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I Spherically arranged polarization

  1. Apr 26, 2017 #1
    If we arrange the polarization spherically, Does it make a uniformly charged sphere radius of R?? If right, How can I find out \vec{P}(\vec{r}) which results in the constant charge density \rho?
     
  2. jcsd
  3. Apr 27, 2017 #2

    Charles Link

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    You can solve the divergence equation in spherical coordinates to produce a ## \rho_p ## that is constant everywhere, where ## -\nabla \cdot P=\rho_p ##, but polarization still creates an electrically neutral system, so that the surface polarization charge density of ## \sigma_p=P \cdot \hat{n} ## will be such that it neutralizes any positive ## \rho_p ## in the interior. Solving the divergence equation, you get ## P_r(r)=-(\rho_p/3)r ##. ## \\ ## ## \int \rho_p dv=+(4/3) \pi R^3 \rho_p. ## ## \\ ## Now ## P_r=-(\rho_p/3)R ## at ## r=R ##, so that ## \sigma_p=-(\rho_p/3)R ##. ## \\ ## ## \int \sigma_p dA=-(\rho_p/3)R(4 \pi R^2) ## precisely neutralizing ## \int \rho_p dv ##.
     
    Last edited: Apr 27, 2017
  4. Apr 28, 2017 #3
    very thanks!!
     
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