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

Recap

Plane waves. Far from its sources every wave is locally plane ( $r > 2 D^{2} / λ$ ). In a source-free region each field component obeys $\nabla^{2} u + k^{2} u = 0$ , with the solutions $e^{\mp j k z}$ in one dimension and $C e^{- j k \cdot r}$ in three, $k_{x}^{2} + k_{y}^{2} + k_{z}^{2} = k^{2}$ . With loss $k = β - j α$ , $γ = j k = α + j β$ , and $e^{- j k z} = e^{- α z} e^{- j β z}$ .
Speed and wavelength: $v_{p} = ω / β$ , $λ = 2 π / β$ , and in a lossless medium $v_{p} = 1/ μ ε = c / μ_{r} ε_{r}$ . Along the axes $λ_{x} = 2 π / k_{x} \geq λ$ .
Uniform plane wave, Eq. (6.1.26): $H = \frac{1}{η} a_{k} \times E$ , $E = - η a_{k} \times H$ , $E ⊥ H ⊥ a_{k}$ , $η = μ / ε$ , $η_{0} = 120 π \approx 377 Ω$ . With loss $η = ∣ η ∣ e^{j θ_{η}}$ and $H$ lags $E$ by $θ_{η}$ .
Power: $P = E \times H$ , $P_{a v} = \frac{1}{2} Re {E \times H^{*}}$ , and for a plane wave $P_{a v} = ∣ E_{0} ∣^{2} / (2 η) a_{k}$ . Poynting's theorem, Eq. (6.1.45): the power supplied by the sources is stored, dissipated, or carried out through the boundary.
Polarisation: the curve traced by $E (t)$ at a fixed point, linear, circular or elliptical. IEEE: right thumb along $a_{k}$ , fingers with $E (t)$ , RH. Test: $(E_{r} \times (- E_{i})) \cdot a_{k} > 0$ means RH. $AR = a / b$ , and any wave is $E_{R} (a_{x} - j a_{y}) + E_{L} (a_{x} + j a_{y})$ with $E_{R, L} = \frac{1}{2} (E_{1} \pm j E_{2} e^{j δ})$ . The received power fraction is $∣ p_{w} \cdot p_{a}^{*} ∣^{2}$ .
Doppler: $f^{'} \approx f (1 + \frac{v}{c} cos θ)$ for a moving source or observer, and $f_{D} = 2 v cos θ / λ$ for a radar echo.

6.1.8 Recap