# Two electrons and a paradox?

## Main Question or Discussion Point

Suppose there are two electrons which are not moving relative to each other. Then in the reference frame of either electron, there is an electrostatic force pushing the two electrons apart, and they will move apart. However, if an observer is in a reference frame moving relative to the electrons, he or she will observe two parallel currents, which will create magnetic fields causing the electrons to attract, according to Ampere's law. Hence such an observer will see the two electrons move closer together. How can this be resolved? Am I simply wrong? Please help me out!

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ghwellsjr
Gold Member
Einstein, in his 1905 http://www.fourmilab.ch/etexts/einstein/specrel/www/" [Broken] introducing Special Relativity, started off by raising a similar question about a conductor and a magnet in relative motion. Rest assured, there is no paradox. It all works out if you care to go into the details.

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Suppose there are two electrons which are not moving relative to each other. Then in the reference frame of either electron, there is an electrostatic force pushing the two electrons apart, and they will move apart. However, if an observer is in a reference frame moving relative to the electrons, he or she will observe two parallel currents, which will create magnetic fields causing the electrons to attract, according to Ampere's law. Hence such an observer will see the two electrons move closer together. How can this be resolved? Am I simply wrong? Please help me out!
Congratulations! You are following in the footsteps of Albert Einstein. It was magnetism like this that caused him to formulate the theory of Special Relativity. You can read all about it in his original 1905 paper at http://www.fourmilab.ch/etexts/einstein/specrel/www/, which explains it much more clearly than I ever could.

jtbell
Mentor
Dale
Mentor
One basic point that you may be missing is that in the frame where they are moving there is still a repulsive electric field in addition to the attractive magnetic field. The net force remains repulsive in all frames, so no observer will see them move together. They will all see them move apart, but at different rates.

Thanks guys! Looks like I've got some reading to do! I also appreciate the hand-waving explanation that there is always some sort of repulsive force caused by the electric fields, though this seems to imply that coulomb's law, ampere's law and/or the magnitude of the charges changes with reference frame... like i said, looks like i've got some reading...

pervect
Staff Emeritus
Gauss's law still works in a moving frame, but it no longer implies coulomb's law, because the problem is no longer symmetrical.

http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_15.pdf has a diagram of the field of a moving charge, and you could look at Professor fields lecture notes for some of the formulae, but you would probably better off getting an E&M textbook, like Griffiths.

https://www.amazon.com/dp/013805326X/?tag=pfamazon01-20

You should be able to get it via an inter library loan, it'll be pricey to buy. Most libraries have a system for gettig books from other libraries if they don't have them.

The other thing you'll need to take into account is to learn how forces trasnsform. Due to time dilation, forces look different from different frames.

I think Space-time Physics by Taylor and Wheeler would have that transformation, if not it woul at least give you the background you need to understand it when you do find it. What you'd look for is a general understanding of 4-vectors, and more specifically the application of 4-vectors to forces, the 4-force.

The first chapter of the older edition of Space-time physics is online at the author's website, but I'm not sure it willl have what you need. http://www.eftaylor.com/download.html#special_relativity You could also take the library loan approach.

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This is from Einstein’s 1905 paper: (halfway down)
If a unit electric point charge is in motion in an electromagnetic field, the force acting upon it is equal to the electric force which is present at the locality of the charge, and which we ascertain by transformation of the field to a system of co-ordinates at rest relatively to the electrical charge. (New manner of expression.)
In short, he says that a magnetic force experienced by a charge travelling through a magnetic field can be seen as an electrical force as viewed by the charge at rest.

How does this relate to the OP? Well from the point of view of someone observing 2 travelling charges in his/her rest frame, we have the opposite case as the one Einstein referred to, ie a travelling electrostatic field can now be seen to generate a magnetic field.

However, the crucial point for a magnetic field to be detected in a rest frame, is that charges have to be travelling in that rest frame.

In a frame which travels with the same velocity as the electrons no magnetic force will be observed only an electrostatic force. This in turn means that the 2 electrons as described in the OP will not experience a magnetic field.

The proof of this is that 2 perfectly isolated, charged spheres in a lab on earth, will always exert the same electrostatic force on each other, no matter what time of day or time of year.