Earlier I had shown that the electric force and magnetic forces acting between two charges moving through our lab balanced out exactly at the speed of light. And, at zero velocity, only the electric force is active as would be expected. Jones came back with a comment that this behavior is well known according to plasma stability when the particles move at near light speeds. Then Harry found a wikipedia article that described the forces acting between charges that are in motion. His find suggested that the force equations varied as the usual SR formulation which is more or less stated as a product of the variable under consideration and 1 - V^2 / C^2. This same basic relationship shows up in every SR(special relativity) function I recall appearing in either the numerator or denominator.
Being a bit lazy, I wanted to take a short cut to save writing space and time and not to reinvent the wheel. For this reason I took a quick look at the force equation that I derived from parallel currents and its implications. An obvious question is how does my formula vary as the observed speed of the charges falls from light speed to zero. The magnetic field due to the first charged particle is directly proportional to the velocity of that particle. So if I choose a velocity that is exactly one half the velocity of light, I calculate that the new value of the field is exactly one half the magnitude found at the balancing value. This is the expected magnetic field strength as seen by my lab appearing in the path of the second charged particle as it passes by. Now, the second charged particle is also moving at one half the velocity of light along a path parallel to its brother charge. According to standard physics, the magnetic force acting upon a charged body is proportional to the charge multiplied by the magnetic field strength multiplied by the velocity of the charge. The direction is along a line between the two brother charges due to the cross product rules which I will not go into here. So, the field has dropped by a factor of 2, the velocity of the second charge has also dropped by that same factor of 2. The product is thus one forth of the original value. If this reduced magnetic force is subtracted from the original value due to a balance with the electric field we have a remainder of 3/4 of the total force, dominated by the electric field. This result is exactly what is expected according to the original SR premise. That is 1 - (1/2)^2 = 3/4. There has been quite a bit of discussion about time dilation resulting in a paradox, and if you use this same type of reasoning that I have for the two charge case you will come to the conclusion that a pair of charged particles moving along with the lab observer relative to the main pair I am assuming would be subject to the same time dilation effects from their point of view. I realize that this equal dilation observation effect seems difficult to understand, but it clearly does appear real. If you accept that there is no velocity in space that can be used as a reference then there is no choice about that conclusion. Special relativity teaches us that no special consideration can be given to any particular location or velocity in space. When we attempt to match clock readings on board space ships to those of stationary twins, a complication arises due to the history of each clock throughout the time period of interest. The one that remains moving at a steady rate will tick at a constant rate to us as observers in that frame. The one on the space ship must undergo numerous accelerations and direction changes before it passes by at a time dilated velocity. Unless someone takes the opportunity to integrate the total dilation effects throughout the entire history of separation and velocity change, they will not be able to make any solid conclusions regarding the matching of the clocks. I have seen examples of this clock behavior and differences expected in the total elapsed time, but at the time I did not engage the example adequately. Dave
