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Two capacitor paradox (wikipedia.org)
438 points by rathel 43 days ago | hide | past | favorite | 169 comments

Its easier to think about this using the water pressure analogy. Imagine a large barrel full of water connected to another identical barrel. The first barrel is filled to the top whilst the second is empty. A valve is opened between the two barrels and they equalise with half of the water in each.

The initial energy is mgh and the final energy is 2 * (m/2 * g * h/2) = mgh/2 so half of then energy has disappeared. It is clear that work could have been done by the water moving between the two barrels (like in a hydro-electric power station).

And the resolutions are the same as in the capacitor example: either friction in the pipes damps the oscillation, or the inertia of the water keeps it oscillating between the two barrels indefinitely and half the energy is kinetic.

Well, technically there would also be some inductance... which as an externally coupled field would radiate, and that inherently causes loss!

I’m curious, how would you describe inductance in the context of the water analogy? I’m not very experienced in electronics so I’m hoping the “water version” of it might help me understand electric inductance better.

There's a property called "inertance" which is analogous to inductance in electrical circuits: https://en.wikipedia.org/wiki/Inertance

and it can cause things like water hammers: https://en.wikipedia.org/wiki/Water_hammer which are analogous to inductor sparks: http://web.physics.ucsb.edu/~lecturedemonstrations/Composer/...

This video explains water hammer really well.


Yes it does! Watched, learned something, grateful.

It is the reason why domotica valves move so slow. If they were to move instantaneously then it would cause water-hammer to the point where it might even dislodge the valve from its connections if the run before it is long enough!

There is also a practical device which is basically a boost converter for water: https://en.wikipedia.org/wiki/Hydraulic_ram

It is the inertia of the water in the connecting pipes. In electrical systems, the inductance is that property of a component which tries to keep current constant. In both forms of the thought experiment, if you shut off the connection instantly, the voltage or pressure build to destroy the thing which stopped the connection.

Water analogies only go so far. You would do well to pick up a book on basic electrical networks to learn this material further. Don't shy away from the math; it's really the only way to build the understanding that leads to intuition.

In electric circuits capacity and inductance make energy oscillate between the electric and the magnetic field. The closest analogy I can think of for water pipes would be the oscillation between the kinetic and the potential energy (or the flow and the pressure). As for better understanding of inductance and magnetic field, imho it's better to stick to Faraday, Gauss and Lenz laws than looking for analogies, as EM field has it's very specific properties.

Might be an unpopular opinion but my advice as an electronics engineer would be to try and move away from water analogies as quickly as possible. While they can be useful at the start for some basic concepts, I feel like trying to relate everything to hydraulic concepts (or springs) makes it harder to understand things in the long run when you start looking at concepts where the analogies don't work as well.

I have only a passing understanding of both, but enough to appreciate the (limited as you say) intuitive resemblances - can anyone enlighten me as to what are the fundamental mathematical differences between modelling electronics and water?

If you keep the water slow enough to stay in laminar flow and the electronic scales large enough to avoid quantum effects, electronic and hydraulic components have an essentially 1:1 correspondence.

The biggest difference is in mechanical effects: in hydraulics, physical force is primarily a function of the system pressure and velocity of the flow rate. For electromagnetics, those are reversed: voltage controls the speed of a motor and current controls the force, for example.

They also deviate from the linear regime in different ways, which means the more interesting components have to be built completely differently to perform the same job. A one-way check valve and a diode are both governed by similar equations on a macro scale, but you’ll never be able to understand the internal structure of the valve by an electronic analogy or the design of the diode by a hydraulic one.

What would be a better visualization?

What I’m saying is that it’s best to get to the point where you can reason more directly about voltage, current, charge, inductance etc.

Part of that is getting comfortable with the maths, and the way that’s visualised is just equations, graphs and sometimes things like diagrams of fields or heatmaps. I work in RF so we do antenna simulations and things like that, and the software generates radiation pattern diagrams in 2D, or in 3D heatmaps showing the energy density. It just doesn’t work to, say, try and think of what would happen if an antenna was spraying water out or something.

The analogy I always used as a kid was a heavy, free-spinning water wheel in the path of the water. It would tend to resist changes in current.

And the loss will be in the form of gravitational waves.

Argh is this real or not? Would energy radiated away as a result of inductance be analogous to energy radiated away by gravitational waves in the "water pressure analogy"?

I think so yes, but the effect would be very small

Probably more simply in kinetic and sound energy with water as it rumbles through the pipes and vibrates the pipes and barrels a bit.

That, along with the parent comment, clinched it for me. Thank you.

I think the Wikipedia article's final sentence is probably the easiest way to visualize the question if you want to stick to electrical terms. If there's just one capacitor and you short it out by closing the switch, it's not hard to see how all of the energy vanishes, rather than complicating the question by referring to half of it.

Energy doesn’t vanish. Part of it is dissipated in the wire’s resistance and the rest is radiated as EM energy in a ring-down oscillation.

Yes, and that's the point; to answer the paradox you have to explain where the seemingly-"vanished" energy goes, and that's relatively straightforward in the single-capacitor case.

It's then equally straightforward to see how the same explanation applies to the more contrived case of two capacitors.

So how do we capture this 'resolution' in actual equations? Why aren't the equations revealing this to us?

You can set this up with superconductors so they have zero resistance, but any wire will still have a trace inductance. (Moving charges must create a magnetic field, so if you're moving charges from one capacitor to another, you need to have some inductance somewhere).

Once you include that, you'll see your LC resonator is perfectly undamped and oscillates the charges back and forth forever, breaking the assumption that a "steady state" would equalize the charge on both capacitors.

The equations will also reveal the problem to you when you try to calculate the current that flows from one capacitor to another, with no inductance or resistance in between. You might try putting in an inductor and looking at the circuit behaviour in the limit as the inductance goes to zero -- you'd see the frequency of oscillation climbs to infinity; the full charge essentially teleports back and forth from one capacitor to the other, but still never settles into a steady state of equal charge on both capacitors.

Once you see this, it's like having a problem set up with a frictionless ball on a hilltop beside a valley, saying "in the steady state, the ball has rolled down and settled in the valley. But there was no friction! Where did the energy go?", and the answer is just that the ball doesn't settle in the valley, but rather continues back up the other side, carried along by the kinetic energy that had been neglected in the problem statement.

(Before the well-actualies point out that an LC circuit will damp itself via radiation, let's just say it's also perfectly shielded).

The actual equations do capture it. It's just the simplified minimal equations that result in 'paradox'.

A full model is complicated by the fact that your capacitors have an internal series resistance and leakage resistance, and that the leads and circuit board traces have resistance and inductance. Just like the pipes and valve has some resistance to flow, and some water might leak out or evaporate, and the water has inertia and nonzero viscosity, and turbulence will turn some of the motion to heat, and depending on the phase of the moon, the time of day, and the compass orientation of the barrels, the water may be pulled into a picometers-higher tide in one barrel. When you say "they equalize with half the water in each" you don't typically mention that the phase of the moon may be a factor.

Still, picometers are very different from "half the energy in the system".

We do store energy in water towers for example, so it is pretty surprising and unintuitive that if you put two large water tanks, one full and one empty, right next to each other, open a valve between them allowing their levels to equalize, then assuming there is no distance and you used teflon coated valves you lose... half of the energy as they equalize!

I certainly wouldn't have thought so. I'd have thought you keep 70%-95+% of the energy.

Actually the oscillation explanation didn't match my intuition at all, because I would have thought the water flows from high to low until the point of equalization and then stops flowing, without oscillation.

I get that this doesn't happen, but I would have thought it would!

> Actually the oscillation explanation didn't match my intuition at all, because I would have thought the water flows from high to low until the point of equalization and then stops flowing, without oscillation.

It might very well do that. You will just have lost half the energy already to heat from friction with the piping and due to water's viscosity.

> it is pretty surprising and unintuitive that if you put two large water tanks, one full and one empty, right next to each other, open a valve between them allowing their levels to equalize, then assuming there is no distance and you used teflon coated valves you lose... half of the energy as they equalize!

If you did that, they would equalize - very briefly - and then the second tank would fill higher and higher, until it's (nearly) full. Then the reverse begins.

This is very similar to a pendulum. Just that our intuition about pendulums is better than for near-frictionless transfer of fluids in connected systems.

> I'd have thought you keep 70%-95+% of the energy.

Well, the potential energy being zero at the bottom of each tank is arbitrary. If both tanks are inside a water tower, you might keep 99% of the "useful" energy even if you let them equalize, because the height above ground is greater than the height above "tank bottom". Maybe this is the source of some confusion here?

You lose 50% of the energy relative to the bottom of the water tanks (actually, the level you're using the water to perform work). If you did this with two 20 foot high tanks, but on towers 40 feet off the ground, then you retain 45 / 50 (the ratio of the changed center of mass) == 90% of the energy.

The oscillation bit happens, but it doesn't dominate. Water analogies can be misleading because water has intrinsic properties (eg turbulence ~= resistance) which aren't always significant in an electronic circuit. To make the equivalent of an LC-dominant circuit with (open) water tanks, you'd need something like a high-momentum turbine in the transfer pipe.

When water equalizes in the "water seeks its own level" demonstration (in a series of connected glass tubes of various shapes, the water is at the same level in the whole system, as here: https://www.physics.purdue.edu/demos/display_page.php?item=2... ), does the system lose a lot of potential energy as the water equalizes?

The demonstration is called Pascal's vases.

In normal situations, you would not lose half the energy, or hydropower would be totally bunk (actually maybe not, considering the efficiency of burning fossil fuels…) But in this degenerate case, you cannot ignore the losses.

In a hydro power plant, the water turbine constitutes the "friction" so, if there were no other losses, the half of the energy "lost" from the system is precisely what your generator gave you.

Are you sure? I was going to add that the intuition comes from demonstrations we might have seen, where in a series of tubes, regardless of shape, water will be at one level. As in this picture:


So is it not accurate to say that water does this by oscillating and throwing away energy as parasitic losses, until it equalizes?

Are you saying in general does a system of connected tubes NOT throw off lots of energy as it gets into the equalized state shown?

If it does, I think this fact should also be mentioned when teaching the "water seeks its own level" demonstrated above. (Called Pascal's vases.)

Here's a video showing water oscillating: https://www.youtube.com/watch?v=cVEbh_COcRY

Cool! Thanks :)

The equations do reveal that! As Wikipedia pointed out in the idealized model the current is infinite and the time to settle is zero. Thus the energy loss is proportional to ∞*0 , that tells immediately that the model is not applicable.

We make an intuitive assumption about the final state and then use the rules we know to apply equations to verify the final state.

In this case, the equations not matching up proved that our initial assumption (steady state) was itself wrong.

If I think straight, the initial potential energy in a column of liquid would be the integration of mgh (actually integration of density * g * section * height), therefore mgh/2 at the beginning of the situation. Or said intuitively : the center of gravity of the column at half its height.

But obviously, even with this 1/2, your relation after dividing between two tanks still holds. :)

I was thinking that too, the initial potential energy should be proportional to h^2/2. However then we still have 2*(h/2)^2/2 = h^2/4.

Yes, one h is hidden inside m... Ending up with energy = something * h^2 actually makes the relationship show the same face as the one for capacitors, where energy = something * v^2.

* where h = half the water level in the first barrel.

Good observation!

Another equivalent analogy is a pendulum. Hold a pendulum with mass m at height h, so its potential energy is mgh. Let it go, and after a long time of swinging back and forth the pendulum eventually stops at the bottom (h=0). Where'd the energy go? It's all lost to heat through friction from the air and tension on the string. The rate of energy lost is just a lot less than the water barrel example. The general name for this kind of system is a harmonic oscillator.

Now I want to try the water barrel thing myself and see how many times the water goes back and forth before it finally stops...

It's also intuitively obvious that energy must disappear, otherwise nothing would happen (the system would already be in its lowest energy state).

Very nice! The hydraulic analogy actually works for electric circuits in general, and can be made precise (you can calculate how much pressure corresponds to one volt etc), though the numbers involved can get a little silly: http://amasci.com/miscon/voltpres.html

Heat transfer as well.

At the first engineering firm I worked for, we had a very good heat transfer solver but no electrical solver since it was an infrequent need in our field.

One of old timers was an expert at reframing the electrical problems into heat transfer, solving in the available tools, then converting back. As he said "it's all unit conversions". I never picked it up beyond simple resistance networks, but it was a cool way to abuse the tools.


I don't think the water analogy is a good fit here. For me the revelation was that energy can also be radiated away. When we did basic circuit exercises we never talked about that. Energy was either stored or turned into heat. Not sure how you would explain electromagnetic radiation with water pipes.

If you actually build this on a PCB with typical ceramic or electrolytic capacitors and a MOSFET or mechanical switch, an insignificant amount of energy will be radiated. Most of the energy does go into heating the capacitors through their internal series resistance.

Radiation could be the sound produced by the rushing water?

A steady current would also produce a rush of water, but I don't think you can get EM radiation with just a steady current.

Well, but maybe... if you equalize two water tanks with a very fat pipe it will swap forth and back, causing the whole assembly, table, and room to rock?

And that is what is called an analogy. Gives a simple and intuitive understanding but is not complete at all.

Gravity waves ;) - if they were many orders of magnitude stronger.

The secondary question though is: Why does the energy always dissipate by 50% in both the electric and hydrodynamic models?

Is there some underlying physical explanation of this? Something that says that "in a dynamical system the maximum efficiency can be at most 50%".

Because in the examples you have two capacitors/barrels, if you had three or four, you would be looking at 33% or 25%.

To say that the energy of the water is mgh is a simplification. It's really 1/2 * h^2 * A * mass density. The water at the top has more energy than the water on the bottom. The energy in the water is square to the water level, but the amount of water is linear. It's the same with a capacitor, the energy is square to the voltage, but the number of electrons is linear.

It works out to the same 50% because the analogy is just very good. Dividing over two barrels/capacitors means halving the potential level (amount of water/load doesn't change), which means 1/4 of the energy in each barrel.

This is wrong. The energy is proportional to the height, not its square. You would get wrong units otherwise.

The energy is proportional to both the height and the mass. The mass is proportional to the height itself. Hence the energy is proportional to the square of the height.

If the fill level is h, so the weighted average of the mass is at h/2,then

E = m * g * h / 2

m = V * r

V = A * h

E = A * r * g * h^2 / 2

m2 * kg/m3 * m/s2 * m2 = kg m2 / s2 = J

Yeah, I totally forgot that there is a height component in the mass as well.

I think it's probably easier to imagine as a pendulum.

And if no work was done that energy turned into heat?

I once got this question on an interview, and they didn’t know I’d heard it before. So I pretended to figure it out during the interview. I gave him a correct answer after just a few minutes, and they cut the interview short and had me directly proceed to the HR office to proceed with onboarding. Two years later I told them how I already knew the question. I’ve been with the company for 6 years now.

A friend of mine had two job interviews lined up back to back for two game studios. On the first interview he bombed pretty poorly, but the interviewers were great guys who explained every solution and helped him work through the problems.

On the second interview they asked the exact same questions and he got the job offer.

He's now got his name credited to several AAA games and moved in with an indie studio to publish his personal project under them.

> He's now got his name credited to several AAA games and moved in with an indie studio to publish his personal project under them.

See? Interview puzzles work!

Reminds of how I dev-tooled my way through a JS test with a nice 666% score and with NaN time left.

"We are pleased to inform you of your top standing on our developer test, with a score of 420/69."

You just gotta Kobayashi Maru your way in.

Hadn't coded much in the 9 months I'd been off. Googled Java interview questions, practiced on the first few results.

Interview question was the first result... 7 years later.

Your interviewer: hadn't coded much in the 7 years, googled Java interview questions...

I got the "manhole cover is round" problem at a interview. I'd already heard it before and admitted so and proceeded to give a number of reasons why it was round. They were still impressed.

> a number of reasons why it was round

Except that question in particular is a fairly silly one. A satirical rendition exists that illustrates why, and happens to have been discussed long ago on HN (https://news.ycombinator.com/item?id=569897) but apparently the original article has moved (https://sellsbrothers.com/12395).

I had this one too. When I told them I already knew it and explained the reasoning I got a new problem instead.

That question seems backward.

Instead they should have asked: design a manhole such that these constraints are satisfied.

Similar story. Got a life-changing job because I figured out the Monty Hall problem. Hadn't heard about it before. Lucky day!

I'm not sure if it was the MH problem--all I can remember for sure was that it was a probability problem that was simple to state and the immediately obvious answer was wrong--but I have a similar story. Except I blew the problem and didn't get the job.

It was 1982. A small company from the Pacific Northwest called Microsoft sent some engineers down to recruit at Caltech. I was a senior that year, and interviewed with them. They asked me their probability problem, I got it wrong, and did not receive an offer.

I've occasionally wondered how different things would have been had I got it right, and so maybe got an offer, and accepted [1]. I think pretty much every engineer that started at Microsoft around that time and stuck with it (which I almost certainly would have done) ended up quite well off.

I've also wondered if any of the engineers that interviewed me are people whose names I would now recognize.

[1] It was not at all certain I would have accepted. I was more interested in bigger machines than the microprocessors MS was focused on, and also had a strong preference for staying in the Los Angeles area.

...in the alt reality you readily passed the interview and accepted the offer. MS tanked, DIGITAL pivoted into PC market and dominated it. At some point alt-you made this post on alt-HN.

We just hope to be certain about the present moment.

I really love the short QA style interviews, they are so accurate and comprehensive and not at all circumstantial.

Well you deserved some credit for having the curiosity and energy to read about these things. Weak candidates know nothing more than what was fed to them in their last class.

Isn't that quite unethical? You are knowingly misleading a potential employer when you play games like that.

I once interviewed fresh out of college for a tech job at a big shop. When the interviewer got to the puzzle questions, they prefaced with ‘And be honest if you’ve heard it before. We want to get one you haven’t heard.’

Seven puzzles later of me saying, “yes, I’ve heard that one” and giving an short description indicating that I understood the solution, the interviewer became visibly upset/annoyed.

And so when they asked the 8th question I suddenly ‘didn’t know this one.’ The interviewer was delighted. Of course I did know it already but pretended to work my way through it.

Got the offer, didn’t take it because that interviewer would’ve been my boss’ boss.

Similar story: I had an interview for an internship at a trading firm, in which they asked me several questions that I wrote (for a math competition at my school). I took the honest route and explained that I had authored all these problems, leading to the interviewer getting so annoyed that they just cut the interview early.

Unsurprisingly, did not get an offer.

How about being interviewed using your own book, but the interviewer didnt notice authors name on the cover, doesnt understand the questions and butchers the problems? ;-) Happened to Elecia White (of https://embedded.fm/)

Lying (misleading) is a complicated subject. Though some use it to excuse too much, lying and misleading can actually help things flow smoothly and justly.

In this case, if the interviewee had been radically honest and interviewer had not had a backup question, the interview might have gone badly, even though the interviewee apparently was a good employ.

In fact, radical/extreme honesty is often seen as a sign of childhood problems or mild autism by mental health practitioners, as people normally learn to white lie at young age.

> If the ... interviewer had not had a backup question, the interview might have gone badly

That seems like a wild assumption to make.

Interviewers will usually tell you straight up to inform them if you already know the question.

What would happen if you didn’t?

You'd be joining the ranks of pod people who real people work day and night to eliminate from among them.

If you are fit for the job role, then no. The employer is misleading themselves by thinking that a silly interview question is a good measure of a person's ability to fill a role.

GP is in fact, doing the employer a favour by bypassing this silly charade and letting them have access to his real talents.

I am not entirely certain I owe my potential future employer a favour which is probably going to turn against me immediately.

Knowing an answer is one thing, being able to explain it is another one. You can easily spot people who "cheat" by rote learning.

As long as you can actually walk them through your thought process on how you came up with the answer, I think you're good.

But a good interviewer should also follow up on questions that can lead to trivia-type answers.

For example, if some interviewer ask you "What is the sum of integers from 1 to 100", a valid answer from you could be

"Well we learned in school that Gauss figured out this problem at early age, and came up with the answer 5050 - so the answer is 5050"

In which case the interviewer has the options of

A) Conclude the question, and move onto something else

B) Try to get you to expand on this theory - and how Gauss came up with that

If he then goes with B, and you start explaining that (1 + 100) + (2 + 99) + ... + (50 + 51) equals 5050, he could ask you to come up with a more general form of this answer - i.e the sum of some arbitrary arithmetic sequence.

And if you then go to expand on this, and arrive at the answer that the sum of the original sequence is sum = s(s+1)/2, then that is IMO very good for both parts.

The key here is that you actually walk them through the steps, and interact with the interviewer. Maybe they introduce constraints, or new rules.

One can take "easy" questions that you've seen thousands of times, and make them into interesting ones, where you spend some time communicating, and showcasing how you solve problems.

The point is not to arrive at "the sum is 5050", but rather to monitor how you think, and solve problems, with people around you.

Well interview's typically have a trick question... Perhaps this was a trick answer

Employers mislead employees all the time. You could say the hiring process was unethical as it rewarded having googled questions instead of actual competence, thus robbing a worthy engineer of their job.

I've been sitting here for 5 minutes feeling horrified and sad that your comment is downvoted to near unreadability for suggesting that lying like that in an interview may be unethical, while a sociopathic-sounding child comment extolling the virtues of lying/misleading – they "can actually help things flow smoothly and justly", without lying "the interview might have gone badly" and lecturing you on "radical honesty" being a sign of problems/mild autism like you're a naive idiot who just doesn't understand the world – isn't downvoted at all. I wouldn't call it "radical honesty" not to lie in a situation like this! Sounds like "radical honesty" has a new meaning "not lying in any situation you could gain an advantage by doing so".

This reminds me of another time on HN someone told a story about how they'd cheated in a hi-tech way in an exam and a dozen comments supported them, not one person suggesting it might be wrong...until me, then I got virulently attacked for doing so. At least there their comment got [dead]ed.

I just wanted to say - you're not alone, that also seems very dodgy to me. Moments like this HN is very disappointing.

I thought "radical honesty" sounded silly. And turning the argument into an attack on the speaker wasn't kosher either.

But I also thought "quite unethical" was a bit silly (though I didn't downvote anyone). Perhaps people didn't like the tone. Also there could be a language issue with the degree implied by the word "quite". In stark black and white terms, it is unethical, yes, as a lie of omission. If there are shades of gray, it is not on the extreme to get lucky and have heard a question before.

I have also noticed there is a contingent of commenters who seem to view stories in terms of the victimized workers versus the evil corporations, and punish people who appear to take the wrong side. Probably a similar bunch shows up when the argument is about students versus teachers. Not necessarily a universal lack of ethics.

If you want to change things, the first step is to recognize the problem lies to those with power, like your manager, who lied their way to the top, not to a fellow employee trying to get a job. Being honest to liars and manipulators to earn enough of their pity to hire you did not suddenly become the peak of morality. The bottom of society is responsible for their unhappiness because they have not risen to end it, not because they have not licked enough boots.

I don't think this should be called a paradox. It's just a case where the limitations of the model (ideal everything) is clear and leads to inconsistent results. Adjusting the model and making it more realistic, quickly clears up the "paradox". To me this seems like something one would use as an example in a physics lecture to show when certain assumptions are necessary and when they aren't.

> To me this seems like something one would use as an example in a physics lecture to show when certain assumptions are necessary and when they aren't.

IMO, that's the perfect use of this thought experiment. In fact, I dare say if there are more examples like this, where laying out basic clear assumptions and following the model through its rules leads to an answer that is clearly wrong, then this example and those other examples of that type should probably make up a series of lectures in a course (or subsection thereof) on model use. It seems to me that plenty of people pick up on what models are and how to use them. It seems less people internalize, "All models are wrong. Some models are useful."

You can easily get "invalid" circuits when using diodes (the standard assumption of 0.6V or 0.7V drop across didn't make sense). I once got one of those on a final exam and waster 40+ minutes trying to figure out why my solution was "impossible." The teacher later cancelled that question, but I didn't get my 40 minutes back. Many lesson learned: A) don't be stubborn, let go; and B) circuit models ≠ circuit reality.

Notice the definitions:

> The two capacitor paradox or capacitor paradox is a paradox, or counterintuitive thought experiment, in electric circuit theory.

> A paradox, also known as an antinomy, is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion.

(emphases mine). It's paradoxical in that it runs counter to expectations or intuition, using apparently valid reasoning. (Of course, maybe it doesn't run counter to your intuition; but that's how correct intuition is developed, by breaking naïve intuition against examples like this!) In fact, the article explicitly acknowledges this:

> Unlike some other paradoxes in science, this paradox is not due to the underlying physics, but to the limitations of the 'ideal circuit' conventions used in circuit theory.

Fair enough, wikipedia does include the reference to expectation in its definition of paradox.

However, I think my main point is still valid that this should not necessarily be called a paradox exactly because it differs a lot from how this term is commonly used in the naming of scientific thought experiments.

My initial gut response was the same - that this isn’t a paradox, but rather a proof by contradiction that one or more of the assumptions are wrong. But after thinking about it a bit more, I don’t think there’s really that much of a difference. It’s just that some people have already learned not to apply those assumptions in this context (or by chance never stumbled into them in the first place).

Perhaps it’s fair to say that all paradoxes are “just” part of a proof by contradiction and the only difference between ones that feel truly paradoxical and those that feel obvious is something subjective about a person’s understanding of the situation. Maybe it’s down to whether someone has enough of an understanding of the conceptual landscape to be able to actually recognize some of the problematic assumptions from the start of the problem statement or not, or maybe it’s something else, but either way I’m suspecting that it’s something subjective related to a person’s state of knowledge and/or approach to organizing their own knowledge.

Model inconsistencies, unreasonable mathematical idealiazations and false assumptions are what are commonly classified as paradoxes in physics. It’s a common use of the term.


Consider a simplified version of the problem, without the second capacitor. When you close the switch, all of the energy in the first capacitor mysteriously disappears! Or not so mysteriously, after the first time you've done it by accident.

There are three tiers of paradoxes in physics:

- these which only contradict "common sense" intuitions/expectations but reality conforms to predictions of the theory instead of "common sense", and so these are usually called not real paradoxes like Pascal's hydrostatic paradox;

- these which show us limitations of the models we use to construct them, like two capacitor paradox or black hole information paradox;

- these true contradictions in the nature of reality which when you discover them the universe disappears in the puff of logic and you wake up as a student in the middle of the physics lecture, and that student was Albert Einstein.

A difference between model which we know how to adjust to make more realistic and one we don't know how is about history of physics and not physics itself.

the birthday paradox is also not a paradox. the namers of such things just like to use the word ‘paradox’ for some reason. it probably sounds more puzzling to initiates.

My gut feel is the same. Ironically, the fact that this usage sounds wrong might, in fact, be a paradox :)

It’s an apparent contradiction between our understanding of the meaning of the word and the actual usage of the word. The resolution of this paradox could be that there is widespread wrong usage of the word. Alternately it could be that our understanding of the meaning is wrong - which itself could be for multiple reasons. Perhaps our understanding of the definition is incomplete. Or it could be that the definition itself is simply wrong because it doesn’t reflect actual real-world usage.

Upon some reflection I think that a core part of the idea of a paradox is that there are many possible solutions to a problem and that some people, but almost certainly not all, will initially fail to even recognize the existence of solutions other than the one they expect, due to unconscious assumptions.

Naturally in many cases there will also be people who do not make those problematic assumptions to begin with, but that doesn’t on its own invalidate the use of the word “paradox” in that situation. The birthday paradox probably fits this definition just fine - ask anyone not trained in statistics about the subject and you’ll likely find a lot of people have unconscious assumptions that lead them to wrong answers, and in some cases thy will even be exceptionally confident in their wrong answers.

You might object to the “not trained in statistics” part but that objection itself would be circular - modern statistics education was designed with paradoxes like this in mind, specifically to lay down a foundation that doesn’t lead you down the wrong paths in cases like that. The solutions to the basic paradoxes of statistics are built into the way statistics is taught, just like the solutions to Godel’s and Russel’s and many others’ paradoxes are built into the foundations of any modern logical system.

Good point. A paradox is something that produces an unexpected contradiction. It must be unexpected, otherwise any trivial false statement such as 2+2=5 would also be a paradox. Russell's paradox is a good example since no one expected a contradiction. The birthday paradox on the other hand does not produce any contradiction, so it's just a case of sloppy labeling.

By that description, the birthday paradox is a paradox for anyone who doesn't already know it. The instinctive answer is generally 1/2 the total.

> the birthday paradox is a paradox for anyone who doesn't already know it

It is a fate of any solved paradox. A paradox is a confusing paradox only to a point when you learn where the contradiction comes from, and how it does it, and how to avoid it.

It's still not a contradiction, unless you first postulate that 1/2 is the answer, for no particular reason at all.

This example violates one of the core assumptions of circuit theory, which is that you cannot have a node with two voltages. As soon as the switch closes, is the voltage at the switch Vi or 0v? It would be both, which is impossible. If you had some kind of component between the two then there would be a voltage drop across it, and you would get realistic results.

It’s insufficiently specified.

There is actually a small resistance in series with an inductor between the nodes. There’s a time varying voltage across those.

Edit: as a follow-on, the paradox is posed as a problem in electrostatics, when it’s actually a problem in electrodynamics.

The system in question is not closed, as the state change requires application of external energy. Thus expecting a conservation of the initial energy only may be unwarranted.

My intuition is that there's a missing counter-action energy which would be required in order to maintain the initial state in such a circuit. Once the state change is being effectuated (switch closed), it would change the balance of action-counter-action which would lead to dissipation in the real world. Not sure how to quantify/formalize it.

I mean, that is a simplification, and in the real world it really follows some sort of differential equation that just happens to be very fast. And the description on Wikipedia does seem to use an explanation based on this differential equation, e.g. see "When a steady state is reached".

The differential equation is still a model and not what's happening in "the real world", remember, all models are false.

It's simpler than that. When you short a capacitor, where does the energy go? An ideal capacitor has an infinite current in that situation. Inductors have a comparable situation - when you open-circuit an ideal inductor, you get an infinite voltage.

In practice, you can get hundreds or even thousands of volts from inductors that way, which is how auto ignitions and boost-type switching power supplies work.

Similar problems come up in the idealized physics of impulse/constraint physics engines. Getting rid of the energy in collisions requires hacks to prevent things from flying off into space, a problem with early physics engines.

That's one way to solve it. More generally, thinking of edge cases or simplifying the problem is a good problem-solving strategy.

Adding the second capacitor is equivalent doubling the area of a single capacitor. If you had a plate capacitor where the plates were extendable you could charge it and decrease the energy by extending the plates, which tells you that there's a force that is pushing to extend the plates. Of course there is--the charges on one plate repel each other and "want" to increase their spacing. If you do extend the plates there is a force times a distance, which equals work done by the charges, decreasing the capacitive energy.

This is a beautiful explanation, thank you!

It's also compatible with barbegal's hydraulic pressure analogy elsewhere in the discussion. A liquid- or gas-filled tank feels pressure (i.e. force per area) on the walls as the fluid "wants" to either (1) flow to a lower gravitational potential energy location, for a liquid tank in a gravitational field, or (2) expand in volume, in the case of a pressurized gas tank.

Here's another one - put an oscilloscope across a capacitor and turn off the circuit it's connected too. Quickly short the capacitor with something, and watch the voltage on the oscilloscope as the capacitor discharges, remove the short and watch it start recharging slightly.

Where does this charge come from?

It's a non ideal behavior of a cap, and some dielectric types have it worse than others. I once made a sample-and-hold for a data acquisition system, and must have tried every type of cap in the bin until I found one that resulted in the least amount of correlation between subsequent readings. I think it was a polypropylene type. Mica was the worst. I don't think I tested an electrolytic.

1. The short is not an ideal short circuit because whatever you use will have some resistance.

2. The shorted capacitor will discharge some current into the short wire and some into the oscilloscope’s capacitor (probably wire capacitance).

3. Once you remove the short wire, some current will flow back into the capacitor under test to equalize the voltage.

Dielectric absorption.

Potentially the scope cable. Cables have capacitance too.

That, or this is the "memory effect" I've heard about in certain dielectrics acted out.

It's in the cap

FYI - this is a popular intro level interview question at semiconductor companies. This mechanism is the whole basis of how DRAM works, as well as the foundation of some cap sense technologies. Likely plenty of applications I don't know about, either!

Explained here:


For an infinitely small resistor the energy is effectively a spark pulse coupled directly to free space, and half is radiated away in EM waves.

Not an electrical engineer, but I wish this was higher. Seeing the integral completely reframes the problem and the paradox vanishes. The intuition is that charge transfer is a linear system (half goes here, half goes there), but seeing the integral form shows otherwise.

Thanks for the great source!

I'm just excited that hyperphysics is still going strong. It looked outdated in 2010!

In the true EE parts reduction fashion, you don't even need two capacitors. One capacitor plus a switch works out similarly - in fact it's an equivalent circuit. Everyone has an intuition for shorting a capacitor (zap!). When there are very few components defining a system, the parasitic components must be significant.

When there are very few components defining a system, the parasitic components must be significant.

Lee Hill, a popular instructor for EMC compliance trainings, has a saying about that: "There's a lot of money to be made understanding the circuit components that aren't captured in the schematic."

The spring/mass model works really well here, too. Connect two identical springs together between a pair of fixed walls. Pull on it so that one spring is stretched and the other is at its natural length....

Or release a ball bearing on one side of a U-shaped track. The ball ends up on the bottom of the track, where did all the potential energy go?

This is effectively the same paradox some friends and I discussed in college. Another way of looking at it is:

If you connect an ideal voltage source through a resistor R to a capacitor C, the amount of energy required to charge the capacitor is CV^2 while the amount of energy that winds up in the charged capacitor is 1/2 CV^2. The other 1/2 CV^s is dissipated across R. This is unaffected by the value of R, even as R -> 0. The only thing that changes is the charge time.

R = 0 is impossible for real circuits, but no matter how close you get you still lose half of that energy in the resistor.

Hearing about this problem while I was taking a stats mech course (online from Leo Susskind) this reminded me more of energy / entropy relationship than a paradox. For example if we have hot and cold reservoirs separated by a divider in a bath, when we remove the divider entropy of the system increases when it reaches equilibrium. After reaching equilibrium, moving the divider back will not cause the two reservoirs to return to the initial hot cold state.

The system with 2 capacitors seems like a good analogy to a heat bath; when the switch is flipped the charge is divided equally between the capacitors when it reaches equilibrium. The entropy of the system has increased, and the potential to do work has decreased. The number of possible states the system can be identified in has decreased by half (it is not possible to know which side had the initial charge after the switch has been flipped). Flipping the switch back will not bring the charge back to only one side. While this line of thinking doesn't explain the physics of what is happening, there is clearly a (statistically) irreversible change going on which seems like the natural language is energy, entropy, and temperature.

Does anyone have a sense of how much the idealization of circuit design limits people's imagination when designing real circuits? Are there fruitful possibilities that could be explored but aren't because the abstraction doesn't contain them?

This question tests understanding of limitations of idealized circuits. Anyone with hands-on experience of shorting a capacitor knows about sparks and arcing that accompanies it.

"While in theory there is no difference between theory and practice, in practice there is."

The effect is smaller than you might imagine, because many of parasitics can be modeled as adding additional basic elements and are understood fairly well. For example here is just some random model you could use for capacitor http://www.iequalscdvdt.com/cap_model.html

If simple lumped element model really fails to model your circuit, there is distributed element model which can analyze all sorts of RF voodoo https://en.m.wikipedia.org/wiki/Distributed-element_model and https://en.m.wikipedia.org/wiki/Distributed-element_circuit

I wouldn't really consider this a paradox. It is a failure mode of the idealization you made in saying that wires and capacitors have zero resistance and inductance.

Even if you work by your idealization, saying wires have zero resistance means that every piece of connected conductor in the circuit is at the same potential (in the absence of a magnetic field) so saying you have two connected capacitors charged to different potentials is already violating your assumption.

You can find similar edge cases in almost every situation and field of study where you try to simplify things with an approximation, and most of them aren't called paradoxes.

Got this question in an interview at National Semiconductor (out of MSEE school). I didn't think it was too paradox-y: we didn't work the math through fully but my answer of "start with the circuit with a resistor with value R and the energy will burn up in R; now recalculate as R→0 and the energy will still burn up in R (even though R is zero-y...". That seemed to satisfy the interviewer (who had not heard that answer before).

I think an interesting question following it up is, how do you transfer energy between capacitors then, without losing so much energy? (someone else asked if this is a fundamental limitation)

There are many alternatives. A good start is noting that the inefficiency is actually lower the lower the starting voltage difference:

V1 = V, V2 = V-dV

V' = (V + (V-dV))/2 = V+dV/2

We can define efficiency as the ratio of energy lost by the first to energy transferred to the other.

dU1 = U1-U1' = CV^2/2-CV'^2/2 = C/2 (V dV-dV^2/4)

dU2 = U2'-U2 = CV'^2/2-C(V-dV)^2/2 = C/2 (-V dV+dV^2/4+2VdV-dV^2) = C/2 (V dV - 3dV^2/4 )

n = (dU2/dU1) = ( V - 3dV/4 ) / ( V - dV/4) ~ 1 - 3/4 dV/V (for small dV)

as dV → 0, n → 1. You lose 3/4 of the fractional difference in efficiency, so for 10% difference your efficiency is ~92.5%, pretty great.

Now, there are indeed devices that change voltages without energy losses! (transformers, for example). So if you plug in a variable transformer that keeps the voltage close to the target, your (dis)charging efficiency can be arbitrarily high.

Of course, if your second voltage is 0, the efficiency must start at 1/3 no matter what (which can seem to imply this cannot be changed) -- but as soon as you have a small voltage you can start tracking it and keep efficiency high.

Challenge to the reader: Use quantum mechanics and thermodynamics to derive a fundamental limit of efficiency (which must be less than 1 at positive temperatures) :)

Inductor in series with a diode. The voltage differential stores energy in the inductor's magnetic field, which comes back out when the destination capacitor is at a higher voltage than the source capacitor. The diode keeps the process from repeating in reverse (oscillation).

Cool, but what if you want the voltages to be exactly equal (while keeping no losses)? :)

Dump the charge into a third capacitor, connect the two you want equal, and dump the charge back.

I remember explaining how an NMR machine works to an EE person. They simply refused to believe me that you could inject current into a supercooled superconductive magnet and have it circulate for months at a time. The only way I could convince him it worked was to point out that slowly the system did loose energy and you had to go back and add more current.

Try it in Spice with realistic small resistances and inductance. Every electrical engineer is taught, in basic theory: The voltage on a capacitor cannot change instantaneously. The current in an inductor cannot change instantaneously. These become much clearer when you write down the INTEGRAL form of the V-I relations.

The problem with this is that once you close the switch, you get an infinite current (aka short circuit). Bad idea!

Yeah, with substantial amount of energy in that capacitor, that is exactly how you weld the contacts in that switch shut.


Why did someone mod you down dead?

- quote - It's an exponential function on the W=CV^2

let's say initially:

C = 1

V = 16

W = 1 * 16^2

  = 256
When the Voltage is split between the two capacitors, the voltage drops in half from 16 to 8, but because there are two capacitors you count them twice:

W = CV^2 + CV^2

C = 1

V = 8

W = (1 * 8^2) + (1 * 8^2)

  = 128
Where did half the energy go? People are saying it's loss in magnetic radiation during the transfer. But on paper it still seems counter intuitive, so it's called a, "paradox," instead of just how an exponential function works. Something in the universe makes the energy levels exponentially higher as voltage increases. More electric pressure (V) = exponentially more energy. Cool! - unquote -

Correction: It's not exponential, it's a quadratic (or polynomial).

(I think the difficult part isn't to accept that some of the energy vanished from the system, it just contradicts our expectation from having supposedly no dissipation elements)

Without reading the solution: wouldn’t this impossible zero resistance zero inductance ideal setup result in infinite frequency infinite magnitude oscillations? Add in some resistance and the missing energy goes into heat generated by the resistance on the way to equilibrium.

Edit: also when I think about it there is a little bit of additional energy in the open switch which is itself a capacitor.

Edit 2: for this circuit to stay the way it is in the initial state, wouldn’t the open switch need to have equal capacitance to the capacitors? Or some kind of voltage generating field applied across it that is removed when the switch is closed?

If inductance is zero, there would be no oscillations.

That would be true if resistance were non-zero.

To get oscillations, you need to shift the phase by 180 degrees. Capacitance alone (without some inductance) can only do 90.

A somewhat analogous effect/paradox in thermodynamics is the Joule expansion:


This is another instance of those situations where "the abstraction is leaky". Relatedly, I've seen (unfortunately) a few textbooks which attempt to teach basics of computing and digital logic by assuming that gates have no propagation delay, or at least that's what their timing diagrams seem to show. It's very puzzling because a lot of sequential circuit elements rely on that in order to work.

A real-world analogy is a brick standing upright on the table. If you topple it and it falls, some of the potential energy is gone.

I knew nothing about electricity until I started studying for my technicians license recently. It was actually very interesting and I learned enough that I could actually understand this wiki article. It made me interested in studying for the general and extra licenses since they have more advanced electrical knowledge required.

Funny thing is - this is exactly what happens many millions times every single clock cycle inside pretty much any modern(-ish) digital CMOS CPU/ASIC, with the right-hand-side capacitor being parasitic gate capacitance in the driven gate.

Which in turn is why your CPU needs a heatsink.

And why reversible computing can be done without using energy ;)

This is an easy one. The assumptions are flawed. W = (0.5) x(C)x(V) is a back-of-the-envelope shorthand for a whole realm of formulas in the study of capacitors, and it doesn't work in edge cases like this.

I learned that this is also why you should never connect batteries in parallel - the power will oscillate in much the same manner, draining away the energy and wearing out the batteries.

The 'paradox' in this experiment lies in believing that you can have a voltage across one of the capacitors and zero volts across the other before you hit the switch. In other words, you can't have two connected locations resting at different voltages.

  --------| |---------------|  |----------
     --- +6 -6 ------------ 0  0 -----

             ^^^^^^^^^^^^^^^^ NOT POSSIBLE!
Two capacitors in series are equivalent to one (combined) capacitor, just like a bunch of batteries can be considered to be one (combined) battery.

You can definitely have a charged capacitor in series with an uncharged capacitor. They only act as an evenly-split combined capacitor if they start at the same charge.

Your diagram is wrong. Let's mark out some contact points:

  ---| |-------| |---
    A   B     C   D
Voltages only exist between points. To measure against ground, part of the circuit has to be connected to ground. Ground is a point like any other.

So when you measure A at 12 volts above B, it doesn't mean B is at any absolute voltage. If only one capacitor is charged, then A can be 12 volts above B while B=C=D. It works out fine.

If you attach ground to B or C, then A is 12 volts relative to ground, and B and C and D are all 0 volts relative to ground.

If you attach ground to A, then A is 0 volts relative to ground, B and C and D are all -12 volts relative to ground.

If you somehow attach ground to the midpoint of A and B, then A is 6 volts relative to ground, and B and C and D are all -6 volts relative to ground.

None of these are impossible. Nothing goes wrong until you try to close the switch.

What experiments were done to settle the question which property of the "non-idealized" models is the dominant one?

My brain hurts, is there a simpler version?

Yes there is, and it's listed at the end.

You have an ideal capacitor, charged to some voltage. You connect both ends, which empties it.

Where did all the energy go?

The answer is that there's no such thing as an ideal capacitor or an ideal wire. There are a few effects you can't avoid, but most importantly there are tiny bits of internal resistance that usually go ignored. In a short circuit, they become important, and will turn all the energy into heat.

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