Before anything else, we need to change to the stored notebook's directory, which contains input files for the PIC simulator.
In this project, we are going to look at the dispersion relation for electromagnetic waves.
The dispersion relation, ω(k), tells us the natural frequencies of oscillations for these waves, and the information contained in this function about the relationship between ω and k can be used to determine the phase and group velocities of these waves. [There will be a subsequent notebook on wave velocities]
For transverse waves have:
∇⋅E⃗=0 -- transverse waves
Te=0 -- cold plasma
B⃗0=0 -- unmagnetized
From Maxwell's Equations we have:
Taking the curl of the first equation and substituting into it the second equation, we get:
Since −∇×∇×A⃗=∇2A⃗−∇(∇⋅A⃗), we have
For transverse waves, ∇⋅E1⃗=0, so
Where in the third line we used Euler's equations. Plugging in our definitions for Ωp and ωp and moving everything to the left hand side, we finally have
Note as in longitudinal waves in cold unmagnetized plasmas the term [Ωp2+ωp2] appears. As before, we note that this is a high frequency wave and hence approximate the ions as fixed due to their large mass, hence [Ωp2+ωp2]≃ωp2, and we write the above equation as
Next, assuming E⃗=E¯⃗ei(k⃗⋅x⃗−ωt), we finally obtain
the dispersion relation for an electromagnetic wave in an unmagentized plasma!
recalling expressions for phase velocity, vϕ=ω/k, and group velocity, vg=dkdω we obtain
Comparing this with the plot, confirm that indeed vg=0 for k=0, and that vg→c as k→∞. Importantly, although vϕ>c, we have vg<c. Thus special relativity is not violated, since information can only propagate at the group velocity and not at the phase velocity.
Also note, if ω<ωp a wave cannot propogate since k becomes imaginary and we get an evanescent wave.
Simulations with a Particle-in-Cell Code
In this project you simulate plasmas with exactly the same conditions as in Project 1a.
Each plasma electron is initialized with positions (only in x or what we call x1) such that the density is uniform. The ions are initialized at the same positions but they have an infinite mass. Each electron is also initialized with velocities (v1, v2, v3) or momentum (mv1, mv2, mv3) from a Maxwellian in each direction. The particles then begin to move in the self-consistent fields that their current and charge density produce.
The length of the plasmas is 50 c/ωp
The simulation will run for a time 400 1/ωp.
The simulation uses 50,000 particles.
You will be looking at plots of the electric field in the x3 direction, E3. In Project 1a you plotted E1.
The following lines must always be executed before running anything else.
Reminder: Hit Shift+Enter to run a cell, or select the cell and click on the "Run" button in the top menu bar
Run a case in which vth1=vth2=vth3=0.02c.