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insynC
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Homework Statement
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)] }
Homework Equations
Maxwell's Equations and ω=kv
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.