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

About the further exercises of Section 6.2

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 6.2.8

The ionospheric range error of GPS

A GPS signal crosses the whole ionosphere, whose electron density $N (z)$ varies along the path, with $f ≫ f_{p}$ everywhere. (a) Show that the envelope arrives later than in vacuum by the equivalent extra distance $Δ R = c \int (1/ v_{g} - 1/ c) d z = K TEC / f^{2}$ , where $TEC = \int N d z$ is the total electron content, and evaluate $K$ . (b) For a daytime $TEC = 5 \times 1 0^{17} m^{- 2}$ find $Δ R$ and the delay at $f_{1} = 1575.42 MHz$ (L1) and $f_{2} = 1227.60 MHz$ (L2). (c) A dual-frequency receiver finds the L2 pseudorange longer than the L1 one by $5.26 m$ . Find the TEC, and the combination $a R_{1} - b R_{2}$ of the two measured ranges that is free of the ionosphere. (d) By how much does the carrier phase path change?

Try it yourself first.

cf. Problem 6.2.3

Exercise 6.2.8: The ionospheric range error of GPS