Could someone me get my mind around this paradox

[aceofspades]

Okay... I am pretty sure this isn't a paradox, so my
reasoning must be flawed somewhere.

Could someone tell me where I am going wrong?

Consider 2 electrons, traveling in straight lines,
parallel to each other, at the same speed v, in
the same direction. An observer in a stationary
reference frame will see them each produce a magnetic
field, and so will notice them being drawn towards
each other.
(Analagous with the force between 2 parallel current carrying wires)

Now consider the case where the observer moves
with the electrons, within their frame of reference.
To him, the electrons will appear stationary, and so
he will only see an electric field between them.
If this was the case, he should see the electrons being
repelled from each other.

Obviously the physics must be the same, irrespective of the
frame of reference, so where have I gone wrong?

Thanks a lot! :tongue:

 

[pervect]

Take a look at

http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_15.pdf

and

http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_14.pdf

The electric field of a moving charged particle is definitely not spherically symmetric, it's given by the formula in the second URL above. (It turns out that the E-field of a moving charge does always point radially outwards, only the scale factor changes with angle).

Thus the observer watching the static particles sees them repelling each other. The observer watching the moving particles sees them repelling each other more strongly than the first observer, but also sees an additional attractive magnetic force that first observer does not see.

I believe one also has to take time dilation into account as well (or use 4-forces, which are invariant for all observers) to make the equations balance.

 

[zefram_c]

I believe one also has to take time dilation into account as well (or use 4-forces, which are invariant for all observers) to make the equations balance.

Indeed one does. The discussion of this problem appears in most undergraduate level books on E&M (i.e. Griffiths or Jackson)

 

[yogi]

aceofspades - The charged particles will repell one another in all frames - the electric force and the magnetic force are only equal when the two electrons are moving at velocity c relative to the frame in which they are observed - but since they cannot move at the velocity of light, the electric force will always predominate.