Homework 10 (Due. Apr. 5)

1. Different kinds of conductors

In Griffiths (and in class), we derived equations for EM waves in conductors, assuming a “good conductor.” Interestingly, it turns out that the formulas we got can be pushed a good deal further than you might naively expect, into regimes where $\sigma$ is not so large (“poor conductors.”) In this case, you will need to use Griffiths more careful results for the real and imaginary parts of the k vector and work with those.

  1. Based on the above comments, show that the skin depth for a “poor conductor” (i.e., $\sigma \ll \varepsilon \omega$) is $d \approx ?? \sqrt{\varepsilon/\mu}$, independent of frequency or wavelength. Work out what the “??” is in this equation. (Also, check the units of your answer explicitly, please!)
  2. Show that the skin depth for a “good conductor” (i.e., $\sigma \gg \varepsilon \omega$) is $d = \lambda/??$, where $\lambda$ is the wavelength in the conductor. (Work out the ?? in that formula.) Find the skin depth for microwaves in Cu ($f$ = 2.5 GHz). Briefly, how do you interpret that answer physically?
  3. About how thick does aluminum foil have to be, to be optically opaque? Comment briefly.
  4. For biological tissues (like skin), $\varepsilon$ and $\sigma$ depend on frequency, you can’t use their free space values. But, $\mu$ on the other hand is close to its free space value. At microwave frequencies (say, about 2.5 GHz), their values are $\sim$ $\varepsilon =47 \varepsilon_0$, and $\sigma = 2.2 \Omega^{-1}m^{-1}$. Is skin (at this frequency) the “good conductor” or “poor conductor” case, or neither? Evaluate the skin depth for microwaves hitting your body.
  5. If the EM wave from part 4 (e.g. from a radar station) hits your body, what fraction of the incident power do you absorb? (Hint: think about “R” first, then you can get “T” easily!)
  6. Let’s think about contacting submarines by radio. For low frequency radio waves (say, $f$=3 kHz) estimate the skin depth in the sea, and comment on the feasibility/issues of such radio communication. What is the wavelength of this same radio wave in free space, by the way? (Hint: Treating seawater as like a human body is ok for this one- just use the given values from part 4 as needed. But, as mentioned there, in reality they’ll be different at this very different frequency and slightly different material. Small bonus credit if you can find more appropriate values, and give the reference!)

2. Wave Packets

We keep saying that you can always sum up plane waves to get real wave packets. Let’s try it! Consider a localized wavepacket that satisfies the one-dimensional wave equation from a sum of sinusoidal waves using Fourier’s integral method:

\[f(x,t) = \int_{-\infty}^{\infty} A(k)e^{ik(x-ct)}dk\]
  1. Show that the $f(x,t)$ satfies the one-dimensional wave equation with wave speed, $c$.
  2. Assume that $A(k)$ is given by the a Gaussian distribution centered at some positive wavevector $k_0$: \(A(k)=\dfrac{1}{\sqrt{2 \pi \sigma^2}}\exp\left(-\dfrac{(k-k_0)^2}{2\sigma^2}\right)\) Sketch this function, Roughly what range of wavevectors $\Delta k$ contribute signficantly to the wave packet?
  3. Calculate $f(x,t)$ from the above $A(k)$. There is a famous and handy Gaussian integral that can be helpful here that works for any $z$ even complex ones! \(\int_{-\infty}^{\infty} \exp\left(-\dfrac{q^2y^2}{2}+zy\right)dy = \sqrt{\dfrac{2\pi}{q^2}}\exp\left(\dfrac{z^2}{2q^2}\right)\)
  4. Describe $f(x,t)$ physically as best you can. How is the $x$-width, $\Delta x$ of the “wavepacket” related to the $k$-width $\Delta k$? Does this relationship between $\Delta x$ and $\Delta k$ remind you of anything from quantum mechanics (PHY 215 or 471)?
  5. Pick a visible wavelength, and plot, using Jupyter, a Gaussian pulse that lasts 1 femtosecond. (When I say plot, I’m thinking of $Re[f(x,t=0)]$, and then think about what happens as time goes by) Try to get as many details reasonably correct as you can.

As usual, you will turn part 5 in using your GitHub repo

3. Paired Project Problem

  1. Read the feedback that you received on your figure/models and think about how you and your partner are going to work through these details.
  2. Project work - Provide some sample calculations and figures produced by your code. This can be a notebook (Jupyter), but the work needs to be explained inline (i.e., what are you doing and why?). This is the work that is the meat of your original contribution. It need not be complete yet.
  3. Self-reflection - Think about how the project is going and how you are both contributing. Write out a document for the last couple of weeks worth of work inclduing this one that describes: Who did what? Hoes does it feel like the contributions for the members of your pair are equal? Regarding the project specifically, what questions do you need to answer to continue to move forward and 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!