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E Field Drop Exponentially

  1. Jan 29, 2008 #1
    1. The problem statement, all variables and given/known data
    Can an electric field drops exponentially? (in free space)

    2. Relevant equations
    Starting from a hypothetical potential, [tex]V = F(x,y,z)e^{-y^2}[/tex] which decays exponentially in the y direction

    3. The attempt at a solution
    [tex]\nabla^2V=e^{-y^2}(\frac{\partial^2F}{\partial x^2}+\frac{\partial^2F}{\partial z^2}+\frac{\partial^2F}{\partial y^2}-4y\frac{\partial F}{\partial y}+(4y^2-2)F)=0[/tex]
    Using separable variable, the general solution of [tex]\frac{\partial^2F}{\partial y^2}-4y\frac{\partial F}{\partial y}+(4y^2-2)F=0[/tex] has the form [tex](A+By)e^{y^2}[/tex], assuming that all general solutions of the above equation can be expressed as a linear combination of the product of the 3 individual solution of the separable variable, we can see that the exponential term will cancel out.

    I am not sure how to proceed from here, is this correct? does it mean we cannot have the exponential term in either potential or e field?
    Last edited: Jan 29, 2008
  2. jcsd
  3. Jan 29, 2008 #2
    You forgot the separation constant, i.e.

    [tex]\frac{\partial^2F}{\partial y^2}-4y\frac{\partial F}{\partial y}+(4y^2-2)F=k^2[/tex]
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