### Jackson Wrong and Hyperbolic Bessels

I'm back in the lab this week working on some numerical models for the solenoid to be used in the H-Ray experiment.  In looking for analytic solutions for the z and rho components of a magnetic field due to a coil, I came across a very interesting reference in an article by Bergeman, Erez, and Metcalf[1].  Take a look

If you're one of the many who after poring over Jackson, thought "This can't be right", in this one case, you were right.  The proper expressions according to Bergeman et al. are

Hyperbolic Bessels
Solving for quantized fields, one is often led to Bessel eigenfunctions in cylindrical coordinates.  Based on some of my recent work, I was left with the feeling that relativistic quantizations of the field in a uniformly accelerated space might give the result in terms of hyperbolic Bessel functions.  Sure enough, the Macdonald function solution that Dr. Fulling[2] finds in his paper "Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time" is also known as the hyperbolic Bessel[3].  Cool!

References
1.Bergeman T., Erez G. & Metcalf H. (1987). Magnetostatic trapping fields for neutral atoms, Physical Review A, 35 (4) 1535-1546. DOI:

2. Fulling S. (1973). Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time, Physical Review D, 7 (10) 2850-2862. DOI:

3.  http://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions_:_I.CE.B1.2C_K.CE.B1

### 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…