Worksheet 5

PHY201, 1999
Donev Aleksandar

Problem IV-Charged Particle Orbit in a Complicated Electro-Magnetic Field

Part i) Cyclotron Motion

The case of a charged particle moving in a uniform magnetic field is a very famous one and known as the cyclotron motion. The orbit of the particle is well known to be a spiral with a radius R and frequency of procession ω determined from the equality of the Lorentz and centripental force and the initial velocity  v:
	m ((v^2)/R) = m (ω^2) R = q v B  = 
            q ω R B
  The code below should not be difficult to understand with your previous experience with Mathematica. Both the electric field and magnetic field are vectors with fixed directions having and angle (π/2) - θ in-between (by choice we set the electric field in the x-z plane. Also, my choice of θ differs by 90 degrees from that in the problem formulation (because of my sloppy reading), with the magnetic field in the z direction.

Remove["Global`*"]
Bfield = B*{0, 0, 1};
Efield = E*{Cos[Theta], 0, Sin[Theta]};
v = {Vx, Vy, Vz};
F = Efield + Cross[v, Bfield]

In this case we solve the equation both analytically with DSolve and numerically with NDSolve. You should be able to understand why the motion is a spiral from the anlytical solution. It is easy to see that ω=1. Does this correspond to the formulas above?

\(\(
DSolve[{\(\(x'\)'\)[t] == F[\([1]\)], \(\(y'\)'\)[t] == F[\([2]\)], 
        \(\(z'\)'\)[t] == F[\([3]\)], x[0] == 0, y[0] == 0, z[0] == 0, 
        \(x'\)[0] == 0, \(y'\)[0] == 1, \(z'\)[0] == 1}, {x[t], y[t], z[t]}, 
      t]\)\)

\(Solution = 
      NDSolve[{\(\(x'\)'\)[t] == F[\([1]\)], \(\(y'\)'\)[t] == F[\([2]\)], 
          \(\(z'\)'\)[t] == F[\([3]\)], x[0] == 0, y[0] == 0, z[0] == 0, 
          \(x'\)[0] == 0, \(y'\)[0] == 1, \(z'\)[0] == 0.1}, {x[t], y[t], 
          z[t]}, {t, 0, 50}, MaxSteps -> 10000]; 

    Sol = 	{x[t], y[t], z[t]} /. Flatten[Solution]; 
	
    ParametricPlot3D[Sol, {t, 0, 25}]; \)

Part ii) Oscillatory Quasi-Static Magnetic Field

The Mathematica code is the same as before, with only a different magnetic field:

Theta = 0;
Bfield = {0, 0, 1 + Cos[0.2*t]};
Efield = 0*{Cos[Theta], 0, Sin[Theta]};
v = {x'[t], y'[t], z'[t]};
F = Efield + Cross[v, Bfield]
Solution = 
        NDSolve[{x''[t] == F[[1]], y''[t] == 
              F[[2]], z''[t] == F[[3]], x[0] == 0, y[0] == 0, 
            z[0] == 0, x'[0] == 0, y'[0] == 1, z'[0] == 0.1}, {x[
              t], y[t], z[t]}, {t, 0, 100}, MaxSteps -> 10000];
\(\(\(\(Sol = 	{x[t], y[t], z[t]} /. 
                Flatten[Solution];\)
	!\)\ ParametricPlot3D[
          Sol, {t, 0, 75}, PlotPoints -> 500];\)\)

It is not too difficult to undestand the interesting graph that we got. When the magnetic field is at its maximum, the cyclotron frequency is largest and radius of gyration is smallest (the small circles in the graph), and when the field is small, its the opposite (the long arcs between the condensed small-circle regions).

Part iii) Perpendicular and Parallel Constant Magnetic Fields

In the first case, we leave the angle θ at zero (meaning perpendicular electric and magnetic field) and just make the amplitude of the electric field unity:

\(Theta = 0; 
Bfield = {0, 0, 1}; 

    Efield = {Cos[Theta], 0, Sin[Theta]}; 

    v = {\(x'\)[t], \(y'\)[t], \(z'\)[t]}; 
F = Efield + Cross[v, Bfield]; 

    Solution = 
      NDSolve[{\(\(x'\)'\)[t] == F[\([1]\)], \(\(y'\)'\)[t] == F[\([2]\)], 
          \(\(z'\)'\)[t] == F[\([3]\)], x[0] == 0, y[0] == 0, z[0] == 0, 
          \(x'\)[0] == 0, \(y'\)[0] == 1, \(z'\)[0] == 0.1}, {x[t], y[t], 
          z[t]}, {t, 0, 100}, MaxSteps -> 10000]; 

    Sol = 	{x[t], y[t], z[t]} /. Flatten[Solution]; 
	
    ParametricPlot3D[Sol, {t, 0, 25}]; \)

The trajectory that you see above is known as the cycloid and is also a well known example. The motion is well behaved and has a drift in the direction perpendicular to both the magnetic and electric field (the y direction). This comes due to the competition between the magnetic and electric field---while the electric field tries to accelerate the charge, the magnetic field bends it backwards.
The case of parallel magnetic and electric field on the other hand is not so well since the electric field will keep accelerating the charge in the z direction, while the magnetic field will have little impact on the motion, less and less so as the charge accelerates. In other words, the path is again a spiral, but with an increasing height of the winding (compare this to the true spiral in part i. To make this more visible on the picture, I reduced the magnitude of the electric field to 0.05:

\(Theta = Pi/2; 
Bfield = {0, 0, 1}; 

    Efield = 0.05*{Cos[Theta], 0, Sin[Theta]}; 

    v = {\(x'\)[t], \(y'\)[t], \(z'\)[t]}; 
F = Efield + Cross[v, Bfield]; 

    Solution = 
      NDSolve[{\(\(x'\)'\)[t] == F[\([1]\)], \(\(y'\)'\)[t] == F[\([2]\)], 
          \(\(z'\)'\)[t] == F[\([3]\)], x[0] == 0, y[0] == 0, z[0] == 0, 
          \(x'\)[0] == 0, \(y'\)[0] == 1, \(z'\)[0] == 0}, {x[t], y[t], 
          z[t]}, {t, 0, 100}, MaxSteps -> 10000]; 

    Sol = 	{x[t], y[t], z[t]} /. Flatten[Solution]; 
	
    ParametricPlot3D[Sol, {t, 0, 20}]; \)