Experimental setup being simulated

Reference:

http://www.ostfalia.de/export/sites/default/de/pws/turtur/DownloadVerzeichnis/Series-english-5Articles.pdf

On pdf-pages 32-48 of the referenced document, an ‘uncomplicated setup’ is described, and a simulation presented which appears to show increased oscillation amplitudes of two charges on springs for no apparent input cost.

The code has been copy-pasted and converted into a Matlab file called ‘SlowEM_v5p0.m’. The behaviour of this simulation will be presented in the following slides.

The basic code of Prof. Turtur was followed, but some algorithmic adjustments were made:The routine to find the ‘time-of-emission’ of the delayed electric field was rewritten to increase

the algorithmic speedThe steps to update the position of the mass was modified in order to preserve conservation

of spring energy better

These two adjustments were checked against the original code to ensure consistency, and everything is fine.

For example, see the oscillation of just the mass on the spring (no EM interactions) below:

The initial conditions need to be chosen so that the mass never moves at speeds greater than c.

Just start with the spring force

Turn on the electrostatic interaction

The mass is held for a short time at the beginning of the simulation in order for the field to travel between masses, before the mass is released. The speed of propagation has been increased back up to 3e8 ms-1 in order to calculate the absolute radiation reaction value later.

One can obtain both increasing and decreasing oscillations - this reproduces Turtur’s claim.

However, there are two things missing from this (and therefore from Turtur’s) model:The electrostatic field strength is modified by the velocity of the emitterRadiation reaction forces have been neglected

The following slides show the implications of these omissions.

Modify the electrostatic strength to account for velocity of the ‘emitter’

According to http://www.mathpages.com/home/kmath576/kmath576.htm, the electric field of a charge at a distance away from it in a straight line along its direction of motion is modified by a factor (c+v2(ts))/(c-v2(ts)) as compared to when the charge is not moving. ‘v2(ts)’ is the velocity of the other charge at retarded time ts (the time of ‘emission’).

Comparing results of not including this effect (left) to including it (right), we can see that it can make the oscillations increase at a faster rate

Hence this effect does not contradict the claim that the amplitudes may spontaneously increase

Include radiation reaction

The final effect to include is called the ‘Abraham-Lorentz’ forcehttp://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force

It was found numerically that calculating the third derivative of position with time was subject to a great deal of noise and made the algorithm unstable. A much cleaner implementation was made by noting that the motion we are simulating is basically simple harmonic motion which can nearly be described by a sine wave. In this case da/dt=-ω2v where ω is the angular frequency of the periodic motion, ω=sqrt(2*D/m1), where D is the spring constant and m1 is the mass. A factor of 2 is needed due to the way the geometry of the spring is simulated. Such a dependence on velocity clearly shows that this force acts like a viscous damping. It was checked numerically that the magnitude was correct, and it was found to be the case.

In the figure below, the first half of the trace does not include radiation reaction, but the second half of the trace does.

It is clear that what was an increasing oscillation when radiation reaction is omitted becomes very quickly damped when it is included. Playing with the parameters, for example by reducing the oscillation frequency, shows no hint that the radiation reaction force can be reduced enough to prevent the oscillation dying. Indeed from ‘classical’ ideas one would not expect the oscillation to spontaneously increase, and so one may presume that no such parameters can be found.

Turn on radiation reaction here

no radiation reaction here radiation reaction included

Summary

Prof. Turtur’s simple model of two masses on springs which have a delayed EM interaction has been reproduced.

The original model neglects two effects – the change in electric field strength due to the velocity of the ‘emitter’, and the radiation reaction force.

When these effects are neglected, the amplitude of the oscillations of the charged masses on springs may spontaneously increase.

However, it was found that the radiation reaction force produces damped oscillations, with a damping rate many times larger than the rate of increase due to the effect of delayed propagation.

Radiation reaction will undoubtedly be present in any real system, hence the proposition that oscillations may spontaneously increase is not supported.