# Physics Paradoxes?

Hi,

I have two questions about electromagnetism that I would like to ask. The 1st one is as follows:

I have two point charged particles A and B a distance x apart. A and B exert a force on each other by Coulomb's Law, but a force is exerted on the particles to keep them apart at exactly x distance, thus making the system remain in equilibrium.

If the particles are in a stationary inertial frame, then there is nothing wrong. However, consider the case where A and B are in an inertial frame moving at constant velocity, and we are viewing the particles at a rest frame. In that case, A and B will produce magnetic fields, and these magnetic fields will also affect each other. If they have like charges, they will move towards each other. If they have unlike charges, they will repel charges. However, if we were to analyse the motion of A and B in the moving inertial frame, then A and B will be stationary and there will be no magnetic field. So A and B will be stationary.

How can I get 2 different answers by just taking a different inertial frame? Shouldn't all inertial frames give me the same answer?

The 2nd question is:

Consider the case where where I have 2 charged point particles. A is at the origin (0, 0) moving in the x-direciton, while B is at (0, a), where a is some arbitrary value. B is moving in the y-direction. We ignore analysis of electric fields. Since A is directly on the line on which B is travelling, A will not experience any magnetic field and thus will move straight. However, B will experience a magnetic field by the moving A. How do you explain these by using Newton's Third Law or something? Why don't the forces balance out?

I got quite confused here :S. Do I have some misconception somewhere?

Thanks in advance.

## Answers and Replies

Mapes
Science Advisor
Homework Helper
Gold Member
Note that a distance moving relative to you may appear to be shorter due to relativistic length contraction (depending its orientation). I seem to recall that this exactly offsets the effects of a magnetic field created by a moving charge, so that the configuration makes physical sense in all frames. See http://en.wikipedia.org/wiki/Moving_magnet_and_conductor_problem" [Broken].

(I'm sure the forum members with in-depth EM and special relativity experience will be able to answer more thoroughly than just a Wikipedia link, but I thought you might be interested in the article.)

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Electromagnetism is Lorentz invariant, and not Galilei invariant. Actually the formulas for Special relativity where known in electromagnetism before Einstein. So you have to transform all the fields when you move into another inertial frame, and also do the length contraction. Then everything fixes itself, sorry that I don't know the details now. Btw are you sure you know how to calculate the magnetic field of a moving point mass? Do you factor in the retarded potentials? I am quite sure that electrostatics is not enough since the charge distribution changes.