Potential near the center of a charged hollow sphere

[Gene Naden]

I worked problem 2.28 of Nayfeh and Brussel's Electricity and Magnetism. The problem asks for the potential near the center of a charged hollow sphere, based on the near-field expansion given by equation 2.62, which is:
$\Phi=\frac{1}{4\pi\epsilon_0}[\frac{dq}{r^\prime}+ \vec r \cdot \int \frac{\vec r^\prime}{{r^\prime}^3} dq + \frac{1}{2} \int (\frac {3 (\vec r \cdot \vec r^\prime)^2}{r^{\prime^5}}-\frac{r^2}{r^{\prime 2}})dq]$

For the first term I get something proportional to the charge. The second integral vanishes by symmetry. I expect the third integral to also vanish. The solutions manual says that it does vanish. But when I do the integration I get a dependence on $\theta$, the angle from the z axis in polar coordinates. I get:
$\frac{\sigma R}{\epsilon_0}\frac{1}{4}R^{-2} r^2(1-rcos(2\theta))$

where $\vec r$ is the vector to the point where the potential is calculated.

This seems physically wrong since by symmetry the potential should not depend on $\theta$. My question is, did I do the integral wrong, or is equation 2.62 wrong/inapplicable, or is my reasoning wrong?

Thanks.

 

[tnich]

It's hard to tell where you went wrong if you don't show your work.

 

[Gene Naden]

Yes I see your point. I see my error. I used a two dimensional diagram for a three dimensional problem.