Lossless dielectric, find ω η E

Given: σ = 0, μ_r = 1, ε_r = 4, and

H = −0.1 cos(ωt − z) a_x + 0.5 sin(ωt − z) a_y A/m.

The phase is (ωt − z), so the wave travels in +z with phase constant β = 1 rad/m.

(a) Angular frequency. β = ω√(με) = (ω/c)√(μ_r ε_r) = (ω/c) × 2.

ω = β c / 2 = 1 × 3×10⁸ / 2 = 1.5×10⁸ rad/s     (f = 23.9 MHz)

(b) Wave impedance

η = η₀ √(μ_r/ε_r) = 377 / 2 = 188.5 Ω

(c) E. For a wave along a_z, E = η H × a_z (equivalently H = a_z × E / η).

a_x × a_z = −a_y,   a_y × a_z = a_x

E = 188.5 [ −0.1 cos(ωt − z)(−a_y) + 0.5 sin(ωt − z)(a_x) ]
  = 188.5 [ 0.5 sin(ωt − z) a_x + 0.1 cos(ωt − z) a_y ]
  = 94.25 sin(ωt − z) a_x + 18.85 cos(ωt − z) a_y   V/m

Check: E·H = E_xH_x + E_yH_y = (94.25 sin)(−0.1 cos) + (18.85 cos)(0.5 sin) = −9.425 sin cos + 9.425 sin cos = 0 ✓ (E ⟂ H). Also 94.25/0.5 = 18.85/0.1 = 188.5 = η for each component pair.

Answer: ω = 1.5×10⁸ rad/s; η = 188.5 Ω; E = 94.25 sin(1.5×10⁸t − z) a_x + 18.85 cos(1.5×10⁸t − z) a_y V/m.