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Potential inside (and outside) a charged spherical shell

  • Thread starter russdot
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[solved] Potential inside (and outside) a charged spherical shell

1. The problem statement, all variables and given/known data
Use the integral (i) to determine the potential V(x) both inside and outside a uniformly charge spherical surface, with total charge Q and radius R.


2. Relevant equations
(i) V([tex]\vec{x}[/tex]) = [tex]\frac{1}{4\pi\epsilon_{0}}\int\frac{\rho(\vec{x'})d^{3}x'}{|\vec{x}-\vec{x'}|}[/tex]
[tex]\rho(\vec{x'}) = \frac{Q}{4\pi R^{2}}[/tex]


3. The attempt at a solution
if the x vector is along the z axis (can exploit spherical symmetry), then |x-x'| = [tex]\sqrt{x^{2} + R^{2} - 2xRcos\theta}[/tex]. (using cosine law)
Then
V([tex]\vec{x}[/tex]) = [tex]\frac{1}{4\pi\epsilon_{0}}\frac{Q}{2}\int_{0}^{\pi}\frac{sin\theta d\theta}{\sqrt{x^{2} + R^{2} - 2xRcos\theta}}[/tex]

V(x) = [tex]\frac{Q}{4\pi\epsilon_{0}x}[/tex]
or
V(r) = [tex]\frac{Q}{4\pi\epsilon_{0}r}[/tex]
Which is correct for outside the spherical shell, but I can't figure out how |x - x'| would be any different (or what else should be different) for inside the spherical shell to get
V(r) = [tex]\frac{Q}{4\pi\epsilon_{0}R}[/tex]

I realize there are better/easier ways of doing this, but the question says to use that integral specifically...
 

gabbagabbahey

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Shouldn't you have:

[tex]V(x)=\frac{1}{4\pi\epsilon_{0}}\frac{Q}{2} \int_0^{x} \int_{0}^{\pi }\frac{sin \theta ' dx' d \theta '}{\sqrt{(x')^{2} + R^{2} - 2(x')Rcos \theta '}}[/tex]

I understand that the integration over [itex]\phi '[/itex] gave you [itex]2 \pi[/itex] , but what happened to the integration over [itex]x'[/itex]?
 
Well [tex]d^{3}x'[/tex] turns into [tex]r^{2}sin\theta dr d\theta d\phi[/tex] and since r is constant at R (spherical shell) then an R^2 comes out of the integral and cancels the R^2 in the denominator from the charge density rho = Q / (4 pi R^2).

I also figured out the problem, after integration:
[tex]V(x) = \frac{Q}{4\pi \epsilon_{0}} \frac{1}{2xR} [\sqrt{(R + x)^{2}} - \sqrt{(R - x)^{2}}][/tex]
and I forgot to consider the different cases for [tex]\sqrt{(R - x)^{2}}[/tex] when x > R (outside spherical shell) and x<R (inside).
 
Last edited:

gabbagabbahey

Homework Helper
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Since your dealing with a surface charge, and not a volume charge, you should have:

[tex] V(\vec{x})=\int \frac{\sigma (\vec{r'}) d^2 x'}{|\vec{x}-\vec{x'}|}[/tex]

where [itex]\sigma (\vec{r'})= Q/4 \pi R^2 [/itex] is the surface charge density.
 
Yes, you're right. The form I used was ambiguous, sorry about that!
I guess I just jumped right to using [tex]R^{2}sin\theta' d\theta' d\phi[/tex] as the surface area element.
I have solved the problem though, thanks for the assistance!
 

gabbagabbahey

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No problem, but I wasn't much assistance:smile:
 

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