E and H ratio (Homework 1-323)

In free space find the ratio of magnitude of the transverse components of E and H; that is find $E_{0t}/H_{0t}$ . What’s its numerical value?

In free space the wave equation gives for E and B vectors:

$∇^2 E ⃗=(1/c^2) *[(∂^2 E ⃗)/(∂t^2)]$   and $∇^2 B ⃗=(1/c^2) *(∂^2 B ⃗)/(∂t^2 )$

If we suppose that the waves are propagating in the z direction the solutions to above equation are

$\widetilde{E}(z,t)=(\widetilde{E_0} ) e^{i(kz-ωt)}$     and $\widetilde{B }(z,t)=(\widetilde{B_0} ) e^{i(kz-ωt)}$    where $E_0$  and $B_0$  are complex numbers
Two of the Maxwell equations are

$∇\overrightarrow{E}=0$  and $∇\overrightarrow{B}=0$   ($div E=0$  and $div B=0$)

Suppose E is purely real (or purely complex). Then in the divergence the first two terms are real and the 3rd term (containing $\widetilde{E}_oz$    is complex). Therefore one has the necessary condition
$\widetilde{E}_0z=\widetilde{B}_0z=0$

And thus the E and B waves are transversal waves (and also transversal on the propagation direction). That is E, B and z directions are all perpendicular to each other.

Now the question is simple to answer. From the 3rd Maxwell equation (called also the Faraday induction law):

$∇×\overrightarrow{E}=-(∂\overrightarrow{B})/∂t$    or $kE_0y=ωB_0x$     and $kE_0x=ωB_0y$
or equivalent   $B_0=k/ω ((\hat{z} ×E_0)$      or   $B_0/E_0 =k/ω=1/c$
where $c=1/\sqrt{(ϵ_0 μ_0 )}=3*10^8 (m/s)$