# Mechanism by which electric and magnetic fields interrelate

I read that force due to electric field on some particle in one reference frame can exhibit itself as force due to magnetic field in some other reference frame and that electric and magnetic fields are two aspects of same underlying electromagnetic field.

My question is what is the mechanism which can explain how an electric field becomes/creates magnetic field in some other reference frame. Is there any such explanation available in relativity theory? I am not looking for mathematics but a physical explanation.

Wikipedia article http://en.wikipedia.org/wiki/Relativistic_electromagnetism explains something about origin of magnetic forces in a wire as a consequence of lorentz contraction and motion of electrons in the wire

Calculation of the magnitude of the force exerted by a current-carrying wire on a moving charge is equivalent to calculating the magnetic field produced by the wire. Consider again the situation shown in figures. The latter figure, showing the situation in the reference frame of the test charge, is reproduced in the figure. The positive charges in the wire, each with charge q, are at rest in this frame, while the negative charges, each with charge −q, are moving to the left with speed v. The average distance between the negative charges in this frame is length-contracted to: where is the distance between them in the lab frame. Similarly, the distance between the positive charges is not length-contracted: Both of these effects give the wire a net negative charge in the test charge frame, so that it exerts an attractive force on the test charge.

But this still does not explain origin of magnetic field in case when there are no positive charges.

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– Qmechanic Feb 20 '12 at 19:07
There is no "mechanism". It is simply your perspective that changes. Think of something looking different from frontal view and side view. – Siyuan Ren Feb 21 '12 at 1:15
Physicists are confused in this matter. Let's say we have two parallel lines of electrons, and an observer moving along the lines. The observer says that the lines repel, because of the charges, and because of length contraction the repelling force is increased. Then the observer says that the electrons move apart at slowed down speed, because of time dilation. Now the observer, who is a physicist, says that the attractive magnetic force that should be there between the lines, is the same thing as the time dilation of the moving apart of the electrons. Particularly the last part sounds silly. – kartsa Feb 22 '12 at 0:24
Thanks kartsa, its good to know the current state of affairs – Nitin Nizhawan Feb 22 '12 at 6:28

There isn't a mechanism. You're trying to find a mechanism for how two abstract objects can exchange identities. Any mechanism involving abstractions must consist of abstractions. So,the only way to explain it is through mathematics.

## Least abstract way to look at it

I feel that the least abstract way to explain it is to look at two stationary charges. They attract via the Coulomb (electrostatic) force. Now run perpenducular to the line joining their centers. Each charge creates a magnetic field as it is moving (moving charge can be thought of in certain cases as current). The magnetic field acts upon the other charge, creating a force. Meanwhile, the electric force has decreased (no longer electtoSTATIC). The net force is the same, but part of it is magnetically caused.

## Relativistic way

Another way to look at it is to remember that EM fields are set up by EM radiation. An EM wave carries oscillating EM fields with it; see pictures here. A point charge radiates EM waves in all directions. The oscillating E field of one of these waves interferes with the E field from a nearby wave constructively, creating a nearly non-oscillating field, which decreases as distance squared (Comes from the fact that intensity of a point source $\propto 1/r^2$), giving us Coulomb's law. The oscillating magnetic fields destructively interfere, so we get no net magnetic field.

Now, if you start moving with respect to the charge, things get interesting. The relativistic doppler effect will act upon the EM waves, altering them (since the speed of light is the same in all frames, we can't apply relative velocities to it). The interferences won't work quite the same, and we'll get a bit of a magnetic field and mainly an electric field. Move faster, and the magnetic field intensity increases, E decreases. Accelerate, and you get complicated stuff. Note that infact em waves are radiated only by an accelerated charge. A sitting charge does not emit em waves. The waves emitted by an accelerated charge produce change in the fields. The easiest way to visualize this is by assuming that the em waves are radiated in all cases.

I think that explains it without too many abstractions..

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Thanks. I would like read more about your relativistic explanation. Is there some article/book where I can find more detail on it? – Nitin Nizhawan Feb 21 '12 at 8:37
Purcell's intro E&M book is IMO fantastic, and deals with E&M in a relativistic way almost right off the bat, something I appreciate a lot. Also, just in case you have any doubts left, there is substantial literature on this subject, and the phenomenon is a classic subject on introductions to relativity, I highly recommend researching it on your own, its a real treat. – kηives Apr 11 '12 at 5:15
@Nitin see above comment. – Manishearth Apr 11 '12 at 5:20

The simplest explanation I know of requires only one test charge and two reference frames with a relative velocity between them.

Frame 1: The charge is at rest. It is the source of a (purely) electric field.

Frame 2: The charge is moving. It is a current, and the source of a magnetic field.

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