Parallel and Anti-parallel Currents

This short analysis offers an interpretation of the magnetic force between two parallel conductors without invoking the magnetic field vector, the right hand rule, or the Lorentz force.  It is in no way intended to minimize the importance of these tools for the practical analysis of magnetic effects.  The intent is to give the reader another perspective and some insight into why the magnetic field is now viewed as a modification of the electric field.  It also reinforces the intricate involvement of the speed of light with both electric and magnetic phenomenon.

The magnetic force per unit length between two long straight parallel
conductors is given by the equation:

where is the permeability of space.

Multiplying both numerator and denominator by the permittivity constant,

Recognizing that;

We can write the force per unit length as:

Charged particles interact, or communicate, at the speed of light.  Further, any current is equivalent to a specific linear charge density moving at light speed.  The charge density of a light-speed current, i, is simply;

(Coulombs per second divided by meters per second yields coulombs per meter.)

Therefore, we can write the magnetic force equation in exactly the same form as the electric force equation between the two parallel charge densities that represent the two light-speed currents.

So why do the charge densities lambda-prime-one and lambda-prime-two create a force between the conductors when they are moving parallel to each other at the speed of light?

Let's consider first the case where both charge densities are negative and their velocities are in the same direction.  Before the current started flowing, the conductors were electrically neutral and there was no net force; i.e., all electric forces were in equilibrium.  The repulsive force between the two like charge densities made up part of that equilibrium.

Now that the two charge densities are moving in the same direction at the speed of light, they can no longer communicate - the electric repulsive force has vanished, upsetting the static equilibrium.  The other electric forces that contributed to the static equilibrium remain unchanged and squeeze the two conductors towards each other in the absence of the repulsive force.

Charges moving parallel to each other at the speed of light cannot interact, since a signal would require a speed greater than c.  (With a longitudinal velocity component of c, there's nothing left for a transverse component.)
Further, a round trip is required for any interaction because of Newton's third law - any action has an equal and opposite reaction.  That is, a charge cannot push or pull on another charge without the other charge pushing or pulling on it.  Photons, including virtual photons, are their own anti-particles - they go in both directions.

Actually, (due to time dilation and length contraction), a photon doesn't really go anywhere in space-time.  It is a true singularity.  In the photon's reference frame there is no time delay and no distance traveled between emission and absorption - whether across the galaxy or across the room.

(By the way, the equivalence of a charge density moving at the speed of light does not mean that electrons migrate through the conductor at the speed of light.  e.g., sound waves propagate at the speed of sound without requiring a Mach-one wind.)

Now consider currents in opposite directions.  As an electron is pulled from one end of a wire by a battery, it leaves a small gap in the electron distribution called a hole.  This hole is filled with another electron slightly further from the battery.  The hole can migrate through the wire at the speed of light while individual electrons barely move at all.  (Just as the sequenced flashing lights on a theater marque cause the dark segments to move.)

Anti-particles are defined by their equivalence to normal particles traveling backwards in time.  i.e. an electron charge density traveling through a conductor is indistinguishable from a positron charge density traveling in the opposite direction.
Thus, current flow in opposite directions is exactly equivalent to opposite charge densities flowing in the same direction.  The analysis is identical to the first case:  The static attraction between the opposite charge densities vanishes when the charge densities are moving in the same direction at the speed of light.  The loss of this attractive force causes an imbalance in the static equilibrium and the conductors experience a net repulsive force.