The purpose of this material is to show that for any type of charged particle with a force field Special Relativity requires the existence of a force field associated with the motion that particle in the same way as a magnetic field is associated with motion of an elecric field. Special Relativity requires that there not be any evidence that one observer is at rest and other observers are in motion.

By a charged particle is meant one which can be either of two types which are denoted as positive and negative.

Consider two adjacent streams of particles. One stream is of positive particles traveling at velocity of v to the right. The other is of negative particles traveling at the same speed to the left. The charge per unit length of the positive particle stream at rest is λ₀. The resting charge per unit length for the negative particle stream is −λ₀. At rest the two-particle stream is charge neutral, In motion at the same speed the two streams experience the same Lorentz contraction and the particle densities become λ=λ₀α where

α = 1/(1−(v/c)²)^½

The magnitudes of the charge densities of the two streams are the same but the charges are of opposite sign so the two-streams are still charge neutral,

Now consider a line parallel to the charge streams at a distance from them of r. On that line consider an observer traveling to the right at a velocity u<v. For that observer the apparent velocities of the positive and negative particles are given by:

v₊ = (v−u)/(1−vu/c²)
 
v₋ = (v+u)/(1+vu/c²)

The apparent charge densities change based the apparent velocities. The apparent velocity of the negative charges is higher so the Lorentz contraction for the negative charges is greater than the one for the positive charges, Consequently the charge density for the negative particle stream is great than that for the positive particle stream. Thus the two-particle stream has a net negative charge. The observer traveling at a velocity u along the line separated from the two-particle stream would observe a force.

For Special Relativity to hold there must be force experienced by an observer on the parallel line at a distance r from the two particle stream for u=0. This cannot be from the particle force field since the two-particle stream is particle neutral. It must be a different force; one that is analogous to the magnetic force for the electrical force field.

Conclusion

Special Relativity requires that any charged particle force must have associated with it a magnetism-like force.