Lightbulbs and the Poynting Vector (Veristasium)

Electro-magnetism (E&M) is genuinely weird. Most people never find this out, because most people never go into electrical engineering or a physics PhD (where you really have to grapple with it).

Most people think of electricity as volts, amps and watts. Maybe ohms. We don't use ohms in a household context.

The initiated talk about capacitance, reactance, inductance, resistance and conductance. They talk about "transmission lines", "skin effects", and "antennas". The rest of us need to be electrical engineers before all that makes sense. (Oh. Wait. I almost was one.)

Here's a way in: metals like copper are often called good "conductors of electricity", as if electricity is something that passes through the metal. Instead, think of metals as good receivers of electromagnetic radiation. Wires do not in some sense channel or concentrate the electromagnetic fields, or act as pipes for electrons to flow along, they respond to the electromagnetic fields. Indeed, everything responds to electromagnetic fields. Mostly not much.

Wires respond by creating their own little electromagnetic field around them. Most materials (because "everything is a capacitor") respond by retaining tiny, tiny amounts of charge which they then eventually let go of. Air does this. So does polyester, which is why it crackles when you take it off.

Mr Veritasium set up a circuit with a battery, a switch, a light opposite the switch and some very long wires connecting everything. The idea was that the wires would be so long it would take a noticeable amount of time for the electricity to "flow" along the wires and power the light.

Except the light comes on instantly.

His explanation uses a thing called the Poynting vector. Do not use those words near physicists, as they may call your bluff.

Electromagnetic waves have an electric field (E) and a magnetic field (B) that are always in phase and at right angles to each other, and to the direction of travel of the wave. (This is why there have to be at least three physical dimensions, or we couldn't have Radio Three.) Since electromagnetic waves carry energy, it makes sense that there should be an energy vector corresponding to the wave in the direction of travel. Poynting proved that this vector (*) S = E x B, where 'x' is the vector cross-product, and I've left out a constant of proportionality. It's the E times B bit that is the achievement, not the direction (cross product), because we got that already from the physics.

So Mr Veritasium said, the electric field is pointing this way (points along wire) and the magnetic field is pointing that way (points upwards) so the S vector must be (pointing at the light bulb). Presto! The energy gets to the light more or less instantly.

Which convinced absolutely nobody, because they piled in to discuss this using anything but Poynting vectors. Transmission lines and displacement currents was a favourite, because, well, engineers. Nobody was doubting the Poynting explanation, because physics > engineering, but because they were engineers, they wanted to explain it in terms of something more familiar and material.

Complete the well-known phrase or saying: cart, horse.

Poynting's insight was that the materials in which the wave moves (e.g. air, wires) do not facilitate the power transmission, rather they modify the electro-magnetic waves, and hence the power transmission. The transmission-line / displacement current explanations are consequences of the transmission of power in the direction of the Poynting vector, not explanations. When modelling a specific setup, the B vector (which is for free space) is replaced by the H vector, which takes into account the effect on the B vector of the materials involved.

What happens is this: when the switch is closed, a voltage pulse starts to travel round the circuit. This creates a magnetic field B through the Maxwell equation (with J = 0) for B

$\nabla \times B = \mu_0 \epsilon_0 \frac{\partial E}{\partial t}$

which creates a Poynting vector S. Behind that pulse comes the first lap of a current J that will be circulating once the circuit is in a steady state. That sustains the B field by the Maxwell equation (with $ \frac{\partial E}{\partial t} = 0$) for B

$\nabla \times B = \mu_0 J$

which sustains the Poynting vector S. (The E field is sustained by the battery voltage). That S field carries the energy that excites the molecules in the wire in the bulb and creates the light. Because the wire in the bulb is a good receiver of electromagnetic radiation.

Not a transmission line in sight.

It's worth noting that if the bulb was put, say 300,000,000 metres away from the switch on the opposite side of a loop, then it would take 1 second for the bulb to light, but that would still be faster than the roughly 1.6 seconds it would take for the voltage / current wave-front to reach it.

This is, of course, handwaving. More precise calculations would take account of the dielectric air between the wires to calculate H and also factor in the displacement currents, but the principle remains the same. That would start to sound like engineering. But the engineering is there to help perform the calculation, not to help understand what's happening.



(*) Strictly E, B and S are not vectors, which are 1-forms, but flux densities, which are 2-forms. This is the only time in your life you will ever read that.

Philosophy of Mathematics - Number Theory

Off in another part of my thoughts, which have been on hold for a while, I have been trying to work out some ideas on the philosophy of mathematics.

I have two theses. One is about the relationship of abstract mathematical ideas to various types of measurement or geometric properties. If you want to know how the various derivatives on curved spaces arise from the simple issues of co-ordinate changes, it's all there. The other is a methodological thesis, that the purpose of mathematics is to provide tools and techniques to solve problems that arise from modelling physical and other processes, and to understand the scope and limits of those techniques. Creating and solving the equations of the mathematical models is what's usually called "applied mathematics", while understanding the scope and limits of the techniques is a lot of what's called "pure mathematics".

And then there's Number Theory. Which is about numbers. Not mathematical models.

You know that Langlands thing that all the Kool Kids are working on?

Yep. Number theory. Finite field number theory at that. Geometric Langlands is even more abstruse.

It takes genius-level insight and technique to understand the more recent developments in Langlands. That's the point: if the specialists can barely follow it, how is it going to be any use to some poor post-grad working on differential geometry at the University of Ennui-sur-Blase?

The social purpose of mathematicians is to teach other people - physicists, statisticians, epidemiologists, computer scientists and programmers for example - how to use the problem-solving techniques mathematics offers. What mathematicians do in their spare time is their business: they need a decent laptop, a whiteboard and some paper and pens: math is cheap compared to fundamental physics.

The Langlands guys can do what they want in their spare time. But it's a rabbit-hole. Maybe it's a big, well-lit rabbit-hole with all the health and safety gear and plenty of mechanical digging tools, but it's still a rabbit-hole. Unlike some of the rabbit-holes mathematicians have buried themselves into (functional analysis, for instance), Langlands is not going to produce anything useful to regular working stiffs (for instance, functional analysis produced the theory of weak solutions to differential equations, which is very useful). I feel confident saying that because Langlands is about structures the rest of mathematics just doesn't use.

(Rabbit-holes are as opposed to specialisms, which are very specific subjects that have useful applications in the real world or other parts of maths with real world applications. Like research in PDEs.)

Maybe "rabbit-hole" should be a term of art in methodology. It's a line of research that has no obvious application to any existing problems or in other branches of maths. The scientific version would be a research programme that was making theoretical progress but no empirical progress (was not making new predictions). A rabbit-hole may branch up to the surface every now and then, as applications to problems in other branches of maths are found, but generally once dug, the researchers dig away happily underground.

In this case I would be saying that Number Theory was a mathematician's pastime, and that other very abstruse, or very off-beat, programmes, are for all the sophistication, esoterica for the aficionados. Which doesn't sound too dramatic.

November 2021 Diary Update

I have "missed" a number of posts this month. I've also missed a number of trips to the gym, and there were whole days when I barely left the house except after dark to have my daily walk.

I stopped going to the gym because I had a problem with my right hip which had reached see-the-osteopath serious. Osteo's don't like it when you come in after a weights session with tight and hard muscles. They can't do those odd manoeuvres to put your spine back in alignment easily or sometimes at all. I've got one more visit left and it should be clear. When I go back to the gym it won't be to do heavy-ish weights as I have been doing. My days of ego-lifting are now well past, and really stopped early in 2019. I'm going to be all about the health-lifting, which is nowhere near as much fun.

The days I barely left the house were about a) the lack of motivation that sweeps over me at the sight of a dull grey sky, b) the fact that 10,000 steps in a day now wipes me out when in 2019 I could do that standing on my head, c) a lack of connection with London and all other places. This is all about me pulling myself together and just f***ing doing it and various other deeply sensitive maxims. There's a thing called "commute hardening" that we never notice because we commute all the time, but all those months working from home have left me and many others "commute soft", and unable to handle the amount of walking and effort needed for a commute. (That's going to be a real issue when getting all those cosseted bankers and civil servants out of their homes and back into their over-crowded open-plan offices.) So I'm working on building up the ability to handle a 10k-step day without feeling exhausted halfway through. Nothing I can do about my reaction to grey skies, except stop being a cissy.

I've missed the posts because I've been caught up in various decisions and other things, none of which I could formulate coherent thoughts about. (Which has never stopped me in the past.)

I am not going to put a curtain against my front wall, as I suggested I might in a previous post about Room Treatment for Small Rooms. Nor am I going to buy some absorbing panels from Ginger White (not actually a lot more expensive than some curtains). I came this close to both.

I decided that a) having 10kg of absorbing panels on my front wall right above all my kit would be disastrous if any screws came loose (you don't know my walls), b) I couldn't man-handle something that size and weight on my own, c) what happened if it didn't work enough, or was more absorbent than I could live with? As for the curtains, it would look odd, but because you should leave space between curtains and the wall, the curtains would be tucked in behind the Kallax units and it would all look silly.

I finally got up the nerve to bust out the drill, measured up, drilled three holes (two into brick, one into plaster - I do not live in a precision-built house) put in Rawlplugs and screws, and hung three of my collages. Perspex has to be as reflective as plaster, so I'm not expecting acoustic improvements, but at least I'm not staring at a blank white wall anymore.

Because for a domestic listening room, sonic treatment screams you can take this too far, you know.

And I've been reading as well. You have no idea how fascinating the theory of antennas is. I started my life as an electrical engineering student, and while I get electric circuits, I've never really grokked electromagnetism in all its weirdness. Antennas are exactly that. And that was just one of the subjects I read.

So there will be the usual gratuitous back-filling, and I will carry on. The self-imposed restriction on writing about "current affairs" (as we used to call it) does remove an easy source of posts, but it also stops me wasting time on nonsense, or at least writing about it.

Living Right Takes Character, Not Purpose

You know what I hear far too much?

That our collective problem is a lack of meaning / purpose / connection with others / (enter something else that post-modern Capitalism doesn't encourage here).

If only we didn't do b*******t jobs for ungrateful bosses, and we had supportive connections with our neighbours and family, didn't eat meat or burn carbon, saved a species every week, planted a tree every day... then we would feel fulfilled and happy and not do sad things like binge eat / drink / watch TV series / play computer games / (enter things you wish other people would stop doing here)

Hah!

It takes real character to accept that your life is insignificant, and yet still behave as if you are a worthwhile human being.

Go to work, exercise, eat right, not drink too much, keep yourself and your digs clean and neat, see your friends and relatives, save for the future, and keep yourself entertained and interested in something.

Live right, even if you don't know why.

Allez en avant, et la foi vous viendra, as D'Alembert is supposed to have said.

Second-Best Housekeeping Productivity Hack

(The best is to pay for a cleaner. Everyone living in a shared flat or house should do that.)

Get a robot vacuum cleaner.

Really.

Vacuuming is a chore. Not everyone feels that way, but normal people do.

How good were these robo-vac things?

I watched the You Tube review videos, and decided that the base-level Eufy 11S Max would suit my needs just fine, as well as being far more affordable than the ones that use GPS tracking to learn about your house so they know what to do in the master bedroom next time. You can YT the review videos using the name.

Prep the room: get all the wires off the ground, and I put small items of furniture on couch / bed. (I went round and used wire ties to lift a lot of wires off the floor. Which of course I should have done anyway.)

There are two basic modes:

a) you put the cleaner in a room, set it off, close the door and get on with something else somewhere else

b) you let it do the vacuuming, and go round with the damp cloth wiping down surfaces and skirting boards

Admit it, did you wipe down the skirting boards when you vacuumed? Thought so.

That's the productivity bit: you can do something else while it does the chore.

It can sense when it's on carpet and turns the vacuuming up a notch, and it can sense it's about to run out of landing, stop and turnaround. It doesn't fall downstairs.

Don't be control-freaky, it wanders about and eventually covers everywhere it can reach. You do have to get into the corners, but you probably missed those when vacuuming and had to do them specially. It has an 'edges' mode where it will go round the walls.

Empty the tray at the end of each session.

Admire your new cleaner digs.

(No. I'm not being paid. Buy a Robo-Vac if you want.)

In Praise of A Well-Chosen Indulgence

I've been watching how-to-record-your-music videos recently and ran across the wonderfully over-the-top Spectre Sound Studios. In this one he talks briefly about debt and how it should be avoided. Turns out he owns his house, studio, and car. How? Because, in his words, he bought what he needed, not what he wanted.

Spectre Sound was saying: don't spend £1,000 on the fancy gear when £200 will get you something that will do the job well enough that the audience won't notice the difference.

Most companies do this. Most companies buy the least-cost, lowest spec-for-what-they-think-is-needed kit. Sometimes they get away with it, and sometimes they lose thousands of hours a week from laptops that take ten minutes to boot up and shut down, and are unusable for an hour when doing weekly updates (ask me how I know). The company doesn't care that their staff feel that they aren't worth decent kit. That's what economic-value-optimisation makes us feel like. Not important enough for the Good Stuff.

In order, there's

a) doing without

b) buying the least-cost, lowest-spec

c) buying nice so you don't buy twice

d) indulgence (buying something that's a little better than "nice" because it makes you feel warm and fuzzy)

e) wasting money buying specs or stuff you are never going to use. 

For example, bread:

a) is not buying any bread because, well, who needs bread? you're not going to die if you don't have it b) is Tesco White Sliced c) is a sourdough from a supermarket d) is a loaf from Paul or some other such brand e) is any loaf from Whole Foods or an "artisan" baker with their own shop in Notting Hill or Greenwich

The base-line for judging indulgence or waste is not "go without". Nor is it Tesco White Sliced, which barely qualifies as bread. The baseline is the cheapest "buy nice" option. It's deviation either way from "Buy Nice" that needs to be justified. We should justify buying low-quality-and-cheap as well as higher-quality-and-more-expensive.

Buying what you want, without wasting money, just makes you feel like, in the words of the ad, you're worth it.

(Now you ask, I buy whatever shampoo is on sale, but not the diluted cheap own-brand stuff.)

This is why I have always had the suspicion that people who buy what they want rather than what they need are a little more fun and little richer and softer in texture. (Assuming they aren't being stupid with the "wants".) By adopting a very narrow definition of "need" - as in "you're not going to die if you don't go on holiday / only drink tap water / never eat chocolate" - one can lead a very miserable life.

Until April 2020, my indulgence was membership at a fancy gym in Soho: there are gyms that cost more, but most cost less. A lot less. But I enjoyed and appreciated it. I joined my local The Gym in April this year. I don't dislike it, but I don't feel a little thrill walking through the door either. It's functional, and that's it. The local fancy gym charges not far short of what I was paying for my previous gym and is over-priced for what it provides. "Need" rules in this case.

I may be missing a well-chosen indulgence.

Holland Park Japanese Garden

 

One fine sunny day in early November, Sis and I wandered round Holland Park. I hadn't been to Holland Park since one of the summer evening concerts back maybe in the Oughties. It's still there and it's much as I remembered it. I hadn't visited the Japanese garden before and was glad we did.