(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Integrate [itex]\nabla^{2}[/itex] ([itex]\frac{1}{|\underline{r} - \underline{r}'|}[/itex]) over the volume of a sphere using the divergence theorem.

2. Relevant equations

[itex]\nabla^{2}[/itex] ([itex]\frac{1}{|\underline{r} - \underline{r}'|}[/itex]) = -4[itex]\pi[/itex][itex]\delta^{(3)}[/itex][itex](\underline{r} - \underline{r}')[/itex] (i.e. it's a dirac delta function, this just tells me that the answer I should get if I integrate over the volume is -4[itex]\pi[/itex])

[itex]\nabla^{}[/itex] ([itex]\frac{1}{|\underline{r} - \underline{r}'|}[/itex]) = [itex]-\frac{(\underline{r} - \underline{r}')}{|\underline{r} - \underline{r}'|^{3}}[/itex]

[itex]\underline{r} = (x,y,z)[/itex]

[itex]\underline{r'} = (x',y',z')[/itex] (note that the dashes do not mean the differential of, just a different point in space from x,y,z).

And the divergence theorem

3. The attempt at a solution

[itex]\int_{V'}[/itex][itex]\nabla^{2}[/itex] [itex](\frac{1}{|\underline{r} - \underline{r}'|} )dV'[/itex]

=

[itex]\int_{S'}[/itex][itex]\nabla[/itex] [itex](\frac{1}{|\underline{r} - \underline{r}'|} ) \circ d\underline{S}'[/itex] (by the divergence theorem)

[itex] d\underline{S}' = \hat{\underline{n}} dS' [/itex]

[itex] dS' = R^{2}sin(\theta)d\theta d\phi [/itex]

I guess R should be the radius of the sphere. In which case I choose r' = R?

= [itex]\int_{S'}-\frac{(\underline{r} - \underline{r}')}{|\underline{r} - \underline{r}'|^{3}} \circ d\underline{S}'[/itex]

Then, putting r' = R

[itex]\int_{S'}-\frac{(\underline{r} - \underline{R})}{|\underline{r} - \underline{R}|^{3}} \circ \hat{\underline{n}} dS'[/itex]

=

[itex]\int^{2\pi}_{0} \int^{\pi}_{0} -(\frac{(\underline{r} - \underline{R})}{|\underline{r} - \underline{R}|^{3}} \circ \hat{\underline{n}}) R^{2}sin(\theta)d\theta d\phi[/itex]

Am I right so far? This looks like a complete mess, and I'm not sure what to do with the dot product of the vector and the normal vector?

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# Homework Help: Integrate this function over the volume of a sphere sphere

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