# Electric field of a uniformly polarized sphere

by stunner5000pt
Tags: electric, field, polarized, sphere, uniformly
 P: 1,443 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 $$V(\vec{r}) = \frac{1}{4 \pi\epsilon_{0}} \oint_{S} \frac{\sigma_{b}}{r} da' + \int_{V} \frac{\rho_{b}}{r} d\tau'$$ 3. The attempt at a solution well obviously there is no volume charge density rho but there is a surface charge density $$\sigma_{b} = P \cos\theta$$ now to calculate the potentail we gotta use that above formula Suppose r > R then $$V(\vec{r}) = \frac{1}{4 \pi\epsilon_{0}} \int \frac{P \cos\theta}{r} da'$$ now the squigly r is found using the cosine law right...? $$r = \sqrt{R^2 + r^2 - 2Rr\cos\theta}$$ and $$da' = R^2 \sin\theta d\theta d\phi$$ 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??)
 P: 73 Squigly r ?? Did you mean $$\tilde{r}$$ ??? Your solution is basically correct, but you have abuse the usage of $$\theta$$. Notice the $$\theta$$ in $$\tilde{r} = \sqrt{R^2 + r^2 - 2Rr\cos\theta}$$ is represecting the angle between r and R. It is not the same $$\theta$$ in the rest of your equations... you should not treat it like a variable and integrate over it....
P: 1,443
 Quote by chanvincent Squigly r ?? Did you mean $$\tilde{r}$$ ??? Your solution is basically correct, but you have abuse the usage of $$\theta$$. Notice the $$\theta$$ in $$\tilde{r} = \sqrt{R^2 + r^2 - 2Rr\cos\theta}$$ is represecting the angle between r and R. It is not the same $$\theta$$ in the rest of your equations... you should not treat it like a variable and integrate over it....