Forces, energy and electromagnetic momentum

In this chapter, we shall go on to discuss the concepts of the energy, the linear momentum and the angular momentum of the electromagnetic field. So far in this book, we have generally tried to avoid methods based on the concept of energy, except, for example, when we discussed electrostatic energy and the definition of the electrostatic scalar potential φ in Section 1.2.10 of Chapter 1. We have been able to do most of what we have done so far using the Lorentz force law to relate the fields E and B to experiments. The Lorentz force law will again be the starting point for all of our developments in this chapter. This is similar to the position in Newtonian mechanics, where it is Newton’s laws of motion that are the starting point for the development of the concepts of energy, linear momentum and angular momentum and the corresponding conservation laws. To quote French [1]:

It is an interesting historical sidelight that in pursuing the subject of energy we are temporarily parting company with Newton, although not with what we may properly call Newtonian mechanics. In the whole of the Principia, with its awe-inspiring elucidation of the dynamics of the universe, the concept of energy is never once used or even referred to! For Newton, F = ma was enough. But we shall see how the energy concept, although rooted in F = ma, has its own special contributions to make.

For example, in Newtonian mechanics it is Newton’s laws of motion that are used to determine the expression for the kinetic energy of a particle. The importance of the concept of energy is that energy is conserved. The law of conservation of energy can often be used to solve problems. For example, if we let a particle fall from a height h in the Earth’s gravitational field, we can determine the velocity with which it hits the ground, either by applying the law of conservation of energy by equating the gain in the kinetic energy of the particle to its loss of potential energy, or by applying Newton’s law of universal gravitation and Newton’s second law of motion to determine the acceleration of the particle, and then using the appropriate kinematic relation to determine the velocity with which the particle hits the ground. It is important to realize that the two approaches are alternatives. In this chapter, what we shall be doing is developing alternatives to the direct application of the Lorentz force law.