Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Thirteen further exercises in the spirit of the problems, the worked examples and the home exercises, graded from easy to harder within each group. Exercises 6.2.1–6.2.3 are about lossy media and the loss tangent, Exercises 6.2.4–6.2.5 about good conductors and the skin effect, Exercises 6.2.6–6.2.8 about plasmas and the group velocity, and Exercises 6.2.9–6.2.13 come from the exercise lecture and mix all three. They use only the equations listed at the head of this subsection, together with $v_{g} = d ω / d β$ and $γ η_{c} = j ω μ$ , $γ / η_{c} = σ + j ω ε$ .

Exercise 6.2.1

One soil, two survey frequencies

Moist sandy soil has $ε_{r} = 12$ , $σ = 0.006 S/m$ , $μ_{r} = 1$ . (a) At what frequency $f_{t}$ are the conduction and displacement currents equal? Over which frequencies is the soil a good conductor ( $p > 100$ ) and over which a low-loss dielectric ( $p < 0.01$ )? (b) A low-frequency induction survey works at $10 kHz$ . Find the skin depth and the depth at which the field is $20 dB$ down. (c) Repeat (b) for a radar at $2 GHz$ with the low-loss formula, and find the wavelength in the soil.

Try it yourself first.

cf. Problem 6.2.1

Exercise 6.2.1: One soil, two survey frequencies