Surface Integral of a sphere

  • #1
TheGreatDeadOne
22
0
Homework Statement
Let a sphere of radius ##r_0## be centered at the origin, and ##\vec{r'}## the position vector of a point p' within the sphere or under its surface S. Let the position vector ##\vec{r}## be an arbitrary fixed point P. With this information, solve the integral, and analyze the cases for ##r\geq r_0## and ##r\leq r_0##
Relevant Equations
$$I=\oint_{s} \frac{1}{|\vec{r}-\vec{r'}|}da'$$
Solving the integral is the easiest part. Using spherical coordinates:

$$ \oint_{s} \frac{1}{|\vec{r}-\vec{r'}|}da' = \int_{0}^{\pi}\int_{0}^{2\pi} \frac{1}{|\vec{r}-\vec{r'}|}r_{0}^2 \hat r \sin{\theta}d\theta d\phi$$

then:

$$I = \dfrac{1}{|\vec{r}-\vec{r'}|}r_{0}^2(1+1)(2\pi)\hat r=\dfrac{4\pi r_{0}^2}{|\vec{r}-\vec{r'}|}\hat r $$

But what I couldn't understand was the final answer which is:

##\frac{4\pi r^{2}_{0}}{r} \quad\text{for} \quad r\geq r_0 \quad and \quad 4\pi r^{2}_{0}\quad for \quad r\leq r_0 ##

What happened to ##\frac{1}{|\vec{r}-\vec{r'}|} \hat r## for both ## r\geq r_0## and ## r\leq r_0 ## ? I know that for r -> 0 the integral explodes -Dirac enters here? -, but I don't know how this can help to reach the final result and I was very confused by the analysis for ## r\geq r_0## and ## r\leq r_0 ## part.
 
Last edited:
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  • #2
You are making mistakes based on not using proper vector notation for vectors. Please indicate vector quantities using \vec as so \vec r = ##\vec r##.
 
  • #3
Also, please give the full problem statement verbatim. Your statement does not actually ask a question so it is unclear wether the integral you present is something given by the problem or something you have arrived at.
 
  • #4
Yes, they are vectors, I thought it was implicit in the question. But I still don't understand your point in saying "mistakes".

That integral at the beginning is given in the problem.
 
  • #5
TheGreatDeadOne said:
Yes, they are vectors, I thought it was implicit in the question. But I still don't understand your point in saying "mistakes".
Please fix the vector notation first. Without it it is impossible to tell exactly which mistakes you are making. No, it is not always implicit and you are sometimes treating what should be a vector quantity as if it was a vector norm. It is therefore difficult to tell if it is a problem of setup or of computation.

Also see #3
 
  • #6
Ok, edited. The vectors within the module would be the separation vector ## R = |\vec{r} - \vec{r'}|##.
 
  • #7
Ok so here are some points:

You are treating ##da’## as a directed area. It is a scalar area so you need to get rid of the ##\hat r##

You cannot integrate like that. ##\vec r’## depends on the angles (so does ##\hat r##, but it should not be there in the first place).
 
  • #8
Orodruin said:
Ok so here are some points:

You are treating ##da’## as a directed area. It is a scalar area so you need to get rid of the ##\hat r##

You cannot integrate like that. ##\vec r’## depends on the angles (so does ##\hat r##, but it should not be there in the first place).
Ohhhhh I focused a lot on the radial (fixed) part that I forgot about the azimuthal and polar dependence.

edit: misspelling
 
Last edited:
  • #9
I'll try to solve it tomorrow and post it here. Corrected some spelling errors (typed on cell phone)
 
  • #10
TheGreatDeadOne said:
I'll try to solve it tomorrow and post it here. Corrected some spelling errors (typed on cell phone)
Good luck! Choose the z-axis of your spherical coordinates wisely!
 
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