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Electric field of a uniformly polarized sphere 
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#1
Dec2106, 11:24 PM

P: 1,440

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??) 


#2
Dec2206, 03:20 AM

P: 73

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.... 


#3
Dec2306, 02:22 PM

P: 1,440

i shouldve put the primes 


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