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 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.
http://amasci.com/emotor/cap1.html