How Does Relativity Explain Magnetic Fields from Moving Charges?

[Swapnil]

We know that moving charges produce magnetic fields.

Now, there are two observers in two different intertial frames: observer A is on the ground and observer B is on a train moving at some speed relative to observer A.

Now suppose that there is stationary charge in the train according to observer B. The problem is that to observer A, the charge would be moving and therefore would be producing a magnetic field whereas to observer B the charge would be stationary and would produce no magnetic field!

How does relativity resolves this apparent paradox??

 

[jtbell]

The electric fields as "seen" by the two observers are such that the net effect of the electric and magnetic forces on another charge is the same according to both observers (after taking into account length contraction, time dilation, etc.).

The Lorentz transformation "mixes" electric and magnetic fields in much the same way that it "mixes" distances and time intervals. The components of E and B together form a "4-tensor" that transforms in a standard way under the Lorentz transformation, just like x, y, z and t together form a "4-vector" that transforms in a standard way under the Lorentz transformation.

 

[Swapnil]

The Lorentz transformation "mixes" electric and magnetic fields in much the same way that it "mixes" distances and time intervals. The components of E and B together form a "4-tensor" that transforms in a standard way under the Lorentz transformation, just like x, y, z and t together form a "4-vector" that transforms in a standard way under the Lorentz transformation.

I am not quite sure what you mean here. But its not you its me, I am not so good at Lorentz tranformations and I know nothing about tensors. Oh well.

The electric fields as "seen" by the two observers are such that the net effect of the electric and magnetic forces on another charge is the same according to both observers (after taking into account length contraction, time dilation, etc.).

I kinda see what you mean, but it would be great if you could please give a simple example.

 

[rbj]

I kinda see what you mean, but it would be great if you could please give a simple example.

see this thread:

How does magnetism occur?

 

[quasar987]

The magnetic field might disapear as we switch frame, but length contraction, time dilatation, etc. will make an electric field appear that is such that the combination of the E & B field in both referentials produce the same effect on the charge.

The effect of the E and B fields change as the referential frame of observation changes, but their combined effect is frame-independent.There is no pradox then, because both frame will produce the same physical reality. The B and E field taken separately are not physically palatable; what is however, is how their combination affects charges, and THAT is frame-invariant. So, no paradox.P.S. I've learned about this only last week so don't take my words too seriously. Nevertheless, I believe I have efficiently conveyed the essence of the "paradox" solution.

 

[pervect]

The magnetic field might disapear as we switch frame, but length contraction, time dilatation, etc. will make an electric field appear that is such that the combination of the E & B field in both referentials produce the same effect on the charge.

The effect of the E and B fields change as the referential frame of observation changes, but their combined effect is frame-independent.


There is no pradox then, because both frame will produce the same physical reality. The B and E field taken separately are not physically palatable; what is however, is how their combination affects charges, and THAT is frame-invariant. So, no paradox.


P.S. I've learned about this only last week so don't take my words too seriously. Nevertheless, I believe I have efficiently conveyed the essence of the "paradox" solution.

Yes, you've got the idea. I'm going to give the OP some references for more reading - I don't know what age or level he's at, the first good treatments usually occur at college undergraduate level, though.

At the undergraduate college level try a good E&M textbook (Griffiths, for example)
https://www.amazon.com/gp/product/013805326X/?tag=pfamazon01-20
or possibly Purcell
https://www.amazon.com/gp/product/013805326X/?tag=pfamazon01-20

Online you can try:
http://physics.weber.edu/schroeder/MRR/mrr.html
http://en.wikipedia.org/wiki/Relativistic_electromagnetism
http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_11.pdf

the later link, you need to read _12.pdf, _13.pdf, etc, it's a good set of lecture notes. The wiki article is probably more convenient.

 

[bernhard.rothenstein]

magnetic field

We know that moving charges produce magnetic fields.

Now, there are two observers in two different intertial frames: observer A is on the ground and observer B is on a train moving at some speed relative to observer A.

Now suppose that there is stationary charge in the train according to observer B. The problem is that to observer A, the charge would be moving and therefore would be producing a magnetic field whereas to observer B the charge would be stationary and would produce no magnetic field!

How does relativity resolves this apparent paradox??

I think that the use of the concept of magnetic field is not compulsory. As many authors say it is no more then an electric field in motion. Please have a look at
Physics, abstract
physics/0607048 arxiv
ine ira et studio

 

[quasar987]

I recommend reading the last section of chapter 13 of volume 2 of the Feynman lectures on physics if you have access to them. It is where I got what I wrote from.

 

[Swapnil]

The magnetic field might disapear as we switch frame, but length contraction, time dilatation, etc. will make an electric field appear that is such that the combination of the E & B field in both referentials produce the same effect on the charge. The effect of the E and B fields change as the referential frame of observation changes, but their combined effect is frame-independent. There is no pradox then, because both frame will produce the same physical reality. The B and E field taken separately are not physically palatable; what is however, is how their combination affects charges, and THAT is frame-invariant. So, no paradox.

Thank you quasar987. Very clear and to the point.

At the undergraduate college level try a good E&M textbook (Griffiths, for example)
https://www.amazon.com/gp/product/013805326X/?tag=pfamazon01-20
or possibly Purcell
https://www.amazon.com/gp/product/013805326X/?tag=pfamazon01-20

I think you mistakenly gave me the reference to Griffiths' book in both links.

Online you can try:
http://physics.weber.edu/schroeder/MRR/mrr.html
http://en.wikipedia.org/wiki/Relativistic_electromagnetism
http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_11.pdf

Thank you very much pervect. I am much clear on these topics now. BTW, I loved that University of Florida link. Its has some sweet pdfs.

 

[pmb_phy]

Just as energy is a frame dependent quantity so too are the electric and magnetic fields. Analogously, energy (as measured in an inertial frame) is proportional to the time component of a 4-tensor (the energy-momentum 4-vector) the electric field is proportional to a component of the Faraday tensor as is the magnetic field. The components of the Faraday tensor (aka "EM tensor") transform as a second rank tensor (since that is one it is). Its easy to see this if you think of an infinitely long charged wire in the inertial frame S (where there is zero current) to S' where there is both a magnetic field and an electric field. Its easier to visualize the physics that way.

Pete