Magnetic Forces: Part 2 - Parallel Motion via 4-Potential

Magnetic Forces: Part I, motion parallel to current (via the 4-potential)

We will now derive the previous result using the 4-potential.  This proves to be much simpler algebraically, and after the previous derivation we can see how the 4-potential calculation reflects the physics revealed in the previous derivation.  The problem set up is exactly the same as in the last post.

This time we calculate the 4-vector potential for both the stationary positive charges and the moving negative charges.  In the observers frame we have:

where X is the length of the wire, assumed very long compared to x.  Since the positive and negative charge densities are equal in this frame, this can be written

Where

This potential transforms as a 4-vector, so in the test charge frame the Lorentz transform gives

The test charge will now respond to the gradient of these potentials as derived previously, resulting in the force:

The test charge is not moving towards or away from the wire in this case, so the time derivatives are all zero and thus the time component of the force (the rate of change of energy) is zero.  Likewise the time derivative of the vector potential in the spatial term is also zero.  This leaves the gradient term:

The only direction with a gradient is towards/away from the wire, which is the x direction, so we get:

Transforming this to the observers frame gives (where r is the radius from the wire)

Which as before can be related to the currents and normal velocities

This is the same result obtained by the physical argument in the last section.  This can also be written in the observers frame as:

This result can be generalized for any motion of a test charge parallel to the vector potential, or to a component of the vector potential:

Note that this is not the complete magnetic force yet.  We still need to calculate the force for a charge moving towards or away from the wire.  We will do that in the next post.