(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

What conditions need to be imposed on [tex]\vec{E}[/tex]_{0}, [tex]\vec{B}[/tex]_{0}, [tex]\vec{k}[/tex] and ω to ensure the following equations solve Maxwell's equations in a region with permittivity ε and permeability µ, where the charge density and the current density vanish:

[tex]\vec{E}[/tex] = Re{ [tex]\vec{E}[/tex]_{0}exp[i([tex]\vec{k}[/tex]⋅[tex]\vec{x}[/tex] - ωt)] }

[tex]\vec{B}[/tex] = Re{ [tex]\vec{B}[/tex]_{0}exp[i([tex]\vec{k}[/tex]⋅[tex]\vec{x}[/tex] - ωt)] }

2. Relevant equations

Maxwell's Equations and ω=kv

3. The attempt at a solution

I know from Gauss' Law for E & B the following is required:

[tex]\vec{k}[/tex]⋅[tex]\vec{E}[/tex]_{0}= [tex]\vec{k}[/tex]⋅[tex]\vec{B}[/tex]_{0}= 0

and from Faraday's Law:

[tex]\vec{k}[/tex] x [tex]\vec{E}[/tex]_{0}= ω[tex]\vec{B}[/tex]_{0}

Now the Ampere-Maxwell law would suggest the following may be a requirement:

[tex]\vec{k}[/tex] x [tex]\vec{B}[/tex]_{0}= -(1/c²)ω[tex]\vec{E}[/tex]_{0}

But is this final one really required, or does it in fact follow from the three previous requirements? I remember being told that the latter was correct, but I have no idea how to show this last requirement from the previous three.

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# Homework Help: Conditions on complex plane wave solutions to Maxwell's Equations

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