Q & A: magnetism and relativity

Two electrons moving parllel to each other in the same direction and with the same speed, as observed by a stationary observer, magnetically ATTRACT each other -- Ampere’s experiment. Now, if a different observer moves along with the electrons, he sees them stationary. Stationary electrons, electrically REPEL each other. How do you resolve this anomaly? I don’t think this anamoly is due to relativistic effects, as Ampere’s experiement can be conducted in low speeds without intrdocution of relativistic effects.
- Mehran (age 53)
Lisle, IL

Mehran- Great question!

Actually, magnetism is ENTIRELY a relativistic effect. Some nice textbooks (like the one by Purcell) actually introduce magnetism by showing that it's what's needed to make the physical behavior of electricity look the same in all the different reference frames.

The force between the two electrons has both an electrical and a magnetic component in the frames which say the electrons are moving. The sum of these two forces isn't exactly the same as the simple electrical force in the frame where they're standing still. That's because ALL forces change in different reference frames. The net force transforms just like any other force. In the case you specify, the total force (electrical plus magnetic), the sum is still a repulsive force, in all frames of reference.

You might find the following example amusing. Say that you have a stationary electron near a wire in which positive and negative charges with equal density are flowing opposite directions at equal speeds. Obviously, there's no force between the electron and the wire, either electrical or magnetic.Now set the electron in motion parallel to the wire. We say there's a force (say in the attractive direction) due to magnetism. What happens in a frame in which the electron is at rest, and cannot feel a magnetic force? The different Lorentz contractions of the positive and negative charges in the wire give them different densities in that frame. So in that frame there's an atrractive ELECTRICAL force. Whether you say the force is electrical or magnetic or some commbination depends on which frame you use.

Mike W.

(published on 10/22/2007)

In your answer about magnetism and relativity you write: "Say that you have a stationary electron near a wire in which positive and negative charges with equal density are flowing opposite directions at equal speeds. Obviously, there’s no force between the electron and the wire, either electrical or magnetic." I would like to ask what happens it the negative charges move to a certain direction (relative to the electron), while the positive charges remain at rest (relative to the electron). According to non-relativistic magnetic theory, the electron should feel no magnetic force because it is at rest, and no electric force since the wire is electrically neutral. However, since the electron sees the positive and negative charges move at diffrenet speeds, the different Lorentz contractions of the positive and negative charges in the wire should give them different densities, so the electron should feel an ELECTRIC force!
- Erel Segal (age 30)
Haifa

Neat question. Let's define the "lab" reference frame to be the frame in which the test electron is at rest. According to the premise of your question, you have set up the wire to be electrically neutral in this reference frame. This statement already includes the effect of Lorentz contraction of the moving charges. If the positive and negative charged particles in the wire have different average velocities, then the charge per unit length of the wire depends on the reference frame, as you point out, due to the different Lorentz contractions of the two kinds of charges.

A perplexing question to ask then, is that since the charge density on a segment of wire depends on how fast an observer is moving with respect to that piece of wire, how come charge isn't created or destroyed, just by looking at the same piece of wire in a different way? As it turns out, all observers must agree on the total amount of charge in a system, but they may disagree on its distribution.

Because total charge is conserved, it flows around typically in loops, in electrical circuits. For every bit of wire which gains an average negative charge under a Lorentz boost, an equal and opposite positive charge must be observed somewhere else in a loop of wire. This positive charge may be farther away from the test charge than the negative one, giving rise to an unbalanced electrical force (in one frame, which is interpreted as a magnetic force in another frame, so long as the test electron is moving).

Now back to your situation -- you've got a wire with current flowing in it and no net charge in the lab, but also a stationary electron floating nearby. It feels no force. If we look at the same system in another reference frame, we've got current flowing, a net charge on the wire, and a moving test electron! It feels both electrical and magnetic forces. But since the electron isn't accelerating in the lab frame it cannot accelerate in any other frame, and so the electrical and magnetic forces have to delicately cancel in this case.

Mike W. and Tom

(published on 08/02/06)