(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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Maxwell equations and wave equation in a medium

**Physics Forums | Science Articles, Homework Help, Discussion**