Maxwell equations and wave equation in a medium

ModusPwnd

1. Homework Statement

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

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 $\vec{J}=\sigma\vec{E}$.

2. Homework Equations

Maxwell equations and the curl

3. The Attempt at a Solution

a)
Maxwell equaitons with $\rho_f=0$ and $\vec{J}=\frac{\vec{E}}{\rho}$.

$$\nabla \cdot \vec{D} = 0$$
$$\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$$
$$\nabla \cdot \vec{B} = 0$$
$$\nabla \times \vec{H} = \sigma \vec{E} + \epsilon_0 \mu_0 \frac{\partial \vec{D}}{\partial t}$$

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

b)
Here I am a little confused. I take the curl of the curl of $\vec{E}$,

$$\nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2 \vec{E} = -\nabla^2 \vec{E} = \nabla \times (-\frac{\partial \vec{B}}{\partial t}) = -\frac{\partial}{\partial t} (\nabla \times \vec{B})$$

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

Any ideas?

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rude man

Homework Helper
Gold Member
First thing you do is assume sinusoids. It's pretty near impossible otherwise. So start with the equations for E and H assuming a sinusoidal plane wave. Use the exponetial form E = E0exp(jwt) and H = H0exp(jwt) if you're an engineer or substitute i for j if you're a physicist. :-)

Wind up eliminating H, and get a partial differential equation for E. Solve it.

ModusPwnd

Im not trying to solve the wave equation, I am trying to derive it.

ehild

Homework Helper
The fourth equation is correctly $\nabla \times \vec{H} = \sigma \vec{E} +\frac{\partial \vec{D}}{\partial t}$

and use also the "material equations" $\vec{D}=\epsilon \vec{E}$, $\vec{B}=\mu\vec{H}$

ehild

ModusPwnd

Can I get $\mu$ and $\epsilon$ from the conductivity I am given?

ehild

Homework Helper
No, they are also characteristics of the medium.

ehild

ModusPwnd

bleh, so the question does not provide enough for an answer? My profs. really suck at writing questions, this is not the first time this has happened...

ehild

Homework Helper
You have the appropriate Maxwell equations, and can write the damped wave equation replacing B=μH and D=εE. ε and μ are constants.

ehild

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