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Plane wave from E and H, find β λ η μr εr

Short answer

From ω=10⁷π rad/s and v=1.4×10⁸ m/s, β=ω/v=0.224 rad/m, λ=v/f=28 m. Intrinsic impedance η=E/H=473.7 Ω. Then ε_r = 1/(ηvε₀)=1.70 and μ_r = η/(vμ₀)=2.69.

Let E(z, t) = 1800 cos(10⁷πt − βz) a_x V/m and H(z, t) = 3.8 cos(10⁷πt − βz) a_y A/m represent a uniform plane wave propagating at a velocity of 1.4×10⁸ m/s in a perfect dielectric. Find (a) β (b) λ (c) η (d) μ_r (e) ε_r.

Given: E = 1800 cos(10⁷πt − βz) a_x V/m, H = 3.8 cos(10⁷πt − βz) a_y A/m, v = 1.4×10⁸ m/s, perfect dielectric.

ω = 10⁷π = 3.1416×10⁷ rad/s, so f = ω/2π = 5×10⁶ Hz.

(a) Phase constant

β = ω/v = 3.1416×10⁷ / 1.4×10⁸ = 0.2244 rad/m

(b) Wavelength

λ = 2π/β = v/f = 1.4×10⁸ / 5×10⁶ = 28 m

(c) Intrinsic impedance

η = E/H = 1800/3.8 = 473.7 Ω

(d) μ_r and (e) ε_r. Use η = √(μ/ε) and v = 1/√(με). Multiplying: η v = 1/ε, and dividing: η/v = μ.

ε = 1/(η v) = 1 / (473.7 × 1.4×10⁸) = 1.508×10⁻¹¹ F/m
ε_r = ε/ε₀ = 1.508×10⁻¹¹ / 8.854×10⁻¹² = 1.70

μ = η/v = 473.7 / 1.4×10⁸ = 3.384×10⁻⁶ H/m
μ_r = μ/μ₀ = 3.384×10⁻⁶ / 1.2566×10⁻⁶ = 2.69

Check: v = c/√(μ_r ε_r) = 3×10⁸/√(2.69 × 1.703) = 3×10⁸/2.14 = 1.40×10⁸ m/s ✓.

Answer: β = 0.224 rad/m; λ = 28 m; η = 473.7 Ω; μ_r = 2.69; ε_r = 1.70.

✓ Answer checked by Claude Opus 5.5 (high effort)

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