Particle defined "at rest" compared to a magnetic field?

[Jarfi]

I am trying to understanding magnetism and I've been running into this thought problem

A particle a in a magnetic field B responds with Force F=|q|v x B.

frame S:
The field is B1, caused by a moving charge/s - current at speed v.

the particle is at rest, F=0.

frame S' moves with speed v/2 relative to frame S:
The field is B2=B1/2, and the current here moves at speed v/2 from our new frame of reference.

now, a moves at speed -v/2 relative to S', so the force would be F=-qv/2 x B2.

And so the particle should accelerate towards the current(or away). Frames S and S' disagree with the acceleration of particle a.

 

[Orodruin]

You will not be able to understand how magnetism behaves in different inertial frames without also taking the electric field into account. Electric and magnetic fields mix under Lorentz boosts. The "magnetic" and "electric" forces are just the two parts of the Lorentz invariant electromagnetic force. Hence, there is nothing saying that you should have the same magnetic force in different inertial systems.

 

[Jarfi]

I know the Lorentz Force already, it would not change anything in this example as far as i can tell. With a current carrying conductor the electric field itself is 0 with 0 net charge, the lorentz force is F=qE+qv x B = qv x B.

 

[Orodruin]

No, you have failed to take into account that the electromagnetic field transforms between inertial frames (the electric and magnetic fields are just components of the rank 2 anti-symmetric field tensor). Also, it is the 4-force that has the appropriate transformation properties, i.e., transforms as a 4-vector.

 

[Jarfi]

No, you have failed to take into account that the electromagnetic field transforms between inertial frames (the electric and magnetic fields are just components of the rank 2 anti-symmetric field tensor). Also, it is the 4-force that has the appropriate transformation properties, i.e., transforms as a 4-vector.

Is there any specific field of physics I can read to understand this?

 

[Orodruin]

Is there any specific field of physics I can read to understand this?

Special relativity.

 

[Dale]

I am trying to understanding magnetism and I've been running into this thought problem

A particle a in a magnetic field B responds with Force F=|q|v x B.

frame S:
The field is B1, caused by a moving charge/s - current at speed v.

the particle is at rest, F=0.

frame S' moves with speed v/2 relative to frame S:
The field is B2=B1/2, and the current here moves at speed v/2 from our new frame of reference.

now, a moves at speed -v/2 relative to S', so the force would be F=-qv/2 x B2.

And so the particle should accelerate towards the current(or away). Frames S and S' disagree with the acceleration of particle a.

A very similar thought problem was used by Purcell in his book and can be used as an introduction to both magnetism and relativity. Here is a good reference: http://physics.weber.edu/schroeder/mrr/mrr.html

 

[jtbell]

For the transformation of electric and magnetic fields between two inertial reference frames:

http://farside.ph.utexas.edu/teaching/em/lectures/node123.html

More details, and some alternative versions of the transformation equations, from Feynman: (scroll down to section 26-3)

http://www.feynmanlectures.caltech.edu/II_26.html