1.1 What electromagnetics is about

Electromagnetics is the study of the effects of electric charges, at rest and in motion. Two charged bodies exert forces on each other even when nothing visible connects them. A current-carrying wire deflects a compass needle, and a changing current in one coil drives a current in a second coil that it never touches. All of these are actions at a distance, and the central idea of the theory is the way in which such actions are described.

Rather than saying that one charge acts directly on another one far away, we say that every charge (and every current) sets up a field in the space around it, and that any other charge reacts to the field at the place where it is. A field is therefore a quantity that has a value at every point of a region of space and, in general, at every instant of time. Temperature in a room is a scalar field: one number at each point. The electric field is a vector field: at each point it has a magnitude and a direction. The description is local: the force on a charge depends only on the field at the position of that charge, not on where the sources happen to be (Figure 1.1.1).

To see two charges acting on each other, drag them around and watch how the force on each one changes with their positions.

The field is not a mere bookkeeping device. When charges move, the change of the field travels outwards with a finite speed, the speed of light, and the field can carry energy and momentum on its own. This is what a radio wave or light is. In this first part of the course, however, the sources do not change in time, and the field simply gives the most economical way of computing forces, energies, potentials and capacitances.

1.1.1 Static and time-varying fields

The subject is naturally divided according to how the sources behave:

• Electrostatics: charges at rest. The only field is the electric field, and it does not change with time.
• Magnetostatics: steady currents (charges moving with constant velocity, so that the current distribution does not change with time). A time-independent magnetic field appears in addition to any electric field.
• Time-varying fields: sources that change in time. The electric and magnetic fields are then coupled to each other and can no longer be studied separately. This coupling is what makes electromagnetic waves possible.

The present notes cover the mathematical tools of vector calculus and then electrostatics, in free space and in material media.