The direction of the force on a moving charge caused by the acceleration of another charge

 It is now time to get on to the final bit: the direction of the force due to acceleration exerted by a charge q₁, moving at the velocity [u] and with the acceleration [a] at the retarded time, on a charge q₂ moving at the velocity v at the current time.

The retarded time is the current time minus the time it took for information on q₁, including [u] and [a], to reach q₂ at the current time.

I am considering this question for a frame of reference Σ in which electric conditions are isotropic for charges at rest. The one-way speed of light is thus c in this frame of reference. The following image presents the basic situation:

Here, q₁* is the position q₁ would be in if it had continued at the constant velocity [u] it had at the retarded time.

In a previous post, I have presented my result for the magnitude of the force on q₂ as measured in the rest frame of q₂. This turns out to be the same as the classical result for relatively small velocities u, as in electric currents, and any velocity v. It is also similar to classical results where the speeds of q₁ and q₂ are roughly < 0.1c. When both speeds are larger, there is a growing mismatch with the classical result.

In this post, I would like to determine the direction of the force on q₂ in Σ. Note that the direction of the force in the moving frame is not well defined because I have not introduced any coordinates for that frame. By contrast, the strength of the force on q₂ as measured in the rest frame of q₂, for example by a co-moving spring balance, does not rely on any time or space coordinates.

I would like to determine the direction of the force on q₂ without using the concept of the magnetic field or the results of special relativity. This is the way I have worked so far in this blog, because the magnetic field and special relativity both cry out for an explanation. My hope is to provide such an explanation. I have already presented my results for the case of charges moving at constant velocities, which is sufficient to provide a partial explanation for special relativity. It also explains magnetic effects on charges moving at constant velocities. This post will fit in the last missing piece, which is the direction of the force exerted by a charge q₁, moving at the velocity u and with the acceleration a at the retarded time, on a charge q₂ moving at the velocity v at the current time in Σ.

Let us take a deep breath after these preliminaries, and then proceed.

Let me first note down the classical result for the direction vector of the force on a moving charge q₂ caused by the acceleration of another charge q₁. The direction I am talking about is the direction of the force in the frame of reference moving at v as seen from the stationary frame of reference Σ.

Classically, we can calculate that direction by first determining the direction vector of the force in the frame of reference moving at v:

We can then multiply the x-component of that direction vector by (1-v²/c²)^0.5 to obtain the direction in Σ:

What do I obtain for the direction in my approach, which does not use magnetism or special relativity?

I think it makes sense to adapt the solution for the case of two charges moving at constant velocities (see Equation 9 in this article) to the case of q₁ being accelerated by [a]. For constant velocities, the direction of the force is given by

As is demonstrated here, this direction turns out to be the same as the classical direction.

Now consider the case of q₁ being accelerated by [a]. We have already seen that in this case we have to find the projection of [a] that is perpendicular to r₁₂*, and to project that onto a vector b_Ʇ that is perpendicular to n. Let the unit vector of b_Ʇ be

It then turned out that we had to use n_Ʇ and – n_Ʇ to calculate the magnitude, so I will use these vectors to calculate the direction, too. Similar to the case of constant velocities, I form

If we multiply this by the product of the two denominators, we get

The direction of this vector corresponds to about half the elements of the direction of the classical solution:

Is there something missing from my solution? I do not know. It is perhaps difficult to see, for example, why there should be a large z-component when neither n_Ʇ nor v contain such a component. Be that as it may, I cannot easily see what I should add to my solution, so I will have to leave it as it is.