Figures for
Cyclotron Heating
These figures demonstrate the behavior of the functions
y(x) for various values of the driving frequency
and cyclotron frequency, as explained in the notes More on Driving at Resonance, an
examimation of the equation
where y is the independent variable and x is
the independent variable.
For the purposes of plotting, the cyclotron frequency (the
lower-case greek letter "omega" in the notes) was taken to be one
(this may always be done by adjusting the time scale),
and the driving frequency is varied from 0.5 to 1.5. Plots for a
driving frequency of less than one and greater than one are presented
separately for clarity. In both plots, the curve corresponding to
driving at resonance is indicated by the blue crosses.
In the above, it is seen that as the driving frequency is increased to
unity, the response approaches that of driving at resonance.
In the above, it is similarly seen that as the driving frequency is
decreased to
unity, the response approaches that of driving at resonance.
In addition a "phase plot" (we will encounter these in
Chapter 7) of y(x) on the horizontal axis
and the derivative of y(x) with respect to
x on the vertical axis shows that driving at resonance
produces a trajectory which spirals outward, with increasing speed.
For the actual case of cyclotron heating, this figure would indeed
represent the path of an electron in a plasma confined by a magnetic
field, until collisions with the more massive nuclei transfer the
electron kinetic energy and the process repeats. Of course, starting
from rest is a gross idealization for an electron in a plasma.
The Maple worksheet which generated the figures on
this page may be downloaded here.