10.1 Physics experiment: simulation of moving electrons, hydrogen and boron ions in a magnetic field

Sitemap    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)
. C
alculate the magnetic field and electric field in this position, caused by all other particles and the fixed magnetic field (with Coulomb's law & Biot-Savart 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) >(1E-18) then multiply the speed of each electrons with 0.999 until it is < (1E-18)
IF ( present kinetic energy all electrons)- ( intial kinetic energy all electrons) < (1E-18) then multiply the speed of each electrons with 1.001 until it is >(1E-18)

---

Now we implemented the more accurate Leapfrog integration :

. At t=i take a particle with position (xi,yi,zi) and velocity (vxi, vyi, vzi)
. C
alculate the magnetic field and electric field in this position, caused by all other particles and the fixed magnetic field (with Coulomb's law & Biot-Savart 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 vi+1 = vi + ―(ai + ai+1).Δt  (= leapfrog method)
The
ai (the force F/m) in position i we already had. To calculate ai+1 we  need the velocity vi+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 time-step is constant and    Δt ≤ 2/w .

 Δt = 1.59E-12 s ;  T= 2pm/Bq = 2p*9.1095E-31/(0,1 * 1,6E-19) = 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:=0.5-random)*sp*0.00000000002; (the initial vertical speed of the electrons varies)
electron[i].z:=(0.5*s/400 -0.00001/2+random*0.00001); (the initial vertical position varies )

dt=1.59E-13

Generated 70 electrons with 100 eV. Then slowly changed B-field from 10 G to 1000 G in small steps.
Beam current 0.01 nA.

Screenshot exp. 10.1  4,24 E-6 s
The electrons spread out (but had an initial random speed in the z-direction).

-----------------------

Experiment 10.11:

electron[i].vz:=0;
electron[i].z:=0.5*s/400;
dt=1.59E-12

Generated 50 electrons with 100 eV. Then slowly changed B-field from 10 G to 1000 G in small steps.
Beam current 0.01 nA.

Before introducing H+ and B+ ions, the electrons did not spread out
After introducing H+ and B+ ions, the electrons did spread out, see screen shot.

Screenshot exp. 10.11  5,22 E-5 s.jpg

---------------------------

Experiment 10.12:

electron[i].vz:=0.5-random)*sp*0.00000000002; (the initial vertical speed of the electrons varies)
electron[i].z:=(0.5*s/400 -0.00001/2+random*0.00001); (the initial vertical position varies )
dt=1.59E-12

Generated 100 electrons with 10000 eV. Then slowly changed B-field from 100 gauss to 1000 gauss in small steps.
Beam current 0.01 nA.

Electrons did spread out.

Screenshots exp. 10.12  6 E-6 s.jpg

Then we changed (decreased) the B-field to 10 Gauss:

--------------------

Experiment 10.13:

electron[i].vz:=(0.5-random)*sp*0.00000000002; (the initial vertical speed of the electrons varies)
electron[i].z:=(0.5*s/400 -0.00001/2+random*0.00001); (the initial vertical position varies )
dt=1.59E-12

Generated 200 electrons with 100000 eV.The  The B-field is constant 1000 gauss.
Beam current 0.1 nA.

It seems to be that the electrons did not spread out.

Screenshots exp. 10.13  1.33 E-5 s.jpg

Looking at the vertical forces, itīs evident that the mayority of the electrons experience a force towards the centre.
It seems to be that the magnetic forces that keep the electrons together are stronger than the Coulomb forces that drive the electrons apart.

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:=(0.5-random)*sp*0.00000000002; (the initial vertical speed of the electrons varies)
electron[i].z:=(0.5*s/400 -0.00001/2+random*0.00001); (the initial vertical position varies )
dt=1.59E-12

Generated 200 electrons with 10000 eV.The  The B-field is constant 1000 Gauss.
Beam current 0.1 nA.

It seems to be that the electrons did spread out.

Screenshots exp. 10.14  5.9 E-7 s.jpg

Looking at the vertical forces, itīs evident that the mayority of the electrons experience a force out from the centre.
It seems to be that the magnetic forces that keep the electrons together are weaker than the Coulomb forces that drive the electrons apart.

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:=(0.5-random)*sp*0.00000000002; (the initial vertical speed of the electrons varies)
electron[i].z:=(0.5*s/400 -0.00001/2+random*0.00001); (the initial vertical position varies )
dt=1.59E-14

Generated 300 electrons with 1000 eV. Then slowly changed B-field from 10 gauss to 1000 gauss in small steps.
The B-field in the centre is smaller than at the sides (see the screenshot).
Beam current 0.01 nA.

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 E-8 s.jpg

Looking at the vertical forces, itīs evident that the mayority of the electrons experience a force out from the centre.
It seems to be that the magnetic forces that keep the electrons together are weaker than the Coulomb forces that drive the electrons apart.

--------------------

Experiment 10.16:

electron[i].vz:=(0.5-random)*sp*0.00000000002; (the initial vertical speed of the electrons varies)
electron[i].z:=(0.5*s/400 -0.00001/2+random*0.00001); (the initial vertical position varies )
dt=1.59E-14

Generated 30 electrons with 1000 eV. Then slowly changed B-field from 10 Gauss to 1000 gauss in small steps.
The B-field in the centre is smaller than at the sides (see the screenshot).
Beam current 0.001 nA.

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 E-7 s.jpg

Looking at the vertical forces, itīs evident that the mayority of the electrons experience a force out from the centre.
It seems to be that the magnetic forces that keep the electrons together are weaker than the Coulomb forces that drive the electrons apart.

--------------------

Experiment 10.17:

electron[i].vz:=(0.5-random)*sp*0.00000000002; (the initial vertical speed of the electrons varies)
electron[i].z:=(0.5*s/400 -0.00001/2+random*0.00001); (the initial vertical position varies )
dt=1.59E-12

Generated 30 electrons with 1000 eV. Then slowly changed B-field from 10 gauss to 1000 gauss in small steps.
The B-field in the centre is smaller than at the sides (see the screenshot).
Beam current 0.01 nA.

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 B-field back again to 10 gauss.

Screenshots\Exp. 10.17  2.3 E-4 s.jpg

Looking at the vertical forces, itīs evident that the mayority of the electrons experience a force out from the centre.
It seems to be that the magnetic forces that keep the electrons together are weaker than the Coulomb forces that drive the electrons apart.

------------------------------------------------

Experiment 10.18:

electron[i].vz:=(0.5-random)*sp*0.00000000002; (the initial vertical speed of the electrons varies)
electron[i].z:=(0.5*s/400 -0.00001/2+random*0.00001); (the initial vertical position varies )
dt=1.59E-12

Generated 20 electrons with 1000 eV. Then slowly changed B-field from 10 gauss to 1000 gauss in small steps.
The B-field in the centre is smaller than at the sides (see the screenshot).
Beam current 0.001 nA.
After a while changed the B-field back to 10 gauss (
(instantaneously)

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 E-4 s.jpg

Looking at the vertical forces, itīs evident that the mayority of the electrons experience a force out from the centre.
It seems to be that the magnetic forces that keep the electrons together are weaker than the Coulomb forces that drive the electrons apart.

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.
But the formulas are still not relativist...  Also, the higher the speed of the electrons, the more the energy loss (see  Larmor Formula  )

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

 

 

 

 

First idea (Jan. 2013)
Second idea (Jan. 2013)
Third idea (Jan. 2013)
Fourth idea (3 Febr. 2013)
Fifth idea (6 Febr. 2013)
Sixth idea (8 Febr. 2013)
Seventh idea (16 Jan. 2014)
Eighth idea (17 Jan. 2014)
Ninth idea (20 Jan. 2014)
Tenth idea ( Febr. 2014- 2016)
Computer simulation tenth idea (2014-2016)
Eleventh idea   (2016.. 2017)

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