### Magnet Design and Sample Size

I've been looking lately at using an already constructed superconducting magnet instead of building my own for the upcoming experiment, (an Experimental Search for the Bremsstrahlung Radiation Predicted by the Hole Theory of Superconductivity)[1].  The issue at hand is that the bore isn't large enough to accept the originally planned 3.8 cm radius spherical Pb sample.  I took a look this morning at what reducing the sample size would do to the energy of the predicted radiation in electron volts as well as what the dependency of the radiation flux would be with respect to sample size.  The two formula for the energy and the flux (pictures 1 and 2) are:

See the aforementioned proposal as well as reference 2 for more details.

Plotting each of these versus R, the radius of the sample gave the following plots, (pictures 3 and 4).  If the radius is reduced all the way down to 2 cm, the fall off in energy isn't unacceptable.  It still lands in the ballpark of 160 keV which should light up the NaI detector just fine.  Where I run into trouble is in the flux.  Because of the R squared dependency it goes down by a factor of three.

For the moment, it looks like a better choice to build a magnet.  That gets us to the fun part!  Superconducting magnets are awesome!  With 153 turns of .0018 in. diameter superconducting wire, I should be able to get approximately, (using the infinitely long solenoid approximation), 1300 Gauss.  The approximation isn't quite valid since my solenoid will be pretty far from being significantly longer than it is wide, however, the required field strength is only 800 Gauss.  For a first cut, things are looking pretty good.

Apologies for the xkcd style graphs.  It occurred to me this morning, that the xkcd package is the only one I know how to label plots in.

References:
1.  Experimental proposal and such
http://copaseticflow.blogspot.com/2013/03/open-science-is-cool-in-concept-but.html

2.  Paper on radiation predicted by the hole theory of superconductivity
http://dx.doi.org/10.1103%2FPhysRevB.68.184502
Hirsch J. (2003). Charge expulsion and electric field in superconductors, Physical Review B, 68 (18) DOI:

### 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…