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??)
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....