great experimenter, as they all tell me, but not too much of a mathematician. Faraday wasn’t the first to connect electricity with magnetism, though. H C Ørsted was the first, I think, to announce, and presumably to discover, that an electric current flowing through a wire produced a magnetic field around it. That was around 1820, which dates the first recognised connection between electricity and magnetism. The discovery was drawn to the attention of Andre-Marie Ampère, who began experimenting with, and mathematising, the relationship. Here’s a quote from Britannica online:

Extending Ørsted’s experimental work, Ampère showed that two parallel wires carrying electric currents repel or attract each other, depending on whether the currents flow in the same or opposite directions, respectively. He also applied mathematics in generalizing physical laws from these experimental results. Most important was the principle that came to be called Ampère’s law, which states that the mutual action of two lengths of current-carrying wire is proportional to their lengths and to the intensities of their currents.

Jacinta: That’s interesting – what does the mutual action mean? So we have two lengths of wire, which could be flowing in the same direction, in which case – what? Do they attract or repel? Presumably they repel, as like charges repel. But that’s magnetism, not electricity. But it’s both, as they were starting to discover. But how, proportional to the lengths of the wire? I can imagine that the intensity of the currents would be proportional to the degree of attraction or repulsion – but the length of the wires?

Canto: You want more bamboozlement? Here’s another version of Ampère’s law:

The integral around a closed path of the component of the magnetic field tangent to the direction of the path equals μ0 times the current intercepted by the area within the path.

The magnetic field created by an electric current is proportional to the size of that electric current with a constant of proportionality equal to the permeability of free space.

Canto: The symbol in in the equation above, (μ_0), is a physical constant used in electromagnetism. It refers to the permeability of free space. My guess is that it wasn’t defined that way by Ampère.

Jacinta: I understand precisely nothing about that equation. Please tell me what an integral is, as if that might provide enlightenment.

Canto: It’s about quantifying areas defined by or under curves. And a tangent – but let’s not get into the maths.

Jacinta: But we have to!

Canto: Well, briefly for now, a tangent in maths can sort of mean more than one thing, I think. If you picture a circle, a tangent is a straight line that touches once the circumference of the circle. So that straight line could be horizontal, vertical or anything in between.

Jacinta: Right. And how does that relate to electromagnetism?

Canto: Okay, let’s return to Ampère’s experiment. Two parallel wires attracted each other when their currents were running in the same direction, and repelled each other when they were running in the opposite direction. It’s also the case – and I don’t know if this was discovered by Ampère, but never mind – that if you coil up a wire (carrying a current), the inside of the coil acts like a magnet, with a north and south pole. Essentially, what is happening is that the current in a wire creates a magnetic field around it, circling in a particular direction – either clockwise or anti-clockwise. The magnetic field is ‘stronger’ the closer it is to the wire. So there’s clearly a relationship between distance from the wire and field strength. And there’s also a relationship between field strength and the strength of the current in the wire. It’s those relations, which obviously can be mathematised, that are the basis of Ampère’s Law. So here’s another definition – hopefully one easier to follow:

The equation for Ampère’s Law applies to any kind of loop, not just a circle, surrounding a current, no matter how many wires there are, or how they’re arranged or shaped. The law is valid as long as the current is constant.

That’s the easy part, and then there’s the equation, which I’ll repeat here, and try to explain: