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.

Room Treatment For "Small Rooms" - Part Three

Welcome to the "small rooms" owners' club.

Once again, pro sound treatment methods (bass traps, reflectors, absorbers and soundproofing) are essential for studios and can be useful for "larger rooms". No-one is saying otherwise.

But one-metre deep bass traps are not feasible in a small room. Nor is six-inch absorption padding all round the walls. Isn't the room small enough already? Nor are serious soundproofing measures, which also require thick lumps of absorbent materials. And have you noticed that the rooms in treatment videos belong to people who don't read, have no art or decoration, and store their CD's and records in another room? No wonder they need absorbers and diffusers. Real rooms have bookcases, shelving, pictures on the wall, and other stuff. Some of that helps. What else can we do?

If you are following the user manual for your speakers, you will be sitting about two-three metres from them. No matter what size your room is. Or what the speakers are. (Have you seen how close people sit to those Wilson towers?)

Contrary to some commentators' sniffy remarks, thick pile carpets and loose hanging curtains a distance from the wall do work: see this table of absorption coefficients. Curtains and carpets are pretty much third for absorption after foams and fibres, and then people.

Am I sure I'm not rationalising my unwillingness to spring for a dozen GiK acoustics panels for £700 or so, plus all that drilling and hanging? Well, that's why I write things like this: to make sure I've got my facts in a row. And I think I have.

There wouldn't be home hi-fi if the first thing you were told by your dealer was "we couldn't sell you any of this with a clear conscience until you've had your listening room re-built by an expert, otherwise you'll just come back and complain it sounds terrible". It has to be pretty good out of the box in nearly all circumstances.

We small-roomers are left with the simple things, which are more about the overall sound of the room than specific flaws.

Rugs for wooden floors

Curtains to the full width of the room so that the corners as well as the windows are covered

Shelves with books, record collections, even storage (as long as it's not a wall of boxes), for dispersion and absorption. Just don't line everything up neatly or you'll lose the dispersive effect

Symmetry: equal spacing between left and right speakers to their near walls, same distance from the front wall, both at ear height; books or records (aka 'damping') to the right wall means books or records ('damping') to the left wall in the same place.

All that work for that conclusion? Hey, I saved a lot of money on those acoustic panels.

So I'm upgrading the carpets and doing something about the (long story) curtains on the windows. The front wall is going to be a curtain with folds hanging from a tension bar. That will do for now. In time I may change the furniture around and get some more natural damping.

Room Treatment For "Small Rooms" - Part Two

Room treatment is mostly about room modes and reflections. (Sound-proofing is taken to be out-of-scope since it needs building work.)

Room modes first. These are sound waves that bounce back and forth between the walls, or floor and ceiling, because the wavelengths fit the dimensions just right. This is where boomy bass comes from.

Small rooms are held to be a lost cause because they have many, many of these resonating frequencies. That may be true, but there's a VERY important qualification: those room modes only matter if any of them correspond to one of the 88 frequencies used in the music you are most likely listening to.

Yep. There are zillions of frequencies that could be used to make music, but almost all Western Music is made with 88 of them. Here's a list. You will notice the only ones that are whole numbers are the A's from 55Hz upwards. All the rest are given to five (5) decimal places, in a scheme called twelve-tone equal temperament (which is a music theory rabbit-hole all its own). Western musical instruments are mass-produced to reproduce those notes. The chances of your room having a resonant frequency corresponding to some random note like F# above low-C (say) are approximately zero.

And if you do, all you have to do is move the speakers either back or forward a couple of centimetres (front-to-back resonance), or closer or further apart a couple of centimetres (side-to-side resonance), and it will disappear. (This is called positional equalisation.) It will not to be replaced by one on another note because a) the difference in wavelengths between any of the 88 notes is more than a couple of centimetres, and b) the resonance is between the speaker and the back or side walls, not between the front and back walls, which would be a room resonance, and your speaker is not mounted in the walls. (And even if it was, the point remains the same!)

If you have a floor-firing subwoofer, you can't fix a room mode like that, because the way the sound waves come from the subwoofer mean the resonance will be from floor-to-ceiling. Should a piece of music contain a hefty belt of 73.4 Hz D or 36.7 Hz D, both of which will pass into my subwoofer, I get a resonance. However, only five-string double-basses and instruments with names starting 'octocontra' ever get down to 36.7 Hz, leaving me with exactly one note that can trigger that resonance, and that's still way down low even for a string-bass. And no, very few pieces of music are written to include octocontra-anythings, and most orchestras would either not perform them, or use the programme or sleeve notes to apologise for the missing instrument. The lower you cross over to your subwoofer, the fewer possibilities for resonance you have.

How about all those reflections? According to the Master Handbook of Acoustics

Our hearing mechanism integrates spatially separated sounds over short intervals, and under certain conditions tends to perceive them as coming from one location.... in an auditorium, the ear and brain have the ability to gather all reflections arriving within about 35ms after the direct sound, and combine...them to give the impression that the entire sound field is coming from the direction of the original source, even though reflections from other directions are involved...This is called the Precedence Effect, Hass effect, or law of the first wavefront.

In more familiar terms, the ear has a buffer about 35ms deep. At the speed of sound that's 12m. I am two metres from my speakers. Any sound along a path strictly less than 14 metres from speaker to ear will have its sound combined with the direct sound from the speakers. That's all the first reflections in my room. So in a "small room", first reflections do not appear as separate sound sources. Instead, those reflections give the sound a sense of spaciousness which is greater as the power of the reflections increases. Reflections have to be quite loud before they are perceived as echoes. (In my "small room", the first reflections are travelling something like 3m to reach me, so they are 4/9 (inverse square law) as powerful as the direct sound, which leads to a drop of 3dB in volume and whatever absorption I get at the wall. Every little helps.)

Reflections good - sometimes. Too many and too loud, and the sound image will lose sharpness or you will get echoes. Too few and too quiet and the sound will feel muffled and dull. The trick is to get the sound quality you like.

Those with "small rooms", read on.

Room Treatment For "Small Rooms" - Part One

Steve Gutenberg says we should try room treatments. John Darko has those GiK boards all over the place. I'm starting to feel I'm not responding to the memo.

Except...

Acoustic treatment for studios is a real thing: studios need all the soundproofing their owner can afford, and a lot of plain old echo-deadening in the recording space. Performers don't like heavily damped studio acoustics, which is why some of them prefer to plug into the board and hear each other over headphones. Then they can perform together in a room that looks as if it was designed for humans.

Audio design for a concert-hall is a real thing. Soundproofing from outside noise, getting a decent quality of sound in all the seats, tuning it to be lively or dry, depending on the taste of the resident orchestra, or the kind of music that it will be played in it, all this is serious stuff. Consult someone with an acoustics degree. Something similar might be said for the listening rooms of millionaires, who can afford to have chunks of their houses re-built or re-furbished.

I can't, and I assume you can't either. I have a rectangular room that's 14L x 10w x 7.7H in feet, just over 1,000 cu ft. In the trade this is called a small room. The Master Handbook of Acoustics(*) dismisses any room of less that 1,500 cu ft as a lost cause. Others define "small" as anything smaller than a classroom.

Watch a few room treatment You Tube videos and you will wonder how on earth you are able to hear anything, let alone identify it as Miles Davis' Sketches of Spain. What with diffusion, room modes, reflections off the floor, the ceiling, the side walls, the back corners, runaway bass that needs to be trapped... it's a wonder you can hear the tuba.

Except...

Once the speakers and listening position are set up right...

...and you've got some carpet on that varnished pine floor...

...and you've accepted, like me, the inevitability of hauling the couch from the back of the room so that my head was the third point of a (nearly) equilateral triangle with the speakers (**)

...you can indeed hear the tuba. And everything else. And no obvious echoes or reflections.

How does this happen?

Read on.

(*) That's what it's called, and you get 10,000 Audiophile Points for reading it.

(**) Which is where speaker manufacturers say to sit. If you don't do this, and put your speakers at different heights and distances from the walls, and have books below them, then the orchestra may well wind up in the upper-right-hand corner of your room. So a friend told me, anyway.