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 $t_0$. Imagine a gigantic sphere, out at radius $r$ (Fig. 11.1).

In this chapter we seek the general solution to Maxwell’s equations. Given the sources $\rho (\mathbf{r},t)$ and $\mathbf{J}(\mathbf{r},t)$, what are the fields $\mathbf{E}(\mathbf{r},t)$ and $\mathbf{B}(\mathbf{r},t)$? In the static case, Coulomb’s law and the Biot–Savart law provide the answer.

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.

To make a current flow, you have to push on the charges. How fast they move, in response to a given push, depends on the nature of the material. For most substances, the current density $\mathbf{J}$ is proportional to the force per unit charge, $\mathbf{f}$.

Classical mechanics obeys the principle of relativity: the same laws apply in any inertial reference frame. By “inertial” I mean that the system is at rest or moving with constant velocity.1 Imagine, for example, that you have loaded a billiard table onto a railroad car, and the train is going at constant speed down a smooth straight track. The game will proceed exactly the same as it would if the train were parked in the station; you don’t have to “correct” your shots for the fact that the train is moving – indeed, if you pulled all the curtains, you would have no way of knowing whether the train was moving or not.

In this chapter, we extend perhaps the most famous law in mechanics, Newton’s Second Law, to study objects and systems of objects executing rotational motion. Emphasis is placed on developing an intuition for the effects of torques on the rotational dynamics of systems by comparing and contrasting them to the effects that forces have on the linear motion of such systems.

In this chapter, we begin by defining the concept of the angular momentum for a point mass, systems of discrete masses, and continuous rigid bodies. We then use the most general form of Newton’s Second Law for rotational motion to study the impulse due to a torque, the angular momentum impulse theorem, and finally the conservation of angular momentum. To develop these theorems, we draw from our understanding of the analogous theorems in linear motion.