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Derivation of Poisson's Equation and Laplace's Equation

  1. Feb 23, 2010 #1
    Hi,

    Can someone point me in the right direction to a derivation of Poisson's Equation and of Laplace's Equation, (from Maxwell's equations I think) both in a vacuum and in material media?

    How does one get from Maxwell's equations to Poisson's and Laplace's?
     
  2. jcsd
  3. Feb 23, 2010 #2

    Nabeshin

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    Science Advisor

    So the two relevant maxwell's equations are:
    [tex] \vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon} [/tex]
    [tex] \vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} [/tex]

    For an electrostatic system, there is no changing B field so,
    [tex] \vec{\nabla} \times \vec{E} = 0 [/tex]
    Which implies E can be written as the gradient of a scalar potential,
    [tex] \vec{E} = - \vec{\nabla} \varphi [/tex]

    Combining this fact with the first equation,
    [tex] \vec{\nabla} \cdot \vec{\nabla} \varphi = \nabla^2 \varphi = - \frac{\rho}{\epsilon} [/tex]

    And of course Laplace's equation is the special case where rho is zero.

    Cheers!
     
  4. Feb 24, 2010 #3
    Ah, thank you very much. That's not so bad after all.
     
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