Maxwell’s Equations

Note: Since this sideways force is perpendicular to the direction of travel, the direction of the charged object’s motion will change but not the speed. Thus, magnetic fields can be used to “steer” charged objects without slowing them down. For example, magnets are used to steer fast-moving protons around a circle in particle accelerators like the Large Hadron Collider, which will be discussed in Chapter 5.

The relationships between electric and magnetic fields are surprisingly complex. For example, when a charged object is in motion with respect to a magnetic field, a mysterious “sideways” force is exerted on it—a force perpendicular to both the magnetic field and the direction of travel. In other words, a charged object “feels” an electric field when it moves through a magnetic field (or when a magnetic field moves with respect to it), even if there are no other charged particles nearby!

The total electromagnetic force on a charged particle—including the “sideways” force caused by motion through a magnetic field, plus the forces exerted by other charges as described by Coulomb’s law—is called the Lorentz force.

Furthermore, electric fields and magnetic fields can influence each other even when no particles are present: a change in the magnetic field induces a change in the electric field, and vice versa. (This is how light propagates through “empty” space, as we’ll see in what follows.) Electric fields and magnetic fields are so closely interrelated that they are regarded as two different aspects of the same fundamental force—the electromagnetic force. The relationships between electric and magnetic fields are described by Maxwell’s equations.

portrait of James Clerk Maxwell
James Clerk Maxwellimage source (public domain)
1831-1879

The actual equations are shown in appendix B [link coming soon], which also provides a brief explanation of what each equation means. For present purposes, however, you don’t have to understand all the details of Maxwell’s equations. The following summary will suffice:

Maxwell’s equations also provide insights into the nature of light, as we will see. First, however, it will be necessary to provide some background information about waves. That is the aim of the next section.