How To Translate Faraday's Law of Induction into Math

I know what you're thinking. What does he get up to that stops him posting promptly and prolifically? I wish it had something to do with Instagram models and / or  staying up late making music via Garageband, but it is much more mundane than that. Here's a short passage about the translation of Faraday's Law of Induction into mathematical notation that I've been working on for far longer than you might think. If I've done my job well, it should seem obvious. (Some of the original \LaTeX has been butchered to accommodate Blogger.

(starts)

Faraday's Law, more or less as stated by Faraday, is: the electromotive force around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path. How does this get translated into mathematical notation? We need to know that the `electromotive force' is, in the case of magnetic induction, the work done on an elementary electric charge (such as an electron) travelling once around the loop. Work done moving along a path is always a line integral of the product of a force and a displacement (since `work = force times distance').

As a first step, we re-name those things as variables or constants:

let $\mathcal{E}$ be the electromotive force

let $B$ be the magnetic flux

let $\partial A$ be the path, enclosing an surface A

let $ds$ be a small displacement along $\partial A$

let $E$ be an electric flux field

We can write down the equations quite easily if we are familiar with the vector calculus. Work done is given by the mantra `work = force times distance'. For a small displacement $ds = (dx, dy, dz)$ and a force $E = (E_x, E_y, E_z)$ the product is $E_x dx + E_y dy + E_z dz$ which is $E \cdot ds$ in vector notation. The work done along a line is the sum of such displacements along it, which is conventionally shown by the integral $\oint_{\partial A} E \cdot ds$, giving us $\mathcal{E} = \oint_{\partial A} E \cdot ds$.

Translating the other side of Faraday's Law, Faraday thought of electromagnetic fields as `lines of force' - the more lines, the more force - and the flux of a field through an area was the number of lines of force through it. This was Faraday's way of thinking about line and surface integrals without having to actually use either. 

The number of lines of force within a path is the integral of the (strength of the) vector field over any smooth surface enclosed by that path. (The `any' has to be proved, but it is becomes intuitively obvious after visualising a few examples.) So if we take a surface $A$, divide it into non-overlapping patches $dA(n)$, calculate $\frac{\partial B}{\partial t}(n)$ for the centre of the $n$-th patch, and add the total, we get an estimate of the electromagnetic field strength. Make the patches smaller, and we get a better estimate, which in the limit is the integral 

$\mathcal{E} = -\iint_{A} \frac{\partial B}{\partial t} \cdot dA$

That can also be turned into a conventional double integral by substituting coordinates. Hence Faraday's Law of Induction is translated into mathematical notation as

$ \oint_{\partial A} E \cdot ds = -\iint_{A} \frac{\partial B}{\partial t} \cdot dA$

The left-hand side is the work done, and the right hand side is the negative of the time rate of change of the magnetic flux enclosed by the path. This completes the translation of Faraday's Law into mathematical notation.

This is no more conceptually complicated than if we had translated, say, a passage of Freud from German to English. There is no word-for-word mapping of the two languages, and there are many concepts for which there is a German word, but not an English one, and one must attempt to explain the German concept in English. Using an integral to denote the result of a limit of finite sums is no more exceptional that using a derivative to denote the result of take rates-of-change over ever small intervals. 

We can use some maths to go further. By Green's theorem, assuming the fields are sufficiently smooth, we have

$\oint_{\partial A} E \cdot ds = \iint_A \nabla \times E \cdot dA$

So we can put

$\iint_A \nabla \times E \cdot dA = -\iint_A \frac{\partial B}{\partial t} \cdot dA$

which gives us immediately one of Maxwell's equations

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

We can prove that, with the rest of Maxwell's equations, this another statement of Faraday's Law of Induction. 

This is no more conceptually complicated than if, having translated the passage of Freud, we then drew a conclusion from the translation and some background knowledge that was not in the original, but helps us understand what Freud was saying. It just looks impressive / mysterious / difficult because it uses undergraduate maths.

(ends)

My thesis is that translating from a natural language into math notation is the same as translating from one natural language to another. It's just that maths is the language in which it is easier to see the patterns and make the deductions.

"Good People" As Useful Idiots

Throughout history groups of people have wanted to identify each other, and to feel superior to everyone else. That's what the "Shibboleth" story (Judges 12:5-6) is about. Not only must the Bad People find it almost impossible not to betray themselves almost immediately, but it must be difficult for the Bad People to turn into Good People.

Good People usually work in the public sector, broadly construed as any occupation financed almost completely by the taxpayer. NHS, universities, schools, the Armed Forces, the Emergency Services, BBC, local council staff and officials, and publicly-funded arts organisations. It's about a third of the UK workforce. It's not everyone in the public sector. Good People are usually managers rather than doers, and policy-oriented rather than operational. They see themselves as being funded by the Government, and prefer political parties that hand out money, which has usually been the "left-wing" parties. They loved the EU, because of its generous grants, hence their existential panic over Brexit.

Their education stressed going along with the prevailing group-think, which can happen in the Science Faculty (think "Climate Science") as well as the Arts. They are usually not engineers or researchers whose work can be disqualified by an experiment. Good People often believe in "experts" rather than "facts" and "experience", not least out of collegial courtesy, since many of them are "experts". This makes Good People sitting targets for frauds, grifters, marketing departments, visionaries, snake-oil salesmen, and pseudo-scientists.

A qualifying Good Belief must meet a number of criteria: it must work against the interests or beliefs of regular people; it must have a very low probability of ever being turned into legislation; and it is must have a very low probability of Capital ever turning into a profitable product or service; and it must not be detrimental to the Good Person. Also, it helps if it is more expensive than the mainstream alternative.

Refusing to use modern medicine is not a Good Person belief, since it can cause harm to the Good Person. Veganism is a Good Person belief, as is keeping kosher. World Peace is not a Good Person belief, because who doesn't want that?

Being "for" homosexuality was a Good Belief. It had resisted changes in legislation for hundreds of years; it is not well-regarded by the common herd; even after the 1967 legislation, Capital has been hesitant to aim products at the Gay market, partly because it is too small to support major spending; and of course, if the Good Person was gay, it was a bonus, and if they were straight, it didn't matter. Until their children came out. By the 2010's the Police across the world were happily taking part in Pride marches, as were lots straight people and politicians. Homosexuality is now nothing remarkable.

Which was an utter disaster for the Good People. For about five decades, the Good People had been able to distinguish themselves by professing a belief that homosexuality should be nothing remarkable. And now the Bad People agree. Which means the Good People needed to find some other cause to distinguish themselves. If that cause was going to be around sexuality, it was going to be pretty far out. And it turned out to be just that.

Climate Change had some things going for it. The facts are iffy as heck, which means that one must believe. Bad People want facts, Good People are Believers. Nothing bad was going to happen because of Climate Change (if was real) in the short-term: climate disaster has been ten years away for the last thirty years. For a couple of decades after the start, there was no hope of it turning into legislation, there were no serious alternative energy sources, and few demands on individuals' behaviour.

Having baited the Good People, the Climate Change con-men pulled the switch. They continued to press for legislation from whichever body might find it a constructive-looking gesture. They let the investment banks in by inventing so-called "carbon offsets". They persuaded Governments to subsidise wind farms, which made alt-energy a better investment. Nobody ever pushed the "energy security" issue, because that smacked too much of Nationalism, and Good People are not Nationalists, but energy security (from the increasingly volatile Middle East) was the unspoken benefit of all this. The con-men flooded the culture with pseudo-science, graphs compiled by methods so dodgy as to make political polls look honest, and frequently-published forecasts of possible disaster and death. The Good People found they had been suborned into a full-fledged activist movement, and it took a few years for them to disassociate themselves from it. People who use phrases like "climate emergency" sound like the cranks they are. Sadly, that crankishness has been embedded into Government policy, and it will take our politicians a while to extricate themselves from it.

"Diversity" is a Good Person belief. They are insulated from it by their postcodes and occupations, and by the fact that Good People are inter-nationally the same. The legislation had long been in place, it cannot be turned into a product or service, and it provided a lot of cheap labour, often to look after Good Children. Furthermore, "diversity" could mean that one's wife was now up for a lucrative part-time Directorship. In the meantime, "diverse" neighbourhoods and organisations have much lower levels of social trust, participation, co-operation and communication. Crime figures are carefully never printed or discussed. It was hi-jacked by a number of interest groups, all of whom measured it by looking at outcomes, while simultaneously denying they were imposing quotas.

"Being nice to illegal immigrants" was briefly a Good Cause in 2016. The Economist - the reference Good Person source - even suggested that a million people was not so many given the population of Europe, and we could absorb them. It never occurred to the witless writer that there were several hundreds more of those millions in the queue. Women in Germany and Sweden felt guilty about having to explain that the men who molested or raped them were, well, not native Swedes. The official obfuscation of the crime figures was driven by one thing: the incumbent Governments were all left-wing socialists, and they were terrified that the right-wing would get support if the truth was made public. Also, the fact that these immigrants seemed to have the phone numbers of lawyers, and detailed instructions about the welfare payments they could get, and of the European rail system, started to seem a little odd.

The immigrants had NGOs helping them, and those NGOs needed money. That money did not come from millions of donors, but from a handful of wealthy activists who have more money than they will ever need, and are looking for elite social recognition, which they get through backing Good Causes. The emergence of that class was something new in the 2000's: there had been rich people before, but they used their wealth to help themselves, even when founding institutions in their names. The new class did not want popular publicity and recognition, but only to belong to an elite. They did not elderly men and women gluing themselves to the M25: they did it through PR companies, NGOs, and lawyers who realised there was a nice living to be made pandering to them. Small in number, these are the most dangerous kind of Good People, as they can and have inflicted huge amounts of harm on ordinary people.

Illegal immigration in the USA is a large-scale political grift: the main beneficiaries are some large religious organisations who are paid to place illegal immigrants, and they do so because the Democratic party uses the immigrants to jerrymander voting districts. It is hidden, badly, behind the facade of a Good Cause, which is used by Democrats to pose as morally virtuous. Good Causes are now often hi-jacked for very different purposes.

Capital has been hi-jacking Good Causes since the start: Good People make up a decent-sized, high-margin market segment. Electric cars. Fake meat. Green electricity. The Whole Foods chain. Feminism was hi-jacked to get women into the workforce (notice how they got the jobs, but never the creches that the original feminists were asking for). MeToo was hi-jacked by managements and used to remove people - usually older white men - from their jobs or contracts on a mere allegation. ESG / DIE scores were invented specifically to whitewash the activities of hedge funds and investment banks. In return it was exploited by the certifying agencies, who realised they could push extremist agendas, and the hedge funds simply would not care.

The ability of Capital to react at speed does not help: no sooner does a cause become Good than marketing departments start to think about how they can hi-jack the cause reflect their customers' concerns in their products and messaging. Those marketing departments are often full of Good People who see themselves as furthering the cause. And hey, if they can get a promotion out of it, that's a win-win. After all virtuous consumption is the easiest virtue of them all.

Happy Birthday To Me

I am now 69 years old. I prefer the round decade birthdays. The 9-ers are a bit of an anti-climax

I have worn discs in my neck vertebrae

I have to take ibuprofen to get a reasonable night's sleep as a consequence

I'm having private osteopathic treatment to help manage it

I take lanzoprosole for a hiatus hernia

Last week, I was putting antibiotic on my lower eyelid to get rid of a sty

My blood sugar is a little iffy

My skin is losing its elasticity

I have a discreet chicken-crop at my throat

It's hay-fever time and I'm fall asleep if I remain in the same position for more than forty seconds

On the plus side...

The osteopathic treatment does seem to be working

Everything else works

I still have all my teeth (*)

I was sober yesterday, and the day before, and the day before...

Proudly representing for lifetime bachelors

I live inside the M25

God bless the Freedom Pass

The car has been thoroughly serviced and repaired and is ULEZ-compliant

I'm getting some decent tones out of the Les Paul + Katana

And when the ****ing sun shines again, I will take some good photos with my Fuji



(*) You need to be pretty old to remember that one.

Frost Spiders

 Taken earlier this year. I knew I must have seen something when I snapped this, but it took me a while to see it.

What Proportion of Your Music Collection Did You Play...

Last week? Last month? Last year?

I have a thousand CDs (give or take) and of course many thousands available via Qobuz.

Suppose I listen for four hours a day - these days that's four to six albums. Twenty-eight to forty-two a week. Say thirty-five a week, or 1,820 a year. Some of that is going to be "new music" from the streaming service. I'd say around five a week or 260 a year.

I have not kept records, but I reckon on any given day, I will be choosing from a pool of about 150 albums or artists from the last two or three months' listening. I reckon I listen to about 300 different albums a year. Not new albums: all albums. Another way of saying that is that I listen to each album six times a year, or once every eight weeks or so. Half of those albums will be streamed.

So why do I need 1,000 CDs in boxes cluttering up my Kallax units? As for the books in the same units, I haven't read most of them a second time.

This is where the minimalist / maximalist thing comes in.

Maximalists love yards of shelves reaching to the ceiling, loaded with LPs, books, CDs, magazines and anything else. It's a record of their life and how it has changed, as well as how much continuity it has. Maximalists live in a present suffused with the past.

Minimalists be like: do I really need to be reminded of something I'm never going to play or read again? It was of its time, and that time has passed. Minimalists live in the now and the past only exists insofar as now reaches back for it.

I'm pretty much a minimalist. Also, I live in a small terraced house and I don't have a lot of room for what amounts to an archive. So it's time to move stuff in and out of archive. (The archive is a bunch of shelves in the box room.) Also, if it's in the archive, I don't need to feel guilty for not playing it.

Staying

My CDs fall into a number of groups:

Miles Davis 

Other Jazz 

Rock / pop / folk / flamenco 

EDM 

Plainchant / early choral music 

Bach

Other Baroque 

Contemporary avant-garde 

Other "classical" (inc Mozart, Beethoven, Mahler etc) 

String Quartets (good for working first thing in the morning while under headphones)

I haven't listened to the early Church music for a long while now. It was a period I went through. The same goes for the contemporary avant-garde stuff. Chalk that up to education.

Miles is staying. The jazz is staying. So is the Baroque, Bach and Handel. Probably the "other classical" as well. I'm never going to play Second Best In the Infants at home because it's just too loud, even if played quietly. It's train music. But I am going to play the less in-your-face EDM I have on CD (not a lot, a chunk of it was bought as train music from Amazon). The rest can go into the archive, until the day I think that I could just do with a blast of Luciano Berio.

Archiving

Upgrading My Music Experience

There's more to the hi-fi listening experience than the sound and the gear and sources that produce it.

Everyone talks about "the room" with its reflections, interferences and standing waves. It's a wonder we can hear anything really. But I'm not thinking about that.

Having a comfortable chair or sofa to sit and listen is important. If you think hi-fi is expensive, try buying a well-made armchair or sofa.

There's also the visual experience: what are you looking at when the music is playing?

And there's another one, which doesn't have a name, but is something to do with not making the best - whatever that means - of one's vast music collection. All those CDs that haven't been played in two or three years, and worse, all those CDs on Qobuz I haven't even found out are there. People with wall-high collections of vinyl will know the feeling of not even remembering they had that album. All that music, and we're ignoring it.

I thought far too much about this stuff (so you don't have to). As a result, I have

a) organised a wi-fi extender for the Hegel H120 so it can use Air Play and I can update it without running LAN cable all over the downstairs 

b) removed all the CDs from the top of the Kallax and put a vase of flowers between the amp and the left speakers 

c) spaced the speakers an extra six inches apart (makes a difference to the sound, really) 

d) turned up the subwoofer a tad (makes even more of a difference to the sound) 

e) accepted that at any given time, I will be playing 1 of about 150 CDs / artists, which is 15% of my collection, hence 

f) I have pruned some of the CDs from the downstairs boxes into the box-room archive 

g) made a directory of the pop / rock / electronic music I have ripped to AAC / MP3 that isn't in the CD boxes, and pointed the iOS Music Streamer at that 

h) resolved to play my Favourites on Qobuz more often 

i) put subscribing to Apple Music (for the classical service) on the to-do list 

j) appreciated that the Sonos app is an excellent consolidator for streaming services and files (the new search looks a lot better)

Which is a lot cheaper than getting a Roon subscription, less time-consuming than ripping everything to FLAC, buying another set of headphones, or upgrading the gear.

See, the problem wasn't the gear. That's as good as I'm going to get for the money.

The problem was the music collection, or rather, the way I was using it. All that music sitting there sulking because I wasn't playing it. No! said Jacob Obrecht, he's not playing Jefferson Airplane again! He did that two weeks ago! What about rotation? Diversity? Equality of playing time?

Anyway, here's a photograph.

Health News

So it's time to follow on with the shoulder problem, which I thought was due to poor posture and adapting to the narrower body of an electric guitar, and or some pulled muscles in my back from doing something silly when turning a heavy mattress.

Yeah. Not so much, it turns out.

I was getting pins and needles down my right arm, waking up in the middle of the night with a shoulder that felt like I'd been abusing it for hours, and otherwise a tonne of pain. This, I was told by my regular osteo, was a trapped nerve at C6/7 (because the pins and needles went to my thumb and forefinger, but not the others).

I went for some regular treatment, and while it improved, I felt it had reached a point where I'd have a good day and two bad ones, and it had settled into a painful equilibrium. After a very uncomfortable and abandoned attempt in the hygenist's chair, I wound up with a recommendation to a different osteo, and Allah or someone be praised, managed to get an appointment two days later. He preferred to get an MRI before starting on something like I had, and the next day I was gritting my teeth in an MRI scanner. He had the results three days later.

West End private medicine. (No, I am not a millionaire. I don't spend money on foreign holidays.)

The radiologist's conclusion was "moderate degenerative disc disease, most marked at C6/C7 with foraminal narrowing and multilevel neural impingement (bilateral C4, C5, C6 and right C7)." So basically my neck vertebrae all all slightly out of whack. Everything else is good: spine is okay, spinal cord is unaffected, and my bone marrow is fine. (MRI can see that?)

I'm now in for a once-a-week treatment, at the moment by electric pads on my back. Pressure-based treatment like massage isn't what's needed. It seems to work. I'm in for probably another three sessions at least and maybe some more after that. But as long as it works... 

I also take the ibuprofen and / or paracetamol. I'm still AA and I don't like to take any kind of drugs if I can avoid it, but a) medical people are suggesting it, and b) it's not mood-altering.

I have to remind myself that the pain is not where my brain / body is telling me it is: there is nothing wrong with my shoulder that moving my neck about carefully won't cure. The nerve is getting pressed and sends all sorts of random signals to the brain, which then thinks I need to massage my shoulder. I don't.

Train, tube and bus seats are especially good at putting me in a posture that impinges the nerve, and I can't sit at a table for long either. I have to sleep on my back with some very carefully set-up pillow support. No raising weights above my head, so shoulder-presses are out.

I can play the guitar, and it is not causing any problems to do so. I have to stand up. Doing that for over an hour a day has worked wonders for my leg strength.

This kind of neck problem happens to much younger people, but as far as I'm concerned, this makes me officially old. Young people don't have to think about their bodies (diet, weight, exercise, sure, but not whether you carry a full watering-can in your left or right hand), old people do.