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### Oscilloscope Peak Via Slope

The first oscillating magnetic field superconductor levitation tests yesterday went great. With the Peavey power amplifier driving the electromagnet, the superconductor levitated at frequencies up to 164 Hz. An interesting measurement problem arose that required the use of a math trick to compensate for the limitations of the oscilloscope. Our oscilloscope has a maximum vertical range of 5 volts per division. There are 8 vertical divisions on the screen, so as long as the oscillating voltage from the amplifier is less than or equal to 40 volts peak to peak, we can read its peak value by counting divisions on the oscilloscope screen. To attain the necessary levitation force however, the oscillating voltage applied to the electromagnet had to be greater than 40 V peak. Instead of readable trace like the one shown above, the oscilliscope looked like:

There aren't enough divisions on the scope screen to read the voltage. This is where the math trick comes in. We know the frequency of the sine wave and we can measure the slope of the wave as it crosses zero. Using that information, we can calculate the peak value. The equation for a sine wave is:

Where A is the peak value of the sine wave that we can't see on the screen.

The derivative, or slope, of a sine wave is:

The cosine of 0, (the value where where the sine wave crosses the x axis), is 1, so we can setup an equation:

Now, if we know the frequency and measure the slope as rise over run on the oscilloscope screen, we can calculate the unknown peak value A.

Picture of the Day:

 From November2006

### 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…