Sources and Fields

The Field Concept: From Electricity to Magnetism

The interaction between two electric charges is understood in two stages:

1. The source charge $Q$ creates an electric field $\mathbf{E}$ in the space around it.
2. A test charge $q$ placed in this field experiences a force $\mathbf{F} = q \mathbf{E}$.

The electric field due to a point charge $Q$ at a distance $r$ is given by:

Here, $\hat{\mathbf{r}}$ is the unit vector pointing from $Q$ to the point of interest, and $\varepsilon_0$ is the permittivity of free space. The field $\mathbf{E}$ is a vector field — it has a magnitude and direction at every point in space.

Crucially, the field is not just a mathematical tool. It has a physical reality: it can carry energy and momentum, and it propagates at a finite speed (the speed of light). This concept, emphasized by Faraday and formalized by Maxwell, is the foundation of field theory.

The Magnetic Field: A New Vector Field

In the same way that static charges produce an electric field, moving charges (or electric currents) produce a magnetic field, denoted by $\mathbf{B}(\mathbf{r})$.

• $\mathbf{B}$ is also a vector field — defined at every point in space and can vary with time.
• In this chapter, we assume the fields do not change with time (static fields).
• The magnetic field obeys the principle of superposition: the total magnetic field due to multiple sources is the vector sum of the fields produced by each individual source.

Key Experimental Evidence: Oersted's Discovery

Hans Christian Oersted (1777–1851) first demonstrated the connection between electricity and magnetism. He observed that a compass needle deflects when placed near a wire carrying an electric current. This was the first empirical proof that electric currents produce magnetic fields. …