I am trying to derive the electromagnetic wave equations from Faraday's law of Induction, and the Ampere-Maxwell law.(adsbygoogle = window.adsbygoogle || []).push({});

But, I am having a problem with the 1/c^2 disappearing.

This is what I am using:

[tex]\nabla\times\vec{B}=\mu\vec{J}+\mu\epsilon\stackrel{\partial\vec{E}}{\partial t}[/tex]

Taking the cross product of both sides, I eventually end up with...

[tex]\nabla (\nabla\cdot\vec{B}) - \nabla^{2} (\vec{B}) = \mu\nabla\times\vec{J}+\mu\epsilon\stackrel{\partial(\nabla\times\vec{E})}{\partial t}[/tex]

Divergence of B is 0, so that goes away.

I'm stuck, though, when it comes to the curl of the charge flow. I know that the charge flow is as follows:

[tex]\vec{J}=\sigma\vec{E}[/tex]

Does the curl of the charge flow vanish? Faraday's law of Induction would imply that I'd be taking the curl of the negative time rate of change of the magnetic field. I'd ideally like to see the curl of the charge flow disappear, unless something else appears when talking about the charge flow through a medium. I understand that space is essentially non-conductive, and if you are deriving these equations in the vacuum of space, you won't have a charge flow. I'd like to not make that assumption, though, and it would be great if I ended up with an extremely generalized form of the wave equation, which could show the speed of light as being different through different media.

Why doesn't mu and epsilon show up in Faraday's law of Induction? I'd imagine that the relationship between the Displacement field and Magnetization field would still end up producing constants. And, if it does, then I'm in big trouble, because my 1/c^2 disappears due to the [tex]\mu\epsilon[/tex] canceling out with the [tex]\mu\epsilon[/tex] in the curl of Ampere-Maxwell's law.

Maybe after more discussion, I can clear my mind a bit on why I'm hung up on this concept.

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# Question regarding Curl of Charge Flow, and the vanishing constants in Faraday's law.

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