Electric field of a uniformly polarized sphere

  1. 1. The problem statement, all variables and given/known data
    Find the electric field of a uniformly polarized sphere of radius R

    2. Relevant equations
    [tex] V(\vec{r}) = \frac{1}{4 \pi\epsilon_{0}} \oint_{S} \frac{\sigma_{b}}{r} da' + \int_{V} \frac{\rho_{b}}{r} d\tau'[/tex]

    3. The attempt at a solution
    well obviously there is no volume charge density rho
    but there is a surface charge density
    [tex] \sigma_{b} = P \cos\theta [/tex]

    now to calculate the potentail we gotta use that above formula
    Suppose r > R

    then
    [tex] V(\vec{r}) = \frac{1}{4 \pi\epsilon_{0}} \int \frac{P \cos\theta}{r} da' [/tex]

    now the squigly r is found using the cosine law right...?

    [tex] r = \sqrt{R^2 + r^2 - 2Rr\cos\theta} [/tex]
    and
    [tex] da' = R^2 \sin\theta d\theta d\phi [/tex]
    is that right???
    and the limits of integrate for the theta would be from 0 to pi
    and for the phi is 0 to 2pi??

    thanks for your help
    (o by the way how do i put the squigly r??)
     
    Last edited: Dec 21, 2006
  2. jcsd
  3. Squigly r ??
    Did you mean [tex]\tilde{r}[/tex] ???

    Your solution is basically correct, but you have abuse the usage of [tex]\theta[/tex]. Notice the [tex]\theta[/tex] in [tex] \tilde{r} = \sqrt{R^2 + r^2 - 2Rr\cos\theta} [/tex] is represecting the angle between r and R. It is not the same [tex]\theta[/tex] in the rest of your equations... you should not treat it like a variable and integrate over it....
     
    Last edited: Dec 22, 2006
  4. sorry about the slopppy notation...

    i shouldve put the primes
     
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