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Laplace Equation on semi-infinite plate

  1. Aug 17, 2009 #1
    Hello all!

    I just finished the following problem:

    Consider a thin semi-infinite plate of negligible thickness made of an isotropic conductive material. A voltage V0=1V is applied at x=0 on the plate (across the short dimension). At a distance x=d=1cm from the end (x=0) V is measured to be .1V. Find the voltage V(x) at an arbitrary distance x from the end.

    In my first attempt I got V(x)=-90*x+1, which is a solution to the Laplace equation in 1D, but does not match the boundary condition at infinity.

    I tried the problem again and got V(x)=V0*10^(-x/d), which matches all boundary conditions and is the correct answer. My question is why doesn't this solution satisfy the Laplace equation? Does it have to? Why/why not?
  2. jcsd
  3. Aug 17, 2009 #2


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    The electrostatic potential only satisfies Laplace's equation in regions where the charge density is zero. The rest of the time it satisfies Poisson's equation.
  4. Aug 17, 2009 #3
    Oh dangit! Cause we're dealing with a conductor gotcha! So I guess the epsilon_0*(ln10)^2*exp(-x/d) would be the charge density as a function of x.


    *Maybe I should rename myself "the phorgetful physicist"...
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