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

6.4.13 Problems and solutions

The problems of this section use Snell's law, the Fresnel coefficients and the power fractions of Table 6.4.1,

θ_{r} = θ_{i}, n_{1} sin θ_{i} = n_{2} sin θ_{t}, Γ_{⊥} = \frac{η _{2} cos θ _{i} - η _{1} cos θ _{t}}{η _{2} cos θ _{i} + η _{1} cos θ _{t}}, τ_{⊥} = 1 + Γ_{⊥}, Γ_{∥} = \frac{η _{2} cos θ _{t} - η _{1} cos θ _{i}}{η _{2} cos θ _{t} + η _{1} cos θ _{i}}, τ_{∥} = \frac{2 η _{2} cos θ _{i}}{η _{2} cos θ _{t} + η _{1} cos θ _{i}}, R = ∣Γ ∣^{2}, T = \frac{η _{1} cos θ _{t}}{η _{2} cos θ _{i}} ∣ τ ∣^{2},

with $cos θ_{t} = - j sin^{2} θ_{t} - 1$ beyond the critical angle $θ_{c} = arcsin (n_{2} / n_{1})$ , the Brewster angle $tan θ_{B} = n_{2} / n_{1}$ of nonmagnetic media, the plane wave $E = E_{0} e^{- j β a_{n} \cdot r}$ with $H = \frac{1}{η} a_{n} \times E$ , and, where a normal-incidence result is needed, the tools of Section 6.3. The reference directions are those of Figures 6.4.2 and 6.4.10, so that $Γ_{∥} = Γ_{⊥}$ at normal incidence. The constants are those of this book, $ε_{0} = 1 0^{- 9} / (36 π) F/m$ , $μ_{0} = 4 π \times 1 0^{- 7} H/m$ , $c = 3 \times 1 0^{8} m/s$ and $η_{0} = 120 π = 377 Ω$ . For nonmagnetic media $η = η_{0} / n$ , and dividing every impedance by $η_{0}$ keeps the numbers simple.

6.4.13 Problems and solutions

In this part

Problems · 6

Home exercises · 4

Further exercises · 10

Worked examples · 1