Towards explaining the forces between accelerated charges

Having provided a new explanation for the forces between charges moving at constant velocity, I thought that perhaps somebody else would take up the case of accelerated charges. But it seems that has not happened. So, I have had a go.

I have been working on 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. The retarded time is the present time minus the time it took for the effect of a and u to arrive at q₂.

I have not used the concept of the magnetic field as it is an unexplained (and I would say unphysical) entity in classical physics, nor have I used the assumption of a uniform two-way speed of light in any inertial frame of reference made in special relativity, which also calls for an explanation.

I have tried to construct the magnitude of the force on q₂ as measured in the frame moving at v. This is because that is the only force which is unambiguously well defined. The direction of the force, on the other hand, can be given in the coordinate system in which q₂ moves at v. This is assumed to be a coordinate system in which electric conditions around charges at rest are the same in every direction.

Let me now first consider the classical result. To begin with, look at this image:

The position of q₁ in this image is the position where q₁ was when a signal was sent out from it at speed c to reach q₂ at the present time. The position of q₁* in this image is the position where q₁ would be at the present time if it had continued at the constant velocity u which it had at the time when it had the acceleration [a]. Without loss of generality, the position of q₁* is in the xy-plane. The vector n is the unit vector in the direction from q₁ at the retarded time to q₂ at the present time. The green circle is the circle around the position of q₁ that goes through q₂ and the straight line through q₁ and q₁*.

It is necessary to consider a coordinate system like the one shown to make the relativistic transformation into the frame moving at v as simple as possible. In addition to the angles shown, we can define an angle α as α: =  (). Then, as per my article on the case of charges moving at constant velocities (copyright: Physics Essays Publications), plus associated calculations, the following relations apply:

and

 

It is thus clear that the only independent angles in this diagram are the angles γ, δ, and θ. We can define the acceleration at the retarded time as:

  

Remembering that

and

and if we define

 

and

then the classical formula for the force on q₂, resulting from the acceleration of q₁ in the frame of reference shown in the figure, gives us (see for example Rosser, 1997):

Classically, the force on q₂ resulting from the acceleration of q₁ as measured in the rest frame of q₂, in other words taking account of electric and magnetic fields and the transformations of special relativity, is as follows:

As I said before, I have chosen this quite lengthy coordinate formulation because it is the only way in which I can hope to deduce the force from first principles. Let me now consider the strength of the force. The magnitude of the vector sum is the square root of the following term:

I have marked some terms in red: they are what I have calculated so far, with a method similar to the one I used for uniform velocities, as a partial solution for the force due to the acceleration of q₁. I do not think it can just be a coincidence that my partial solution yields a subset of the terms of the classical solution. But before I consider this further, I should set out the method used to arrive at my partial solution. In the next few posts.