Chapter 11 Galilean Relativity in Electrodynamics

That a charge must move with a definite velocity to experience a magnetic force raises an interesting question. What happens if we transform to a reference frame in which the charge is at rest? Apparently, there is no magnetic force exerted on the charge in the rest frame of the charge. Yet the principle of relativity, first described by Galileo Galilei and then incorporated by Isaac Newton into his laws of motion, assures us that the laws of physics do not change whether you are traveling at a constant velocity or standing still.

Galileo used the principle of relativity in order to illustrate how the earth can move without our feeling the motion. As he (roughly) put it, imagine that you and a friend lock yourselves in a windowless cabin below the deck of a docked, motionless ship. Now look about the cabin. You observe water dripping from bottle that is suspended upside-down, fish swimming in all directions in an aquarium, and butterflies fluttering randomly. Throw something to your friend. It will require the same force no matter what direction you throw it in, so long as the distances of the throw are equal. Now imagine that the ship is cruising along at a constant velocity in calm waters. The water continues to drip as before. The fish and butterflies are equally content swimming or flying in all directions. There are no new challenges in throwing the ball. In fact, there is no way for you or your friend to tell whether or not you are stationary or moving uniformly with respect to the shore.[1] The earth is our cabin and though we are hurtling some 100,000 kilometers per hour relative to the sun, the butterflies do not seem to care.

Galileo’s analysis of projectile motion can also be understood in light of the principle of relativity. Consider, once more, a ship sailing along a smooth sea at constant velocity. If one drops a heavy stone (so that air resistance is not a factor) from the top of the ship’s mast, an observer on the ship would see it accelerate straight downward and hit the deck at the base of the mast. An observer standing on the shoreline sees the same stone move in a parabolic arc. In addition to the vertically accelerated motion of the stone, that person would also see the ship (and everything on it, including the falling stone) move horizontally together with the same speed.

Both observers, of course, are correct in describing the different scenes they saw, but the physics remains the same. For example, the time of flight for the stone—which we can derive by noting only the forces on the stone—remains the same, despite the different apparent flight paths. The laws of physics are consistent regardless of the frame of reference in which you choose to describe them.

As with falling bodies and butterflies, the physics for our test charge also will not change depending on the reference frame. Yet no magnetic force exists in the rest frame of a charge in the presence of a magnetic field, even though a magnetic force is exerted on that charge according to someone who moves uniformly with respect to it. How can this fact be reconciled with Galileo’s principle of relativity? The ultimate answer to this conundrum comes from Einstein’s special theory of relativity, which demonstrates that electric and magnetic fields are aspects of a more fundamental entity—the electromagnetic field. What appears to be only a magnetic field in one frame of reference will be observed as separate electric and magnetic fields in other frames of reference.

Einstein’s theory has its roots in electricity and magnetism; in fact, his first paper on the subject published in 1905 was titled On the Electrodynamics of Moving Bodies. It was his concern with questions related to the moving charge that led Einstein to develop this theory, and we will see just how successfully he clarified our understanding of electromagnetism later. We do not need the full machinery of special relativity to obtain at least a partial answer to the puzzle raised here, however. We will see that electric and magnetic fields transform in a simple way when measured by different moving observers and can be approximately related by the Galilean transformation from pre-Einstein physics. One consequence of this interrelationship of electric and magnetic fields observed in different frames of reference is that an electric field can be induced when a neutral metal bar moves through a given constant magnetic field.

We no longer think of electric and magnetic fields as separate entities, but instead as the unified electromagnetic field. Today, physicists adopt this concept of a field to mediate interactions among particles when describing all fundamental forces of nature. In light of this, we shall introduce the idea of electromagnetic fields in this chapter with an emphasis on the response of charges to already existing electric and magnetic fields.

[1] Galileo Galilei, Dialogue Concerning the Two Chief World Systems: Ptolemy and Copernicus, (1632); (Translated by Stillman Drake, University of California Press, 1953), 186 – 187; Second Day).