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.

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.

Towards explaining the forces between accelerated charges – 3

In my previous post, I partially presented the first steps towards explaining the force, resulting from acceleration, exerted by a charge q₁ on a charge q₂ where q₂ moves at the constant velocity v and q₁ moved at the velocity [u] and with the acceleration [a] at the retarded time. Here, I will conclude this explanation. Let me first repeat the basic image that describes the situation.

The goal is to derive the ‘frequency’ with which [a] is applied to q₂, to be able to compare a situation in which both q₁ and q₂ are static and one in which q₁ moved at u at the retarded time, when it had the acceleration [a], and q₂ moves at v. I derived a vector  and a number d_1,2 that will help me to derive this frequency (see my previous post for more details). First, I will consider a plane vertical to  and at a distance d₁ from (0, 0, 0), which is the current location of q₂:

I will now work out in what time t₁ that plane and q₂ meet if the plane moves at c towards (0, 0, 0) and q₂ moves at v along the x-axis. The equation for the plane S₁ is

or

We can now insert

as well as what we previously calculated for and d₁:

We obtain

Hence

Now consider that the equivalent time for a=v=u=0 is

and we obtain the first frequency factor

We now have to perform the same calculation for a time t₂, where we use d₂ and . This is a bit different from the case of uniform velocities because n^> does not feature here. We obtain:

This results in

We finally obtain:

The result corresponds exactly to the red terms in the classical solution specified two posts ago. That is certainly a good beginning. However, even if the classical solution is not quite correct, clearly more work needs to be done to bring my solution closer to the classical result. Not to mention the fact that I have so far only investigated the ‘frequency factor’ and not any other elements of the force that determine its magnitude and its direction. Perhaps I will pursue this in the months and years to come, but maybe there is somebody else out there who can follow this up sooner than I can.