I am integrating a hollow cone (point at origin) to get the electric potiential at a point b at the center of the cone (at a height h --also assume the radius of the circular top to be length h at this height). Assume a uniform surface charge density $\sigma$ I want to find the electric potiential at point b.
I am aware of the "ring slice method" and my issue is not with what is the final value but why is it that using spherical coordinates for integration does not work? I have tried converting the position vector to using cartesion base vectors and using still spherical coordinate variables but I still do not get the right answer. To make things clear I have tried utilizing the following integrals using spherical coordinates;
$\frac{\sigma \sin (\theta ) \left(\int \frac{r'}{r-r'} \, dr\right)}{2 \epsilon }$
where $\theta$ is 45 degrees and r=h (the point b)
*note some simplification took place, I thought this would be fine since the integrand is not a vector but it appears not to be so, since the answer does not come out right. So next I thought maybe the vector nature of "R=r-r'" is somehow affecting the so thats when I tried,
putting things in terms of the cartesion base vector form, that is,
$r \sin (\theta ) \sin (\phi ) \hat{x}+r \sin (\theta ) \cos (\phi ) \hat{y}+r \cos (\theta ) \hat{z}$
So that my integral becomes
$\frac{\sigma \sin (\theta ) \left(\int_0^{\sqrt{2}h} \frac{r}{\sqrt{\left(r-r'\right)^2+\left(r'\right)^2}} \, dr\right)}{\sqrt{2} \epsilon }$
which is pretty close to a step in the solution manual however not quite, the funny thing is that if I use the substitution $r'=l/\sqrt{2}$ I get the same form as the solution value. This does not make sense though because I am already integrating the hypotenuse variable which is the longest side of the triangle, so what could $l$ represent!
I am very confused and could use some help. I hope I explained things clearly enough but if need be I can attach pictures.
P.S. sorry I did not include the limits of integration, my knowledge on latex is not very good and this is the best I could do. The first integral I already have calculated (the 2*Pi result) and simplified, the limits in the remaining integral should be from 0 to $\sqrt{2}*h$