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Homework Help: Maxwell equations and wave equation in a medium

  1. Sep 24, 2011 #1
    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?
     
  2. jcsd
  3. Sep 25, 2011 #2

    rude man

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    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.
     
  4. Sep 25, 2011 #3
    Im not trying to solve the wave equation, I am trying to derive it.
     
  5. Sep 25, 2011 #4

    ehild

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    The fourth equation is correctly [itex]\nabla \times \vec{H} = \sigma \vec{E} +\frac{\partial \vec{D}}{\partial t} [/itex]

    and use also the "material equations" [itex]\vec{D}=\epsilon \vec{E}[/itex], [itex]\vec{B}=\mu\vec{H}[/itex]

    ehild
     
  6. Sep 25, 2011 #5
    Can I get [itex] \mu [/itex] and [itex] \epsilon [/itex] from the conductivity I am given?
     
  7. Sep 25, 2011 #6

    ehild

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    No, they are also characteristics of the medium.

    ehild
     
  8. Sep 25, 2011 #7
    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...
     
  9. Sep 25, 2011 #8

    ehild

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    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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