Electric field of a uniformly polarized sphere

stunner5000pt

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??

(o by the way how do i put the squigly r??)

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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....

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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....

i shouldve put the primes

"Electric field of a uniformly polarized sphere"

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