Friday, October 14, 2016

John David Jackson (1925-2016)

John David Jackson died on May 20 of this year. I am familiar with Jackson mainly through his book Classical Electrodynamics. Russ Hobbie and I cite Jackson in Chapter 14 of Intermediate Physics for Medicine and Biology.
"The classical analog of Compton scattering is Thomson scattering of an electromagnetic wave by a free electron. The electron experiences the electric field E of an incident plane electromagnetic wave and therefore has an acceleration −eE/m. Accelerated charges radiate electromagnetic waves, and the energy radiated in different directions can be calculated, giving Eqs. 15.17 and 15.19. (See, for example, Jackson 1999, Chap. 14.) In the classical limit of low photon energies and momenta, the energy of the recoil electron is negligible."
Classical Electrodynamics is usually known simply as "Jackson." It is one of the top graduate textbooks in electricity and magnetism. When I was a graduate student at Vanderbilt University, I took an E&M class based on the second edition of Jackson (the edition with the red cover). My copy of the 2nd edition is so worn that I have its spine held together by tape. Here at Oakland University I have taught from Jackson's third edition (the blue cover). I remember my shock when I discovered Jackson had adopted SI units in the 3rd edition. He writes in the preface
"My tardy adoption of the universally accepted SI system is a recognition that almost all undergraduate physics texts, as well as engineering books at all levels, employ SI units throughout. For many years Ed Purcell and I had a pact to support each other in the use of Gaussian units. Now I have betrayed him!"
Jackson has been my primary reference when I need to solve problems in electricity and magnetism. For instance, I consider my calculation of the magnetic field of a single axon to be little more than a classic "Jackson Problem." Jackson is famous for solving complicated E&M problems using the tools of mathematical physics. In Chapter 2 he uses the method of images to calculate the the force between a point charge q and a nearby conducting sphere having the same charge q distributed over its surface. When the distance between the charge and the sphere is large compared to the sphere radius, the repelling force is given by Coulombs law. When the distance is small, however, the charge induces a surface charge of opposite sign on the sphere near it, resulting in an attractive force. Later in Chapter 2, Jackson uses Fourier analysis to calculate the potential inside a two-dimension slot having a voltage V on the bottom surface and grounded on the sides. He finds a series solution, which I think I could have done myself, but then he springs an amazing trick with complex variables in order to sum the series and get an entirely nonintuitive analytical solution involving an inverse tangent of a sine divided by a hyperbolic sine. How lovely.

My favorite is Chapter 3, where Jackson solves Laplace's equation in spherical and cylindrical coordinate systems. Nerve axons and strands of cardiac muscle are generally cylindrical, so I am a big user of his cylindrical solution based on Bessel functions and Fourier series. Many of my early papers were variations on the theme of solving Laplace's equation in cylindrical coordinates. In Chapter 5, Jackson analyzes a spherical shell of ferromagnetic material, which is an excellent model for a magnetic shield used in biomagnetic studies.

I have spent most of my career applying what I learned in Jackson to problems in medicine and biology.

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