Exploring the Effects of Length Contraction on Electromagnetic Forces

[djsourabh]

I was trying to understand how the electric & magnetic forces are related.
I was going through the equations & the simple explanation of relativistic magnetic field given everywhere. Which goes like this...
There is a current carrying conductor & a charge capable of moving outside it. The current is set up (velocity of electrons =v). Then the charge which is outside the wire moves with the same velocity.
They say that due to length contraction, the force in charge's frame is electric force. (Charge density of positive charges increases & hence the net electric force).

Suppose the charge is at rest & there's no current in the wire, will there be any electromagnetic force?
Of course not.
Now the current is set up..charge is stationary wrt to positive charges in the wire, does the charge experience a force?,well no..

But the electrons are moving.
Their length has to contract.
Giving rise to higher charge density & net electric force on the outside charge..but the charge doesn't experience a force...
My question is where does this force go??

 

[Ibix]

But the electrons are moving.
Their length has to contract.

No (and note that it's the space between the electrons that's the issue here, not the electrons themselves). There's nothing constraining the electrons to be a fixed distance apart in their rest frame (unlike the protons). So if we (crudely) model the wire initially as a line of electron/proton pairs separated by a distance $l$, then start up the current then in the rest frame of the wire the electrons are still $l$ apart, but in their own rest frame they are now $\gamma l$ apart.

This might look like it violates charge conservation, but if you take into account that a current needs a loop to flow then take into account the other side of the loop it all works out. The argument is usually due to Purcell - if you search for Purcell and in this forum there are other threads on this topic, including diagrams that may be helpful.

By the way, for future reference I think this thread should be an I thread not an A thread - the latter means you expect postgraduate level maths to be thrown at you, and I don't think you do.

 

[Nugatory]

By the way, for future reference I think this thread should be an I thread not an A thread - the latter means you expect postgraduate level maths to be thrown at you, and I don't think you do.

Fixed.

 

[vanhees71]

Have a look at the mechanics section. There I discuss, how one can make and educated guess based on symmetry arguments about the motion of a charged particle in an external electromagnetic field, using the Lagrange formalism of analytical mechanics:

http://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[djsourabh]

No (and note that it's the space between the electrons that's the issue here, not the electrons themselves). There's nothing constraining the electrons to be a fixed distance apart in their rest frame (unlike the protons). So if we (crudely) model the wire initially as a line of electron/proton pairs separated by a distance $l$, then start up the current then in the rest frame of the wire the electrons are still $l$ apart, but in their own rest frame they are now $\gamma l$ apart.

This might look like it violates charge conservation, but if you take into account that a current needs a loop to flow then take into account the other side of the loop it all works out. The argument is usually due to Purcell - if you search for Purcell and in this forum there are other threads on this topic, including diagrams that may be helpful.

By the way, for future reference I think this thread should be an I thread not an A thread - the latter means you expect postgraduate level maths to be thrown at you, and I don't think you do.

Dear Sir,
I couldn't exactly understand your answer. I did not find any satisfactory explanation of Purcell argument that you have mentioned. Will you please direct me to the thread?

 

[djsourabh]

http://physics.weber.edu/schroeder/mrr/MRRtalk.html

please explain l_=l*sqrt
(1-v2/c2)