A positive charge of 5,57E-7 C is placed in the points (x,y,z): (Fig._2)

(0,4,0,4,0,1),  (0,6,0,4,0,1),  (0,6,0,6,0,1),  (0,4,0,6,0,1), 
(0,4,0,4,0,9),  (0,6,0,4,0,9),  (0,6,0,6,0,9),  (0,4,0,6,0,9)

In the program dt=1.59E-12

A (solitary) sphere with diameter 10 cm and a voltage of +1kV (to infinity) has a charge of: qi= 1000/(9E9)* 0.05 = 5.56E-09 Coulomb.
100 kV -> 5.56E-7  Coulomb. .

Bfield: (variable)
xc:=0.5*s/400; {if s=400 then xc=0.5 mtr}
yc:=0.5*s/400;
zc:=0.5*s/400;
r1:=sqrt( sqr(xc-x) + sqr(yc-y));
r2:=sqrt( sqr(zc-z) ); {abs value}
{if r1=0,5s and r2=0, then Bfield=1.75*B at the side and 1B in the centre, if r1=0 and r2=0,5s then Bfield=1.75*B in Gauss}
Bh:=B+(rcxy*r1+rcz*r2)*B;Bfield:=Bh;

B(0.5,0.5,0.5) = 100 gauss
B(0.5, 0,5, 0.0) = 2 T (20000 gauss)
B(0.5, 0,5, 1.0) = 2 T
B(0.5,0.0, 0.5) = 2 T
B(0.0,0.0, 0.0) = 4.8 T
Bfield is in the centre weak and near the sides/top/bottom stronger.

50 H+ ions were generated:
hydrogen[i].x:=0.5 + ( - 0.5 + random) /50; {so they start not in exactly the same point}
hydrogen[i].y:=0.5 + ( - 0.5 + random) /50;
hydrogen[i].z:=0.25 + ( - 0.5 + random) /5;

50 B+ ions were generated:
boron[i].x:=0.5 + ( - 0.5 + random) /50;
boron[i].y:=0.5 + ( - 0.5 + random) /50;
boron[i].z:=0.75+ ( - 0.5 + random) /5;

Screenshots\Exp 11.1 screenshot.jpg

After 2,02E-5 sec a few ions have escaped, see screenshot. (Bfield in the centre is only 100 gauss)

A positive charge of 5,57E-7 C is placed in the points (x,y,z): (Fig._2)
(0,4,0,4,0,1),  (0,6,0,4,0,1),  (0,6,0,6,0,1),  (0,4,0,6,0,1), 
(0,4,0,4,0,9),  (0,6,0,4,0,9),  (0,6,0,6,0,9),  (0,4,0,6,0,9)

In the program dt=1.59E-10 (a lot faster, but the simulation is less precise)

Bfield: (variable, in the centre less; above, under and near the sides stronger)
xc:=0.5*s/400; {if s=400 then xc=0.5 mtr}
yc:=0.5*s/400;
zc:=0.5*s/400;
r1:=sqrt( sqr(xc-x) + sqr(yc-y));
r2:=sqrt( sqr(zc-z) ); {abs value}
Bh:=B+(rcxy*r1+rcz*r2)*B;

B(0.5,0.5,0.5) = 1000 Gauss
B(0.5, 0,5, 0.0) = 20 T
B(0.5, 0,5, 1.0) = 20 T
B(0.5,0.0, 0.5) = 20 T
B(0.0,0.0, 0.0) = 48 T

50 H+ ions were generated:
hydrogen[i].x:=0.5 + ( - 0.5 + random) /50; {so they start not in exactly the same point}
hydrogen[i].y:=0.5 + ( - 0.5 + random) /50;
hydrogen[i].z:=0.25 + ( - 0.5 + random) /5;

50 B+ ions were generated:
boron[i].x:=0.5 + ( - 0.5 + random) /50;
boron[i].y:=0.5 + ( - 0.5 + random) /50;
boron[i].z:=0.75+ ( - 0.5 + random) /5;

Screenshots\Exp 11.2 screenshot.jpg  After 6,21E-5 sec all ions are still confined.

1,25 T = strength of a modern neodymium–iron–boron (Nd₂Fe₁₄B) rare earth magnet
36 T = Strongest continuous magnetic field produced by non-superconductive resistive magnet.
See: https://en.wikipedia.org/wiki/Orders_of_magnitude_(magnetic_field)#cite_note-14
The applied magnetic field is quite strong.... (could be difficult to realize in the real world..)

Screenshots\Exp 11.2 screenshot 8.09E-5 s.jpg  All ions are still confined. But 48 T is a very strong B-field and difficult to achieve.

A positive charge of 5,57E-7 C is placed in the points (x,y,z): (Fig._2)    (5,57E-7 C = 200 kV on a sphere with diameter 5 cm)

(0,4, 0,4 , 0,1),  (0,6, 0,4, 0,1),  (0,6, 0,6  ,0,1),  (0,4, 0,6 ,0,1), 
(0,4, 0,4, 0,9),  (0,6, 0,4, 0,9),  (0,6, 0,6, 0,9),  (0,4, 0,6, 0,9)
(0,5,  0 , 0,5    ), (0,5, 1 , 0,5), ( 1, 0,5, 0,5), (0, 0,5 , 0,5)
(4 above, 4 under, and 1 in the middle of each vertical side)

In the program dt=1.59E-11

Bfield: vertical constant 1 T

50 H+ ions were generated:
hydrogen[i].x:=0.5 + ( - 0.5 + random) /50; {so they start not in exactly the same point}
hydrogen[i].y:=0.5 + ( - 0.5 + random) /50;
hydrogen[i].z:=0.25 + ( - 0.5 + random) /5;
ve:=1000000 =1E6 m/s;
hydrogen[i].vx:=( - 0.5 + random)*ve;{they have some initial horizontal speed }
hydrogen[i].vy:= ( - 0.5 + random)*ve;
hydrogen[i].vz:=0;

50 B+ ions were generated:
boron[i].x:=0.5 + ( - 0.5 + random) /50;
boron[i].y:=0.5 + ( - 0.5 + random) /50;
boron[i].z:=0.75+ ( - 0.5 + random) /5;
 ve:=1000000; (1E6)
boron[i].vx:=( - 0.5 + random)*ve; {they have some initial horizontal speed }
boron[i].vy:= ( - 0.5 + random)*ve;
boron[i].vz:=0;

Exp 11.3  screenshot 1.92 E-5 s.jpg (one ion did escape)
Exp 11.3  screenshot 6.13 E-5 s.jpg  (two ions did escape, although the escape trail is not seen)
Exp 11.3  screenshot 2.1 E-4 s.jpg
Exp. 11  50 H+ 50 B+ 1 T constant 12 point charges 2E-4 s.mp4

Remarks:

The light H+ ions seem to be quite well confined by the magnetic field (during the short time lapse of the simulation)
The heavier B+ ions (11 x heavier than the H+  ions) are spreading out a bit, and some of them escaped (suppose it were B+ ions that escaped; cannot determine this).

The ions seem not to interact with each other, but their dimensions, their amount and their charge is relavively very small.
When their charge is increased (10000..0x) , then they do interact (collide etc.), so the formulas in the program seem to be correct.

 

(Fig._2)

The same as exp. 11.3, with Bfield = 1 T (constant, vertical), but the ions did not have an initial speed.

dt=1.59E-11

hydrogen[i].vx:=0; {initial speed}
hydrogen[i].vy:=0;
hydrogen[i].vz:=0;

hydrogen[i].x:=0.5 + ( - 0.5 + random) /50; {so they start not in exactly the same point}
hydrogen[i].y:=0.5 + ( - 0.5 + random) /50;
hydrogen[i].z:=0.25 + ( - 0.5 + random) /5;

boron[i].vx:=0; {intitial speed}
boron[i].vy:=0;
boron[i].vz:=0;
boron[i].x:=0.5 + ( - 0.5 + random) /50; {so they start not in exactly the same point}
boron[i].y:=0.5 + ( - 0.5 + random) /50;
boron[i].z:=0.75 + ( - 0.5 + random) /5;

Exp 11.4 screenshot 1.9E-4 s 1 T.jpg

Exp.11.4 2E-4 s.mp4

The H+ and B+ ions did not escape.

 (Fig._2)

dt=1.59E-11

Bfield: (variable)
xc:=0.5*s/400; {if s=400 then xc=0.5 mtr}
yc:=0.5*s/400;
zc:=0.5*s/400;
r1:=sqrt( sqr(xc-x) + sqr(yc-y));
r2:=sqrt( sqr(zc-z) ); {abs value}
{if r1=0,5s and r2=0, then Bfield=1.75*B at the side and 1B in the centre, if r1=0 and r2=0,5s then Bfield=1.75*B in Gauss}
Bh:=B+(rcxy*r1+rcz*r2)*B;Bfield:=Bh;

B(0.5,0.5,0.5) = 100 Gauss
B(0.5, 0,5, 0.0) = 2 T (20000 Gauss)
B(0.5, 0,5, 1.0) = 2 T
B(0.5,0.0, 0.5) = 2 T
B(0.0,0.0, 0.0) = 4.8 T
Bfield is in the centre weak and near the sides/top/bottom stronger.

ve:=1000000 = 1E6 m/s
hydrogen[i].vx:=0 + ( - 0.5 + random)*ve; {they have an initial horizontal random speed}
hydrogen[i].vy:=0 + ( - 0.5 + random)*ve;
hydrogen[i].vz:=0;
hydrogen[i].x:=0.5 + ( - 0.5 + random) /5; {so they start not in exactly the same point}
hydrogen[i].y:=0.5 + ( - 0.5 + random) /5;
hydrogen[i].z:=0.5 + ( - 0.5 + random) /1.5; {initial vertical position quite spreaded}

boron[i].vx:=0;//( - 0.5 + random)*ve; {they have an initial horizontal random speed}
boron[i].vy:=0;// ( - 0.5 + random)*ve;
boron[i].vz:=0;
boron[i].x:=0.5 + ( - 0.5 + random) /5;
boron[i].y:=0.5 + ( - 0.5 + random) /5;
boron[i].z:=0.5 + ( - 0.5 + random) /1.5; {initial vertical position quite spreaded}

Exp 11.5 screenshot 3.67E-6  s var B.jpg  

In this time period no ions did escape.

The same experiment, but with a constant Bfield of 2 T.

Exp 11.5 screenshot 1.56E-6  s  constant B 2 Tesla.jpg

There seems to be not a lot of difference in applying a variable B-field or a constant B-field.
In the real world I suppose it will be easier to apply a constant magnetic field..

The same experiment, with a constant Bfield of 2 T, but with 300 H+ and 300 B+ ions.

Exp 11.5 screenshot 1.59E-11 s  B  2 T 300 H+ and 300 B+ ions.jpg

Exp 11.5 screenshot 1.59E-11 s  B  2 T 300 H+ and 300 B+ ions with explanation.jpg

No ions did escape in this time period.

With a constant magnetic field of about 1 or 2 T (should be possible to realize..), eight positive charges placed up and down (corresponding to four  round conductors with a diameter of 10 cm and a voltage of 100 kV, also possible to realize?) and a same positive charge placed in each of the sides, the positive ions are confined in the simulation program (applying Coulomb force and Biot Savart, non-relativistic); at least during the (short) time period of the simulation.

The magnetic field is important: if it is decreased the ions escape away to the sides.

No interactions (colisions) between the ions are observed. The reason of this  is maybe because the ions are relatively very small and there are only a very few (in reality there would be millions..). When the charge of the ions is increased a 10000000 times, then yes interactions between them are observed. I let one H+ and one B+ ion collide with each other. If the charge is only increased 100000 times, then they did not collide (dt= 1.59E-14). With dt=1.59E-15 and  100000 times more charge they do collide.

The total energy ( = kinetic energy of all particles + potential energy of all particles ) is calculated.

dt=1E-10 s

Bfield:  1 tesla (constant)

ve:=1000000= 1E6  m/s;

hydrogen[i].vx:=0 + ( - 0.5 + random)*ve; {they have an initial horizontal random speed}
hydrogen[i].vy:=0 + ( - 0.5 + random)*ve;
hydrogen[i].vz:=0 + ( - 0.5 + random)*ve;
hydrogen[i].x:=0.5 + ( - 0.5 + random) /5; {so they start not in exactly the same point}
hydrogen[i].y:=0.5 + ( - 0.5 + random) /5;
hydrogen[i].z:=0.5 + ( - 0.5 + random) /1.5; {initial vertical position spreaded}

boron[i].vx:= (- 0.5 + random)*ve; {they have an initial horizontal random speed}
boron[i].vy:=( - 0.5 + random)*ve;
boron[i].vz:=( - 0.5 + random)*ve;
boron[i].x:=0.5 + ( - 0.5 + random) /5;
boron[i].y:=0.5 + ( - 0.5 + random) /5;
boron[i].z:=0.5 + ( - 0.5 + random) /1.5; {initial vertical position spreaded}

The voltage of the top and bottom charge is 200 kV, so the positive charge of each point charge is:  5.56E-7 Q (if each point charge should be a sphere wit diameter 10 cm).
There are four point charges above and four point charges under.
In the sides there are no charges.

See also experiment 9

Because there are now also fixed point charges, the potential energy of the moving particles relative to this point charges is also calculated.

We started a experiment with 200 H+ and 200 B+ ions.

Exp 11.6  screenshot 3.8 E-5  s B 1 T constant.jpg

The total energy (= potential+kinetic energy) stayed constant: 6,0239668..  E-12 J. 
This should be so; it´s an indication that the formulas in the simulation program are correct.

Changed dt=1E-9 s. The simulation is less precise, but faster.
At elapsed time = 0,000168 s the the potential energy stayed constant: 6,023966...  E-12 J. 
At elapsed time = 0,000383 s the the potential energy stayed constant: 6,023966...  E-12 J. 

Changed dt=1E-8 s.
At elapsed time = 0,000537 3 s the potential energy: 6,0240..  E-12 J.   (a very tiny change).
One ion escaped.

Changed dt=1E-7 s.
At elapsed time = 0,00204 3 s the potential energy: 1,738..  E-11 J.  .
This dt is too big: the simulation is not anylonger precise.
More ions escaped.

Changed dt=1E-6 s.
All ions escaped.

Note: with a speed of 1E6 m/s and a height of 1 mtr of the simulation space (cube) ,  a particle will travel in  1/(1E6) = 1E-6 s from one side to another side. So it is obvious that we cannot take dt=1E-6 s, and even dt=1E-7 is quite big.

dt:=1E-9 s
ve:=1000000 = 1E6 m/s;

boron[i].m:=mb;
boron[i].vx:=0 + ( - 0.5 + random)*ve;
boron[i].vy:=0 + ( - 0.5 + random)*ve;
boron[i].vz:=0 + ( - 0.5 + random)*ve;
boron[i].q:=qe;
boron[i].x:=0.5 + ( - 0.5 + random) /5;
boron[i].y:=0.5 + ( - 0.5 + random) /5;
boron[i].z:= 0.5+ ( - 0.5 + random) /1.50;

idem for hydrogen

B=1,4 tesla (constant)
Top and bottom voltage = 150 V
Sides voltage = 0
time= 0,00408 s
total energy = 1,79871 +/- 0,000002 J
20 H+ and 20 B+
All ions confined

change dt=1E-8 -> total energy = 1,798 +/- 0.0002, all ions still confined after 0.0044 s

change dt=1E-7 -> total energy = 1,63008 +/- 0.00002, all ions still confined after 0.0076 s

change dt=1E-6 -> total energy = 1,6389 , all ions fly away rapidly