Homework 11 (Due. Apr 12)

1. Waves in plasmas

Consider a neutral plasma (gas of electrons and positive ions), with electron density $N_e$. At high frequencies, because the ions are very heavy, we can consider them to be essentially fixed and any current due solely to the light electrons. The total charge density can be set to zero for an electrically neutral gas.

1. Use Maxwell’s Equations to derive the wave equation for the electric field: $\nabla^2 \mathbf{E} -\dfrac{1}{c^2}\dfrac{\partial^2}{\partial t^2}\mathbf{E} = \mu_0 \dfrac{\partial}{\partial t}\mathbf{J}$
2. For a monochromatic wave, $\mathbf{E} = Re(\widetilde{\mathbf{E}} e^{−i\omega t})$, and ignoring any collision between electrons (mass $m_e$), use Newton’s law to relate $\mathbf{E}$ and $\mathbf{J}$ and show that $\left(\nabla^2 + \dfrac{\omega^2}{c^2}\right)\widetilde{\mathbf{E}} = \dfrac{\omega_p^2}{c^2}\widetilde{\mathbf{E}}$ where $\omega_p^2 = \dfrac{N_e e^2}{\varepsilon_0 m_e}$ is the plasma frequency.
3. Derive the dispersion relation $k=\sqrt{\omega^2-\omega_p^2}/c$. Sketch the graph of $\omega(k)$.
4. Give the real elecrtic field when $\omega < \omega_p$.

2. Star Wars with relativity

Space Probe #1 passes very close to earth at a time that both we (on earth) and the onboard computer on Probe 1 decide to call t = 0 in our respective frames. The probe moves at a constant speed of 0.5$c$ away from earth. When the clock aboard Probe 1 reads t = 60sec, it sends a light signal straight back to earth.

1. At what time was the signal sent, according to the earth’s rest frame?
2. At what time in the earth’s rest frame do we receive the signal?
3. At what time in Probe 1’s rest frame does the signal reach earth?
4. Space Probe #2 passes very close to earth at t = 1sec (earth time), chasing Probe 1. Probe 2 is only moving at 0.3c (as viewed by us). Probe 2 launches a proton beam (which moves at v = 0.21c relative to Probe 2) directed at Probe 1. Does this proton beam strike Probe 1? Please answer twice, once ignoring relativity theory, and then again using Einstein!

3. Muon calculations

The mean lifetime of muons is 2 $\mu\text{s}$ in their rest frame. Muons are produced in the upper atmosphere, as cosmic-ray secondaries.

1. Calculate the mean distance traveled by muons with speed $v = 0.99c$, assuming classical physics (i.e. without special relativity).
2. Under this assumption, what percentage of muons produced at an altitude of 10 km reach the ground, assuming they travel downward at $v = 0.99c$? Careful here, you will have to think about the distribution of lifetimes given that particle decay is a Poisson process.
3. Calculate the mean distance traveled by muons with speed $v = 0.99c$, taking into account special relativity.
4. Under this assumption, what percentage of muons produced at an altitude of 10 km reach the ground, assuming they travel downward at $v = 0.99c$?

4. Paired Project Problem

At this point, you should produce draft figures for poster with captions. You should have pressed onward with your calculations and produced appropriate figures for your poster with captions. The “graphic” and caption should be turned in. These need not be complete in the sense that you should continue working on your calculations and models until the poster is turned in. By draft, I do mean draft, so any figures that you are thinking about putting in the poster, please include them in their current form. Make sure to give me enough information to:

1. judge what the figures are,
2. where they will go in the poster, and
3. what part of the poster story they will be telling.

1. Who did what?
2. What questions were you able to answer last week?
3. What questions do you need to answer to continue to move forward?
4. What help do you need from me or others?

You will turn in both your “notebook” and your self-reflection using the same GitHub repository you started for Project 2. Make sure that you sync your repository first to get the new feedback!