# Maxwell equations and wave equation in a medium

1. Sep 24, 2011

### ModusPwnd

1. The problem statement, all variables and given/known data

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. Relevant 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?

2. Sep 25, 2011

### 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.

3. Sep 25, 2011

### ModusPwnd

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

4. Sep 25, 2011

### 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

5. Sep 25, 2011

### ModusPwnd

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

6. Sep 25, 2011

### ehild

No, they are also characteristics of the medium.

ehild

7. Sep 25, 2011

### 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...

8. Sep 25, 2011

### 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