The strength of the force an accelerated point charge exerts on another point charge – 2

I have made a bit more progress on this question, and this post will just serve to pull everything together that I know.

As I’ve said before, my intention is to derive the force an accelerated point charge q₁ exerts on another point charge q₂ without any reference to magnetism or special relativity. So far, I have looked into the magnitude of the force on q₂ as measured in the rest frame of q₂. 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 formula for the magnitude of the force is:

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

The frequency factor describes the frequency of the interaction compared to the frequency when both q₁ and q₂ are at rest. In the two equations above, if the acceleration at the retarded time is

  

and if we define

then

is the component of [a] that is vertical to r₁₂*, and

The question that still needs to be answered is the origin of the expressions

and

in the formula for the frequency factor, and of

in the initial denominator.

The following image may be helpful:

The vectors  and  and the length d are all explained in a previous post. We can define a vector

We can then establish a plane equation for a plane that is perpendicular to [r₁₂] and, at the time t = 0, at a distance d = d₁ from q₂, which is located at (0, 0, 0):

At t = 0, the plane can be visualised as follows:

The black dot lies in the plane, which moves at the speed  towards (0, 0, 0). We can now set

and set t = t₁ in the plane equation to calculate at what time t₁ the plane meets q₂, which moves at v along the x-axis.

We can then determine

where t₀ is the corresponding time for u=v=0.

We can set up a similar plane equation for the other solution of d = d₂ (see this previous post):

We can again set

and set t = t₂ in the plane equation to calculate at what time t₂ the plane meets q₂, which moves at v along the x-axis. We can then determine

where, again, t₀ is the corresponding time for u=v=0. From this, (2) follows. We can also begin to see the reason for (3) and (4), even if the explanation is not complete since I cannot at present give full explanations for the plane equations (6) and (8).

I have also yet to give a fuller explanation for the term sin²β in the denominator of (1). This term also appears in the denominator of my version of the equation describing the forces between charges moving at constant velocities, where it is introduced in section ‘IV. THE ANGLE FACTOR’. I think it may have the same origin in (1).

Equation (1) turns out to be very similar to the classical solution, but it is not identical. As can be seen in detail here, 35 terms of the expression under the square root are the same in both versions, but a few terms with exponents of c ≥ 4 are not the same. In more detail, two terms with ‘the exponent of c‘ = 4, two terms with ‘the exponent of c‘ = 5, and five terms with ‘the exponent of c‘ = 6 are not the same. That is a total of 9 terms. Provided all my calculations are correct.

I will now try to establish the direction in which the force acts according to my solution, and compare it with the classical result. I do not anticipate complete equivalence.