Basic Concepts: Ohm’s Law

Scott Aaronson is explaining “Physics for Doofuses,” and has started with electricity. He’s got a nice breakdown of the basic quantities that you need to keep track of to understand electricity, leading up to Ohm’s Law. He asks for a little help on this point, though:

Well, as it turns out, the identities don’t always hold. That they do in most cases of interest is just an empirical fact, called Ohm’s Law. I suspect that much confusion could be eliminated in freshman physics classes, were it made clear that there’s nothing obvious about this “Law”: a new physical assumption is being introduced. (Challenge for commenters: can you give me a handwaving argument for why Ohm’s Law should hold? The rule is that your argument has to be grounded in terms of what the actual electrons in a wire are doing.)

As it turns out, this is the topic of tomorrow’s lecture for my class, and one of the things I need to do today is to dust off my slides for that lecture. Scott’s post provides a nice way to turn that productive activity into bloggy procrastination, and help me avoid grading papers while jet-lagged…

So, Ohm’s Law in terms of the motion of electrons. The key to the whole thing is a miocroscopic picture of a solid as a regular array of atoms, with electrons moving about inside. In the absence of any external forces, the electrons will sort of noddle around aimlessly, always in motion, but never really going anywhere. Any given electron is as likely to be headed to the right as to the left, and every so often, they’ll hit an atom, and change direction more or less at random. The average velocity of all the electrons in the material is zero, even though the average speed of an individual electron is pretty high.

This all changes if you put a voltage across the material, though.

When you put a voltage across the material, you’re essentially establishing an electric field inside the solid. Any electron in that field will experience a force that pushes it in the opposite direction from the field lines (since electrons have negative charge). A force, as we all know, produces an acceleration, so the electrons will accelerate in that direction. If you apply an electric field pointing from right to left, the electrons will feel a force accelerating them from left to right. An electron headed to the right will start moving faster, while an electron headed to the left will slow down, and eventually turn around.

Of course, there are still the ions making up the solid lattice to worry about, and the electrons will still collide with them from time to time. After a collision, the electron velocity will be redirected more or less at random, so the effect is really to limit the amount of time that an electron spends accelerating in the field. On average, an electron will travel for a time tavg before hitting something, so the average velocity for all the electrons can be written as:

vavg = – (e/m) E tavg

where e is the electron charge, m the electron mass, and E the magnitude of the electric field. The electric field is related to the voltage V by V = E/L (where L is the length of material in the direction of the voltage), so we have a relationship between the velocity of the electrons and the voltage across the material, which is a good start.

Now, we need to relate that velocity to the current. Current is, as Scott notes, basically a measure of the number of electrons that pass a given point in a given amount of time. You can easily relate this to the velocity of the electrons (making the simple approximation that every electron in the material is moving at the average velocity found above). The number of electrons passing a given point in any time interval Δ t is equal to:

N = n A vavg Δ t

where A is the cross-sectional area of the material, and n is the density of electrons in the material.

You can understand this by thinking of the electrons like water in a pipe. If you want to figure out how many liters of water will come out the end of a pipe in a given amount of time, you determine the length of pipe that can be emptied in that amount of time, which is just the velocity of the water in the pipe multiplied by the time. The volume contained in that length of pipe is just the length multiplied by the area of the pipe. The only thing that changes when we’re talking about electrons is that the electrons, unlike the water, do not completely fill the volume, so we need the extra density factor.

The current through the material is the charge per second passing through, but we can easily account for that by multplying the number by the charge, so:

I = e N/Δ t = e n A vavg

Combining this with our earlier result, we get:

I = e n A (e/m) (V/L) tavg

A bunch of these things are just properties of the material: the charge density, the average time between collisions, the effective charge and mass of the electrons. We can save a bunch of writing by grouping this together into a quatity called the “conductivity” σ :

σ = n q2 tavg / m

and this lets us relate current and voltage by:

I = σ (A/L) V

or, putting it in the usual form for Ohm’s Law:

V = I (L/A σ)

The resistance, then, is the stuff after I: the length of the material that the electrons flow through divded by the cross-sectional area of the material and the conductivity (1/ &sigma is often given the symbol ρ, and called the resistivity of the material, so you might see it written that way).

What does this tell us? Well, it explains a bunch of basic rules about resistors. Thin wire has higher resistance than thick wire, for example, because the cross-sectional area is smaller. It also tells you that longer wires have higher resistance. So, if you want to make a heater, use a long piece of thin wire. If you want to connect two electronic devices and not lose any voltage along the way, use a short piece of thick wire.

It also explains the rules for adding resistances in series and parallel. If you’re adding resistors in series, you’re essentially increasing the length of the material that the electrons have to flow through, so the resistances just add. If you’re connecting resistors in parallel, you’re effectively increasing the cross-sectional area, so the total resistance should decrease.

And all of that stuff comes from looking at the microscopic behavior of electrons in a solid responding to an external voltage across that solid.

(I should note that this treatment is largely lifted from the Six Ideas That Shaped Physics book, and also Principles of Physics by Serway and Jewett, which is the textbook we actually use for the class…)

Comments

  1. #1 bigTom
    April 15, 2007

    You have one important (that doesn’t mean unreasonable) assumption, that the thermal electron velocity is much higher than the average
    (bias) velocity induced by the E-field. Just for fun let us assume that the opposite were true. We assume each electron travels a fixed distance L before being stopped, now its travel time will be proportional to sqrt( L/E ), and the current would scale as the sqrt of the applied voltage. That would be quite different from what we actually observe. Do we have any example of materials which obey this alternate law?

  2. #2 Jonathan Vos Post
    April 15, 2007

    Ohm’s law violation in spin glasses

    A A Lisyanskii et al 1979 J. Phys. F: Met. Phys. 9 1629-1641 doi:10.1088/0305-4608/9/8/014

    PDF (649 KB)

    A A Lisyanskii and G I Trush
    Donetsk Phys.-Tech. Inst., Acad. of Sci., Donetsk, Ukrainian SSR, USSR

    Abstract. Electrical resistivity rho s of metal containing ordered (spin glass) magnetic impurities is calculated. Two limiting cases are considered:
    (i) tau e/ tau i<<1 ( tau e and tau i are the times of collisions of electrons with each other and with nonmagnetic impurities);
    (ii) square root ( tau i/ tau ph)<<1
    ( tau ph is the electron-phonon collision time). In both cases at solution of the kinetic equation it was assumed that at the first stage magnetic impurities do not take part in the conduction electron relaxation. It is shown that dependence of rho s on electric field (or on current density) is strongly nonlinear irrespective of the relaxation regime, and derivatives of rho s with respect to field or current contain a characteristic cusp which can be easily recorded experimentally. The calculated effect can be observed at relatively small current densities j approximately
    10^5-10^6 A cm^-2.

    There are other conditions under which Ohm’s Law is violated.

    But you’re right, students must start with what almost always works.

    Print publication: Issue 8 (August 1979)

  3. #3 agm
    April 15, 2007

    Other details which are extremely important but not touched upon so often. This treatment assumes negligible magnetic fields, which have a fairly important effect in the places you’d most likely use this sort of treatment. To really see what’s doing with the resistivity, you have to think about changing frames. If there’s any magnetic field around at all, and very often there will be, then your description holds for the rest frame of the electrons, thus picking up forces not aligned to the imposed field when boosting into the rest frame for the, oh, say, Hall effect devices. In the view where the electrons are a plasma floating around in a background potential set up by a bunch of ions which can’t move around as much as or as quickly as the electrons, this approximation would apply.

    It’s definitely worth talking about why conductivities are what they are, even if it’s from the point of view where one discusses resistivities in terms of scattering of electrons off an ion in the lattice. Mmmm Rutherford scattering. There’s also room for discussing that these are empirical numbers measured in a set of parameter regimes, maybe point out that the interaction between the electrons and the ions can be pressure dependent (e.g., stress on a crystal altering distances between ions), or thermal interactions (heat the sucker up, more electrons free to move around the cabin, and also more energy available for scattering interactions to knock an electron off an atom — hmmm, need to look into electrically-enhanced Auger spectroscopy…)

    Really, it seems to me that the best way to do this (in the approximation that the current set up isn’t big enough to significantly alter the fields), but not necessarily the most obvious way at first, is to start from Maxwell’s equations to see what the fields are doing, add assorted ancillary electrodynamics relations as one derives through to see what the fields do to the charges (polarization, magnetization, other sources of current in the medium). Of course, given what I study, I would see it that way :-). It’s not that long a derivation, and if I weren’t exhausted I’d go rifling through my piles of notes to find it for ya, or maybe recreate it quickly.

  4. #4 Perry
    April 15, 2007

    don’t sell Ohm’s law as an “assumption”. Sell it as an empirical fact, do the experiment! Also you can start with J=sigma E, that seems to make sense to them. Even freshmen are OK with the fact that its a curve fit that works over some limited range of variables. They are more likely to accept a nonlinear behavior in anything than 1-2 generations ago, its easy to see that J (or i) saturates if you run out of electrons to move. Also J and E don’t have to be in the same direction. Just have a material with a different sigma in two different directions (say x and y). So put a field along x, get one J. put the same field along y get a different J. Now put the x and y fields on at the same time, get a resultant at 45 degrees to the x axis. The resultant J will NOT point in the same direction. Then you can sell J=sigma*E as a matrix equation pretty easily. Then talk about quantum wells, and sigma depends on where you are in the material. Even freshman should be aware that the whole world is not entirely composed of linear isotropic homogeneous materials!

  5. #5 John Sidles
    April 15, 2007

    Over on Scott Aaronson’s blog, as an exercise in practical applications of quantum information theory (QIT), I posted an explanation of Ohm’s Law that is based entirely on the physics in Nielsen and Chuang (“Mike and Ike”) Quantum Computation and Quantum Information.

    Well whadda-ya-know, QIT turns out to be a pretty good way to explain Ohm’s Law. :)

  6. #6 Pseudonym
    April 15, 2007

    Expanding on what Jonathan Vos Post said:

    In our lab, we make nano-scale devices. One of the substances that we have is a smart material that is, conceptually, a bunch of conductive nanoparticles in suspended in an insulating substrate.

    This material acts like a resistor (it “conducts” by quantum tunneling), but depending on how we build it, it doesn’t always obey Ohm’s law.

  7. #7 adam
    April 15, 2007

    That treatment is pretty similar to how I’ve taught it. It’s similar how it’s done to the (now defunct) Nuffield ‘A’ Level physics course, although deriving the average velocity expression isn’t a required part of that syllabus, as electric fields are in module E and basic electricy is in module B, if I recall correctly. It’s the sort of thing that in that course could be revisited when they have a better understanding of fields after module E.

    I have found that, of basic physics, electricity is the hardest to get across to all the students so that they understand it in what I might call a ‘comfortable’ way. I don’t know if it’s because they’ve absorbed so much misleading ‘folk wisdom’ by the time that they get taught about it, or what it is.

  8. #8 CCPhysicist
    April 15, 2007

    As posted in the original blog, but to reinforce Perry’s comment: I’d unask the question, because R = V/I is the definition of resistance. R is basically the linearization (sometimes a local one) of V(I,T,etc) based on what some physicists call data. You know who they are. If your students know what a Taylor series is, so much the better, but it is also what they will do in the lab.

    And, for your level course, the main thing they need to grasp is that R is independent of I under a set of restrictive assumptions not unlike the ones used to say friction is a constant multiple of Fn. (This is also the place where you can relate it to dynamic equilibrium, and the assumptions made about that in the problems being considered in that part of the text.) The main difference is that they are often good assumptions for R, but your calculator would not work if they were always true. Neither would that AC/DC converted that uses diodes.

    On the quantum side, I always point them to the temperature dependence of rho (or sigma) for our favorite metalloid, Si. Why is it “wrong”? The key is the charge density (in the conduction band) that appears in your expression for sigma.

  9. #9 Scott Aaronson
    April 16, 2007

    Thanks so much, Chad! I was travelling and sleeping all day, but I finally had a chance to work through your argument, and it’s exactly what I was looking for. The key point I was missing is that the electrons don’t just move through the wire; they accelerate until they hit an obstacle, then re-accelerate, etc. Furthermore, the rate of acceleration is proportional to the voltage divided by the length of the wire. Once you make those two assumptions, everything else can be worked out from first principles.

    There seems to be one error: you write V=E/L, then later use E=V/L. Did you mean V=EL?

  10. #10 David Phillips
    April 16, 2007

    “It also tells you that longer wires have higher resistance. So, if you want to make a heater, use a long piece of thin wire.”

    To make a former lab-rat comment, that’s only true if you have a constant current source. More generally, you’d like to impedence match your load to your source. If you’ve got a 20 Volt/5 amp power supply, then its source impedence is R=V/I=20/5 = 4 Ohms. Therefore, you’d like a 4 Ohm load (the heater wire) to deliver the most power from the power supply to the thing you are trying to heat up (20 Volts * 5 amps =100 Watts). In practice, 4 Ohms is a lot of resistance for a piece of wire so it will need to be long and thin, as Chad was saying, to get that much resistance.

    However, if you happen to have a 100 amp / 1 volt current source (still 100 watts) lying around that you’d like to use for your heater, you will optimize the heater wire differently to get the same 100 Watts out of the heater.

  11. #11 David Phillips
    April 16, 2007

    I’d also like to agree with Chad’s overall presentation and disagree with the “It’s a linearization” perspective. While it certainly is true that Ohm’s Law is a linearization, there is important and relatively simple physics in the drift model of conduction that allows this linearization to work. As various others have pointed out, it doesn’t have to be this way.

    * In a simple, old-fashioned, vacuum tube, we get very different behavior, again because the electrons are allowed to acquire an energy proportional to the voltage rather than a velocity proportional to it.

    * In 2-D, cold conductors we get the quantum-hall effect which looks nothing like Ohm’s law.

    In these examples, there will still be small-signal linearizations that one can use in calculating the steady-state behavior of some device, but the qualititative behavior is very different.

    <grumpy> Sometimes, it is important to pay attention to the physics and not just to the algebra we used to derive a result. </grumpy>

  12. #12 Torbjörn Larssona
    April 16, 2007

    Do we have any example of materials which obey this alternate law?

    Yet another situation to add to the long list where the classical current model breaks down is in ballistic devices.

    As various others have pointed out, it doesn’t have to be this way.

    Certainly not, but a linear term is natural. It is more interesting when it isn’t a good approximation.

    a constant current source

    Shoot! I discussed the difference in current and voltage dependent active loads on Scott’s thread in response to which is looked at when, but for some reason I forgot to mention the two idealized sources we can use. Ah, well.

  13. #13 Torbjörn Larsson
    April 16, 2007

    a linear term is natural

    Make that “a leading linear term”. And hopefully my name will be correct now.

  14. #14 KeithB
    April 16, 2007

    This isn’t exactly on topic to this, but I have been wondering lately:

    The General Theory of Relativity says that masses are attracted to each other because mass curves space.

    What do Electric fields warp in order to attract/repel other charges? Or do they work in some other way that works out to an exact mathmatic equivalent? (See, I said it was marginally on topic.)

  15. #15 SMC
    April 16, 2007

    Personally, I think the teaching of electricity is a conspiracy by Godless Newtonists, and we should give equal time to the “invisible little green men” theory.[1]

    Seriously, though – the main question *I* have that has never been addressed well in my presence is this:

    Electricity (or more accurately, “current”) is a flow of electrons, induced by a sort of metaphorical pressure differential (the “electric field”), right? And electrons are, somewhat arbitrarily, defined as having a “negative” charge, right?

    Why, then, is electricity entirely defined in terms of imaginary “positive” charge? It’s like defining pressure in terms of how much vacuum there is…

    (“The world’s last remaining stock of Ideal Gas™ is placed into 5.3MegaLiter sealed, perfectly insulated container and shot into space. A meteorite makes a 1.5mm radius perfectly circular hole in the container. At what rate will vacuum leak into the container?”)

    [1] Amdahl, Kenn: “There Are No Electrons: Electronics for Earthlings”. 1991, Clearwater Publishing Company

  16. #16 Devin Smith
    April 17, 2007

    SMC:

    The short answer is that Ben Franklin guessed wrong on which way the charge carriers flowed in a metal. Oops. This wasn’t sorted out until the Hall effect was studied, really. The situtation is not that analagous to studying not-vacuum, though, since pressures can’t go negative.

    Also, the definition of the direction of current flow, as well as the charge of the electron are /entirely/ arbitrary. You could just as well sort it out so that electrons were positive, and so on: the only problem would be that no one would have the foggiest idea what you were on about. The amount of work needed to get all technical people to agree on a sign flip here is astronomical, and for very little gain, so things will stay the way they are.

    However, it is important to note that not all flowing charge carriers are negative: many important semiconductor devices have holes as the charge carriers.

  17. #17 Daniel Black
    April 19, 2007

    Excuse me if I’m obtuse here, but shouldn’t

    I = σ (A/L) V

    become

    V = I (L/(A σ))

    not

    V = I (L/A σ)
    ?

    Hopefully I didn’t miss that if it’s already been discussed. Maybe it’s the space after A that’s throwing me.

  18. #18 Chad Orzel
    April 19, 2007

    I think the space is throwing you off. The A and the σ are both intended to be in the denominator of the fraction L/A σ, so what I wrote is, at least in my mind, the same as (L/(A σ )). The space is in there because I usually do these thigns in LaTeX, and it automatically removes the spaces…

  19. #19 Martin R
    May 9, 2007

    Thanks! I was taught Ohm’s Law in school at about the age of 13, and I remember wondering at the time why it was so. No, finally, I know!

    What I’d like to know now is how “voltage” is communicated to the individual electron…

  20. #20 Norris Preyer
    May 17, 2007

    The *really* interesting question, IMHO, is how the electric field just happens to be parallel to the wire (even around bends!) and have the right magnitude so the same amount of current moves everywhere in the circuit.

    Very few textbooks talk about the surface charges on the wires that make all this happen — see “Matter and Interactions II” by Ruth Chabay and Bruce Sherwood for a good discussion. On my website are some papers and simulations (with movies) showing the feedback that makes circuits behave.

  21. #21 sai
    September 29, 2007

    hi
    can any body help me in deriving ohms law from its first principle

  22. #22 Chad Orzel
    September 30, 2007

    can any body help me in deriving ohms law from its first principle

    If you read this post, and still left that comment, probably not.