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 6.2.4

Induction hardening of a steel shaft

A steel shaft is surface-hardened by an induction coil. Below the Curie temperature (about $770^{\circ} C$ ) the steel has $σ = 5 \times 1 0^{6} S/m$ and $μ_{r} = 100$ , above it $μ_{r} = 1$ and $σ \approx 1 \times 1 0^{6} S/m$ . (a) Find the skin depth in cold and in hot steel at $10 kHz$ . (b) Which frequency gives $δ = 1 mm$ in the hot steel, and what is $δ$ of the cold steel at that frequency? (c) Find both skin depths at $50 Hz$ and comment.

Try it yourself first.

cf. Problem 6.2.4

Exercise 6.2.4: Induction hardening of a steel shaft