Special relativity and magnetism

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I've read in various places that magnetism can be explained in terms of the effects of special relativity. However, all of the explanations of this only mentioned the case of current flowing in a wire. Can special relativity explain the magnetism of free flowing electrons and other moving charged objects?
 

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Orodruin
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In order to answer your question, you must specify what you mean by "explained in terms of". The entire framework of Maxwell's equations was instrumental in the development of special relativity. Magnetic and electric fields mix under Lorentz transformations due to being different components of the same anti-symmetric rank two tensor.
 
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In order to answer your question, you must specify what you mean by "explained in terms of". The entire framework of Maxwell's equations was instrumental in the development of special relativity. Magnetic and electric fields mix under Lorentz transformations due to being different components of the same anti-symmetric rank two tensor.
Okay, "explained in terms of" is probably the wrong way to put it then, what I mean is, the special relativity picture of magnetism says that the magnetic field in one reference frame is actually an electric field in another reference frame (in the current in a wire situation). I'm asking, can the magnetic field of a moving electron or other moving charge in free space be explained as an electric field in another reference frame?
 
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Okay, "explained in terms of" is probably the wrong way to put it then, what I mean is, the special relativity picture of magnetism says that the magnetic field in one reference frame is actually an electric field in another reference frame (in the current in a wire situation). I'm asking, can the magnetic field of a moving electron or other moving charge in free space be explained as an electric field in another reference frame?
Yes. That's basically an English-language restatement of how the Faraday tensor Orodruin referred to transforms from one frame to another.

You're not likely to encounter this treatment at the undergraduate level though. There you'll find the special case of the current-carrying wire because that case is good enough to get the general idea across and can be taken on by students who have been through only an intro course on special relativity.

You will find much more discussion here if you look for threads referencing "Purcell", who will be the Edward Purcell who wrote an intro E&M textbook that has been widely used for decades and which uses the wire example.
 
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Orodruin
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Okay, "explained in terms of" is probably the wrong way to put it then, what I mean is, the special relativity picture of magnetism says that the magnetic field in one reference frame is actually an electric field in another reference frame (in the current in a wire situation). I'm asking, can the magnetic field of a moving electron or other moving charge in free space be explained as an electric field in another reference frame?
Yes and no. As I already said, the components of the electric and magnetic fields mix under Lorentz transformations. However, it is not certain that you can always find a frame where the electric field is zero, or where the magnetic field is zero.

You can easily find the electric and magnetic fields of a moving electron by performing a Lorentz transformation of the field configuration for the stationary charge (for which there is no magnetic field (if you disregard the magnetic moment of the electron)).
 
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Yes and no. As I already said, the components of the electric and magnetic fields mix under Lorentz transformations. However, it is not certain that you can always find a frame where the electric field is zero, or where the magnetic field is zero.

You can easily find the electric and magnetic fields of a moving electron by performing a Lorentz transformation of the field configuration for the stationary charge (for which there is no magnetic field (if you disregard the magnetic moment of the electron)).
In the case of current in a wire there are positive charges in the wire that are needed for the effect to occur. Given that there are no positive charges near or around the electron as it moves through free space I can't see how its magnetic field can be attributed to an electric field in another reference frame. Is there any explanation that involves positive charges??
 
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Orodruin
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In the case of current in a wire there are positive charges in the wire that are needed for the effect to occur. Given that there are no positive charges near or around the electron as it moves through free space I can't see how its magnetic field can be attributed to an electric field in another reference frame. Is there any explanation that involves positive charges??
Any moving charge implies a current regardless of whether the overall charge is zero or not. This goes into Maxwell's equations and generally results in a magnetic field. I do not see why you think a positive charge is necessary. The positive charges are not needed.
 
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Any moving charge implies a current regardless of whether the overall charge is zero or not. This goes into Maxwell's equations and generally results in a magnetic field. I do not see why you think a positive charge is necessary. The positive charges are not needed.
Consider moving a negative charge in the direction of current of a wire. This will cause the stationary positive charges in the wire to appear to move in the reference frame of the moving negative charge, this will cause length contraction on the positive charges and creates a net positive charge to appear on the wire in the reference frame of the negative charge and will cause the charge to attract to the wire. In this example the positive charges are required for any magnetism to occur
 
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In this example the positive charges are required for any magnetism to occur
No they are not. The magnetic field would be there regardless of the positive charges. The positive charges are necessary only to make the electric field of the wire zero (in the wire frame).
 
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Consider moving a negative charge in the direction of current of a wire. This will cause the stationary positive charges in the wire to appear to move in the reference frame of the moving negative charge, this will cause length contraction on the positive charges and creates a net positive charge to appear on the wire in the reference frame of the negative charge and will cause the charge to attract to the wire. In this example the positive charges are required for any magnetism to occur

Consider moving a negative charge in the direction of current of a wire. This will cause the stationary positive charges in the wire to appear to move in the reference frame of the moving negative charge, this will cause length contraction on the positive charges and creates a net positive charge to appear on the wire in the reference frame of the negative charge and will cause the charge to report that it feels a force F.

The motion of the charge has one more effect: We will transform the force reported by the charge, using a Lorentz transformation formula, which is: ## F'= \frac {F}{\gamma}## , when the force is perpendicular to the motion. The force in our frame is the force measured by the charge divided by gamma.
 
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I've read in various places that magnetism can be explained in terms of the effects of special relativity. However, all of the explanations of this only mentioned the case of current flowing in a wire. Can special relativity explain the magnetism of free flowing electrons and other moving charged objects?
Maxwell's equations, which include magnetism, are perfectly adequate to handle currents flowing in a wire, and also moving charges that are not in a wire. Maxwell's equations are also perfectly compatible with special relativity.

It's probably over-simple to say that magnetism "can be explained in terms of special relativity", though it is motiviationally helpful to realize that Maxwell's equations "with magnetism removed" would not compatible with special relativity. I say "motivationally helpful" because there is a certain lack of rigor in what it might mean to "remove magnetism from Maxwell's equations".

If one desires a purely classical (non-quantum) theory that will handle currents in wires, and also handle moving charges that are not in wires, in all generality, then theories one wants to sue are Maxwell's equations and relativistic mechanics.

Hopefully this answers your question? Or are you concerned with the issues that would arise if you tried to use Maxwell's equations with Newtonian mechanics (i.e. without special relativity)?
 
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Consider moving a negative charge in the direction of current of a wire. This will cause the stationary positive charges in the wire to appear to move in the reference frame of the moving negative charge, this will cause length contraction on the positive charges and creates a net positive charge to appear on the wire in the reference frame of the negative charge and will cause the charge to report that it feels a force F.

The motion of the charge has one more effect: We will transform the force reported by the charge, using a Lorentz transformation formula, which is: ## F'= \frac {F}{\gamma}## , when the force is perpendicular to the motion. The force in our frame is the force measured by the charge divided by gamma.

Special relativity's distance transformation has an effect on forces.
Special relativity's force transformation has an effect on forces.
Magnetism includes all special relativity's effects on forces.
 

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