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9.4 GHz plane wave in dielectric

Short answer

Using refractive index n=√(ε_r)=1.5, v=c/n=2×10⁸ m/s; λ=v/f=2.13 cm; phase constant β=2π/λ=295.3 rad/m; intrinsic impedance η=η₀/n=251.3 Ω; electric field magnitude E=ηH=1.76 V/m.

A 9.4 GHz uniform plane wave is propagating in a medium with ε_r = 2.25 and μ_r = 1. If the magnetic field intensity is 7 mA/m and the material is loss less, find (i) the velocity of propagation (ii) the wavelength (iii) the phase constant (iv) the intrinsic impedance (v) the magnitude of electric field intensity.

Given: f = 9.4 GHz, ε_r = 2.25, μ_r = 1, H = 7 mA/m, lossless (σ = 0).

n = √(ε_r) = √2.25 = 1.5 (refractive index of the medium).

(i) Velocity of propagation

v = 1/√(με) = c/√(μ_r ε_r) = 3×10⁸ / 1.5 = 2×10⁸ m/s

(ii) Wavelength

λ = v/f = 2×10⁸ / 9.4×10⁹ = 0.02128 m = 2.128 cm

(iii) Phase constant

β = 2π/λ = 2π / 0.02128 = 295.3 rad/m
(check: β = ω/v = 2π × 9.4×10⁹ / 2×10⁸ = 295.3)

(iv) Intrinsic impedance

η = √(μ/ε) = η₀ √(μ_r/ε_r) = 120π / 1.5 = 251.3 Ω

(v) Magnitude of E

E = η H = 251.3 × 7×10⁻³ = 1.759 V/m

Answer: v = 2×10⁸ m/s; λ = 2.13 cm; β = 295.3 rad/m; η = 251.3 Ω; E = 1.76 V/m.

✓ Answer checked by Claude Opus 5.5 (high effort)

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