# Deviation of compass in another frame

Consider an inertial frame where an electric current is flowing in upward direction, a magnetic field is created and its direction is determined by the right hand rule.
In the frame of reference of the electric charges which they are at rest, how can the observer in that frame explains the magnetic field that developed in another frame moving relative to his frame? For example, how can he explain a deviation of a compass in the other frame that is moving relative to him?

## Answers and Replies

cosmosmike
If the compass is moving past the electrical charges, the compass sees a current in his frame.

If the compass is moving past the electrical charges, the compass sees a current in his frame.

yes I know. My question, how the observer riding over one of the charge can explain the deviation of the compass?

mattt
The electromagnetic field tensor is the same in both frames (but its coordinates are different, in general, in two different frames).

Mentor
Consider an inertial frame where an electric current is flowing in upward direction, a magnetic field is created and its direction is determined by the right hand rule.
In the frame of reference of the electric charges which they are at rest, how can the observer in that frame explains the magnetic field that developed in another frame moving relative to his frame? For example, how can he explain a deviation of a compass in the other frame that is moving relative to him?
Here is a good reference for how electromagnetic fields transform between inertial frames.
http://farside.ph.utexas.edu/teaching/em/lectures/node123.html

Note, if your wire is uncharged then there is no frame where there is no current. If the electrons are at rest in a frame then the protons are moving. Either way there is a current and therefore a magnetic field.

Note, if your wire is uncharged then there is no frame where there is no current. If the electrons are at rest in a frame then the protons are moving. Either way there is a current and therefore a magnetic field.

so there is a current seen by an observer moving with electrons because of the protons moving in the opposite direction.

How if there is no protons, in other words, how if an electrons gun is emitting electrons, will be still any magnetic field in the frame where electrons are moving? and how the observer in the rest to the moving electrons can explain the magnetic field B?

Now back to the current again. I read the link of transformation equation and I attached a PDF file of 1 page., so please go through it.

By the way, how to attach a picture directly in the text instead of thumbnail.

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• If the magnetic field B in the frame where the current carries a flowing charges equals to.pdf
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How if there is no protons, in other words, how if an electrons gun is emitting electrons, will be still any magnetic field in the frame where electrons are moving? and how the observer in the rest to the moving electrons can explain the magnetic field B?

A frame in which the electrons are moving has a magnetic field precisely because the electrons are moving. A frame co-moving with the electrons, IE a frame where they are stationary, sees no magnetic field, and instead sees an electric charge from all those electrons.

A frame in which the electrons are moving has a magnetic field precisely because the electrons are moving. A frame co-moving with the electrons, IE a frame where they are stationary, sees no magnetic field, and instead sees an electric charge from all those electrons.

Fine, so how the observer comoving with electrons explain the deviation of a compass in another frame which is moving relative to him?

cosmosmike
Does anyone know where I can find a derivation of the magnetic field of a moving point charge from Ampere's Law? (not the Biot Savart law, but directly from Ampere's law).

Fine, so how the observer comoving with electrons explain the deviation of a compass in another frame which is moving relative to him?

Drakkith already answered this. In that frame the electrons are moving and thus produce a magnetic field so a compass at rest in that frame will be deflected. A compass at rest in the comoving electron frame will not be deflected because there is only an electric field in that case.

Does anyone know where I can find a derivation of the magnetic field of a moving point charge from Ampere's Law? (not the Biot Savart law, but directly from Ampere's law).

You need all of Maxwell's equations to get the magnetic field of a moving point charge because this is not a stationary charge configuration. Ampere's law by itself is not enough.

If the charge is moving at uniform velocity in some inertial frame then there is a much easier way to get its magnetic field in this frame: just take the Coloumb field in the charge's rest frame and Lorentz boost to the original frame, thus Lorentz transforming the Coulomb field and obtaining the electric and magnetic fields of the uniformly moving charge in the original frame.

Drakkith already answered this. In that frame the electrons are moving and thus produce a magnetic field so a compass at rest in that frame will be deflected. A compass at rest in the comoving electron frame will not be deflected because there is only an electric field in that case.

Is the physical phenomena explained only in the frame where it happens?
I know that the explanation is sound in the frame of the compass. I am asking about the way an observer riding on the current can explain it?

for example, in the "moving magnet and conductor problem", the same physical phenomena could be explained in both the conductor frame as well as the magnet frame. http://en.wikipedia.org/wiki/Moving_magnet_and_conductor_problem

I am asking about the way an observer riding on the current can explain it?

Yes certainly. What happens to a magnetic dipole moving through a static electric field?

Yes certainly. What happens to a magnetic dipole moving through a static electric field?

I don`t know the answer?

Mentor
so there is a current seen by an observer moving with electrons because of the protons moving in the opposite direction.
Yes.

How if there is no protons, in other words, how if an electrons gun is emitting electrons, will be still any magnetic field in the frame where electrons are moving? and how the observer in the rest to the moving electrons can explain the magnetic field B?
This is a physically different scenario from the first. I would strongly recommend staying with one scenario until you understand that one scenario before jumping to another. You have already gotten some good replies, but I think that it is a mistake to pursue it.

Now back to the current again. I read the link of transformation equation and I attached a PDF file of 1 page., so please go through it.
Equation 1 is incorrect in general. In general r≠r' and I≠I', and therefore B≠B' despite the fact that the EM field follows Maxwell's equations in both frames.

Equation 2 is correct for this scenario. E=0 because the wire is uncharged in its rest frame in this scenario.

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Mentor
Is the physical phenomena explained only in the frame where it happens?
There is no such thing as "the frame where it happened". Every physical phenomenon happens in every frame. Different frames are simply different ways to describe the same "happenings". The explanations may be different, but the phenomena are the same.

I know that the explanation is sound in the frame of the compass. I am asking about the way an observer riding on the current can explain it?
It is unclear what scenario you are asking about. I really wish you had just stuck with one.

All classical EM scenarios in all frames are explained by Maxwell's equations. The fields can be transformed according to the link I sent, and nearby you can also find links to see how the sources transform. And space and time transform according to the Lorentz transform.

There is no such thing as "the frame where it happened". Every physical phenomenon happens in every frame. Different frames are simply different ways to describe the same "happenings". The explanations may be different, but the phenomena are the same.

That is fine. So how the observer riding on the current can explain the force acting on the compass which is moving relative to him? What formula described the force affecting a moving magnet dipole n the field of a stationary static electric field?

Equation 1 is incorrect in general. In general r≠r' and I≠I', and therefore B≠B' despite the fact that the EM field follows Maxwell's equations in both frames.

I understand that r is the perpendicular distance from the wire. So it should not be affected by length contraction.

I for an observer where the compass is at rest depends on Q and t. for an observer co-moving with the charge, he still considers the current moving in the same direction with the same velocity. So the same calculation is still hold for him as in the first observer.

There is no such thing as "the frame where it happened". Every physical phenomenon happens in every frame. Different frames are simply different ways to describe the same "happenings". The explanations may be different, but the phenomena are the same.

I posted a different post under the name "time to train stop" in the SR and GR forum. The discussion went fine until the last page where I still did not find a reply on a thought experiment I proposed. In that one, the decrease in the length of the train when it stops relative to the train observer may cause different physical phenomena as compared with another observer in the platform. So would you please have a look at it and your kind reply is appreciated.

Yes certainly. What happens to a magnetic dipole moving through a static electric field?

So why static then? As long as there will be a current even for the riding observer?

Mentor
That is fine. So how the observer riding on the current can explain the force acting on the compass which is moving relative to him?
OK, I am not sure which situation you are asking about so I will re-state the one I think you are asking about for clarity. I believe that you are asking about the case where there is an uncharged and current-carrying wire at rest relative to a compass whose needle experiences a restoring force if it deviates from a certain position as analyzed in the frame where the electrons in the wire are at rest.

In the electron's rest frame there is a current from the moving protons which produces a magnetic field. There is also a net charge on the wire due to length contraction which produces an electric field. There is a restoring force on the moving needle due both to the electric field and the magnetic field.

What formula described the force affecting a moving magnet dipole n the field of a stationary static electric field?
Maxwell's equations and the Lorentz force, in every frame (assuming no quantum effects are important).

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Mentor
I understand that r is the perpendicular distance from the wire. So it should not be affected by length contraction.
I was speaking in general. The velocity need not be along the wire. It is best to actually transform the space and time coordinates according to the Lorentz transform since if you simply make assumptions you will get them wrong sometimes.

I for an observer where the compass is at rest depends on Q and t. for an observer co-moving with the charge, he still considers the current moving in the same direction with the same velocity. So the same calculation is still hold for him as in the first observer.
Here is how current and charge densities transform in general:
http://farside.ph.utexas.edu/teaching/em/lectures/node116.html

For the specific case of an uncharged wire carrying a current density J in the x direction, transforming to a frame moving at velocity v in the x direction gives:

##J'=\gamma J##
##\rho'=-\beta \gamma J/c##

Note that J' is not equal to J in any frame other than the wire's rest frame.

cosmosmike
Does anyone know of a reference that uses only Maxwell's equations to solve the problem of a moving point charge (non relativistically) and the corresponding Magnetic field? I want a solution that does not involve doing a Lorentz transformation. Beginning to think it cannot be done!

Mentor
Does anyone know of a reference that uses only Maxwell's equations to solve the problem of a moving point charge (non relativistically) and the corresponding Magnetic field? I want a solution that does not involve doing a Lorentz transformation. Beginning to think it cannot be done!
Why would you think that? Google "Lienard Wiechert", the first 7 links all do not involve a Lorentz transformation. Including Wikipedia.

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