### Measuring Magnetic Field vs. Frequency Using a Transformer

Alternating current power can be reflected by a reactive load, (like an electromagnet). Ham radio operators are familiar with this concept and measure the amount of reflected power as the standing wave ratio, (SWR). As the frequency driving a ham radio antenna is changed away from the antenna's resonant frequency, the SWR increases, less power is driven into the antenna and more power is reflected back into the radio's amplifier.

On Friday, I observed that the levitation height of the superconductor decreased when the frequency of the current driving the levitating electromagnet was increased. This is the expected result based on the experience of the other teams that tried to replicate Podkletnov's experiment. However, the same behavior would have resulted if power from the amplifier was being reflected without entering the electromagnet.  The amount of reflected power is usually measured with a directional wattmeter, or with an SWR meter.  I don't have either of those instruments.  While I could measure the voltage on the terminals of the electromagnet, that would only give me information on the standing wave at the terminasl, not necessarily how much of the wave was being accepted by the magnet.  Current that was driven into the electromagnet's create a magentic field, while current that was reflected would not.  So, what  I needed was a way to measure the magnetic field produced by the electromagnet.  I have a Gauss meter sitting on my bench, but it only works for DC magnetic field.  The AC magnetic field averages to 0as it changes direction several hundred times per second.

The trick to measuring the current delivered was to use a pick-up coil, (or transformer), to measure the magnetic field produced and then read it as a waveform on the oscilloscope.  Faraday's law of induction states that a changing magnetic field will create a proportional changing electromotive force.  If a coil of wire is placed in the region of the changing magnetic field, (and therefore the electromotive force), a current will flow in it.  Using a spare coil of wire that was laying around, I setup the measuring instrument:

The coil of wire is sitting directly on top of and aligned with the electromagnets pole piece.  The two red circles highlight where the oscilloscope probes connect to the two leads of the coil of wire.  Using this simple instrument, I was able to get oscilloscope waveforms that I could use to check the dependency between the magnetic field produced, (proportional to the peak value of the waveform), and the frequency fo the current drvining the electromagnet:

There was no variance between the peak value of the waveform and the frequency of the driving current.  At these frequencies, all the current is getting into the electromagnet.  This indicates that the observed variation in levitation height was due to change in frequency of the magnetic field and not due to a change in its amplitude.

### 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…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…

### 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…