10.1 Physics experiment: simulation of moving electrons, hydrogen and boron ions in a magnetic field
Computer simulation of idea 10
Experiment 10
In the former experiments we used in the simulation program the following integration method: (See also Computersimulation)
. At t=i take a particle with position
(xi,yi,zi)
and velocity (vxi,
vyi, vzi)
. Calculate the magnetic field and electric field in this
position, caused by all other particles and the fixed magnetic field (with
Coulomb's law
&
BiotSavart Law
).
. Calculate with
the
Lorentz force
the force on this particle and so it's new postion
(xi+1,yi+1,zi+1)
and velocity (vxi+1,
vyi+1, vzi+1).
This seemed to be called the Euler method and is of first order. What happened in the simulation program is that the particles got all the time more speed and "flew out of orbit". To avoid this a procedure was introduced in the program that kept the kinetic energy of all electrons constant by multiplying the speed of the electrons with a factor 0.999 or 1.001 if it deviated too much from the initial kinetic energy:
IF ( present kinetic energy all electrons)
( intial kinetic energy all electrons) >(1E18) then multiply the speed of each
electrons with 0.999 until it is < (1E18)
IF ( present kinetic energy all electrons) ( intial kinetic energy all
electrons) < (1E18) then multiply the speed of each electrons with 1.001 until
it is >(1E18)

Now we implemented the more accurate Leapfrog integration :
. At t=i take a particle with position
(xi,yi,zi)
and velocity (vxi,
vyi, vzi)
. Calculate the magnetic field and electric field in this
position, caused by all other particles and the fixed magnetic field (with
Coulomb's law & BiotSavart Law ).
. Calculate with
the Lorentz force the force
on this particle and so its new postion (xi+1,yi+1,zi+1).
. In this new position (xi+1,yi+1,zi+1)
calculate again the magnetic and electric field.
. Now calculate velocity (vxi+1,
vyi+1, vzi+1)
with v_{i+1} = v_{i
}+ ―(a_{i}
+ a_{i+1}).Δt
(= leapfrog method)
The
a_{i}
(the force F/m) in position i we already had. To calculate
a_{i+1
}we
need the velocity v_{i+1},
just the one we want to calculate.... But we proceed and get three equations
with the three unkowns vxi+1,
vyi+1
and vzi+1
.These three equations we solve with
Cramer's rule and so we obtain the
velocity (vxi+1,
vyi+1, vzi+1).
A desadvantage of this method is that is about two times slower (in each step we calculate now two times the magnetic and electric field, both in position (xi,yi,zi) as in position (xi+1,yi+1,zi+1).
According Wikipedia it is stable for oscillatory motion, as long as the timestep is constant and Δt ≤ 2/w .
Δt = 1.59E12 s ; T= 2pm/Bq = 2p*9.1095E31/(0,1 * 1,6E19) = 3,58 E 10 s.
w = 2p/T = 1,76 E10 s^{1}> 2/w = 1,14 E 10 s
Experiments with Leapfrog but with an error in the program
Testing the simulation program on different computers etc.
Using the Leapfrog method Experiment 10.1: Discovered an error in the Leapfrog equations/variables in the program and corrected this!
electron[i].vz:=0.5random)*sp*0.00000000002; (the
initial vertical speed of the electrons varies) dt=1.59E13
Generated 70 electrons with 100 eV. Then slowly changed
Bfield from 10 G to 1000 G in small steps.
Screenshot exp. 10.1 4,24 E6 s  Experiment 10.11:
electron[i].vz:=0;
Generated 50 electrons with 100 eV. Then slowly changed
Bfield from 10 G to 1000 G in small steps.
Before introducing H+ and B+ ions, the electrons did not spread out Screenshot exp. 10.11 5,22 E5 s.jpg  Experiment 10.12:
electron[i].vz:=0.5random)*sp*0.00000000002; (the
initial vertical speed of the electrons varies)
Generated 100 electrons with 10000 eV. Then slowly changed
Bfield from 100 gauss to 1000 gauss in small steps. Electrons did spread out. Screenshots exp. 10.12 6 E6 s.jpg Then we changed (decreased) the Bfield to 10 Gauss:  Experiment 10.13:
electron[i].vz:=(0.5random)*sp*0.00000000002; (the
initial vertical speed of the electrons varies)
Generated 200 electrons with 100000 eV.The The
Bfield is constant 1000 gauss. It seems to be that the electrons did not spread out. Screenshots exp. 10.13 1.33 E5 s.jpg
Looking at the vertical forces, itīs evident that the
mayority of the electrons experience a force towards the centre. BUT... the speed of the electrons is 1,9 E8 m/s, which is near the speed of light (3 E8 m/s). The formulas in the program are (still) not relativistic!  Experiment 10.14:
electron[i].vz:=(0.5random)*sp*0.00000000002; (the
initial vertical speed of the electrons varies)
Generated 200 electrons with 10000 eV.The The
Bfield is constant 1000 Gauss. It seems to be that the electrons did spread out. Screenshots exp. 10.14 5.9 E7 s.jpg
Looking at the vertical forces, itīs evident that the
mayority of the electrons experience a force out from the centre. The speed of the electrons is 0.,6 E8 m/s. The formulas in the program are (still) not relativistic.  Experiment 10.15:
electron[i].vz:=(0.5random)*sp*0.00000000002; (the
initial vertical speed of the electrons varies)
Generated 300 electrons with 1000 eV. Then slowly
changed Bfield from 10 gauss to 1000 gauss in small steps. The speed of the electrons is 0.2 E8 m/s. The formulas in the program are (still) not relativistic. Screenshots\Exp. 10.15 7.24 E8 s.jpg
Looking at the vertical forces, itīs evident that the
mayority of the electrons experience a force out from the centre.  Experiment 10.16:
electron[i].vz:=(0.5random)*sp*0.00000000002; (the
initial vertical speed of the electrons varies)
Generated 30 electrons with 1000 eV. Then slowly
changed Bfield from 10 Gauss to 1000 gauss in small steps. The speed of the electrons is 0.2 E8 m/s. The formulas in the program are (still) not relativistic. Screenshots\Exp. 10.16 7.15 E7 s.jpg
Looking at the vertical forces, itīs evident that the
mayority of the electrons experience a force out from the centre.  Experiment 10.17:
electron[i].vz:=(0.5random)*sp*0.00000000002; (the
initial vertical speed of the electrons varies)
Generated 30 electrons with 1000 eV. Then slowly
changed Bfield from 10 gauss to 1000 gauss in small steps. The speed of the electrons is 0.2 E8 m/s. The formulas in the program are (still) not relativistic. After 9,7E 5 s changed (instantaneously) the Bfield back again to 10 gauss. Screenshots\Exp. 10.17 2.3 E4 s.jpg
Looking at the vertical forces, itīs evident that the
mayority of the electrons experience a force out from the centre.  Experiment 10.18:
electron[i].vz:=(0.5random)*sp*0.00000000002; (the
initial vertical speed of the electrons varies)
Generated 20 electrons with 1000 eV. Then slowly
changed Bfield from 10 gauss to 1000 gauss in small steps. The speed of the electrons is 0.2 E8 m/s. The formulas in the program are (still) not relativistic. Screenshots\Exp. 10.18 4.5 E4 s.jpg
Looking at the vertical forces, itīs evident that the
mayority of the electrons experience a force out from the centre. Curious that the hydrogen ion above did not bouch back against the boron ions, and that the boron ions did not move.

A conclusion: When the speed of the electrons approaches the speed
of light, the magnetic forces become bigger than the electric forces and
the electrons, moving in a circle, stay together. Let's try a complete new approach, combining electric fields and magnetic fields to accelarate positive H and B ions towards the centre of a vacuum chamber. See Experiment 11 
2016 by Rinze Joustra www.valgetal.com


