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Why is electricity so hard to understand? (1989) (amasci.com)
388 points by Tomte on Nov 8, 2016 | hide | past | web | favorite | 202 comments

I think more than half the time I spent earning my EE degree has been spent unlearning the intuitive (but wrong) things that I was liberally taught. I was fortunate enough to have an exceptional Physics teacher in high school, who managed to avoid a lot of the bullshit that less fortunate students were fed; sadly, I compensated that with some of my own (misguided) self-study.

This experience also taught me to actively mistrust "intuition" and "common sense". Not that there isn't value in intuition (quite the contrary, it's priceless, from design to maintenance), but it's only useful when it rests on a solid theoretical foundation. Prior to earning that foundation, it's nothing but a (very bad) shortcut that people take because they want to feel knowledgeable, but don't want to invest the effort of going through all that math mumbo jumbo.

From the linked notes:

I never really understood capacitors until I started trying to construct proper water-analogies for them. Then I discovered that my electronics and physics classes had sent me down a dead-end path with their garbage about "capacitors store electric charge." Since my discovery, I've gained significantly more expertise in circuit design, which leads me to a sad thought. Maybe the more skilled of electrical engineers and scientists gain their extreme expertise not through classroom learning. Instead they gain expertise in spite of our K-12 classroom learning. Maybe the experts are experts only because they have fought free of the wrong parts of grade school science, while the rest of us are still living under the yoke of the many physics misconceptions we were carefully taught in early grades.


People say "capacitors store electric charge" because that's how the integral form of the capacitor equation works. [1] When tutoring EEs in university I spent a lot of time getting them to unlearn water analogies. They break down in some very fundamental ways. [2] I agree the analogies are useful when introducing the concept of capacitance and inductance but to me the analogies are the dead-end path.

[1] http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capeng2....

[2] https://en.wikipedia.org/wiki/Hydraulic_analogy#Limits_to_th...

Not really sure I like that link. He seems to suggest that capacitors store "energy" instead of charge, which is just as ambiguous really. It's not like there is some sort of energy particle either. Of course what's really happening is that you are creating an electric potential between two plates. It's true that the net charge is the same, but you are moving electrons from one plate and forcing them (doing work) into the other. Then when you remove the battery and have an open circuit, the potential remains because they have no way to move back to the other plate until you close the circuit again. Also, I don't think the water analogies work very well because water does not attract other water in any way, whereas in a capacitor, the electrons attraction to the electron holes in the opposing plate is an essential part of how a capacitor works and explains why the distance between the plates and the surface area are important.

I think you are missing the point of the water analogies. It is what made electricity make sense to me as well. But don't take it too literally, it's an analogy, not an identity.

If you start with things like "a motor is like a turbine, a generator is like a pump, a battery is like an elevated tank", a lot of things can fall into place. The point is you can visualize it.

It's hardly fair to say that water analogies are good but "capacitors store charge" is bad, they are both weak analogies

There's nothing weak about saying capacitors store charge. That's exactly what they do. You measure charge in coulombs, which is how you count electrons. Capacitance is just the value that relates stored coulombs of charge to available voltage.

I didn't agree with what you wrote and maybe I can explain why. Springs store inches. You measure displacement in inches. Spring constant is just the value that relates stored inches to available force.

(You can swap roles of force and displacement if you wish, the point is the same. It sounds bad to say springs store force or displacement, to me.)

But inches are an abstract measurement of distance. Electrons are a thing.

(Well, depending on who you ask... no one has ever seen one, and some people have claimed half-jokingly there's only one electron in the entire universe: https://en.wikipedia.org/wiki/One-electron_universe .)

So, you're saying that capacitors store electrons?!

Debunking this particular conception was the whole point of my capacitor article http://amasci.com/emotor/cap1.html

Capacitors "store" electrons, like springs "store" steel, or rubber bands "store" rubber. A charged capacitor has exactly the same number of electrons as an "uncharged" capacitor.

When "charging" a capacitor, charge is forced into one terminal, and exactly equal charge comes out of the other terminal. No electrons build up inside, nor are placed into it. They've just been moved around inside, same as the steel spring, or the spherical tank in the water analogy.

What then do capacitors store? EXACTLY! That's the questions that students should be asking. They won't think to ask it, if they've been taught that capacitors are like buckets full of electrons. Well, what does a steel spring store? Or a stretched rubber band? KG of steel or rubber? Nope. Capacitors store joules, not coulombs.

The above concepts open the way to unifying several ideas: capacitors store charge in the same way that inductors store charge! In both components, energy is stored, as e-fields in the case of capacitors, b-fields in the case of conductors.

Capacitors store joules, not coulombs.

Fair enough -- a two-terminal capacitor that stored electrons supplied via one terminal could be charged without drawing any corresponding current at the other, violating Kirchoff. I do like your water-filled sphere analogy, and I agree that the word "charge" is an overloaded term.

But what would you say is happening at the top electrode of a Van de Graaff generator? It represents a reservoir of stored (positive) charge. Electrons have been physically moved outside the device, and we use the same language to describe this process -- that of capacitance.

I guess the argument would be that the objects in the room constitute the other terminal of the capacitor, with the intervening empty space forming the "dielectric," and that the electrons removed from the sphere aren't associated with the sphere at all, but have just been moved from one region of the dielectric to another?

Yep, a VDG machine is not a single-ended device. I tell people that there are always two spheres involved, although usually the second sphere is below our feet: planet Earth. Charge conservation says that, with a VDG, the e-field flux extends between the upper metal sphere and the ground below it. So, to concentrate attention on just the charged sphere, while ignoring the oppositely-charged ground surface, is much like concentrating on just one plate of any capacitor.

Better: hang many different metal spheres from insulating threads, then use a HV supply to deposit various charges upon them. "Capacitor" is always taken to mean a pair of opposite-charged objects. But miscellaneous "charged objects" aren't necessarily capacitors.

Also, this:

ENGINEER'S CAPACITOR, not physicists'


While employed at MOS in Boston I temporarily threw together a floating, double-ended VDG with a battery/motor inside one sphere. Like this: http://amasci.com/emotor/vdgdesc.html#diff

I thought it would much better communicate the true nature of electrostatic generators, but it never ended up in our exhibit. VDGs are just constant-current high-voltage power supplies. A long enough chain of 9V batteries would produce all the same phenomena ...aside from the 10amp short circuit current, and the megawatt arcing!

PS, weirdness

With VDGs I was triggering three separate kinds of spark. I've not seen this discussed anywhere. We have the usual kind, the thin straight "needle" that jumps between smooth spheres. Then we have the violet fractal tree. Attach a 1cm ball to a VDG sphere and watch in a darkened room. It periodically spits foot-wide lightning networks, just like the miles-wide kind. And third: occasionally I was getting "silent purple sausage" discharge about an inch thick and a couple feet long. In a lighted room they make a slight "thump" sound, so if you hear that noise from a VDG, try observing in total darkness. Sometimes the "sausage" would even produce branching (possibly nanosecond wave effects,) when it would leap out 1ft, then split into five branches from the tip, then proceed to the adjacent metal wall as five fuzzy pathways. Perhaps the particular "seed" at the micro-scale will determine the type of spark which propagates? Or maybe the "sausage" discharge was actually a relativistic effect seeded by MeV cosmic rays.

Wheeler didn't really claim that the one-electron universe was literally true or even a useful model, but the observation that one cannot draw a clear difference between a particle in a (classical) field in flat spacetime with an enormously complicated worldline and many indistinguishible particles in the same field with straightforward timelike worldlines is a fairly deep insight into the symmetries of flat spacetime, particularly the translation symmetries. They were also on the cusp of spontaneous symmetry breaking while wondering about the missing positrons.

Given that this was decades before the Standard Model was formalized (one-electron, ca. 1940; Glashow electroweak spontaneously broken symmetry, 1967), I think that the one-electron thinking was incredibly productive (especially since Feynman credits it with some insights into what became his path integral formalism).

It's not that one-electron was (or even could be) fully in line with available evidence that was important, but rather that it connected the full symmetries of the Poincaré group (the isometry group of Minkowski spacetime, which is the spacetime of Special Relativity, particle indistinguishability, and representation theory.

The results of the this excited and informal conversation are still found in particle paradigms of quantum field theories (e.g. the Standard Model).

"Electrons are a thing" gets much trickier outside of Minkowski spacetime, however. In non-flat spacetime, the Unruh effect "is a thing", and one consequence is that different observers will disagree on particle count, and even on the interpretation of quantized excitations in the fields as (asymptotic) particles. Unless general covariance is abolished, which seems really hard to do, none of these observers is any more right than any of the others; the number of particles is simply not well-defined locally. Worse, a generally covariant formalism exposes that this is the case in flat spacetime too (e.g. Rindler observers of a patch of a quantum field are not "less right" than another observer at a constant interval from that patch, even if one sees a huge lake of energetic particles and the other sees no particles there at all).

An everywhere-in-spacetime electron field thing is probably a thing in our universe, though. But there are several different descriptions of it... :)

What?? Okay then count by lengths of the spacing of carbon molecules in graphite, or Plank lengths, or whatever.

I don't see why your objection is on inches or otherwise any unit I chose for dimension of length.

If you compress a spring, and then heat it, what happens to the stored inches?

Can you clarify what your point is? I don't know how to answer the question, because springs don't store inches. If you are making a point about why the spring/inches analogy was bad, I'd like to hear it.

> water does not attract other water in any way

Um ... once we've cleared up all the misconceptions about electricity, we might want to move on to this other thing called gravity :-)

The usual reason that people are mislead by "water" analogies—more precisely, the analogy between height and electric potential—is because they misunderstood gravity to start with. If you start by writing Newton's law and Coulomb's law side by side, and develop the analogy in a precise way, smart students will be able to debug their own fallacies.

For example, a capacitor does have a precise gravitational analog, where you have two tanks floating in outer space, and water gets sucked into them by gravitational attraction. Once you understand that, you can think about the effect of removing the minus sign from the gravitational energy law, and about the things you can do with two types of charge, but can't do with only one type of mass.

where you have two tanks floating in outer space, and water gets sucked into them by gravitational attraction

As a person who has always struggled to understand electricity (but understand Newtonian gravity well enough), please tell me more! Water is going from where to where?

Picture a pair of tanks shaped like a parallel plate capacitor, a circuit shaped pipe connecting them, and a mixture of water and air filling the pipe. (The air lets the water move around without creating a vacuum.) All other things being equal, the water gets sucked into the plate shaped tanks to minimise the gravitational energy; the resulting gravitational field is identical to the electric field you would get if you forced positive charge onto both plates of a capacitor.

Some of the differences with electricity are:

Like charges repel, so you have to force the positive charge to go where the water would pull itself.

Mass is always positive (dark energy aside), but charge can be negative, so you can have "negative mass pipes" that cancel out the mass of the water.

You can squash positive charge into one tank, and negative into the other. Unlike the parallel water tanks, this takes work, because the charge on each plate repels itself-mass can't do that. But less work than when there is positive charge on both plates. The resulting dipole field is something that you can't get with gravity. This is how real capacitors work.

Yes as I said in a later comment of course cohesion and gravity exist, but they are not relevant in this analogy.

My water analogy for a capacitor is a piston with pipes attached to both ends, with a spring system that pushes the piston towards the center position. Is that not a mathematically correct equivalent? (in an idealized system with no water resistance/inertia and disregarding that the piston is of fixed length - not that real capacitors have zero resistence, inductance or can store infinite charge)

That's what I started out with! Fill the entire universe with solid rock (since air and vacuum are insulating.) Bore out a cylinder, fill it with water (electricity), and add a piston and spring. Wires are water-channels added to either end.

But note that, if we use a constant-force spring, then the voltage remains the same until just before the "capacitor" is totally discharged. So, it acts like a battery! To get a "capacitor," the spring must have an unchanging spring-constant, so that the potential-difference rises in proportion to how much water has been pumped from one terminal to the other.

Its risky to speculate on fuzzy analogies stacked on fuzzy analogies but another one I've seen is the water balloon, or balloon over an open piece of pipe. So some voltage pressure pushes the current into the balloon and the balloon kinda sorta works to keep a constant-ish pressure/voltage on the pipes.

Now if that was too easy try an inductor analogy. Something like a waterwheel hooked up to a very big flywheel so it works hard to keep the watercurrent flow constant.

The best thing about learning by analogies is eventually they get so hairy and crazy that reality is simpler in comparison...

Or just a membrane between the two sides?

Yeah, but maybe it's trickier to relate the restoration force of a membrane with displacement of water.

From an energetic standpoint, a capacitor is something that takes a trickle over a long time, and releases a flood over a short time. So it would be a water tower with a small input and a large gated output.

But a water tower only has one terminal. A better "capacitor" would be a pair of water towers side by side.

To "charge" this double-water-tower capacitor, pump some water from one to the other. And, when the water-tower capacitor is entirely "discharged," the towers both have the same water level inside. (As with a real capacitor, the total amount of water never changes.)

It's just not clear to me how to think of the water tower as being an element of a hydraulic _circuit_.

Capacitors do store energy, in their electric field. Potential energy in general, as far as I'm aware, is stored in fields of some sort -- water turning a wheel (gravitational), tension in a spring (electromagnetic), capacitors (electromagnetic), chemical energy in gasoline (electromagnetic), nuclear energy (weak/strong fields I believe, not as well versed here) and so forth.

i agree, I linked more for the similarity in experience between the gp and the author in regard their unlearning

I prefer your clarifications when considering the function of capacitors

That said, water does attract water.. the term used to describe the phenomena is cohesion


Don't forget about gravity as well.

Interestingly, cohesion itself is caused by electric fields between water molecules due to their polarity (uneven distribution of charge, oxygen is electron-greedy). Much like a capacitor, separating the molecules (plates of the capacitor in the analogy) requires work which gets stored in the electric field between them.

What if the iron sphere contained an osmotic membrane?

i think you could have some fun with this idea

the op author likens voltage to pressure(o)(i) within the water analogy and your osmotic membrane(ii) functions as a sort of pressure responsive valve

i'm sure there are some mental acrobatics to describe a capacitor's relationships: Q=CV; between charge(Q), capacitance(C), and pressure..er, voltage(V) using osmotic principles in place of the capacitance variable addressing the permittivity of the dialectric(iii)

but you'd have to describe osmotic pressure while waving away incongruencies between the two concepts and i think you'd end up so deep into enervated analogies it would just be better to explain in direct language ;P

(o) http://amasci.com/miscon/voltpres.html

(i) http://amasci.com/miscon/voltage.html

(ii) https://en.wikipedia.org/wiki/Osmotic_pressure

(iii) https://en.wikipedia.org/wiki/Capacitance

Cohesion is not an inverse-square field

gp> because water does not attract other water in any way

though i was merely addressing this quote your comment confuses me both in fillip and content

from cohesion wiki(o):

Water, for example, is strongly cohesive as each molecule may make four hydrogen bonds to other water molecules in a tetrahedral configuration. This results in a relatively strong Coulomb force between molecules.

from coulomb force wiki(i):

Coulomb's law or Coulomb's inverse-square law, is a law of physics that describes force interacting between static electrically charged particles.

(o) https://en.wikipedia.org/wiki/Cohesion_(chemistry)

(i) https://en.wikipedia.org/wiki/Coulomb%27s_law

Yes I knew water attracts water, but it is not relevant to the capacitor analogy. You might as well have said water attracts water via gravity, it's just not relevant in this situation.

Gravity does have an inverse-square attraction. Though obviously it is several orders magnitude less than electromagnetism, in fact the analogy holds.

I still remember struggling with fractions when I was young. The funny thing is that I ended up majoring in math in college (at a highly ranked one at that), and I did very well, even publishing research in differential topology as an undergrad. How can someone who struggled to even grok fractions end up being successful in pure mathematics?

I think it's because all my teachers operated off the assumption that kids can't possibly understand abstract definitions, so everything must be explained via analogies. So nobody could tell me that fractions are equivalence classes. Everything was instead explained in terms of pies, which certainly seems like it would be intuitive, and there's really nothing technically wrong with it, but it encouraged me to think of, say, 1/2 and 2/4 as different entities instead of two representations for the same entity (since, after all, cutting a pizza into two slices instead of four is just not the same thing). I didn't "get" fractions until on my own I came to the conception that these are actually kind of abstract concepts that go by many different names.

For me, I was taught too much to identify numbers with concrete objects in the real world. This sort of works for integers (as long as you ignore negative integers and even zero to some extent), but it basically breaks down for all other types of numbers, including rational numbers. So it's a really bad expectation to set in my opinion. But you can hardly find an elementary school teacher who will dare approach this topic.

I honestly think that if someone had just explained that "1/2" is a symbol we've devised to represent the solution to the equation 2x = 1 -- an equation which otherwise has no solutions at all in the integers -- I would've grokked it a lot faster. It would've been clear to me that we're introducing an entire new set of numbers distinct from the integers. These new numbers do not have a direct correspondence with physical objects in reality. And it would've been easier to see that the same solution must go by many other names as well. For instance, multiplying that equation by 2 on both sides gives us a new equation which must have the same solution: 4x = 2. So "2/4" must represent the same entity as "1/2". And so on.

But because US educational culture has decreed that basic algebra is more advanced than fractions, the very idea of discussing solutions to an equation will never be allowed in an elementary school classroom. It is assumed as a basic fact of human life that kids cannot understand the concept of solving an equation before learning about fractions. We'll teach kids to perform mechanical manipulations of decimal expansions before we ever begin explaining what any of those digits actually mean.

I am all for introducing things the correct way, but this example of yours just does not work for me. Fractions as equivalence classes? or as a solution to an equation. No. That might work for you. In fact I might even argue that it only works for you - the child idealization created by "current you". There are many things wrong with US education. But I do not think introducing fractions using concrete examples is one of the problems. Of course for each his own and in all fairness it is also how the analogy is worded exactly. If I explain 1/2 and 2/4 using pie or pizza slices and ask you how "much" pizza did you get, I think the equivalence idea clears up quite nicely.

It's fine if this wouldn't work for you. Nothing works for everyone, and that's sort of the point. Kids are funneled through an extremely rigid sequence of topics focusing entirely on rote application of manipulative techniques. Anyone who doesn't conform to this sequence is just left to painfully deal with it on their own.

I remember a bit later in school I discovered algebra on my own and became really interested in it. I asked if I could take a class on algebra ahead of schedule and was virulently refused by the school counselor, who scoffed at the very notion and declared it to be impossible. Her response was so indignant that it offended me very deeply and caused me more or less to rebel against school in general. I ended up turning away from math completely for many years and didn't come back to it with any real interest or passion until I was about to graduate high school.

> If I explain 1/2 and 2/4 using pie or pizza slices and ask you how "much" pizza did you get, I think the equivalence idea clears up quite nicely.

Sort of. But childhood me would have just told you that, no, in one case I got one big piece and in the other case I got two smaller pieces. Combined they might have the same mass or the same number of calories or the same area, but the numbers you just gave me don't have units and can't be measures of area since they're independent of the radius of the pizza.

> > If I explain 1/2 and 2/4 using pie or pizza slices and ask you how "much" pizza did you get, I think the equivalence idea clears up quite nicely.

> Sort of. But childhood me would have just told you that, no, in one case I got one big piece and in the other case I got two smaller pieces.

Why are we using pizza pies in the first place? I vaguely remember being taught something about pie as well. Instead how about this:

You have a piece of string 1 m in length. You cut it into 2 equal pieces. We can denote the length of one piece as 1/2 m.

You have a piece of string 2 m in length. You cut it into 4 equal pieces. We can denote the length of one of these pieces as 2/4 m.

These pieces of string are the same length, therefore 1/2 m is the same as 2/4 m.

Although in my head this feels more like a nice way of demonstrating they are the same, I'm not so sure if it helps a lot with reasoning about fractions. But that's fine, like the article said it's good to be able to explain something in multiple ways.

Personally that's one of the things that absolutely fascinated me about arithmetic when I was young; That you can play around and have multiple ways of arriving at the same answer, and that if you follow the rules of math right, they always end up as the same answer, sometimes even unexpectedly so (for young me).

I like your analogy a lot. It's hard to me to say definitively but I think myself as a toddler would have found it more compelling than pizza and pie analogies. Of course, the same sort of manipulation can be done with pizza/pie slices, but it's a much more natural and straightforward manipulation with strings.

[I would edit this onto my comment above, but it's too late so I'm just adding it as a reply.]

I realize this is going to go far beyond the thoughts of a toddler learning about fractions for the first time via the pizza analogy, but I think it demonstrates that even the seemingly obvious pizza analogy is more nuanced than it seems.

I've never bothered thinking about this before, but it seems obvious to me now. Using just a straight-edge and compass, it is not always possible to cut a pizza into N standard-shaped slices for all positive integers N. Doing so is equivalent to constructing a regular N-gon, and this is only possible when N takes on the special form

N = 2^k * p_1 * ... * p_M

for some nonnegative integer k and some sequence of distinct Fermat primes (that is, primes of the form 2^(2^j) + 1).

Thus, you cannot slice a pizza into 7 slices using a straight-edge and compass. So in the pizza analogy the number 1/7 corresponds to something that is not physically achievable with standard implements.

Children take a long time to realise that one half of pizza is the same as two quarters. In general you can give a child less pizza and as long as it's the same number of pieces, they can't tell the difference. The well-known experiment is with presenting two sets of pieces of chocolate, one with less chocolate but more pieces and most children say the one with more pieces has more chocolate.

I'm not sure I believe this. My experience with children is that they're almost always smarter than I expected.

I think I've found the Wikipedia article about the effect, and the criticisms section [1] matches my bias, in that it suggests the children are confused because they are thinking not just about the material presented to them, but also why they're being asked.

[1] https://en.wikipedia.org/wiki/Conservation_(psychology)#Crit...

Depends on what you mean by children. Most of these concepts appear as a child ages but I think they get out of the simple ones pretty early.

I know before a certain developmental state children will say a taller thin glass holds more then a short wide glass even if you pour them back and forth to show they are equivalent. Once they "get" the concept it is for life but before that it is impossible, logic be dammed.

A practical impact of this - tool users from metric countries don't think in fractions. The notion that the next larger wrench after 1/8 is 5/32 is alien to people brought up with the metric system. Metric bolt heads are integral numbers of millimeters.

The irrational part of this is that the 'wrench system' is actually sized in 32nds of an inch, and thus would be much easier to comprehend with 'improper fractions'.

4/32, 5/32, 6/32, 8/32 ... etc

Or just "size 4, size 5", etc, with the "/32" as overall metadata

Except that there are wrench sizes in the /64 range as well.


Rather than 4/32, 9/64, 5/32, you'd have 4, 4.5, 5. That actually seems even better.

And then you can just use millimeters as the unit instead of the awkward "inch/64".

The irrational part is using "inch/32" as the unit instead of, say, millimeters.

Yep, there are many counter-intuitive aspects to the rational numbers. They're ordered just like the integers, and yet given any fraction there isn't a well-defined "next" fraction. But there are still "gaps" (irrational numbers). The same fraction has infinitely many representations. Despite that there are no more rational numbers than integers, yet the number of "gaps" far exceeds either of these. Not every fraction even has a finite decimal expansion. And fractions allow you to specify a level of precision for measurements that could never be achieved in the real world under any circumstances, which immediately raises questions about what these extraordinarily precise fractions actually refer to in the real world.

I honestly believe that lots of kids detect these issues on some level, but their curiosity and confusion remain unresolved, possibly for life.

I have some memories from first grade in a California public school (in Santa Cruz) of "finding a number we can put in place of x so that x + 2 = 4" and how "at some point you will learn how to deal with x + 4 = 2". In 3rd grade we explored prime factorizations, and I also remember an emphasis on clearing out greatest common divisors for fractions -- so, while it didn't encourage thinking of fractions as equivalence classes, it at least encouraged comparing things by canonical forms. I even remember around then an introduction to base four representations, probably to illustrate the "carry the ten" when adding.

My experience, in contrast to yours, was well-meaning teachers drilling the algorithms while hinting at some underlying structure. I suppose it was enough inspiration to figure out what math was about, and I eventually found myself in a math PhD program. (Despite the fact I had a hard time remembering the so-called "math facts." During competitions, I would try to calculate, slowly, something like 7 x 9 using something like (8 - 1) x (8 + 1) = 8 x 8 + 8 - 8 - 1, since the only "facts" I ever remembered were multiples of five, 6 x 8 and 8 x 8. Those competitions did make me think I was a bit bad at math!)

Though, let me share my story about fractions. In fifth grade, I got a day planner because the school pushed for everyone to get one, and in the front leaves there were various references, like a periodic table, lists of equations, etc., and one which caught my eye was "a/b + c/d = (ad+bc)/bd". For some reason I thought "oh, that is what fractions are" and I tried to explain to a teacher how this defines fraction addition, how finding a common denominator is just a way to calculate this, but instead I was told I was incorrect, and you just find common denominators. I can't say I didn't feel somewhat vindicated when I learned, much later, about the Grothendieck construction.

I also remember trying to find a formula for triangular numbers in sixth grade, and I will never forget the look of horror on my "home-room" teacher's face as she backed away while I explained what I was trying to do.

I've heard that the people in the US who have developed a math phobia or who think that they have an inability to understand math self-determine this fact after particularly tough math concepts are being taught. IIRC, learning fractions is one of the first of those, and creates the largest cohort.

I believe it. I've known a lot of adults who don't quite get fractions. They generally also have no trouble working with money and giving change, but no ability to understand interest, especially compound interest. Somehow, the educational system is failing here, not the kid.

You're pinning too much on supposed bad teachers, and not enough on your past-self's youthful incomplete understanding.

1/2 of a pie is 2/4 of a pie, 2 slices of four is the same fraction as 1 slice of 2. It's exactly the same idea as the equivalence classes, where the ratio matters but the description does not. You just didn't understand that at the time.

> the very idea of discussing solutions to an equation will never be allowed in an elementary school classroom.

My kid is in first grade and his homework is to solve equations of the form "7+2=__" and "9-__=7"

With all due respect, that sounds like straight-up circular logic to me. You're arguing that 1/2 = 2/4 because "1/2 [...] is [...] 2/4". You're just restating the conclusion as an explanation of itself, which is completely unsatisfactory.

Another way of putting it:

> 2 slices of four is the same fraction as 1 slice of 2.

The phrase "is the same fraction as" is obviously inappropriate and explicitly circular and self-referential in any definition of "fraction". I don't think you've thought about this as deeply as you think you have.

Similar problem from a local school: "__ - 9 = 9"

The teacher would accept 9 or 0 as an answer, not 18. A parent (with math degree even) who was volunteering pointed out that 18 is correct. The teacher ruled against this, arguing that 18 was unacceptable because math with 2-digit numbers hadn't been covered yet.

This was at a "good" school, in a county full of nerds.

Such logic. Poor kids... and people wonder why kids hate math and give up on it.

How do they justify 0 or 9 as an answer? 9 - 9 is not 9, neither is 0 - 9.


I suppose the "logic" is that stuff involving 9 could connect up with stuff involving 0 or 9. Since all other 1-digit answers seem more distantly related, and the answer has to be 1-digit because 2-digit hasn't been covered, 0 and 9 are the most attractive answers and therefore correct. ???

Remember, that was a class with smart kids in a school in a good area. Imagine what it might be like in not-so-good areas or in the classes that aren't for smart kids.

People who truly hate math are teaching. My sister-in-law is an example. She got a "degree" in "early childhood education". It's a joke. The hardest math she had to pass was algebra, as is taught in 7th grade. Boy did she complain about that class. She thought it was really hard to pass algebra. I'm horrified she got more than halfway through high school without it, yet there she was, taking it in college.

Teachers have serious credential inflation. Many have Master's degrees... but are those degrees worth anything?

I think the tricky thing is that they are going to encounter fractions in everyday language before they are at the stage of discussing different fractions as being equivalent. I've got a 3 year old and 5 year old. The 3 year old uses 'half' whenever something is divided in two, regardless of the size of the pieces, even if I correct him. The 5 year old on the other hand is capable of doing addition/subtraction with halves, quarters etc. but hasn't encountered that at school yet and hasn't encountered fractions like 2/4 because people never refer to that in real life. By the time they learn properly about fractions in school, he's already going to have had a lot of experience of using them in a non-formal setting which is inevitably linked to concrete concepts. On the other hand, he has got a mother with a PhD in maths, so I suspect he might get introduced to the concept of equivalence classes rather earlier than most kids :-)

(By the way if anybody out there does have children that age, I recommend the game Fraction Formula which has tubes in which you put cylinders in to representing different fractions).

Your desired explanation also would prime kids for introduction to other similar number sets, like the complex plane.. easing understanding significantly in comparison to referring to them as 'imaginary numbers'

+1 for drainpipe theory! As a ChemE grad student I took an EE control class after basically forgetting all the circuits knowledge I learned as an undergraduate (my high school didn't teach electronics and magnetism!). Water analogies were the only way I could do the problems with circuits. Drainpipe theory actually made me wonder what was ever so hard about that part of physics class...

I suspect that the top search result for "water analogies" was hugged to death because of this comment.

I got it taught about 3 or 4 times, and every time I understood it less.

In middle school they told me, it's electrons moving at the speed of light through a conductor.

In high school they told me, no no, they don't move at the speed of light, just when one electron enters the conductor, another one on the other side will leave the conductor, like with peas in a straw. And this enter/leaf is at the speed of light.

At university they told me, no no, you got it all wrong, it's about the electro static and electro dynamic fields which stand orthogonal on each other and produce electro magnetic waves... what?!

I seriously have no idea anymore.

As far as circuits go, this is about the closest high school Physics gets to truth (and it's decently close, compared to how wrong it gets e.g. capacitors):

> In high school they told me, no no, they don't move at the speed of light, just when one electron enters the conductor, another one on the other side will leave the conductor, like with peas in a straw. And this enter/leaf is at the speed of light.

although "entering" and "exiting" the conductor isn't really a good description.

A better way to think about it is that "conductor" is really just slang for "material that has a bunch of free electrons and very little room for more". This is the "sea of electrons" that Beaty talks about. It makes sense that, if all this sea flows steadily (i.e. at the same rate throughout the conductor), forcing a change in the rate of flow in one part of the conductor will be felt almost immediately in another part of the conductor, no matter how distant, even if the flow itself is extremely slow.

(If you're thinking that this is only true if the "pipe" through which the "sea" flows is full, you're right, this is how free electrons behave in conductors; that's why Beaty insists that a good analogy for a conductor is "like a pipe which is already full of water".

> At university they told me, no no, you got it all wrong, it's about the electro static and electro dynamic fields which stand orthogonal on each other and produce electro magnetic waves... what?!

If they taught you that, they are most definitely wrong, it sounds loosely like induction, but with two strange names instead of "electric" and "magnetic" :-).

This is a better description of the physical reality, and it does help you understand circuits better if you think about it, but not in a very practical manner.

> If they taught you that, they are most definitely wrong, it sounds loosely like induction, but with two strange names instead of "electric" and "magnetic" :-).

I don't think "definitely wrong" is a good description. electrostatic and electrodynamic are perfectly good, if somewhat archaic, terms for the electric and magnetic fields. (E and H fields)

I probably just translated them wrong from German.

But as I said, since I didn't understand it, I probably don't recall correctly what they told me.

I just remember that the "particles" I had in mind somehow stopped being "the thing" and now it was all abount strange fields that worked without a physical medium and formed waves by being aligned in a specific way.

Ah. That's a terminology fuck-up on my side, then. I've never encountered this convention, but classical electromagnetism has almost two centuries of history behind it. A lot of weird names have been used for a lot of things.

For what it's worth, though, these are terrible names, archaic or not :-D. If they ever were in use, I'm glad we moved on.

I'm not sure they are terrible. Electrostatic describes charges at rest, capable of inducing a voltage across a dielectric. Electrodynamic describes charges in motion, capable of inducing a current across a conductor. They seem like apt descriptions to me.

It doesn't sound too bad when you put it that way, and it certainly made sense back when Ampere introduced the term, but:

* It doesn't match the way we define electrostatics, magnetostatics and electrodynamics. What defines electrodynamics isn't the fact that charges are moving (they're moving if the currents are constant, too, but the magnetic fields produced by steady currents are in magnetostatics' yard) but the interaction of charges and currents (in more formulaic terms, when both charge densities and current densities are present, not only do you get both electric and magnetic fields, as in magnetostatics, but they also vary in time).

* Charges in motion still produce a voltage across a dielectric. Calling the electric field they produce "electrostatic" when the charges are moving and the field is certainly not static.

Thanks for writing that out! Very interesting.

I guess it all comes down to the fact that a magnetic field does not exist without an electric current. One way of thinking about it sees it as charges moving, and the other way of thinking about it sees it as a static magnetic field.

I guess that's why the terms are archaic!

Indeed. That's why I think it might have made sense back in Ampere's time. The classification of these regimes (electrostatic, magnetostatic, electrodynamic) is more recent, and Ampere's own theory of electrodynamics deals more with what we term "magnetostatic" today.

I suspect that it corresponds to Newtonian Statics, the study of mass and forces. It's a subject area; an ignoring of changes. Not a state of nonmoving. The Newtonian Statics viewpoint involves summation of forces. It may also involve taking a snapshot at one point in time.

E.g., when a mass above Earth is in free fall, it still obeys Newtonian Statics: the weight/attraction force, easily analyzed from moment to moment. The resulting acceleration and trajectory then falls under "Dynamics."

In other words, electrostatics applies to capacitors and to the mechanical forces produced by electric fields. Even if currents are also present, and even if the e-fields are changing with time, electroSTATICS still applies. (A high voltage, high-amperes power line is very "electrostatic," because of the significant e-fields and resulting phenomena.)

Static Electricity then is a chapter title, with no existence in the real world. Neither can we fill a box with Newtonian Statics. To be consistent, we wouldn't say "electrostatic motor," instead call it a capacitor-motor, or an e-field motor. (Heh, a stretched spring is statically charged! Full of Newtonian-static energy!)

I start to think that the less you make sense of a topic, the closer you are from real understanding.

ps: this comes from years of self learning music, every now and then my incomplete abilities go through a complete crash, only to reveal a critical misunderstanding, lack of sensitivity in my perception or action, that leads to a sudden and very blissful resolution (your own private eureka moment)

Well someone fucked you up because current is defined as the flow of positive charges just like the electric field direction is defined as the action of a positive test charge. Thanks Ben (Franklin)

lol, seems to be.

I read a few articles on that page and it's like magic, it makes not much sens to me after all I learned at school.

The first chapter explained it well.

There are three different concepts that were all thrown under the umbrella term "electricity".

>..little use by educators of the wind/sound electrical analogy:

>AIR is a physical substance.

>SOUND is a wave that propagates rapidly through a volume of air.

>WIND is a flowing motion of air already present.

>ELECTRIC CHARGES are a physical substance.

>ELECTRIC ENERGY is a wave that travels via a column of charge.

>ELECTRIC CURRENT is a flowing motion of the charge already present.

> ">ELECTRIC ENERGY is a wave that travels via a column of charge."

I'm not an expert when it comes to electricity, so someone correct me if I'm wrong, but one thing that has made sense to me when it comes to trying to understand electricity is that not all electrons have equal potential for work.

When studying electricity, you're often told the charge of an electron as a fixed quantity. However, if I've understood correctly, the work that an electron can do in conducting electric energy is not wholly described by the charge of the electron, it's also important to know the relationship that the electron has with the nucleus of the atom it orbits (i.e. the 'shelf' it's on).

To explain in another way, this analogy may be 'wrong' but I think it helps to think of electrons as capable of more work when they are less tightly coupled from the nucleus of an atom. The electrons that can do the most work are those in the outermost orbit of an atom. Whilst it may be wrong to say outer electrons are more charged than inner electrons, I think it might help in terms of visualisation. Again, correct me if I'm wrong.

How significant is this when you work at the level of Coulombs (high count of charges) ?

Sorry, I don't think I fully understand your question. Again, what I say could be wrong, but if there's a flow of charge it could mean the outer electrons have more charge than they can contain whilst remaining part of an atom, resulting in either electrons breaking free of atoms, or the energy being passed on to another atom (causing a chain reaction as each atom does the same). I'd suggest a higher count of charges just means this is happening more.

I think another useful mechanism to consider is that atoms want to return to an electrically neutral state. If they have energy that differs from this neutral state, they will either give out energy or take in energy to reach the neutral state. The movement involved in rebalancing the atoms can be thought of as electric current. Again, happy to be corrected if I'm wrong.

I meant such precise physics(that I find passionating btw), no matter how true at that level, might be of lower significance for basic electric/eletronics thinking.

Let's start with a familiar example. The speed of sound is basically fixed, and signals caused by perturbing air molecules will propagate with this speed. This is true regardless of the gross wind conditions at play -- the speed of sound is much greater than the propagation speed of any single air molecule.

Similarly the speed of light is fixed, and is really the speed of propagation of electomagnetic waves. When you perturb electrons, other electrons will react to that change in the fields as it arrives at them with the speed of light. So even though the flow of electrons down the wire is very slow, the flow of energy or information may be very fast. In fact, you don't need electrons in between to move at all -- you can simply have empty space in between your antenna and the electrons you are moving around ;).

As it happens, these electromagnetic waves have components that stand orthogonal to eachother. If you take relativity into account, you can find a frame of reference where the magnetic field disappears -- really the magnetic field is just a relativistic correction of the electric field and explains why charged particles attract eachother differently when moving versus at rest. In any case, it is really just one field that is called electromagnetic because we used to think it was two completely separate fields!

It actually is the electrons. But they never touch each other. Instead they push upon each other across empty space, by using e-fields and b-fields.

If there wasn't any chain of electrons inside the conductor, then the EM fields would just fly off into space, like with a transmitting antenna.

Wires can guide the EM energy because each electron can push the next one in sequence. But also, one electron doesn't just push on the next one. Instead, it pushes on a huge number of electrons far upstream and down the long chain of mobile charges going off into the distance. That's why the EM energy can "leapfrog" across the movable charges, at the speed of light. If each electron could only push upon its nearest neighbor, then electrical energy would travel at about the speed of sound.

Your last two answer are the same. It's just that the last one is much easier to calculate.

Intuitive but wrong (and doggedly persistent) idea #1: "electric current is moving electrons". It is moving photons, exciting (largely) stationary electrons. I can't stress how crucial overcoming that misconception was when I was doing EE.

Actually, it's electric wattage which is the moving photons, not electric current. Energy flow. Joules per second, same as with laser outputs.

So, the electric companies are selling us 60Hz photons.

But at the same time, the AC electricity just vibrates back and forth inside the wires, without any net forward motion.

About the only accurate diagram of EM spectrum I've ever encountered was the one made by R. Oppenheimer and sold as a classroom poster by The Exploratorium in SF. At the bottom, below VLF radio, we find audio-freq phone lines, and below that, 60Hz power lines. "Electric power" is on the EM spectrum, down below radio waves.

And yes, if we have a big enough antenna tower, then we can hook it up to a 60Hz dynamo, and spew 60Hz EM radiation out into space. Actually that concept was one of N. Tesla's great breakthroughs: hooking a steam-powered 50KHz alternator directly to an antenna, and broadcasting a silent carrier: CW. A couple years after his patents on this ran out, GE suddenly announced the "Alexanderson Alternator!" Finally making music/voice broadcasting possible! The public wondered how Alexanderson managed to come up with such an amazing idea. (If investors had bought Tesla stock, we'd have had AM radio ~20 years earlier. Tesla clearly grasped the concept that "radio" was the same thing as "electric power," just higher in frequency.)


I remember that poster! I actually got a copy from the Exploratorium...

Ehhhhh. Not really. It is the rate of charge passing through a cross-section. That's how it's defined, and as an EE student, what you're saying made little-to-no sense to me.

And the boson that mediates "charge" is....?

Note that photons only travel along FO cable because of the charge polarization of the silica. This is a direct analog to 60Hz "electric power" traveling along a column of mobile charges in a long conductor. In both cases, the electrons wiggle back and forth, while the photons couple to them, and are guided along. But charge is not energy, in exactly the same way that amperes are not watts, and current is not EM energy-flow. Coulombs wiggle within the waveguide, while the Joules proceed continuously forward as propagating waves.

The EM energy is in the form of propagating waves, while electric charge provides the medium: the waveguide.

Photons don't mediate charge. The mediate the electric and magnetic fields.

And the field that is quantified as "charge" is...?

An electron? I don't know what you're getting at here. An electrostatic field is produced with strength in proportion to the charge. However, that is not the only way an electric field can exist. For instance, they can be produced by changing magnetic fields, and when these two fields interact in a way that they mutually generate one another, that's what we call an electromagnetic wave, or "photon." Photons don't mediate charge, they simply are the carrier for the information of a changing electric or magnetic field.

It seems like you have a fundamental misunderstanding of how photons are produced, or what they do. At least in DC conditions, no photons are produced at all (since there is no changing magnetic or electric field), yet there is still current.

Current is the movement of charge. Photons are irrelevant, besides being the messengers for the electric and magnetic fields, which still only applies if those fields are changing, as no information is changed otherwise, so no bozon is necessary.

What you may be thinking of is in an AC case, where current is constantly changing as the electric field wave propagates down the transmission line. This does necessitate photons. But even then, the current is still the flow of charge, and oscillates between negative and positive.

It may be useful to understand where this error is coming from. Back when the theory of electrical current was developed, we had no good model of the atom. We had no idea about electrons, protons and neutrons, no idea about photons, quarks, nothing. Kirchoff developed his equations without knowing anything about electrons. When Ohm discovered his law, he knew nothing about electrons. Classical electromagnetism, in its entirety, is formulated without any knowledge of electrons (except with some of the late stuff that was developed in order to extend it to relativistic cases, but at that point we were already operating on "legacy code"). The modern idea of the atom, as we know it now, was developed more or less at the same time with classical electromagnetism; various theories about matter were being thrown around at the time, and for a long time, it was thought that electricity would be caused by some sort of charged fluid. Only beginning with the 1870s, at which time Maxwell was already revising his all-encompassing treaty, did the idea that it's all about particles really begin to take off.

The electrical charges in the classic theory of electromagnetism are not electrons, nor ions, nor any other particles that we know of. They're ideal models of "something" that carries charge, but they do not model all the inherent behaviour of electrons (e.g. the electrical charges of classical electromagnetism do not have any magnetic moments, unlike real-life electrons). They are somewhat like the point particles in kinematics, in that they capture an essential feature (electrical charge) of an object while dispensing with other features that are not essential for the study of some (but not all!) phenomenons.

It's ok to model electric current as "the flow of charge", as long as you don't give this model more physical meaning than it's due and attempt to equate the charges of classical electromagnetism with electrons.

Certain phenomenons can be studied within this frame: for instance, the motion of an electron's motion through static electrical and magnetic fields is generally (low enough frequencies, strong enough magnetic fields etc.) OK to do while equating the electron with the charge carrier of classical electromagnetism.

But back when the concept of electrical charge was elaborated, we knew nothing of electrons. The phenomenons gave enough clues for us to hypothesize that whatever "supports" electricity has some of the properties that "real" charge carriers have, but that's all there is to it.

Is this why my EE prof once said, a metal cabel is like a fiber cabel, it trasfers EM energy but at a different wave length?

This sounded like he went mad to me...

Think of the behavior of RF transmission lines. RF is after all just long wavelength light or EM waves.

So obviously a piece of pipe is a circular waveguide and it works more or less like optical fiber.

It helps if you know how optical fiber works, across a boundary with a big enough difference in speed of light in the material, you get total internal reflection and it bounces back in.

Now people are pretty chill with circular waveguide as a transmission line, but there are numerous other schemes and eventually you end up with microstripline or twin-lead that TVs used to use or eventually one wire Goubau line.


Once you're chill with a zillion small steps from circular waveguide to G-line, wait, G-line is what your prof said that initially sounded ridiculous, but its not so ridiculous after all, with some new perspective.

But there is a reason why it sounds so confusing even to an advanced student -- even though EM radiation is at the heart of this science it is often surprisingly difficult to find correct and conclusive information on related topics.

For example, finding answers to questions like "Why does absorption of EM waves in matter rise with lower wavelengths, but at the wavelengths of light matter starts to become more and more transparent again" isn't easy. But so are the involved phenomenons, after all.

Holy crap. It all make sense now. Thank you !

Great point that's really deep.

One of the devilish things about science is that the mental abstraction model that works for a layperson doesn't really scratch the surface.


It took me a huge amount of time to get rid of all the BS learned in school. In some area I'm even still working on it (eg. analysis).

When you say analysis, do you mean you are working on unlearning the stuff you learned in it, or learning it to replace something else?

unlearning mostly.

It looks like interesting content, but I got a little confused by the writing style and general layout.

A more succinct, clearly defined structure would help me a lot. Each topic could be clearer on what is wrong and how is the right way.

It would help a lot just to start with a clear: "What is the right definition of the word electricity?". Then, maybe a "What is "electrical phenomena" and some examples". "Correct definition and differences of electric charge, energy and current".

For me personally, a structure that explains what is the right thing first, then goes on about misconceptions and consequence of misconceptions is much better.

I became lost while reading this and finished with basically the same confused and mistaken understanding of what is electricity as I began.

It turns out that this is not a good link to be starting with: see https://news.ycombinator.com/item?id=12901346

Much better. I'll spend some on it later

Could someone write the same kind of article for chemistry ? That would help me to accept the fact that I didn't get it at all :-) Oh, while you're at it : accounting :-)

Accounting in 8 steps,

1. Memorize this: Assets + Expenses = Liabilities + Capital + Income

2. Everything is positive, no negative numbers!

3. For every transaction, Total Debits = Total Credits

4. "Credit" is source of money, "debit" is destination of money

5. Assets and expenses increase with debits

6. Liabilities, capital, and income increase with credits

7. Expenses and income may only be increased (debited & credited respectively)

8. At the end of the accounting period, distribute income less expenses to capital accounts


Your business buys a property for 90k with 50k cash down and 10k closing costs.

Assets: 90k house (debit), 50k cash (credit)

Expenses: 10k closing cost (debit)

Liabilities: 50k mortgage (credit)

Total Credits: 100k

Total Debits: 100k

> "Credit" is source of money, "debit" is destination of money

But it gets even more confusing when they say "we credited your account" or "your account was debited".

Yes, that is a prime source of confusion for those terms! The problem is all about perspective.

From the perspective of the bank, your account is a liability, so when they deposit into it, then its value increases, hence "credit".

When they say "your account" they actually mean "the account where we track how much money we owe you".

From your business's perspective, your bank account is an asset, so when they deposit into it, then its value increases, hence "debit".

> > But it gets even more confusing when they say "we credited your account" or "your account was debited".

> From your business's perspective, your bank account is an asset, so when they deposit into it, then its value increases, hence "debit".

But what you're describing (a business depositing money into your account) is not what I, as a non-business owner, understand by "your account was debited". Businesses with whom I do (well) business use this terminology to mean that they have removed money from my account. (I'm not arguing with you—I've probably just misunderstood your terminology, and am seeking clarification.)

For the "we debited your account" terminology to be correct, you must consider it from the perspective of the other party.

For example, suppose you go into the bank and withdraw $10 and they charge a $1 fee. The transaction looks like this:

* Credits: $1 (bank income), $10 (bank assets)

* Debit: $11 (customer liability)

So when you get your account statement, it will say "$11 debit", because from their perspective, it was a debit, i.e., their liability went down. If that doesn't make sense, think about how you pay your own credit card bill (a liability): you take money from (credit) your checking account and send it (debit) to your creditor.

I know that this isn't the point of this thread, so I'll stop after asking one more question. (This may be a terminally wrong-headed question, and I apologise in advance if so.)

Despite your very clear explanation, I'm still confused by:

> > But it gets even more confusing when they say "we credited your account" or "your account was debited".

> From your business's perspective, your bank account is an asset, so when they deposit into it, then its value increases, hence "debit".

Is it correct that the 'you' of the first post, in "your account was debited" (me the non-business owner), is the 'they' in the second post, of "when they deposit into it" (written from the perspective of the business owner, so that 'you' is no longer me)?

The main source of confusion here is that "your bank account" is actually two different entities whose numeric value happens to be the same:

1) On your books, it is an asset account. 2) On the bank's books, it is a liability account.

When a bank says "your account was debited", they are talking about entity #2. That is, "the account on our books associated with 'you'".

But when you yourself think of the account, you think of thing #1.

Let's say you deposit some cash into your account at the bank. That means you are lending the bank the money (extending it credit). The bank records this as two transaction entries:

* A debit (receiving money) transaction which places more money into their general "money we have" pool: you gave them money.

* A credit (owing more money) transaction: they now owe you more money. This increases the amount of what the bank thinks of as "your account".

The statement they send you at the end of the month only shows the second transaction, because that's the one relevant to "your account" in the sense of #2 above.

If you were keeping books on your side, you would likewise record this as two transaction entries:

* A debit (receiving money) transaction for your bank account (now in the sense of #1 above).

* A credit (having less money in an asset account) transaction for your wallet.

So the upshot is that in the two entities that "your bank account" corresponds to, a debit for #1 is a credit for #2 and vice versa. And when the bank sends you a statement, it describes #2, not #1. Since most people don't interact with the terms "debit" and "credit" normally, this is the only exposure they have to those terms, so they learn them backward....

No, "they" is always the bank (or whatever business manages the account). By "they deposit into it", I meant the bank adds funds to your account.

It doesn't matter whether you are a business owner; your bank account is still an asset to you and a liability to the bank.

Personal & business accounting are the same, except your only investor is yourself, so you don't have to manage capital accounts.

shouldn't the cash be listed as capital instead of asset?

So the basic equation is "assets = capital + liabilities".

Capital+liabilities explain how your assets are "covered". Either you owe someone for the assets (liabilities), or you own the assets yourself (capital).

Sometimes it's easier to think of this as

"assets - liabilities = capital".

I.e. whatever is left after considering what you owe, you own.

If you have some cash (an asset), then you need to also list it either as a liability or capital, simply because you must always be able to answer "where did this cash from? did we borrow it (liability), or do we own it (capital)?"

Thanks for explaining it well. I was caught up in the parent post's example and tried to equate it with the equation given in it. How would the parent's example look like in balance sheet? I looked at small business example given in wikipedia, but failed at writing down parents example into assets and liabilities balance sheet. (https://en.wikipedia.org/wiki/Balance_sheet).

Capital is a liability (something you owe to investors). Cash is an asset.

When you receive capital in the form of cash, you debit your assets (cash) and credit your liabilities for the same amount (capital). It's "double-entry" accounting, total debits should equal total credits for every entry so that it balances out.

Capital is what you owe your investors, eg "startup capital". Sometimes this is called equity.

That may well be true in some startup contexts but, more generally, capital (more typically called equity) is effectively whatever is left over after you subtract your liabilities (accounts payables, short- and long-term debt, etc.) from your assets. This number shows up on the same side of the balance sheet as liabilities.

A balance sheet doesn't really speak to who might have claims to ownership of the company under what terms and the dollar amount of equity on the balance sheet doesn't imply that investors are owed that specific dollar amount.

I suspect I'm about to regret this, but here comes the electricity analogies for accounting, the appendix to your 8 lines.

A balance sheet is like charge on an array of loosely connected capacitors (or batteries). I know I got a pile of electrons (and holes) stacked up somewheres, and the balance sheet shows where. All "accounting circuits" are electrically neutral and the number of electrons and holes on your balance sheet MUST match.

A income sheet is like looking at the individual cell results from a solar array in parallel. So you got 10 aH out of that entire array, now which cells contributed more or less of their share, and which battery cells soaked up more or less than their share of charge?

The cash flow sheet tells you how fast electricity energy moved, essentially a power. So your 99 watt-hour laptop battery holds 99 watt-hours, but how many times did you fill and empty it in a year, how many times did you turn over the energy in the battery?

Once you learn op-amps you can do some hideous analog computing analogies, but don't call up what ye can't put down, so I'm not even trying that. So a financial derivative is like a sample and hold ckt connected to a four quadrant analog multiplier and a log/antilog ckt, or maybe this is just too far of an analogy not to be nonsense.

The purpose of accounting (aside from mere control fraud prevention, at least optimistically) is to squirt out some ratios to help make management decisions. Much like the transistor collector current is not terribly interesting nor is the emitter current at a large enough scale, but the ratio is exciting because back in the old days people made management decisions to select one transistor over the other based on the ratio of those currents, which is essentially how good of an amplifier it is. Much as income statment vs cash flow ratio tells you a lot about a retail establishment compared to its peers, how long "stuff" is sitting on shelves before getting sold. That current ratio is a bipolar transistor alpha ratio which no one uses anymore. Kind of like how people used to make investment decisions based on the ratios in the famous Graham and Dodd book, but no one has invested on fundamental ratios in, gosh I donno, 30 years? Its been a long credit bubble and fundamentals don't matter in a credit bubble.

I have no idea what a credit bubble is in EE terms. Some twisted analogy of trapped charge on a Teflon dielectric resulting in an integrator getting saturated eventually, but until it does the ride is pretty exciting.

I would extend this post with my traditional HN automobile analogy but I'm not sure there's enough liquor in the world to achieve that level of debauchery. So ... Keynesian economics policy sees the role of the government as like an electronic speed control on the automobile, uh, kinda.

Some people like http://martin.kleppmann.com/2011/03/07/accounting-for-comput...

I don't, I think it stays too much on the surface, in order to make the belabored analogy work.

This may also be interesting to you: http://www.dwmbeancounter.com/tutorial/Tutorial.html

The issue with chemistry is that you need to develop a "feel" (chemical intuition) for a lot of topics. A lot of chemistry taught in undergrad, like Organic, cannot always be neatly explained or solved with an equation. On top of that there seems to be as many exceptions to rules as there are rules themselves.

A lot of stuff taught in undergrad is also flat out wrong, especially anything to do with orbitals.

Exactly what I was thinking at that time. My conclusion was that they were forcing me to admit tons of things while at the same time confusing what could be understood logically and what had to be accepted as is.

I feel better :-)

What aspects of orbitals do they get wrong?

Physics and math also lean heavily on intuition; you might have a clear cut path through the equations when you have the solution in front of you, but there's often no "playbook" that you can follow mechanically to get what you want from the fundamental equations.

Electromagnetics becomes much more elegant when cast in the language of differential forms. See e.g. [1].

[1] http://www.uio.no/studier/emner/matnat/fys/FYS4160/v08/under...

You can also go the fields route like Feynman would:


In fact, it isn't so hard that they would like you to believe - it just seems that way when they remove the best parts due to national security :D

You would have to go back to James Clerk Maxwell's original 20 equations to see what it's all about. Okey, quaternions are kind of hard, I'll admit to that, but all in all it makes much more sense.

Like QM, doing it analytically clears away misconceptions and gives a consistent view - but it isn't exactly something you can teach to children.

It's useful to be able to explain things to laypeople and the not mathematically inclined, which means tightening up the metaphors.

Also Maxwell's equations only give you part of it - for the question of why an atom's electrons don't just fall into the nucleus you need QM. And then you get into the self energy of the electron which is not explained by any present physics (http://www.feynmanlectures.caltech.edu/II_28.html).

So you can't just do the math because we don't have all the math.

For electrical phenomena (baring very very small scale, semiconductors) Maxwell's equations give you a huge amount of insight, especially the macroscopic variant. You can see things like plasmons and polaritons (though obviously not their quantization).

You won't really run into the atom stuff until you try to do something like calculate dielectric constants analytically, though self-energy is definitely a pretty easy trap to fall in even for macroscopic stuff.

You have a good point, but still, it's easier to explain to children stuff that makes sense.

I teach my children that we live in a sea of energy and that you can feel it when you move. It makes much more sense to them than trying explain inertia.

I tell them that of you rotate something, it will make an energy vortex making stuff seem heavier - and I show them with a gyroscope.

Then I explain that electrons are just like little gyroscopes, but of only energy. And it all makes sense to them. They don't think it's hard at all.

Hm. Children can learn the right terminology for things. Those words don't seem any easier to swallow than 'inertia' and 'charge' to me.

Those are excellent metaphors which I've never heard anywhere else.

They're reminiscent of Feynman's anecdote about his father explaining inertia:


"That - he said - nobody knows". It's kind of different from "we live in a sea of energy". I honestly think inertia is a lot easier to grasp than energy. It's a phenomenon you can experiment, describe and name, energy is more abstract.

Can you explain results of double slit experiment (when single particle creates interference pattern) to us, mere mortals, please?

I often think about how Maxwell's equations were simplified because quaternions were hard to deal with by hand. Heaviside reformulated them into calculus, which was easier work with and accurate enough for pretty much everything. But now that we have computers, which can trivially deal with quaternions, (probably even more easily than calculus) why don't we go back to using those formulae?

> You would have to go back to James Clerk Maxwell's original 20 equations to see what it's all about. Okey, quaternions are kind of hard, I'll admit to that, but all in all it makes much more sense.

Not ... really. The problem is that we teach electromagnetics using a 19th century pedagogy that assumes the existence of the Ether.

This works well for some things and makes them nicely closed form and simple. Of course, it breaks down for other things (motors are one of the big ones) where you really want a field description.

Of course, a field description makes nothing simple and closed form. So, it isn't good for making exams out of.

If you can deal with 4-vectors, "Collective Electrodynamics" is a really good book.

Carver Mead! He's one of us, ONE OF US (physics crackpots, like JG Cramer here at UW.)

The real question is, why is this website so hard to understand? Publishing what looks like someones personal notebook isn't helping the cause here.

Poor selection of link. Fortunately the author saw us and pointed us toward the articles instead of the raw notes: http://amasci.com/miscon/whyhard1.html#def

I am not sure that is a big improvement. I dipped in here, and what I found was mostly a repetitive litany of complaints.

The entire essay-collection is a critique of grade-school electricity textbook chapters, in the USA. (I don't know if the texts in other countries are promoting the same misconceptions.) To avoid reading critiques/corrections, go to any other site except mine. Also see:

Am I just a pedantic nitpicker?


To deeply grasp physics, often one must UN-learn the common misconceptions which were acquired in our early school careers. The misconceptions in this list are the ones believed by educators, and taught to students.

For those who don't like raw notes, why not read the finished articles instead?

The index to the large collection is here: http://amasci.com/ele-edu.html

This should be the top comment.

Like incompressible fluids, electricity is hard to reason about because we work in a limit that does not obey local causality.

The entire system must be solved simultaneously, and there is not a well defined "input" or "output". This can be seen from the fact that when we analyze a circuit we also include the model the power source and the load.

The combination of non-locality and complex boundary conditions makes it hard to apply ordinary intuition. We would like to say that the power source "does something" to the circuit but causality actually runs both ways. E.g. if we have a constant voltage source, then the circuit will determine how much current flows through the source. If we have a constant resistance load, then the load will determine how much power the circuit applies to it.

Thanks+++ I think you nailed my frustrations with electricity!

Maybe this is the right occasion to re-ask a question I asked some years ago on stack exchange, but (despite several people trying their best to explain) still failed to understand the answer to.

So: does electricity have 'mass'? What I mean is, when current flows, is there a transfer of electrons (or something else) from the power source to whatever it is send to? And is there a difference between AC and DC?

The context of this question is that the core argument of a legal paper (of all things) I was reading at the time (on property rights of virtual goods) hinged on there being a transfer of mass, however small. I wasn't so sure.

(my SE question devolved into a semi-legal argument - I have a law degree, the context of the question is a lot more nuanced still than the above description, and is very much tied to some specific 80 year old Dutch case law - just saying that to point out that a legal argument on whether or not the question matters won't add much to the discussion.)

Information has mass. Not much. Surely charge on a dram capacitor is s bit (oh the pun) of energy and energy is mass.

An excellent example of electron movement is a DC current in a metal plating tank or refining tank. Every atom of aluminum or copper or plated anything took the movement of precisely one electron (simplification because there are some non-electroplating methods for some base/plate combos, but yeah pretty much aluminium is a block of solidified electricity)

So take one RAM stick and another, with the exact same physical makeup (I know this is impossible, but for argument's sake). Now I put some information on the one, and random noise on the other. Both will have different mass, right (because there are different amounts of electrons in the capacitors of both)? Now I put valuable information on one, and I charge all capacitors of the other (so the second one has very information density, because it can be 'compressed' trivially by saying '4GB of high bits'). The second one will be 'heavier' than the first, right? And then I copy something over that second one. It's not like there is a 'transfer' of mass from the other medium to the RAM stack, because there is a net outflow of electrons, right?

Yes, there is a transfer of electrons from one place to another. It's not to the thing that does work, but through the thing that does work.

With a battery (DC/direct current), there's a surplus of electrons available at the negative end, a deficit at the positive end. The electrons flow through a circuit (from - to +) to perform work.

With AC/alternating current, the electrons flow in one direction, then the reverse direction several times per second. The purpose of this is to push energy further down transmission lines with less loss. Pushing DC from a plant to everyone's houses is very lossy.

Do electrons have mass? Yes. Does the circuit or device increase in mass when current is involved? No. You're actually not introducing additional electrons into the circuitry. The conductors and semiconductors already have electrons on the outsides of their atoms. We introduced an electron at one end, it hops onto an atom, pushing an existing electron over to the next atom and so on. It's the movement that produces "energy" and allows work to be performed.

Strictly speaking, you can hold extra electrons for short periods of time in capacitors, but the mass is so small as to be negligible.

Right, so, in an AC circuit, there is a movement of information without a net movement of the same elektrons, right? And it's (at least partially) the same electrons that jiggle left to right, right? So is it fair to say that in a DC circuit, there is movement of mass (although net, it balances out, because what goes in out on the left comes in on the right); whereas in an AC circuit, information is transferred just through the movement of the elektrons, but not through the elektrons themselves?

In 1975, our physics teacher said he'd been a consultant to the international group who was working on electric utilities and import/export of energy across the US/CA border. Trying to define what "electricity" actually was!


Electric circuits are much like drive-belts, but using charged particles rather than rubber or leather. If a drive-belt extends across the border between two countries, what is being transported? It's not mass, since for every KG of rubber belt going out, an exactly equal amount is coming back. (Go further and use rotating shafts. Then the mass just spins in place, and doesn't actually cross the border at all.)

With circuits, with AC, the charges are just wiggling back and forth, and not proceeding forward. But with DC the charges are taking a closed circular path, so still are not proceeding forward. The thing which proceeds forward is radio waves. Sixty-cycle radio waves, forced to follow a waveguide composed of two or three conductors. EM waves are being sold to us by the utility companies. We live in a "radio-powered" civilization, but where the 60Hz EM waves are forced to follow waveguides, rather than leaping across empty gaps.

Humorous "legal" chapter in Feynman: IS ELECTRICITY FIRE?


Not really.

Electrons do have mass, but a device powered by an electric current will generally have the electrons flowing out at the same rate that they flow in.

Very comprehensive!

I've occasionally tried to clear up misunderstandings of this kind on https://electronics.stackexchange.com/ , they're a common stumbling point for beginners. Especially speed of electrons vs. speed of signals.

I wish academia took more time to explain things in digestible way instead of dumping loads of dry papers in rigorous form. For instance, my life would be so much easier if I saw just this thing below [1]. Alternating current described in complex numbers has so much sense after going through that material.

[1] https://acko.net/blog/how-to-fold-a-julia-fractal/

Also (same site): Misconceptions – http://amasci.com/miscon/elect.html

It gets even more fun in semiconductors when the 'particles' become virtual but no one bothers to mention that part.

The way holes move in the semiconductor is the pointers of EE.

Adding very little to the conversation here but having done a Ph.D. in high-temperature semiconductors this made me chuckle and is nicely apt. Just like pointers, though, you can kind of get by with only a very thin understanding of them for a surprisingly long time.

I've always been bemused by EE wallet cards that contain:

    V = IR
    I = V/R
    R = V/I
If an EE does not know this formulas, he isn't an EE. If he understands so little about algebra that he needs the three forms, he's going to be misusing the formula.

Guess like all engineering jobs, sometimes your job is so far from practice and so involved in regulations, meetings, standards, bureaucracy, reports, a little refcard with the most basic things can help to quickly reset your brain for the rare occasion where you need to actually do something.

As a hobbyist and a complete beginner, you can do surprising amount of stuff just by memorizing U=RI and P=UI. Through a closed loop current is always the same. Through a closed loop voltage drops must equal the voltage source.

What?! No power factor formula?

those cards are jokes, right?

Hmm. I learned about E&M theory from Halliday and Resnick, and the practical stuff from Horowitz and Winfield (The Art of Electronics) and they were pretty precise about distinctions between electrons, holes, charge, flow of charge, and so forth. Sounds like I dodged a bullet by not being drawn too much to electronics as a youth, waiting instead to learn about it as a college physics major.

Ah, memories of those books too. My Dad was an EE and so I also have memories of him, on parent-teacher evenings at school, arguing with the physics teacher about whatever was on the blackboard from the day's lesson..

I think the main reason is that we can't physically interact with it with our hands. We don't play with it as children, the same way we do running water, throwing rocks, etc. Anything that you can't manipulate or play around with is very hard to develop an intuitive sense of. It takes a lot of study and imagination.

I a sense, I DID play with electricity as a kid. My father was an Electrical Engineer and a Ham Radio operator, so we had "stuff" around the house. I learned to solder at a young age. Some time ago, I realized that one of the best toys I had as a kid was the controller for my HO train set. It had a Speed (voltage) lever and a Forward/Reverse lever. Basically, it was a safe, variable voltage DC power supply. I could play with switches and flashlight bulbs and small motors and learn how they work. I've recently gotten back into Electronics as a hobby and I'm having all sorts of fun with Arduinos and Pis and such.

But even then it's not the same because you can't actually see the electricity. You can only see second order effects.

> We don't play with it as children, the same way we do running water

Navier-Stokes equations are not easy to understand.

That's true, but people have been swimming, boating, and sailing for thousands of years before those equations were invented. We can develop an intuitive sense for that, because we have direct and visible interaction with those fluids.

This article is amazing. It helped me to understand things that I were confused about years ago. Whole website is filled (!) with these kind of articles http://amasci.com/ele-edu.html !

Electricity doors mean one thing: the movement of electrons. There are a lot of significant ways it manifests, like in static charge and power systems, and there are a lot of insignificant ways it manifests, but it's still just electrons moving around.

I actually like the water model. Here is a transistor:


It breaks down as an analogy as soon as magnetic fields come in to play, which is pretty much any AC circuit.

What's the difference between ionized hydrogen and electricity?

As mentioned in the article, referring to "electricity" is kinf of vague. Ionized hydrogen floating about is a plasma; a collection of particle with a positive net charge. Therefore if some H+ ions move in a particular direction, that will be a flow of charge: an electric current.

Similarly H+ ions in aqueous solution can carry a current, as in electrolysis.

No, it doesn't have a positive net charge. In a plasma, the electrons are still part of the substance, but they are no longer bound to the nuclei, and are free to move about. Similar to current in an electrolyte, current in a plasma is the result of electrons moving one direction, while the nucleii are moving in the opposite direction.

Hydrogen gas is insulating matter. Ionized hydrogen is conductive matter.

Too bad we don't have metallic hydrogen. It would be a solid conductor, just like any other metal.

"Conductor" actually means "contains mobile charges." Conductor doesn't mean "a hollow pipe which electricity flows through." Conductors are more like long, narrow ponds. They're made of 'electric fluid,' so if we have a ring-shaped pond, we can push the water along so it starts moving in a complete circuit, like a drive-belt.

We have: On October 5th 2016, Ranga Dias and Isaac F. Silvera of Lyman Laboratory of Physics, Harvard University released the first experimental evidence that solid metallic hydrogen has been synthesized in the laboratory.

It took 495 GPa pressure to create. The sample is being held in the cryostat in liquid nitrogen.


They can then check the possibility that, once formed, it remains stable at low pressure.

All those words and it doesn't even entertain my favorite if-i-had-a-time-machine scenario: EE would be a bit easier if Franklin had swapped positive and negative.

Sure it does. You just have to look at the list of articles, not the giant pile of random notes.

BEN FRANKLIN SHOULD HAVE SAID ELECTRONS ARE POSITIVE? Wrong. http://amasci.com/miscon/eleca.html#frkel

EE would be much HARDER if Franklin had swapped positive and negative, since then our confusion wouldn't lead us to shatteing our own misconceptions. We'd never sit down and figure out what "conventional current" actually is. No, it's not backwards. And no, electricity is not made of electrons. In acids, the electric current is entirely a flow of protons. In dirt, oceans, and human bodies the current is at least two separate flows: clouds of positive ions passing forwards through clouds of negative ions travelling backwards. With two opposite charge carriers, what then is the "true" direction of electric current? What if there are five: +Na, +K, +H, -OH, -CL ?

Cute notion: Ben Franklin's kite string was an acidic conductor, a piece of twine which becomes insulating in dry conditions, so it's an electrolyte. And acid conductors have mobile +H ions to carry the current. (What's a hydrogen atom, with one missing electron?)

In other words, Ben Franklin's kite string is a Proton Conductor.



He only was wrong in the case of metal wires. In his day, a typical "conductor" was a small boy hired to hang from silk ropes, to connect the Leyden Jar to the "Electrical Machine." Or rather than commoners, sometimes they used chains of Elizabethan royalty, all standing upon insulating stools.

http://amasci.com/ele-edu.html There's more, lots more. ;-)

Perhaps Quantum Mechanics has an answer to that question. And if we can deeply understand our body, we perhaps have understood electricity.

wish there was a similar website I knew of when I was a EE student -- them misconceptions were crazy enough that they led me to drop out of EE and move into comp. science :)

Whaaat? Electricity does not flow at the speed of light?


EM radiation moves at the speed of light; electrical fields and signals move near the speed of light, depending on the surrounding dielectric properties; electrons in conductors move fairly slowly. Yes, this is kind of confusing.

When I step into the pond, the ENTIRE WATER LEVEL RISES AT THE SAME TIME! Water must travel instantly?

So, a hydro dam is actually a method for sucking the energy out of the entire surface of a lake, all at the same time! (Actually the pressure-waves travel at the speed of sound in water, a few thousand MPH.) It doesn't happen instantly, but damn close.

Electric potential travels at nearly the speed of light, the electrons themselves do not (they flow at a rate proportional in some way to the amount of potential, iirc).

Dear author of the website: Nice article. Please fix the TOC links on "whyhard2.html" to point to "whyhard2.html" and not to "whyhard1.html", and remove the huge whitespace between TOC and content. This was a very confusing experience and likely others will fall in this trap. Thanks!


Fixed now.

I messed it up last time I was getting slashdotted by reddit and ycombinator, and everyone was complaining that the raw unedited notes were just a bunch of raw unedited notes.

Heh, it's still a 1995 gopher-era design, back when we put all our chapters on a single long page, to compensate for 300/2400b modem speeds. If it was broken up into many separate pages, you'd be waaaaaaiting for the text to finally appear.

Thank you very much :)

You might have more luck here http://amasci.com/amateur/amfrmH.html#urls

Wires may not be hollow tubes but surely as a EE he knows that most current flows through the outer shell of the wire?

E: Yes, I was thinking standard AC for household appliances, my bad

This is not true for low frequency current.

What kind of current are you putting into them? And what kind of wires?

For DC, no it does not.

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