The other part of electrodynamics that I'd like to discuss is radiation. If
you're familiar with Purcell's book you'll remember that he draws some beautiful
illustrations of the field lines of a point charge that suddenly starts moving
or comes to a sudden stop. I prefer to consider a bounce, where the particle is
first moving to the right, say at 1/4 the speed of light, then bounces off a
brick wall and moves back to the left at the same speed.
Near the particle, its electric field points directly away from where the particle actually is. But far away, the news that the bounce has occurred hasn't arrived yet, so the field points away from where the particle would be had there been no bounce. As David Griffiths likes to say, "electromagnetic news travels at the speed of light." So there's an inner region of field lines pointing away from the particle, and an outer region of field lines pointing away from the imaginary point where it would be. These two regions are separated by a transition region--a spherical shell that expands outward at the speed of light.
The next question is, what does the field look like in the transition region? I think most people would guess that you just connect up the corresponding field lines, and this is correct, because field lines can't start or end in empty space. To be rigorous, you can apply Gauss's law to the dashed surface shown above, and then to a more complicated surface that actually follows the field lines through the transition region. A complete sketch of the field around the bouncing point charge is shown below. Notice that the transverse field is weakest to the right and left, and strongest in the perpendicular directions. As time passes, the shell expands at the speed of light. The transverse field also gets stronger because the gap, between the particle's actual location and the point where it would be, expands.
(There exists software for drawing and animating pictures like this. An old DOS program that does this is included in the Physics Simulation Programs package available from Physics Academic Software. A somewhat prettier Macintosh program has been created by Blas Cabrera, Sha Xin Wei, and Jim Terman. This program is now in the public domain; to download it, click here.)
When teaching this material I always assign homework problems in which students are asked to sketch the field lines for charges undergoing various instantaneous accelerations. Shown at left below is the somewhat complicated example of a point charge bouncing vertically between two walls. If you just round off the corners, you get something like the illustration on the right, for a point charge undergoing simple harmonic motion as in a radio transmission antenna. Notice that the waves are strongest in the horizontal directions, perpendicular to the particle's acceleration.
