Electric field of a uniformly polarized sphere

[stunner5000pt]

Homework Statement

Find the electric field of a uniformly polarized sphere of radius R

Homework Equations

$$V(\vec{r}) = \frac{1}{4 \pi\epsilon_{0}} \oint_{S} \frac{\sigma_{b}}{r} da' + \int_{V} \frac{\rho_{b}}{r} d\tau'$$

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 got to 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??)

 

[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...

 

[stunner5000pt]

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...

sorry about the slopppy notation...

i shouldve put the primes