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

Let us assume we are in an inertial frame of reference in which light propagates in equal conditions in all directions. To calculate the force exerted by a point charge q₁ on another point charge q₂ due to the acceleration of q₁, we need to consider the acceleration [a] of q₁ at the retarded time. At that time, information about [a] spread out from q₁ in all directions at the speed c. The information reaches q₂ after a time t = |r|/c, where |r| is the distance between q₁ at the retarded time and q₂ at the current time:

The velocity [u] of q₁ at the retarded time and of v of q₂ at the current time, which may be oriented in any direction in space, is also relevant:

So far in this blog, I have calculated the ‘frequency factor’, which describes the frequency at which information about the acceleration [a] of q₁ moving at [u] reaches q₂ moving at v, compared to the frequency when both q₁ and q₂ are at rest (see my last few posts, here, here, here, and here). That was rather hard work, and my result is provisional.

In this post, I will consider the remaining factors that influence the magnitude of the force. First, let us consider the direction of [a] which has an effect on the force on q₂. If [a] is oriented in the direction of r, in other words from q₁ at the retarded time to q₂, then there is no such effect because the shells around q₁ widen and narrow accordingly. This is why q₂ experiences exposure to the same kind of force as when a = (0, 0, 0):

If, on the other hand, [a] is vertical to r, then more ‘charge’ than before enters the immediate vicinity of q₂ and causes a force on q₂:

So far, I have just considered the case of [u] = v = 0. The situation becomes a little bit more complicated when [u] ≠ 0. I have found that in that case I need to consider the part of a that is vertical to n – u/c (some details on n and r* can be found in a previous post):

In that case, the speed at which ‘charge’ passes over q₂ is

It does so for a time

The path in the direction of a_Ʇ to be considered is thus

The effective ‘additional charge’ that is acting on q₂ can now be calculated as follows:

Bearing in mind the frequency factor (f_a^<f_a^>)^0.5 from my previous posts and the factor of sin²β, which is the same as in the result for constant velocities, we obtain:

This result is the same as the classical result apart from some differences in the rather complex expression (f_a^<f_a^>)^0.5, which are particularly relevant at speeds of u and v > 0.01c.

What I have done here is sketching out an explanation of the magnitude of the force on q₂ due to the acceleration of q₁, in the rest frame of q₂, without any reference to magnetism or special relativity. All these results are provisional and will have to be confirmed, especially since I have not yet found any experimental findings for speeds > 0.01c. And then, of course, the direction of the force will have to be established as well. So, there is a lot that remains to be done.

However, I remind my readers that the essential finding regarding special relativity – that equal time coordinates in special relativity do not imply a relationship of simultaneity and that electric length contraction and time dilation are consequences of purely electric (rather than ‘electromagnetic’) forces between charged particles – already follows from the case of constant velocities (detailed calculations can be found here). The full treatment of accelerated charges is the icing on the cake, and you will have to wait a little bit longer to see it completed.