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- May 2nd 2008, 03:08 PMnairbdm.............
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- May 2nd 2008, 11:19 PMIsomorphism
$\displaystyle \nabla\cdot \textbf{D} = 0$

$\displaystyle \nabla\cdot \textbf{B} = 0$

$\displaystyle \nabla \times \textbf{E} = -\frac{\partial \textbf{B}}{\partial t}$

$\displaystyle \nabla \times \textbf{H} = \frac{\partial \textbf{D}}{\partial t}$

$\displaystyle \nabla \times (\nabla \times \textbf{E}) = \nabla(\nabla\cdot \textbf{E}) - \nabla^2(\textbf{E})$

Now $\displaystyle \textbf{D} = \epsilon \textbf{E}$ since empty space is an isotropic medium.

$\displaystyle \nabla\cdot \textbf{D} = 0 \Rightarrow \nabla\cdot \textbf{E} = 0$

So

$\displaystyle \nabla \times (\nabla \times \textbf{E}) = - \nabla^2(\textbf{E})$

But $\displaystyle - \nabla^2(\textbf{E}) = \nabla \times (\nabla \times \textbf{E}) = -\frac{\partial(\nabla \times\textbf{B})}{\partial t} = -\mu \frac{\partial(\nabla \times\textbf{H})}{\partial t} = -\mu\epsilon \frac{\partial^2 \textbf{E}}{\partial t^2} $

Now use $\displaystyle \frac1{c^2} = \mu\epsilon$ to obtain $\displaystyle \nabla^2(\textbf{E}) = \frac1{c^2}\frac{\partial^2 \textbf{E}}{\partial t^2} $ - May 3rd 2008, 08:14 PMIsomorphism