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

Consider an isotropic medium with constant conductivity [itex]\sigma[/itex]. There is no free charge present, that is, [itex]\rho = 0[/itex].

a)What are the appropriate Maxwell equations for this medium?

b)Derive the damped wave equation for the electric field in the medium. Assume Ohm's law is of the form [itex]\vec{J}=\sigma\vec{E}[/itex].

2. Relevant equations

Maxwell equations and the curl

3. The attempt at a solution

a)

Maxwell equaitons with [itex]\rho_f=0[/itex] and [itex] \vec{J}=\frac{\vec{E}}{\rho}[/itex].

[tex]

\nabla \cdot \vec{D} = 0

[/tex]

[tex]

\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}

[/tex]

[tex]

\nabla \cdot \vec{B} = 0

[/tex]

[tex]

\nabla \times \vec{H} = \sigma \vec{E} + \epsilon_0 \mu_0 \frac{\partial \vec{D}}{\partial t}

[/tex]

Its simply a matter of putting a [itex] \sigma \vec{E} [/itex] in place of the displacement current [itex] \vec{J} [/itex] right? hmmm...

b)

Here I am a little confused. I take the curl of the curl of [itex]\vec{E}[/itex],

[tex] \nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2 \vec{E} = -\nabla^2 \vec{E} = \nabla \times (-\frac{\partial \vec{B}}{\partial t}) = -\frac{\partial}{\partial t} (\nabla \times \vec{B}) [/tex]

Now here Im not sure if I am correct in assuming that [itex] \nabla \cdot \vec{E} = 0 [/itex] and I'm not sure what [itex] \nabla \times \vec{B} [/itex] in this case, since its not in fee space...

Any ideas?

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# Maxwell equations and wave equation in a medium

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