Further progress

I have made some more progress in deriving the magnitude of the force caused by the acceleration of a first charge q₁, moving at u and accelerated by a at the retarded time, on a second charge q₂ moving at v at the present time, without any reference to special relativity or magnetism. I have determined this magnitude as measured in the rest frame of q₂.

The latest addition to the formula derived so far comprises just one factor: cos²β. As I will indicate below, this produces essentially the same result as the classical equation if q₁ moves rather slowly, as in an electric current through a wire, and q₂ moves at any speed. It also produces roughly the same result if both q₁ (at the retarded time) and q₂ move more slowly than about 0.1c. If both charges move faster than that (at the retarded time for q₁), then there are some differences between the classical result and my own result.

Let me first repeat the basic image that describes the situation:

Here, q₁ is shown at the retarded time, in other words the time before the current time when information about the velocity and acceleration of q₁ was sent out to q₂ at the speed c to reach q₂ at the current time.

My revised formula for the magnitude of the force is:

with the frequency factor (originally developed here, here, here and here)

In the equation for the magnitude of the force, if the acceleration at the retarded time is

and if we define

then

and

The only difference to the previous result is the addition of the factor of cos²β.

There are still some differences to the classical result, but they are confined to the enumerator of f_a^<f_a^> and to powers of c ≥ c⁴. A precise comparison of the enumerator with the classical result shows the following, bearing in mind that cos²β = 1 – sin²β:

This means the differences between the two solutions are negligible if u is the speed of a current in a wire and v is any speed, and they are also very small for v and u ≤ 0.1c. For example, the result for u = v = 0.1c and γ = 60°, δ = 30° and θ = 45° for both solutions, rounded to the fourth digit, is:

0.8738a₂₁² + 0.0131a₂₁a₃ + 0.8711a₃²

The differences are greater for larger v and u. For example, the results for u = v = 0.9c and γ = 60°, δ = 30° and θ = 45° are:

0.988a₂₁² – 0.634a₂₁a₃ + 0.675a₃²

for the classical solution and

0.491a₂₁² – 0.374a₂₁a₃ + 0.626a₃²

for my own solution.

It may be that there are no experimental findings for such high speeds that would confirm one or the other solution as more correct.

There are a few parameters in my own solution that I have so far explained only very briefly or not at all:

The factors of 

in the denominator before 

I will next try to explain these terms a bit more fully, before I try to deduce the direction of the force.