Maxwell’s Equations

As Physical Science advances we see more and more that the laws of nature are not mere arbitrary and unconnected decisions of Omnipotence, but that they are essential parts of one universal system in which infinite Power serves only to reveal unsearchable Wisdom and eternal Truth. … While we look down with awe into these unsearchable depths and treasure up with care what with our little line and plummet we can reach, we ought to admire the wisdom of Him who has so arranged these mysteries that we find first that which we can understand at first and the rest in order so that it is possible for us to have an ever increasing stock of known truth concerning things whose nature is absolutely incomprehensible.

- James Clerk MaxwellMaxwell, “Inaugural lecture at Marischal College, Aberdeen, 3 November 1856,” in P. M. Harman (ed.), The Scientific Letters and Papers of James Clerk Maxwell, Volume 1 (Cambridge: Cambridge University Press, 1990), 426-427.
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).

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)

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 helped to explain a phenomenon previously discovered by Michael Faraday, who noticed that a changing magnetic field could induce an electric current in a coil of wire, as illustrated in the simulation below. Charged particles like electrons feel a sideways force when they move through a magnetic field, as described above, and the same thing happens when the magnetic field moves past the electrons. So, when a magnet moves through a coil of wire, the electrons in the wire are pushed sideways (i.e., around the loop of wire, perpendicular to the magnet’s motion), generating an electric current along the wire.

Simulation of Faraday’s Law

Move the magnet back and forth through the coil of wire. Also experiment with moving it various directions around the wire. Notice that an electric current is produced whenever the magnetic field lines move perpendicular to the wire. (Check the box labeled “Field Lines” to see them.)

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.