About a month ago I was fortunate enough to hear a talk by Klaus von Klitzing, the Nobel Prize-winning discoverer of the quantum Hall effect.  The talk was mostly from a historical perspective: he recounted the story of how the quantum Hall effect was discovered (at 2am in a lab in Paris — apparently they had to work at night because electricity was too expensive during the day) and he described the many things that it has been used for since its discovery.  There have, in fact, been thousands and thousands of academic studies into the quantum Hall effect during the last thirty years, and all sorts of funny interactions and strange quasiparticles have been found to be associated with it.

The thing that excited von Klitzing the most, though, was the fact that the quantum Hall effect is a phenomenon that is so universal and can be reproduced so exactly that its result is now used to define the SI unit of resistance.  In other words, the Ohm is now defined as exactly $1/25,812.81$ times the quantized resistance observed in the quantum Hall effect (which can be measured to an accuracy of ten digits).  I can certainly see how that would be exciting: you discover something which nature reproduces so exactly that you can set your measurement devices to it.

von Klitzing’s discussion of setting the standard for resistance led him through a brief digression about how various physical units are defined.  It turns out that all but one of of the seven basic units are defined based on similar, highly-reproducible experiments.

The kilogram (housed inside three bell jars, for some reason).

For example, the “second” is defined as 9,192,631,770 periods of microwave oscillations emitted from a certain electronic transition in the cesium atom (the “atomic clock“).  The “meter” is defined as the distance traveled by a ray of light in vacuum during one 299,792,458th of a second.  In fact, the only one of the basic units which isn’t defined based on a very precise experiment is the kilogram.  The kilogram is defined as the mass of a particular block of metal (a platinum-iridium alloy, shown on the right) that sits in a special vault at the International Bureau of Weights and Standards in Paris.

Given its privileged position as standard-bearer among the units, “the kilogram” is about as close to royalty as an inanimate object can be.  Once a year there is a sort of silly procession that begins with three officials from the Bureau of Weights and Standards meeting at the vault which houses the kilogram.  These three officials, who arrive at the vault from their homes in three different countries, open the vault door by putting their three unique keys into three unique locks and then turning their keys in unison.  They then proceed through the vault to the safe which houses the kilogram (as well as seven or eight carefully-made copies).  The combination to the safe, of course, is known by only one living person (the president of the Bureau of Weights and Standards).  After ascertaining that the kilogram is, in fact, where they left it, the officials put the kilogram through a process of inspection, cleaning, and weighing.  The kilogram is then photographed and returned to its place in the safe to await next year’s procession.

It’s quite a lot of pomp and circumstance for a little metal cylinder, especially when you consider that it isn’t actually a very accurate way of setting the standard for mass.  The data from the last 120 years of annual weighings suggest that kilogram is actually slowly evaporating.  Apparently, the kilogram has lost almost 100 micrograms in the last century.  This may not sound like a big deal, but it makes the definition of the kilogram significantly less accurate than that of its other SI counterparts.  Plus, it sounds a little embarrassing to have to say “the kilogram is slowly evaporating”.

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Anyway, the point of this post was not to poke fun at the kilogram, which has its problems and which will probably be redefined in the next decade or so.  Instead, I want to poke fun at the next-oldest of the seven basic unit standards: the Ampere.  The official definition of the Ampere is as follows:

The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10−7 newton per metre of length.

This definition essentially comes from the 1820s, when André-Marie Ampère was among the very first scientists studying the phenomena of electricity and magnetism.  He found that when a current was passed through two parallel wires, they attracted each other.  The strength of this attraction is still the official way to determine how much current is passing through the wires.

The question of why there is a force between the two conducting wires is a standard high school physics problem.  Basically, when a current is passed through one of the wires a magnetic field is created around it which pulls on the moving electrons in the adjacent wire.  You could diagram the situation like this:

An applied voltage V induces a current I in two parallel wires and they attract each other with a force F.

What I want to show in this post is that the setup dictated in the definition of the Ampere — two long, straight, parallel, conducting wires with a current 1 Ampere separated by 1 meter — can produce a force that is very different from 2 × 10−7 newton per meter.  In fact, these two wires can repel each other instead of attracting.  In that sense, the standard for the Ampere may be even more flawed that that of the kilogram.  After all, the kilogram may be evaporating, but there’s no way that you could try to measure one kilogram and actually get negative one kilogram.

So let me ask the following question: what conditions are necessary for the two wires in the picture above to repel each other rather than attract?

This is a question that my advisor posed to me as part of my “training” during the first year of graduate school.  I, of course, solved it a long and difficult way.  But it turns out there is an easy way, and it begins with turning the question on its head: imagine that instead of carrying parallel currents, the two wires constitute opposite sides of the same circuit and therefore carry anti-parallel currents.  Then you can ask “what conditions are necessary for these two wires to attract each other rather than repel?”  Like this:

These two wires are supposed to repel each other.

According to the standard arguments about magnetic fields, these two wires should repel each other.  But there are, in fact, other forces at work here.  Notice that the top wire is connected to the positive terminal of the voltage source (battery), which means that the positive charges inside the battery are likely to bleed over onto the top wire and make this wire positively-charged.  Similarly, the bottom wire is next to the negative terminal, which means that it might accrue some negative charges on it.  Therefore, if enough positive/negative charge has accumulated on the top/bottom wires, these wires might attract each other by the electric force instead of repelling by the magnetic force like Mr. Ampere said they should.

The question of how the positive/negative charge distributes itself along the wire can be a little tricky, so let’s make one big simplification for illustrative purposes.  Namely, we can imagine that all the resistance in these two wires is located in a single place in the middle of the circuit (instead of distributed all across the wires).  That way we can imagine that there is no change in electric potential anywhere along the circuit except across the resistor:

From here it’s a pretty simple problem that proceeds in four steps.  Feel free to skip over everything between the horizontal lines if you’re not interested in the details of the calculation.

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Step 1:  What is the electric field created by one of the wires a distance $r$ away from its center?

Answer: use Gauss’s law to get

$E(r) = Q/2 \pi \epsilon_0 L r$,

where $L$ is the length of the wire.  The electrostatic force between the wires is $F = Q E(d) = Q^2/2 \pi \epsilon_0 L d$, where $d$ is the distance between the wires.

Step 2: How is the charge $Q$ related to the voltage $V$ between the wires?

Answer: integrate the electric field from one wire to the other to get

$V = Q /\pi \epsilon_0 L \times \ln(d/b)$,

where $b$ is the radius of the wire.  So the force from the electric field is $F_E = \pi \epsilon_0 L V/d \ln(d/b)^2$.

Step 3: What is the magnetic field surrounding one of the wires as a function of the current $I = V/R$?

Answer: use Ampere’s law to get

$B(r) = \mu_0 I /2 \pi r = \mu_0 V/2 \pi R r$.

The resulting force $F_B = I L B(d) = \mu_0 L V^2 /2 \pi R^2 d$.

Step 4:  When is $F_E > F_B$?

$R > \sqrt{\mu_0/\epsilon_0} \times \ln(d/b) / \sqrt{2 \pi^2}$.

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So there is the answer.  If the resistance of the wire is greater than $\sqrt{\mu_0/\epsilon_0}$, then the parallel wires will repel each other instead of attracting.  And your standard for an Ampere will, in fact, be so wrong as to give you a negative value when you apply a positive current.  (Don’t worry, by the way, about the factor $\ln(d/b) / \sqrt{2 \pi^2}$.  In almost any practical situation, this is not too far from $1$. )

It turns out, by the way, that $\sqrt{\mu_0/\epsilon_0} \approx 377 \Omega$ has a special meaning.  It’s called the “impedance of free space”, and it’s a value that shows up when you have problems about transmitting electromagnetic waves.  Basically, it is an effective resistance of open space.  If you make a device to transmit electromagnetic waves (like a satellite dish or a radio antenna), you need to tune the internal resistance of your device to about $377 \Omega$ or you will find that open space is reflecting part of your signal back to you.  Strange but true: empty space behaves like some kind of transmitting substance with its own characteristic value of electrical resistance.  And, the question of whether your wires attract or repel depends on whether their resistance is greater or smaller than this resistance of space itself.

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Now, I’ve probably gotten a little carried away in this post about the strangeness of the SI units, and I have perhaps been a little bit of a sensationalist.  In case you have become convinced that the international definition of electric current is so fatally flawed that the world of modern electronics is teetering precariously at an abyss of inaccuracy, let me calm you a bit.  The experiments used to establish the Ampere are generally done with copper or silver wire, which have a tiny resistance.  For example, ten meters of copper wire with a 1 centimeter diameter would only have a resistance of one thousandth of an Ohm.  So there is no reason to be a doomsayer.

But of course, if you really did have an “infinite length” of wire with “negligible cross-section”, as the definition of the Ampere calls for, then the resistance would certainly be larger than $377 \Omega$.  And the wires would repel rather than attract.  It just strikes me as strange and amusing that the technical definition of the Ampere is, in fact, not an Ampere.

June 21, 2010 10:46 am

Well , i think it’s odd that the definition of time is based on the oscillation of a Cs-133 atom at zero K.
Since zero K is unachievable , and should mean there is no motion at all.

June 21, 2010 10:59 am

It is actually a strange consequence of quantum mechanics that “zero temperature” does not mean “zero motion”. Any object that is forced into a confined space will have a characteristic quantum oscillation that has nothing to do with thermal motion (see https://gravityandlevity.wordpress.com/2009/04/03/how-to-trap-a-particle-quantum-wells-in-1-and-3-dimensions-part-1/).

The electrons of a cesium atom are stuck in the trap created by the cesium nucleus, and therefore they are always jittering back and forth against the “walls” of the trap with a characteristic frequency.