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We know that parallel current in two parallel wires causes them to attract. The explanation is that moving charges generate attractive magnetic fields.

https://youtube.com/watch?v=43AeuDvWc0k

If I have two charged pith balls in a space ship, then they should attract more the faster the space ship travels. But relativity says you cannot know your velocity in an inertial frame. In other words you cannot know how fast you are going unless you look outside the space ship. So that means the attraction of two moving charges should only be visible from outside the moving spaceship - which does not make sense to me.

How can a magnetic field not happen in a spaceship but be observed from outside?

There are many questions about the relativistic effects of electrons in a wire, for example

Relativistic explanation of attraction between two parallel currents

But they all hinge on the velocity difference between the moving electrons in the wire vs the static protons. In my example I am just considering the behavior of two charged pith balls through space. Moving charges should generate a magnetic field, no? Or do moving charges only generate a magnetic field when they are moving relative to protons?

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    $\begingroup$ The charges are not moving with respect to each other. "Or do moving charges only generate a magnetic field when they are moving relative to protons?" moving charges only generate a magnetic field when they are moving relative to something. all motion is relative. $\endgroup$ Sep 13 at 13:10
  • $\begingroup$ The answer in the last link is wrong! Just saying ;) There is no magnetic force between static protons and moving electrons. Moving charge is a magnet. Still standing charge is not a magnet. A magnet and a non-magnet don't exert any magnetic force on each other. Very simple. If we ask a moving electron, electron says there is extra Coulomb force. If we measure, there isn't any extra force, for the simple reason that I explained. $\endgroup$
    – stuffu
    Sep 14 at 11:31
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So that means the attraction of two moving charges should only be visible from outside the moving spaceship - which does not make sense to me.

You are right, that does not make sense. What relativity actually says is more subtle.

Relativity says that both frames will agree on all observables. They will agree on the readings of any strain gauges or accelerometers or whatever. But they will attribute those readings to different effects.

For example, suppose you attach needles and a scale to the pith balls to measure the amount of attraction. Then, regardless of the ship's speed, all frames will agree on how much the needles and scale read. In the ships frame the attraction will be due entirely to the E field, and the needles and scale will correctly measure that. In an external observer's frame, regardless of the ship's speed, the attraction will be due to both the electric field and the magnetic field and the needles and scale will be time dilated and length contracted, such that the external frame will also predict the same reading on the needles and scale. Thus regardless of the speed, the reading is the same and cannot be used as a speedometer.

In the ship's frame the charged balls have an E field and the measured forces are caused by those E fields. In the outside observer's frame there is an E field and also a B field. The measured forces are caused by both the E and B field. Furthermore, the accelerometers or strain gauges are subject to time dilation, so their values are somewhat off compared to the coordinate values.

How can a magnetic field not happen in a spaceship but be observed from outside?

In a coordinate-independent sense there is only the electromagnetic field. The electromagnetic field is a tensor which has the electric and the magnetic fields as different components of the tensor. Different frames will partition that into electric field and magnetic field somewhat differently. But intrinsically, physically, the electric and magnetic fields are two parts of the same overall electromagnetic field.

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  • $\begingroup$ So you are saying that the person looking at the two pith balls in the spaceship WILL see them attract due to a B field but the person outside the spaceship (not moving with it) will see a much slower attraction due to time dilation. In any case there should be a B field between the two pith balls when the spaceship is moving. I should then be able to attach needles and a scale to the device to act as a speedometer. The faster I go, the more the should attract. So my speedometer should tell me how fast I am going even without any external reference $\endgroup$ Sep 13 at 15:43
  • $\begingroup$ I added a paragraph specifically addressing this suggestion. $\endgroup$
    – Dale
    Sep 13 at 17:11
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    $\begingroup$ Thank you for the time you took to explain it. The way I begin to see a solution is by using an operational definition of a field. Only by having a magnet outside the spaceship can I measure a magnetic field generated by the charge inside the spaceship which happens to be moving. In the absence of a measurement, we cannot say that a field exists. Inside the spaceship , a magnet cannot detect any field as it is moving along with the charges. The abstract nature of Maxwell's equations sometimes is too concise and small details like 'what is moving and what is not' are sometimes lost to me. $\endgroup$ Sep 14 at 21:08
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If you can make a force measuring device that does not change when its velocity changes, then you can measure velocity dependent forces changing inside an accelerating spaceship.

I happen to have quite a lot of trust on the principle of relativity. So I'm pretty sure that you can't make such device.

For example a device that compares those changing forces to gravitational attraction between two spaceships does not work.

For another example a device that measures accelerations caused by those changing forces does not work, because that device includes a clock which changes when its velocity changes.

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If I have two charged pith balls in a space ship, then they should attract more the faster the space ship travels.

The electric repulsion between the balls is always larger than the magnetic attraction, so they repel less the faster the ship travels.

The rate at which they repel as a function of the speed of the ship matches the time dilation formula. This example is sometimes used to illustrate that special relativity requires/predicts magnetism, although the version with the neutral wires is more common.

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