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. 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.
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
history of Radley College, 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". In his letter on the Coriolis force, he thanks Sir Denys Wilkinson for stimulating converstaions. 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).
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