# 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