On Friday, January 20, 2023 at 2:43:03 PM UTC-6, Richard Hertz wrote:
> On Friday, January 20, 2023 at 5:24:02 PM UTC-3, Volney wrote:
> > On 1/20/2023 9:38 AM, Richard Hertz wrote:
> <snip>
>
> YOU CAN'T DISMISS THE EFFECT OF ELECTROSTATIC REPULSION, AS IT CAN BE MUCH HIGHER THAN
> THE MOMENTUM OF THE COLLIDING ELECTRON!!!
> Trajectories follow an asymptotic path and DO NOT VERIFY 90".
On Friday, January 20, 2023 at 2:43:03 PM UTC-6, Richard Hertz wrote:
> On Friday, January 20, 2023 at 5:24:02 PM UTC-3, Volney wrote:
> > On 1/20/2023 9:38 AM, Richard Hertz wrote:
> <snip>
>
> YOU CAN'T DISMISS THE EFFECT OF ELECTROSTATIC REPULSION, AS IT CAN BE MUCH HIGHER THAN
> THE MOMENTUM OF THE COLLIDING ELECTRON!!!
> Trajectories follow an asymptotic path and DO NOT VERIFY 90".
This post replaces a deleted post for which I provided an incorrect link
(the second one)
Because of other activities (including taking my wife to see The Lion King
and various personal projects) I haven't had a chance to work on my
simulation. What I -have- done is do a few math calculations to get an idea
of what I may expect when I finally get around to programming this thing.
I'll start by using a simple Euler method for numerically solving the
differential equations. A fundamental rule of computer programming that
I always follow is to avoid premature optimization. I am perfectly capable
of using more sophisticated methods including fourth and sixth-order
Runge-Kutta, and I've done some exploration of symplectic integrators,
but going to those methods prematurely would be a waste of time until
I get the basic algorithm right.
I am presuming that the total acceleration on each electron of an
interacting pair is the sum of acceleration due to a force directed along
the line between the two electrons, a_n, and deceleration of each
electron due to loss of energy by Bremsstrahlung, b_n. In implementing
the Euler method, I calculate the n+1 step from the n step as follows:
https://drive.google.com/file/d/1K5TeRETg3veTs5H34TppdcTShJCGYX_I/view?usp=share_link
I calculate a_n from Coulomb's law, and I calculate b_n using the non-
relativistic Larmor formula. As I've explained before, I am avoiding the
correct relativistic expression because the whole point of this exercise
is to see whether Champion's results can be explained as a result of
non-relativistic effects. Can you check over my math? Thanks!
https://drive.google.com/file/d/14OhPPsV5IR22PpxQ6sY6w5nrSqM5jLMc/view?usp=share_link
I can directly compare a_n versus b_n. It turns out that b_n is usually
much less than a_n except when when the two electrons approach to
within two classical electron radii from each other, i.e. when they actually
collide. When they actually collide, the Coulomb and Larmor accelerations
are comparable.
https://drive.google.com/file/d/1MzNc6ojitWBims3FUULtukVf5xixrKjc/view?usp=share_link
I would expect from these preliminary calculations that the angle between
two electrons in Champion's cloud chamber experiment should deviate
perceptibly from 90 degrees as a result of Bremsstrahlung. That is why the
Akerlof et. al experiment that I presented later is important. Since protons
are -vastly- less subject to the effects of Bremsstrahlung than electrons,
their results -cannot- be "explained away" in the manner than you would
want to "explain away" Champion's results.
It's apparent from these results that I need not follow the "asymptotic path"
of the electrons further than, say, 100-1000 times the classical radius, say
to around 10e-12 m. On the other hand, from the dramatic way that the
accelerations change as the electrons approach to within 2 r_e of each other,
it is evident that I want to use small step sizes, no more than than 0.001 r_e
so that I'll typically be integrating over a million steps. This means a fair
amount of accumulated error in the Euler solution. In other words, I may
wish to go to Runge-Kutta.