Progress

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 f_a^< times f_a^> (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 small for speeds of u and v smaller than, say, 0.01c. If we set, for example, u = v = 0.01c and γ = 90°, δ = 30° and θ = 45°, we obtain for the classical solution, rounded to the fourth digit:

0.9860a₂₁² – 0.0001a₂₁a₃ + 0.9859a₃²

For my own solution, we get a rather similar result:

0.9859a₂₁² – 0.0001a₂₁a₃ + 0.9859a₃²

If the speed of u and v is 0.9c, on the other hand, we obtain for the classical result:

1.1843a₂₁² – 1.6482a₂₁a₃ + 1.1442a₃²

And for my own solution:

0.5861a₂₁² – 0.5987a₂₁a₃ + 0.7125a₃²

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.