Simple Nature - Light and Matter

chosen moment. Unadorned with the ∂Φ E /∂t term, Maxwell’s equationfor Γ B reads as c 2 Γ B = 4πkI through , which is Ampère’s law. Ifthe electron is a pointlike particle, then we have an infinite currentI through at the moment when it pierces the imaginary surface, andzero current at all other times. An infinite magnetic circulation Γ Bcan only be produced by an infinite magnetic field, so without the∂Φ E /∂t term, Maxwell’s equations predict nonsense: the edge of thesurface would experience an infinite magnetic field at one instant,and zero magnetic field at all other times. Even if the infinity didn’tupset us, it doesn’t make sense that anything special would happenat the moment the electron passed through the surface, because thesurface is an imaginary mathematical construct. We could just aswell have chosen the curved surface shown in figure e, which theelectron never crosses at all. We are already clearly getting nonsensicalresults by omitting the ∂Φ E /∂t term, and this shouldn’tsurprise us because Ampère’s law only applies to statics. More tothe point, Ampère’s law doesn’t have time in it, so it predicts thatthis effect is instantaneous. According to Ampère’s law, we couldsend Morse code signals by wiggling the electron back and forth, andthese signals would be received at distant locations instantly, withoutany time delay at all. This contradicts the theory of relativity,one of whose predictions is that information cannot be transmittedat speeds greater than the speed of light.Discussion QuestionsA Induced magnetic fields were introduced in the text via the imaginarylandscape shown in figure b on page 695, and I argued that the magneticfield could have been produced by a positive charge coming from behindyour head. This is a specific assumption about the number of charges(one), the direction of motion, and the sign of the charge. What are someother scenarios that could explain this field?11.6.2 Light wavesWe could indeed send signals using this scheme, and the signalswould be a form of light. A radio transmitting antenna, for instance,is simply a device for whipping electrons back and forth at megahertzfrequencies. Radio waves are just like visible light, but with alower frequency. With the addition of the ∂Φ E /∂t term, Maxwell’sequations are capable of describing electromagnetic waves. It wouldbe possible to use Maxwell’s equations to calculate the pattern of theelectric and magnetic fields rippling outward from a single electronthat fidgets at irregular intervals, but let’s pick a simpler exampleto analyze.The simplest wave pattern is a sine wave like the one shownin figure f. Let’s assume a magnetic field of this form, and seewhat Maxwell’s equations tell us about it. If the wave is travelingthrough empty space, then there are no charges or currents present,f / A magnetic field in the form ofa sine wave.g / The wave pattern is curly.For example, the circulationaround this reactangle is nonzeroand counterclockwise.Section 11.6 Maxwell’s Equations 697