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Homework Help: Polarized sphere field

  1. Jun 17, 2012 #1
    1. The problem statement, all variables and given/known data
    A sphere with radius R carries a uniform polarization in the radial direction P = k/r2
    Find the bound charges and calculate the field they produce outside the sphere.

    2. Relevant equations
    ρ = -∇[itex]\bullet[/itex]P
    σ = P[itex]\bullet[/itex][itex]\hat{r}[/itex]

    3. The attempt at a solution
    We find σ = k/r2
    ρ = 0
    So the field they produce is E = kR2/(ε0r2)

    But this can't be true! How can a polarization create a field outside the sphere? Something is definitely wrong. Because we also have:
    D = ε0E
    And using a Gaussian surface you find:

    And I am confident in the last result. There is something disappearing in the above - where have all the charges of opposite sign gone? I believe it has to do with the mathematics behind ∇[itex]\bullet[/itex]k/r2 = 0. For that the pesky delta function is involved in that I believe. If ρ=0 we also find no field inside the sphere. Can that really be true?
  2. jcsd
  3. Jun 17, 2012 #2


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    Homework Helper

    Let me check a couple of things: The polarisation is k/r2 in the radial direction, so how is it uniform polarisation? Also, I agree with your answer E = kR2/(ε0r2) due to the bound charges. Why do you think it is wrong?
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