### Electromagnet Impedance

The data taken last week showed a linear dependence between the voltage measured in the pick-up coil when the superconductor is levitated and the frequency of the current driving the levitating electromagnet.

While reading an article on a susceptometer for superconductors, I came across the graph shown below that shows the decrease in the magnetic field of a solenoid driven at 5 V rms as frequency is increase.  A solenoid is an inductor with an impedance that is linearly dependent on the frequency of the current flowing through it.  The drop in the magnetic field is a result of of the impedance of the solenoid increasing with increasing frequency and reducing the current trough the coil.

I'd like to see if the linear increase in the voltage required to attain levitation is just a result of the increasing impedance of the electromagnet.  My first task was to determine a relationshiop between the pick-up coil voltage and the voltage driving the electromagnet.  To do that, I attached one channel of the oscilloscope to the supply leads of the electromagnet and another channel to the pick-up coil leads.  With the oscilloscope in x-y mode, the first channel is used as the x sweep voltage and the second channel is used as the y sweep voltage.  This resulted in the waveform shown below:

Taking the slope of the loop, gives about 26 mV on the pick-up coil for every 5V on the electromagnet supply.  Using this slope, the data taken last week can be related back to the electromagnet's supply voltage:

The data point near 180 V on the calculated graph seemed somewhat unrealistic at first because the amplifier docuemntation specifies a maximum voltage swign of 93 V into 8 ohms.  The electromagnet, however, has an impedance of about 28 ohms at these frequencies.  Just to perform a sanity check, I'm plugging an estimated rms output power of 1000 watts into  the following equation:

which can be rearranged to give an rms voltage of:

Using a power estimate of 1000 Watts and a driving frequency of 230 Hz, I wind up with a peak voltage of roughly 229 volts, so my data fits within the estimated maximum output voltage from the amplifier.

If the voltage-frequency line in the levitation data was just due to impedance effects I'd expect to see a flat line when graphing the current through the electromagnet vs. frequency.  In other words, the voltage just had to be increased in order to keep the current constant.  I'm modelling the electromagnet as the following circuit based on the reading of an impedance meter:

This gives me a calculated current vs. frequency that looks like:

The current line isn't flat, so there are other things going on.  Bean's model of type II superconductors in AC magnetic fields predicts a power loss that is linear with increasing frequency.  I'll take a look at that tomorrow.

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…