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Homework Help: Inhomogeneous Electromagnetic Wave Equation

  1. Jan 25, 2007 #1
    1. The problem statement, all variables and given/known data

    Consider a medium where [tex]\vec{J_f} = 0[/tex] and [tex]{\rho_f}=0[/tex], but there is a polarization [tex]\vec{P}(\vec{r},t)[/tex]. This polarization is a given function, and not simply proportional to the electric field.

    Starting from Maxwell's macroscopic equations, show that the electric field in this medium satisfies the non-homogeneous wave equation:

    [tex]{\nabla}^2\vec{E} - \frac{1}{c^2}\frac{{\partial}^2\vec{E}}{{\partial}t^2} = -\frac{1}{{\epsilon}_0}{\vec{\nabla}}(\vec{\nabla}\cdot\vec{P})+{{\mu}_0}\frac{{\partial}^2\vec{P}}{{\partial}t^2}[/tex]

    2. Relevant equations

    Maxwell's equations:

    [tex]\vec{\nabla}\times\vec{E}=-{\mu}\frac{{\partial}\vec{H}}{{\partial}t}[/tex] ........(1)
    [tex]\vec{\nabla}\cdot\vec{E}=\frac{\rho}{\epsilon}[/tex] ........(2)
    [tex]\vec{\nabla}\times\vec{H}=\vec{J}+{\epsilon}\frac{{\partial}\vec{E}}{{\partial}t}[/tex] .......(3)
    [tex]\vec{\nabla}\cdot\vec{H}=0[/tex] .......(4)

    [tex]\vec{D}={{\epsilon}_0}\vec{E} - \vec{P}[/tex] .......(5)

    In the notes J and [tex]/rho[/tex] did not have the f subscript, but there was a note saying it was dropped for simplicity. In the problem, the f subscripts were there. I am going to assume they are applying directly to the symbols in the Maxwell equations (that [tex]J = J_f and \rho={\rho}_f[/tex]).

    3. The attempt at a solution

    My first confusion lies in the fact that when you simplify the Maxwell equations using the two knowns above, you end up with:

    [tex]\vec{\nabla}\times\vec{E}=-{\mu}\frac{{\partial}\vec{H}}{{\partial}t}[/tex] ........(1.b)
    [tex]\vec{\nabla}\cdot\vec{E}=0[/tex] ........(2.b)
    [tex]\vec{\nabla}\times\vec{H}={\epsilon}\frac{{\partial}\vec{E}}{{\partial}t}[/tex] .......(3.b)
    [tex]\vec{\nabla}\cdot\vec{H}=0[/tex] .......(4.b)

    But using these exact equations and the identity:

    [tex]\vec{\nabla}\times\vec{\nabla}\times\vec{E} = \vec{\nabla}(\vec{\nabla}\cdot\vec{E}) - {\nabla}^2\vec{E}[/tex] .......(6),

    the homogeneous wave equation is found:

    [tex]{\nabla}^2\vec{E} - \frac{1}{c^2}\frac{{\partial}^2\vec{E}}{{\partial}t^2}=0[/tex] ........(7)

    So I am kind of confused what I am supposed to be doing here. Did I mess up the assumption and reduce the maxwell equations too much? If not, am I trying to equate the RHS of the inhomogeneous equation to 0? I tried rearrange equation (5) for E, then plug it into the LHS of (1) and got:

    [tex]\frac{1}{{\epsilon}_0}{\nabla}^2(\vec{D}-\vec{P})-\frac{1}{{{\epsilon}_0}c^2}\frac{{\partial}^2({\vec{D}-\vec{P}})}{{\partial}t^2}[/tex]

    I separated out the terms with D and terms with E in them, using the fact that the operators are linear. And therefore had:

    [tex]\frac{1}{{\epsilon}_0}{\nabla}^2\vec{D} - \frac{1}{c^2}\frac{{\partial}^2\vec{D}}{{\partial}t^2}-\frac{1}{{\epsilon}_0}{\nabla}^2\vec{P} + \frac{1}{c^2}\frac{{\partial}^2\vec{P}}{{\partial}t^2}[/tex]


    and then in the notes, the parts with D equaled 0. It seemed to be a consequence of equation (7), and would just make is so that I had to satisfy the condition: [tex]{\nabla}^2\vec{P}={\vec{\nabla}}(\vec{\nabla}\cdot\vec{P})[/tex]

    but I couldn't get that to work out either, using the earlier vector calc identity (equation 6). So basically I am either approaching this wrong, or have made a bad assumption.
     
    Last edited: Jan 25, 2007
  2. jcsd
  3. Jan 28, 2007 #2
    Still wondering about this problem..Haven't really worked on it, since i'm kinda stumped.

    EDIT: Figured out that one of my initial equations had a slight error. Problem solved!
     
    Last edited: Jan 29, 2007
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