Plane wave in free space, find β and H

Given: E = 2 cos(10⁷πt − βz) a_x V/m in free space. The wave moves in +z with E along a_x (the question's "z-polarized" refers to travel along z).

Phase constant

ω = 10⁷π rad/s
β = ω/c = 10⁷π / (3×10⁸) = π/30
β ≈ 0.105 rad/m

(Frequency f = ω/2π = 5 MHz, λ = 2π/β = 60 m.)

Magnetic field

For a uniform plane wave, H = (1/η) a_k × E, and in free space η₀ = 120π ≈ 377 Ω. The direction of travel is a_z:

a_z × a_x = a_y
H = (E₀/η₀) cos(ωt − βz) a_y
  = (2/377) cos(10⁷πt − 0.105z) a_y
  = 5.31 cos(10⁷πt − 0.105z) a_y   mA/m

Check: E × H = a_x × a_y = a_z, so power flows along +z, as it should.

Answer: β = 0.105 rad/m; H = 5.31 cos(10⁷πt − 0.105z) a_y mA/m.