Phasor E in free space, find f η Hs

Given: free space, E_s = (100∠45°) e^{−j50z} a_x V/m. So β = 50 rad/m, E₀ = 100 V/m, phase 45°, and the wave travels in +z.

(a) Frequency

ω = β c = 50 × 3×10⁸ = 1.5×10¹⁰ rad/s
f = ω/2π = 1.5×10¹⁰/6.2832 = 2.387×10⁹ Hz ≈ 2.39 GHz

(b) Wave impedance

η₀ = 120π = 377 Ω

(c) H_s. For travel along +z with E along a_x, H points along a_y (a_x × a_y = a_z):

H_s = (E₀/η₀)∠45° e^{−j50z} a_y
    = (100/377) ∠45° e^{−j50z} a_y
    = 0.2653 ∠45° e^{−j50z} a_y   A/m

(d) |E| at z = 10 mm, t = 20 ps.

E(z,t) = 100 cos(ωt − βz + 45°)
ωt = 1.5×10¹⁰ × 20×10⁻¹² = 0.3 rad
βz = 50 × 0.01 = 0.5 rad
45° = 0.7854 rad
phase = 0.3 − 0.5 + 0.7854 = 0.5854 rad
E = 100 cos(0.5854) = 100 × 0.8335 = 83.3 V/m  (a_x direction)

Answer: f = 2.39 GHz; η = 377 Ω; H_s = 0.265∠45° e^{−j50z} a_y A/m; E(10 mm, 20 ps) = 83.3 a_x V/m.