Maxwell equations and wave equation in a medium

[ModusPwnd]

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

Homework Equations

Maxwell equations and the curl

The Attempt at a Solution

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

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

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 I am 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?

 

[rude man]

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]

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]

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]

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