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Homework Help: Finding a solution to Maxwell's equations from initial datas

  1. Apr 15, 2010 #1

    fluidistic

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    Gold Member

    1. The problem statement, all variables and given/known data
    Suppose we know that [tex]B(\vec x ,t)[/tex] is a solution to Maxwell's equations in vacuum and furthermore we know that [tex]E(\vec x , 0)=E_0[/tex].
    How do we find [tex]E(\vec x , t)[/tex]?

    2. Relevant equations
    [tex]\nabla \cdot E = 0[/tex].
    [tex]\nabla \cdot B =0[/tex].
    [tex]\vec \nabla \times \vec B = \left ( \frac{-1}{c} \right ) \cdot \frac{\partial E}{\partial t}[/tex]
    [tex] \vec \nabla \times \vec E = \left ( \frac{1}{c} \right ) \cdot \frac{\partial B}{\partial t}[/tex].
    I'm using Gaussian's units.
    3. The attempt at a solution
    I think I could work with the 2 lasts equations I posted to find E but I don't reach anything. I'd like a very small guidance like if I'm in the right direction + a hint if possible.
    Thanks.
     
  2. jcsd
  3. Apr 15, 2010 #2
    Think about integrating the 3rd equation with respect to time.
     
  4. Apr 15, 2010 #3

    fluidistic

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    Thanks for the tip.
    I reach [tex]\vec E=-c \int \vec \nabla \times \vec B dt[/tex]. I think of using the initial condition. So [tex]E_0=-c\int \vec \nabla \times \vec B (\vec x ,0) dt[/tex].
    I'm stuck here.
     
  5. Apr 16, 2010 #4
    The integral will have bounds.
     
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