### Of Charged Discs, Trig Substitutions, Birds, and Fireballs

In studying form my EM midterm, I came across a practice problem involving finding the potential along the z axis due to a charged disc centered on the same axis.  After thinking about the problem a bit, I turned to what's becoming one of my favorite online references, Dr. J.B. Tatum's text on electricity and magnetism[2].  Sure enough, there was a solution that could be adapted to the task at hand.  It involved several 'clever' trig substitutions one of which were not immediately clear to me, so I've expanded upon it here.  After the clever trig trick, read on to find out more interesting stuff about Dr. Tatum, an emeritus professor at the University of Victoria[1].

The Basic Problem
The practice problem mentioned above is described by the following diagram from Dr. Tatum's text.  The first and handiest innovation in Dr. Tatum's treatment is to parameterize the problem using the angle marked as theta and the limit of that angle labeled as alpha.

To find the potential, we need to integrate the uniform charge density over the entire disc.  After diagramming the problem, the text suggests the following substitutions to make the integral more tractable:

The first and second of these made a lot of sense.  They're essentially just relations between the hypotenuse of the triangle shown above and our z axis, (the diagram's x axis).  The third one for delta r, (the width of the circular track shown on the disc above), gave me reason for pause however.  Secant squared?  Really?  Here's how.  First, a more detailed picture will get us most of the way there (picture 3).

The thickness of the small circle, delta r, corresponds to a small angle swept out by theta, d theta.  We want to get the thickness in terms of d theta.  As the diagram shows, the angle that the hypotenuse makes with the disc is equal to 90 minus theta.  As theta increments, it sweeps out an infinitesimal segment with length equal to the radius from the point where theta is measured, (the intersection with the z axis), times the increment of theta, (d theta).

That infinitesimal segment becomes the hypotenuse of a new triangle that has delta r as it's adjacent side, and that's all we need to get the rest of the way.

Much More About J.B. Tatum
Dr. Tatum, shown to the left, (he's the guy in the center), in a picture from the history of Radley College[3], not only taught electricity and magnetism, but also worked in astrophysics and was something of an expert on birds!  It had never occurred to me that a physicist could work in the field of ornithology, although it makes perfect sense.  Dr. Tatum authored articles on the bird navigation and the Coriolis force as well as an article on calculating the volume of a birds egg.  His articles and letters appeared in ornithology journals such as "The Auk", "The Condor", and "Bird Banding"[5][6][7].  In his letter on the Coriolis force, he thanks Sir Denys Wilkinson for stimulating converstaions[4].  Sir Denys is a knighted nuclear physicist!  He's the rather dapper looking physicist shown below.

As a final aside, Dr. Tatum also wrote about the accoustics of meteor fireballs and about locating fragments of meteors.  See the second list of references for several of these articles, (many of which are free from NASA).

References:
1.  http://www.astro.uvic.ca/~tatum/index.html

2.  http://www.astro.uvic.ca/~tatum/elmag/em01.pdf

4.  http://en.wikipedia.org/wiki/Denys_Wilkinson

5.  http://www.jstor.org/action/showPublication?journalCode=birdband

6.  http://www.jstor.org/action/showPublication?journalCode=auk

7.  http://www.jstor.org/action/showPublication?journalCode=condor

Other articles by and about Dr. Tatum
Fireballs: Interpretation of Airblast Data

A hitherto unrecorded fragment of the Bruderheim meteorite

Tracking a Fireball from Eyewitness Accounts II

Asteroid named after Dr. Tatum
http://en.wikipedia.org/wiki/3748_Tatum

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

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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 differe…

### Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…