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The fundamental problem electrodynamics hopes to solve is this (Fig. 2.1): We have some electric charges, (call them source charges); what force do they exert on another charge, (call it the test charge)? The positions of the source charges are given (as functions of time); the trajectory of the test particle is to be calculated.
Remember the basic problem of classical electrodynamics: we have a collection of charges (the “source” charges), and we want to calculate the force they exert on some other charge (the “test” charge – Fig. 2.1). According to the principle of superposition, it is sufficient to find the force of a single source charge – the total is then the vector sum of all the individual forces.
If you ask the average person what “magnetism” is, you will probably be told about refrigerator decorations, compass needles, and the North Pole – none of which has any obvious connection with moving charges or current-carrying wires. And yet, in classical electrodynamics all magnetic phenomena are due to electric charges in motion; if you could examine a piece of magnetic material on an atomic scale you would find tiny currents: electrons orbiting around nuclei and spinning about their axes.
In this chapter we study conservation of energy, momentum, and angular momentum, in electrodynamics. But I want to begin by reviewing the conservation of charge, because it is the paradigm for all conservation laws. What precisely does conservation of charge tell us? That the total charge in the Universe is constant? Well, sure – that’s global conservation of charge. But local conservation of charge is a much stronger statement: if the charge in some region changes, then exactly that amount of charge must have passed in or out through the surface. The tiger can’t simply rematerialize outside the cage; if it got from inside to outside it must have slipped through a hole in the fence.
What is a “wave”? I don’t think I can give you an entirely satisfactory answer – the concept is intrinsically somewhat vague – but here’s a start: A wave is a disturbance of a continuous medium that propagates with a fixed shape at constant velocity. Immediately I must add qualifiers: in the presence of absorption, the wave will diminish in size as it moves; if the medium is dispersive, different frequencies travel at different speeds; in two or three dimensions, as the wave spreads out, its amplitude will decrease; and of course standing waves don’t propagate at all. But these are refinements; let’s start with the simple case: fixed shape, constant speed, one dimension (Fig. 9.1).
If you walk 4 miles due north and then 3 miles due east (Fig. 1.1), you will have gone a total of 7 miles, but you’re not 7 miles from where you set out – only 5. We need an arithmetic to describe quantities like this, which evidently do not add in the ordinary way. The reason they don’t, of course, is that displacements (straight line segments going from one point to another) have direction as well as magnitude (length), and it is essential to take both into account when you combine them.
In this chapter, we shall study electric fields in matter. Matter, of course, comes in many varieties – solids, liquids, gases, metals, woods, glasses – and these substances do not all respond in the same way to electrostatic fields. Nevertheless, most everyday objects belong (at least, in good approximation) to one of two large classes: conductors and insulators (or dielectrics).
When charges accelerate, their fields can transport energy irreversibly out to infinity – a process we call radiation.1 Let us assume the source is localized2 near the origin; we would like to calculate the energy it radiates at time . Imagine a gigantic sphere, out at radius (Fig. 11.1).
All of our cards are now on the table, and in a sense my job is done. In the first seven chapters we assembled electrodynamics piece by piece, and now, with Maxwell’s equations in their final form, the theory is complete. There are no more laws to be learned, no further generalizations to be considered, and (with perhaps one exception) no lurking inconsistencies to be resolved. If yours is a one-semester course, this would be a reasonable place to stop.