When freshman physics models fail

When we teach basic ac circuits in second semester freshman physics, or for that matter in intro to electrical engineering, we introduce the idea of an impedance, \(Z\), so that we can make ac circuit problems look like a generalization of Ohm's law.  For dc currents, we say that \(V = I R\), the voltage dropped across a resistor is linearly proportional to the current.  For reactive circuit elements and ac currents, we use complex numbers to keep track of phase shifts between the current and voltage.  Calling \(j \equiv \sqrt{-1}\), we assume that the ac current has a time dependence \(\exp(-j \omega t\).  Then we can say that the impedance \(Z\) of an inductor is \(j \omega L\), and write \(V = Z I\) for the case of an ac voltage across the inductor.

Where does that come from, though?  Well, it's really Faraday's law.  The magnetic flux through an inductor is given by \(\Phi = LI\).  We know that the voltage induced between the ends of such a coil is given by \(-d\Phi/dt = L (dI/dt) + (dL/dt) I\), and in an ordinary inductor, \(dL/dt\) is simply zero.  But not always!

Last fall and into the spring, two undergrads in my lab (aided by two grad students) were doing some measurements of inductors filled with vanadium dioxide powder, a material that goes through a sharp first-order phase transition at about 65 \(^{\circ}\)C from a low temperature insulator to a high temperature poor metal.  At the transition, there is also a change in the magnetic susceptibility of the material.  What I rather expected to see was a step-like change in the inductive response going across the transition, and an accompanying step-like change in the loss (due to resistive heating in the metal).  Both of these effects should be small (just at the edge of detection in our scheme).  Instead, the students found something very different - a big peak in the lossy response on warming, and an accompanying dip in the lossy response on cooling.  We stared at this data for weeks, and I asked them to run a whole variety of checks and control experiments to make sure we didn't have something wrong with the setup.  We also found that if we held the temperature fixed in the middle of the peak/dip, the response would drop off to what you'd expect in the absence of any peak/dip.   No, this was clearly a real effect, requiring a time-varying temperature to be apparent, and eventually it dawned on me what was going on:  we were seeing the other contribution to \(d\Phi/dt\)!  As each grain flicks into the new phase, it makes a nearly singular contribution to \(dL/dt\) because the transition for each grain is so rapid.

This is analogous to the Barkhausen effect, where a pickup coil wrapped around a piece of, e.g., iron and wired into speakers produces pops and crackling sounds as an external magnetic field is swept.  In the Barkhausen case, individual magnetic domains reorient or domain walls propagate suddenly, also giving a big \(d\Phi/dt\).  In our version, temperature is causing sudden changes in susceptibility, but it's the same basic idea.

This was great fun to figure out, and I really enjoy that it shows how the simple model of the impedance of an inductor can fail dramatically if the material in the coil does interesting things.  The paper is available here.