# Q & A: magnetism and relativity

Q:
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
A:

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 attractive ELECTRICAL force. Whether you say the force is electrical or magnetic or some combination depends on which frame you use.

Mike W.

(published on 10/22/2007)

## Follow-Up #1: magnetism and Lorentz contraction

Q:
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
A:
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 10/22/2007)

## Follow-Up #2: relativity and charge density

Q:
I'm not sure you answered the questions, though I'm also not sure I understood the explanation so I apologize if you end up repeating yourself. Here is the situation. A stationary negative charge is near a wire with no current flowing through it. It obviously feels no electric or magnetic force since the positive and negative charges in the wire cancel out and their is no current. I now hook up the wire to a voltage source, and set the electrons moving with regard to the positive charges within the wire and the stationary negative charge near the wire. From the stationary charge's perspective, wouldn't the distance between the moving electric charges in the wire be contracted by the Lorentz transformation? And, therefore, wouldn't the negative charge density in the wire be higher than the positive charge density, causing the stationary charge to be repelled by electrostatic forces? This is obviously not he case, but it seems to be the logical conclusion if this explanation is to be self consistent. Thanks for the great feedback!
- Justin Collinger (age 29)
Pittsburgh, PA, USA
A:

That's a nice question. Remember that the stationary charge and the lab are using the same reference frame. So the stationary charge won't see a net charge on the nearby wire unless the lab also sees it. Generally, it won't.

Why not?  One doesn't start with a rigid chain of electrons, then set them in motion getting a simple Lorentz-Fitzgerald contraction. The electrons are flowing about, and free to adjust their spacing. In the lab frame, easy to work with here, you can see that the electrostatic energy is minimized by having the electron charges just cancel the positive charges in the wire. So if we ignore minimizing the magnetic part of the energy,  it remains neutral. What about the magnetic part of the energy? There's still the constraint that the overall circuit is neutral. So to the extent that the circuit is a symmetrical loop, the constraint of neutrality applies locally, regardless of the magnetic field energy.  I guess that if there are kinks in the circuit loop where the effect of the magnetic Lorentz force in helping electrons turn the corners varies, you can get slight charges in the lab frame, still adding up to zero around the loop.

What about in the frame of the electrons in our little region of the wire, or to be more precise in the frame of a non-accelerating observer travelling along with them? This observer will see them stretched out and the positive charges contracted, compared to the lab values for the densities. So they'll see that region as positive, making an electrical force on that other charged particle. Of course, they also see that particle as being in motion, and thus experiencing a magnetic force as well. The two forces must exactly cancel.

Mike W.

(published on 09/26/2013)

## Follow-Up #3: rotating frames and current loops

Q:
It is surprising how you came up with the same Q/A I gave at stackoverflow, http://physics.stackexchange.com/a/64196/16114 The problem is the charge conservation and distribution in the loop with current. We have considered that in order to stay neutral in the stationary (lab) frame, electrons of current must "stretch out" in their accelerated frame. So, resolved the charge conservation in the lab frame, we get the ladder paradox in the moving electron frame: you have a expanded chain of electrons looping along a shrinked chain of rigid nuclei. Yet it is different from the classical ladder paradox. Here we have a "circular ladder paradox". How does stretched out chain of electrons fit into the shrunk loop of parent matter?
- Valentin (age 34)
Tallinn, Estonia
A:

I'm not surprised that this question has shown up on different sites. Even if it turns out to have come from different people, it's a pretty natural thing for people to worry about.

It sounds as if you're attempting to use a rotating frame in which all the electrons are at rest rather than an inertial frame moving along with one little segment of the electrons. Rotating frames have centrifugal forces, coriolis forces and various general relativistic effects. I'm too lazy to work out the resolution of the problem in that sort of frame.

In the inertial frame, the explanation is simple, at least to lowest order in v/c. The electrons are stretched on one side of the loop and shrunk on the other side, compared to the positive charges. So that gives an electric field, whose effects on the test charge just cancel those of the magnetic Lorentz force. The ellipticity of the loop due to Lorentz-Fitzgerald contraction along one axis in the "moving" frame may play a role in cancelling terms of higher-order in v/c.

In the rotating frame I guess you need some net force on the test particle to cancel the pseudo-forces. For a symmetrical loop (rotation at least preserves that symmetry, unlike in our "moving" inertial frames) that force won't be electrical, so I suppose it's magnetic.

Mike W.

(published on 09/30/2013)