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Plane wave parameters from Hs

Short answer

Phase constant β=350 rad/m. Frequency f=ω/2π=βc/2π≈16.7 GHz; intrinsic impedance η₀=377 Ω. With H along a_x and propagation +a_z, E must point along −a_y: E_s = −94.25∠30° e^{−j350z} a_y kV/m. At z=25 mm, t=4 ps, |H| = 11.9 A/m.

A uniform plane wave in free space is given by H_s = (250∠30°) e^(−j350z) a_x V/m. Determine the phase constant, frequency of the wave, intrinsic impedance, E_s and the magnitude of H at z = 25 mm and t = 4 ps.

Given (the unit of H_s is A/m; V/m in the question is a misprint): H_s = (250∠30°) e^{−j350z} a_x A/m in free space (ε_r = μ_r = 1).

Comparing with e^{−jβz}: wave travels in +z, and H_s amplitude is 250 A/m with phase 30°. Take c = 3×10⁸ m/s.

1. Phase constant

β = 350 rad/m

2. Frequency

ω = β c = 350 × 3×10⁸ = 1.05×10¹¹ rad/s
f = ω / 2π = 1.05×10¹¹ / 6.2832 = 1.671×10¹⁰ Hz ≈ 16.7 GHz

3. Intrinsic impedance

η₀ = 120π = 376.99 ≈ 377 Ω

4. E_s. The wave travels along +a_z, and E × H must point along +a_z. With H along a_x, E must point along −a_y, since (−a_y) × a_x = a_z. Magnitude E = η H = 377 × 250 = 94 248 V/m.

E_s = −94.25∠30° e^{−j350z} a_y kV/m
    = 94.25∠210° e^{−j350z} a_y kV/m

5. |H| at z = 25 mm, t = 4 ps.

H(z,t) = 250 cos(ωt − βz + 30°) a_x
ωt = 1.05×10¹¹ × 4×10⁻¹² = 0.42 rad
βz = 350 × 0.025 = 8.75 rad
30° = 0.5236 rad
phase = 0.42 − 8.75 + 0.5236 = −7.806 rad
cos(−7.806) = cos(7.806 − 2π) = cos(1.523) = 0.0476
H = 250 × 0.0476 = 11.9 A/m

Answer: β = 350 rad/m; f ≈ 16.7 GHz; η = 377 Ω; E_s = −94.25∠30° e^{−j350z} a_y kV/m; H(25 mm, 4 ps) ≈ 11.9 a_x A/m.

✓ Answer checked by Claude Opus 5.5 (high effort)

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