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Electric field inside a polarized sphere

  1. Nov 26, 2012 #1
    1. The problem statement, all variables and given/known data

    A sphere of radius R carries a polarization
    [itex]\vec{P}[/itex]= k[itex]\vec{r}[/itex],

    where k is a constant and [itex]\vec{r}[/itex] is the vector from the center.

    Find the field inside and outside the sphere.

    In solution, the field outside sphere is 0.

    I interpreted that as the field produced by the polarization charges.

    Their sum is 0. So, the outside field is zero.

    My problem is inside sphere.

    Can i consider always the electric field inside a sphere as -[itex]\frac{1}{3εo}[/itex]*[itex]\vec{P}[/itex]

    Because the formula -[itex]\frac{1}{3εo}[/itex]*[itex]\vec{P}[/itex] is obtained by a uniformly polarized sphere.

    In the exercise [itex]\vec{P}[/itex] is not uniform.
  2. jcsd
  3. Nov 26, 2012 #2

    rude man

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    Since there is no free charge anywhere inside the sphere, I would think the E field, by Gauss, would be zero everywhere inside the sphere.
  4. Nov 26, 2012 #3


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    Not sure where you're getting that expression. The polarization produces a bound charge density inside the sphere: ##\rho_b = -\vec{\nabla}\cdot \vec{P}##

    From the bound charge you can get the field using Gauss' law.
  5. Nov 27, 2012 #4
    So you are telling me that [itex]\vec{E}[/itex](A) = [itex]\frac{∫-∇.\vec{P}dv}{εo}[/itex]
    Last edited: Nov 27, 2012
  6. Nov 27, 2012 #5


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    Yes (although I'm not real clear on your notation in this equation). See what you get for the bound charge density as a function of ##r## and then use it in Gauss' law to find the magnitude of the field inside, ##E(r)##, as a function of ##r##.
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