I have thought a bit more about the magnitude of the force due to the acceleration a of a first charge q1 at the retarded time, when it moved at the speed u, on a second charge q2 moving at v. In particular, the speed c in the equation for the moving plane S1 from my previous post seemed suspect to me:
Why should the plane be moving at c rather than a different, modified speed? I tried the speed
This is in fact the speed of the effect of q1 on q2 for the case of charges moving at uniform velocities. What would happen if I used this speed rather than just c? The equation for fa< times fa> (see my previous post) then becomes somewhat similar to the classical result.
To be more precise, here is an overview of the two results – classical and my own – for part of the magnitude of the vector describing the force, given separately for a212, 2a21a3 and a32 (see the first post on explaining the forces between accelerated charges):
The differences between the classical
solution and my own solution are relatively small for speeds of u and v
smaller than, say, 0.1c. If we set, for example, u = v = 0.1c
and γ = 60°, δ = 30° and θ
= 45°, we obtain for the classical solution, rounded to the third digit:
0.874a212 + 0.013a21a3 + 0.871a32
For my own solution, we get a rather
similar result:
0.868a212 + 0.013a21a3 + 0.871a32
If the speed of u and v is
0.9c, on the other hand, we obtain for the classical result:
0.988a212 – 0.634a21a3 + 0.675a32
And for my own solution:
–0.095a212 – 0.020a21a3 + 0.586a32
There are thus some real differences, which I have yet to resolve. I think I will have to consult the literature to see to what extent the classical solution has been confirmed experimentally. Next, however, I will present my explanation of the factor outside the square root term in the classical equation for the magnitude of the force applied by q1 on q2, which is a lot easier to explain than what I have attempted here over the last four blog posts.
