Computer program to calculate the net force on a conducting tube inside a solenoid carrying alternating current.
The angle Chi is between the smaller circle centre and the arrow head and the angle Psi is between the larger circle centre and the arrow tail. Z is the extension out of the drawing plane.
The larger circle is the outside diameter and it represents the coil and the inner circle is the plasma referred to as the inside diameter. It is shown displaced from the equilibrium position.
The alternating current is carried in a coil making it a current sheet and the inside conductor is assumed to be an ionised gas with most of the current flowing in the surface.
Result of one calculation. (Version 1.3)
Private Sub Command1_Click()
Const pi = 3.1415926
Dim OutsideDiameter, InsideDiameter, Displacement, ThetaDeg, PsiDeg, ChiDeg As Double
Dim Ri, Ro, D, Z, ThetaRad, PsiRad, ChiRad, DeltaTheta, DeltaPsi, DeltaChi As Double
Dim Pressure, Current, Length, Force, deltaZ, TotalForce, Sign As Double
Current = Val(txtCurrent)
quit = False
ProgressBar1.Value = 0
OutsideDiameter = Val(txtOutsideDiameter)
InsideDiameter = Val(txtInsideDiameter)
Displacement = Val(txtDisplacement)
Length = Val(txtLength)
deltaZ = Length / 1000
If InsideDiameter >= OutsideDiameter Then
ret = MsgBox("Inside bigger that outside", vbCritical, "Error")
If Displacement >= (OutsideDiameter - InsideDiameter) / 2 Then
ret = MsgBox("Displacement too large", vbCritical, "Error")
If Displacement > 0 Then Sign = 1 Else Sign = -1
DeltaPsi = pi / 180 * 10
DeltaChi = pi / 180 * 10
For Z = -Length / 2 To Length / 2 Step Length / 1000
For ChiDeg = 0 To 350 Step 10
For PsiDeg = 0 To 350 Step 10
ret = DoEvents()
ChiRad = ChiDeg * pi / 180
PsiRad = PsiDeg * pi / 180
Force = Current ^ 2 * (OutsideDiameter / 2) * DeltaPsi * (InsideDiameter / 2) * DeltaChi * deltaZ
Force = Force * Sin((ChiDeg + PsiDeg - 180) * pi / 180)
Force = Force / ((InsideDiameter / 2 * Sin(ChiRad) + Displacement - OutsideDiameter / 2 * Sin(PsiRad)) ^ 2 + (InsideDiameter / 2 * Cos(PsiRad) - OutsideDiameter / 2 * Cos(PsiRad)) ^ 2 + Z ^ 2)
Force = Force * 4 * pi * 0.0000001 * Sign * -1
Pressure = Pressure + Abs(Force)
TotalForce = TotalForce + Force
If quit Then Exit Sub
If 100 * (Z + Length / 2) / Length Mod 5 = 0 Then
ProgressBar1.Value = 100 * (Z + Length / 2) / Length
Pressure = Pressure / (pi * InsideDiameter * Length)
txtTotalForce = Str$(TotalForce)
txtPressure = Str$(Pressure)
The result of the calculation indicate that the force is towards the centre and increases as the displacement increases. This would indicate that an ionised gas would be contained near the middle line of a solenoid carrying an alternating current. Extending this idea to a torus, indicates that the plasma would be stably contained within.
Do download the above application to install on your machine. The application has been virus checked with the current version of Norton anti-virus. This is version 1.3 and is for test only I do not promise that it calculates the correct answer.