Electric field of a uniformly polarized sphere

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:
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:
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....
sorry about the slopppy notation...

i shouldve put the primes
 

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