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A Physical Understanding of Kirchhoff's Laws

  1. Mar 13, 2017 #1
    < Mentor Note -- thread moved to the EE forum from the Homework forums since it is a more general set of questions >

    So, we are going into Kirchhoff's Laws in class, and my entire understanding of circuits, which took me a while to form, is again falling apart.

    A physical understanding is absolutely essential to everything I do in physics, and I want to be as strict with it as I can while studying classical mechanics.

    My new issue is with Kirchhoff's rules. I definitely understand the reasoning behind them, matter is conserved, as well as energy, so of course the junction rule describes matter, and the loop rule describes the conservation of energy through voltage. However, the instant I see this intuitive law in action, I fail to picture what is actually occurring on the atomic level.

    To understand circuits, I had to learn electro-chemistry to try and understand the inner workings of a battery. I understand at this point that all batteries are effectively two separated electrodes, an anode and a cathode. By connecting the two together with a wire, their chemical composition (e.g, zinc vs copper) results in a redox reaction in which the atoms of the anode release their electrons into the wire, which travel around the new circuit, and on the way interact with various resistors.

    One important point I inferred from this understanding is that electrons can NOT flow from the cathode to the anode (at least for commercial, non-rechargeable batteries).

    Returning to the above circuit, let me quickly solve it so I have the reality of what is happening, and use it to describe my confusion.

    Find the current through the circuit.

    https://physicsforums-bernhardtmediall.netdna-ssl.com/data/attachments/97/97581-1aad8b8e3ec1d4ec37153371c25a5d91.jpg [Broken] [If not visible, its attached below. Sorry, new to the forum :) ]

    ε1 = 150 V ε2=50 V

    R1 = 3 Ω R2 = 2 Ω

    Using the loop rule, taking positive to be a counterclockwise current:

    ε1 - ε2 - iR1 - iR2 = 0

    Plugging in some numbers:

    i = 20 A

    Now, the reason why I am confused.

    Initially, it makes sense that it should be a current in the clockwise direction, the first battery pushes harder than the second, which results in a net push. However, when looking at what is happening atomically, I become confused. Firstly, lets assume that these are fresh batteries, in which the cathode has had no time to accumulate significant negative charge from the reaction.

    When a circuit is connected, there should be a potential difference supplied through a wire and resistor, etc. However, a battery consists of two completely separate electrodes, in which there is no means for an electron to flow in between the two ends apart from through the wire itself. So, when this particular circuit is connected, both batteries should have no "wire" to impart this potential, as they are cut off from a complete loop by another gap between electrodes.

    I began to think that perhaps its the fact that there is a potential difference between the two negative ends of the batteries, resulting in a current anyway. But let me provide a counterexample which messed that up for me, using the above problem. Using the ground as a reference point, let's say the first battery's negative terminal is practically grounded (0 V potential difference to the ground), and the positive terminal has a very large positive charge relative to the ground. This results in a potential difference of 150 V between the negative and positive terminals of the first battery. The second battery once again has a grounded negative terminal, and a large positive charge on its positive terminal, resulting in a 50 V potential difference between the electrodes. Of course, this is not necessarily the way the battery actually is made, but it is a valid possibility. Once the above circuit is connected, there is no potential difference between the negative terminals, as they are both grounded, but there is a 100 V difference between the positive terminals. Electrons would be biased towards flowing towards the first battery's positive terminal as it has a higher voltage, and the wire connecting the positives would develop a current. As electrons flowed into the first battery's positive terminal, the voltage of that battery would decrease, as the difference in charge between the battery's electrodes decreased. The other battery would experience the opposite, gaining a little more voltage, as difference in charge increased. However, the relative potentials of the negative terminals to the ground would still remain zero, despite changing with respect to the positive terminals. As there are no new electrons being supplied by the imbalance on the positive terminals, it would quickly normalize to no current and no potential difference. By my logic, which must be flawed somewhere of course, this is not the correct theory for how the battery works.

    Could someone please help me out here? I love thinking about this stuff, but circuits have thrown me into a world of solid state and quantum stuff which is a lot harder for me to know for sure I am on the right track.

    Thanks in advance,

    Attached Files:

    Last edited by a moderator: May 8, 2017
  2. jcsd
  3. Mar 13, 2017 #2
    Just a small nitpick; it isn't about conservation of matter as you said here, but charge conservation. Probably a thinko, but just wanted to clarify that.
  4. Mar 13, 2017 #3
    Shouldn't this translate to conservation of matter, since electrons are the carriers? (Ignoring the obviously important fact that they can technically be destroyed)
  5. Mar 13, 2017 #4
    Well, the actual physical law behind the junction rule is charge conservation; now, you can say that wherever there is a charge, there is mass/matter, that's something else.
  6. Mar 13, 2017 #5
    Oh, as in that it is a consequence of ∂p/∂t + ∇⋅J = 0 ?
  7. Mar 13, 2017 #6
    Or equivalently, you can say it is a consequence of Maxwell's equations. The equation you wrote, the equation of continuity, is a mathematical consequence of Maxwell's equations, as you can see here: http://maxwells-equations.com/equations/continuity.php

    That you can derive Kirchhoff's laws from Maxwell's equations can be shown here: http://physics.stackexchange.com/questions/102458/how-can-kvl-kcl-be-derived-from-maxwell-equations

    What is most curious, however, is that charge conservation proves to be much more general and robust than Maxwell's classical electrodynamics.
  8. Mar 13, 2017 #7


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    I hate to rain on your parade, but your approach to try to visualize electricity like little billiard balls filed with energy is wrong, You already displayed serious misconceptions in your post.

    If you want to understand deeper than circuit analysis, the next step down is Maxwells Equations. Trying to reason at intermediate levels between those two leads to inconsistencies.

    My advice : cut it out. Study circuits, then study Maxwells Equations.
  9. Mar 13, 2017 #8
    Yes, I am definitely aware of my disgusting simplification of what is actually occurring, that's why I'm having so much difficulty :/

    But there should be a really obvious explanation to how current is actually able to flow despite my misunderstanding about how there is no way for them to flow right?

    I mean clearly there's no "tunneling" through the electrodes or some dumb explanation from the me who has no training in QM, would you mind exposing me to what I don't know that is preventing me from understanding this?
  10. Mar 13, 2017 #9


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    You ignored what I said. Stop trying to visualize it at the atomic level.
  11. Mar 13, 2017 #10
    Could you explain it on a macroscopic level then?
  12. Mar 13, 2017 #11
    I just did with Maxwell's equations before. Did you not see my message?
  13. Mar 13, 2017 #12
    Well yes, I understand the intuition behind why the laws themselves exist, but as far as their application to a problem like this, I don't really see why they are applicable, because I don't see how it is a circuit in the first place.
  14. Mar 13, 2017 #13


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    Maybe you should hit the books instead of posting questions looking for easy answers.
  15. Mar 13, 2017 #14
    Wait what? Did I say something offensive?

    I referenced Halliday, Griffiths, and Purcell's textbooks looking for an explanation before coming to the forum... I'm not passively looking for some easy solution.

    Look, maybe I can generalize my question.

    How can there be a current through a battery? Not as in through the wires connecting the battery, but literally through the battery, as is happening in the problem I used above?
  16. Mar 13, 2017 #15
    Scratch that! Looking through some engineering websites, batteries can't have reversed current, at least not realistically.

    All the problems I'm attempting are using theoretical voltage sources, in which electrons can go either way without running into issues like the fact that its impossible in a real battery.

    I always run into this issue with theoretical vs real explanations, like how voltage is constant in a wire, it made me believe that there should be no current, but in real life that is not true.

    Thanks for the help guys!
  17. Mar 13, 2017 #16
    The current between the electrodes of a battery is carried by various ions, not by electrons, so you get a closed circuit and kcl is valid.
  18. Mar 13, 2017 #17
    Do you mean specifically without a wire? Like if it were a cathode carrying current to an anode it would be through an ion of its chemical composition, rather than the case with a wire in which electrons are sent through?
  19. Mar 14, 2017 #18


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    No, it is not.
    Trying to understand circuit theory by thinking about electrons is a bit like trying to understand projectile motion by studying general relativity; it is simply not very helpful.

    It is important to realize that the "real" physics behind transport through even something as simple as a metal conductor is extremely complicated; you need lots and lots of quantum mechanics and many-body physics, very few people ever reach the point where they "understand" what is going in detail (the only ones who do are theoretical physicists specializing in condensed matter theory). Fortunately, most of us do not need to understand things in that much detail.

    There are only a few systems (that are hard to make) where you can think of a current as electrons travelling like small "ball" and even then you need at least some basic quantum mechanics (the electrons behave like wavepackets). There is no simple "physical" model that works in the general case.

    At the level you are at it is usually best to use Kirchhoff's laws as a "given". Once you've understand how they are used you can move on to Maxwell's equations; and once you've understood those you can move on to quantum mechanics and condensed matter theory :smile:
  20. Mar 14, 2017 #19

    jim hardy

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    I suffer the same impediment. Been thinking about your plight ince i saw this post:

    Yes. Ions can move and carry charge. That's how we electroplate.

    OH ? How then is it we can recharge them ? Are you old enough to remember when cars had ammeters with zero in the center? They showed whether current was flowing into or out of the battery, and how much.
    62Chrysler dash.jpg
    I used to have one of those Chryslers . Check out that dash at night here: http://www.californiaclassix.com/images2/62C-NIGHTdome-remote.jpg

    At the beginner's level you presented in that image of first post
    where you are learning to apply laws of Kirchoff and Ohm
    it is quite helpful to treat circuits as if they were pipes carrying the mysterious fluid "Electric Charge"
    and since metal wires are effectively atoms awash in a sea of loosely bound electrons, which indeed transport the charge,
    (and i'll probably get hammered for this,)

    for purposes of analyzing circuits i
    t is fair to say that pushing an electron in one end of a wire causes one to be pushed out the other end.
    That causes trouble when not used carefully because it leads people to oversimplifications.

    Careless instructors do not point out that there is a time delay between the electron entering one end and an electron exiting the other, so students assume it's instantaneous.
    Absolutely not so.
    It's not instantaneous. It proceeds at a speed determined by the characteristics of the wire, mostly its insulation, and is usually something like 2/3 c(speed of light) . So it's close to instantaneous and at beginner level we treat it as so.

    That leads students to assume the electron raced down the wire at 2/3c .
    Absolutely not so. If electrons move even a cm/sec the wire will melt.
    It's more akin to pushing marbles through a tube that's already full of them. The marble you push in one end pushes all of them along, and the one that pops out the far end is not the same marble. That one was pushed in a long time ago.

    Whatever Charge is, it progresses along the line of marbles(or electrons) like gossip , way faster than the carriers themselves are moving.

    If you adhere rigorously to those concepts you can go a long way in basic "How Things Work" circuit analysis and avoid the pitfalls of "Electrons whizzing around circuits".
    It then becomes useful to imagine yourself very small and walking along inside the circuit , writing down terms for your KVL equation as you encounter them..
    And imagining ourselves small has the side benefit it keeps us humble.

    Seems to me the exercise OP posted at beginning intends to teach the mechanics of using Kirchoff's laws to solve DC circuits and that's an absolutely necessary skill ..

    Maxwell comes later.

    old jim
    Last edited by a moderator: May 8, 2017
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