Problem 4: Not so good conductor (25 points) In Griffiths (and in class), we derived equations for EM waves in conductors, assuming a "good conductor" (where σ »ew). Interestingly, it turns out that the formulas we got can be pushed a good deal further than you might naively expect, into regimes where e.g. σ is not so large ("poor conductors".) In this case, you will need to use Griffiths's more careful results Eq. (9.126) for the real and imaginary parts of the k vector and work with those. (a) Based on the above comments, show that the skin depth for a "poor conductor" (i.e., σ << ew) is d ??√√μ, independent of frequency or wavelength. Work out what the "??" is in this equation. Then, show (using Griffiths Eq. (9.126)) that the skin depth for a "good conductor" (oew) is dλ/2л, where X is the wavelength in the conductor. (b) For biological tissues (like skin), e and σ depend on frequency, you can't use their free space values (μ on the other hand is close to its free space value.) At microwave frequencies (say, about 2.5 GHz), their values are = 47€0, and σ = 2.22¹m¹. Is this the "good conductor" or "poor conductor" case, or neither? Evaluate the skin depth for microwaves hitting your body. If such an EM wave (e.g. from a radar station) hits your body at this frequency, roughly what fraction of the incident power do you absorb? Hint: Use Griffiths's Eqns. (9.146) and (9.147) and compute the reflection coefficient R = |EO,R|2/|Ẽ0,1|². Then argue that whatever is not reflected must be absorbed. (c) It is difficult to communicate by radio with submarines because it is not easy to make radio waves that can penetrate sea water. For radio waves with frequency of say, 3 kHz, evaluate 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: You can use given values from Part (b) where needed, although as mentioned above, in reality they'll be different at this different frequency.)