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an interminable conversation 8: eddy currents, Ampere’s Law and other physics struggles

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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.

\int \mathrm{B} \cdot \mathrm{d} \mathrm{l}=\mu_{o} I
Jacinta: Right. Why didn’t you say that before? Seriously, though, I do want to know what an integral is. I’m guessing that ‘tangent to’ means ‘perpendicular to’?
Canto: Not quite. Forget the above definition, though it’s not wrong. Here’s another definition:
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.
Jacinta: No, sorry, that’s  meaningless to me, especially the last bit.

Canto: The symbol in in the equation above, (μ0), 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:

\int \mathrm{B} \cdot \mathrm{d} \mathrm{l}=\mu_{o} I
So, that first symbol represents the integral, and B is the magnetic field. Remember that the integral is about the area of a ‘loop’, so the area of B, multiplied by the cosine of theta (don’t ask) with respect to distance (d), is equal to a constant, (μ0), multiplied by the current in the loop (I).
Jacinta: Hmmm, I’m almost getting it, but I’ve never really met trigonometry.
Canto: Well the video I’m taking this from simplifies it, perhaps: ‘the total magnetic field along the loop is equal to the current running through the loop times a constant number’. So, it’s an equation of proportionality, I think. And the constant – mu0, aka the magnetic constant – has a numerical value which I won’t spell out here, but it involves pi and newtons per amps squared.
Jacinta: So you’ve used a ‘crash course physics’ video for the last part of this conversation, which is useful, but assumes a lot of knowledge. Looks like we may have to start those videos almost from the beginning, and learn about trickonometry, and integers, and so much els
Canto: ……..

Written by stewart henderson

August 30, 2022 at 7:56 pm

what is electricity? part 4: history, hysteria and a shameful sense of stupidity

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to be explored next time

Canto: So we’re still trying to explore various ‘electricity for dummies’ sites to comprehend the basics, but they all seem to be riddled with assumptions of knowledge we just don’t have, so we’ll keep on trying, as we must.

Jacinta: Yes, we’re still on basic electrostatics, but perhaps we should move on, and see if things somehow fall into place. Individuals noted that you could accumulate this energy, called charge, I think, in materials which didn’t actually conduct this charge, because they were insulators, in which electrons were trapped and couldn’t flow (though they knew nothing about electrons, they presumably thought the ‘fluid’ was kind of stuck, but was polarised. I presume, though, that they didn’t use the term ‘polarised’ either.

Canto: So when did they stop thinking of electricity as a fluid?

Jacinta: Well, a French guy called du Fay postulated that there were two fluids which somehow interacted to cause ‘electricity’. I’m writing this, but it doesn’t make any sense to me. Anyway this was back in 1733, and Franklin was still working under this view when he did his experiments in the 1740s, but he proposed an improvement – that there was only one fluid, which could somehow exist in excess or in its opposite – insufficiency, I suppose. And he called one ‘state’ positive and the other negative.

Canto: Just looking at the Wikipedia article on the fluid theory, which reminds me that in the 17th and early 18th century the idea of ‘ether’, this explain-all fluid or ‘stuff’ that permeated the atmosphere somehow, was predominant among the cognoscenti – or not-so-cognoscenti as it turned out.

Jacinta: Yes, and to answer your question, there’s no date for when they stopped thinking about ether or electrical fluid, the combined work of the likes of Coulomb, Ørsted and Ampère, and the gradual melding of theories of magnetism and electricity in the eighteenth and nineteenth centuries led to its fading away.

Canto: So to summarise where we’re at now, Franklin played around with Leyden jars, arranging them in sets to increase the stored static charge, and he called this a battery but it was really a capacitor.

Jacinta: Yes, and he set up a system of eleven panes of glass covered on each side by thin lead plates, a kind of ‘electrostatic’ battery, which accumulates and quickly discharges electric – what?

Canto: Electrical static? Certainly it wasn’t capable of creating electrical flow, which is what a battery does.

Jacinta: Flow implies a fluid doesn’t it?

Canto: Oh shit. Anyway, there were a lot of people experimenting with and reflecting on this powerful effect, or stuff, which was known to kill people if they weren’t careful. And they were starting to connect it with magnetism. For example, Franz Aepinus, a German intellectual who worked in Russia under Catherine the Great, published a treatise in 1759 with translates as An Attempt at a Theory of Electricity and Magnetism, which not only combined these forces for the first time but was the first attempt to treat the phenomena in mathematical terms. Henry Cavendish apparently worked on very similar lines in England in the 1770s, but his work wasn’t discovered until Maxwell published it a century later.

Jacinta: Yes, but what were these connections, and what was the mathematics?

Canto: Fuck knows. Who d’you think I am, Einshtein? I suppose we’re working towards Maxwell’s breakthrough work on electromagnetism, but whether we manage to get our heads around the mathematics of it all, that’s a question.

Jacinta: To which I know the answer.

Canto: So let’s look at Galvani, Volta and Coulomb. Galvani’s work with twitching dead frogs pioneered the field of bioelectricity – singing the body electric.

Jacinta: Brainwaves and shit. Neurotransmitters – we were electrical long before we knew it. Interestingly, Galvani’s wife Lucia was heavily involved in his experimental and scientific work. She was the daughter of one of Galvani’s teachers and was clearly a bright spark, but of course wasn’t fully credited until much later, and wouldn’t have been formally educated in those Talibanish days. She died of asthma in her mid-forties. I wish I’d met her.

Canto: So what exactly did they do?

Jacinta: Well they discovered, essentially, that the energy in muscular activity was electrical. We now recognise it as ionic flow. Fluids again. They also recognised that this energy was carried by the nerves. It was Alessandro Volta, a friend and sometime rival of the Galvanis, who coined the term galvanism in their honour – or rather in Luigi’s honour. Nowadays they’re considered pioneers in electrophysiology, the study of the electrical properties of living cells and tissues.

Canto: So now to Volta. He began to wonder about Galvani’s findings, suspecting that the metals used in Galvani’s experiments played a much more significant role in the activity. The Galvanis’ work had created the idea that electricity was a ‘living’ thing, and this of course has some truth to it, as living things have harnessed this force in many ways throughout their evolution, but Volta was also on the right track with his skepticism.

Jacinta: Volta was for decades a professor of experimental physics – which sounds so modern – at the University of Pavia. But he was also an experimenter in chemistry – all this in his early days when he did all his practical work in physics and chemistry. He was the first person to isolate and describe methane. But here’s a paragraph from Wikipedia we need to dwell on.

Volta also studied what we now call electrical capacitance, developing separate means to study both electrical potential (V) and charge (Q), and discovering that for a given object, they are proportional. This is called Volta’s Law of Capacitance, and for this work the unit of electrical potential has been named the volt.

Canto: Oh dear. I think we may need to do the Brilliant course on everyday electricity, or whatever it’s called. But, to begin – everyday light bulbs are designated as being 30 amps, 60 amps and so forth, and our domestic circuits apparently run on 240 volts. That latter is the electric potential and the amps are a measure of electrical output? Am I anywhere close?

Jacinta: I can’t pretend to know about that, but I was watching a video on neuroanatomy this morning…

Canto: As you do

Jacinta: And the lecturer informed us that the brain runs on only 20 watts. She was trying to impress her class with how energy-efficient the human brain is, but all I got from it was yet another electrical measure I need to get my head around.

Canto: Don’t forget ohms.

Jacinta: So let’s try to get these basics clear. Light bulbs are measured in watts, not amps, sorry. The HowStuffWorks website tells us that electricity is measured in voltage, current and resistance. Their symbols are V, I and R. They’re measured in volts, amps and ohms. So far, so very little. They use a neat analogy, especially as I’ve just done’s section on the science of toilets. Think of voltage as water pressure, current as flow rate, and resistance as the pipe system through which the water (and effluent etc) flows. Now, Ohm’s Law gives us a mathematical relationship between these three – I = V/R. That’s to say, the current is the voltage divided by the resistance.

Canto: So comparing this to water and plumbing, a hose is attached to a tank of water, near the bottom. The more water in the tank, the more pressure, the more water comes out of the hose, but the rate of flow depends on the dimensions of the hose, which provides resistance. Change the diameter of the hose and the outlet connected to the hose and you increase or reduce the resistance, which will have an inverse effect on the flow.

Jacinta: Now, to watts. This is, apparently, a measure of electrical power (P). It’s calculated by multiplying the voltage and the current (P = VI). Think of this again in watery terms. If you increase the water pressure (the ‘voltage’) while maintaining the ‘resistance’ aspects, you’ll produce more power. Or if you maintain the same pressure but reduce the resistance, you’ll also produce more power.

Canto: Right, so now we’re adding a bit of maths. Exhilarating. So using Ohm’s Law we can do some calculations. I’ll try to remember that watts are a measure of the energy a device uses. So, using the equation I = P/V we can calculate the current required for a certain power of light bulb with a particular voltage – but using the analogy of voltage as water pressure doesn’t really help me here. I’m not getting it. So let me quote:

In an electrical system, increasing either the current or the voltage will result in higher power. Let’s say you have a system with a 6-volt light bulb hooked up to a 6-volt battery. The power output of the light bulb is 100 watts. Using the equation I = P/V, we can calculate how much current in amps would be required to get 100 watts out of this 6-volt bulb.

You know that P = 100 W, and V = 6 V. So, you can rearrange the equation to solve for I and substitute in the numbers.

I = 100 W/6 V = 16.67 amps

I’m having no trouble with these calculations, but I’ve been thrown by the idea of a 6-volt light bulb. I thought they were measured in watts.

Jacinta: Okay, so now we’re moving away from all the historical stuff, which is more of our comfort zone, into the hard stuff about electrickery. Watts and Volts. Next time.


Written by stewart henderson

December 19, 2021 at 8:33 pm