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.
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.
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.
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 .)
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.
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.
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... :)
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.
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.
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...
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.)
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.
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.
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
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.
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.
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.
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'.
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.
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.
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 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?!
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.
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.
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)
> ">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.
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.
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.)
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.
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.
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.
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.
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.