How Do You Calculate the Electric Potential of a Polarized Half-Cylinder?

[ShayanJ]

Consider a cylinder(height $l$, radii a) cut in half by a plane parallel to its axis(z-axis). Now imagine it has a polarization density $\vec P=P_0 \hat z$ and so its two semi-circular surfaces have bound charge densities $\sigma_u=-\sigma_d=P_0$. I want to calculate the electric potential of these charge densities at an arbitrary point of space.
The charge elements is clearly (using cylindrical coordinates) $dq=\sigma \rho d\varphi d\rho$(with $0\leq \rho \leq a$ and $0\leq \varphi \leq \pi$).
Now I should find out the distance between an arbitrary point on the semi-circular surface of the cylinder and an arbitrary point in space($(R,\phi,Z)$) which can be written as $\sqrt{R^2+Z^2+\rho^2+(\frac l 2)^2-2\sqrt{(R^2+Z^2)(\rho^2+(\frac l 2)^2)} \cos\gamma} \ \$ where $\gamma$ is the angle between the two position vectors. We can write(using spherical coordinates) $\cos\gamma=\cos \theta \cos\vartheta +\sin\theta \sin\vartheta \cos(\varphi-\phi)$. So I can put this into the distance formula and write the integral I should calculate. But the problem is, now I have an integral w.r.t. cylindrical coordinates that contains some spherical coordinates. So I should either transform the integral to spherical coordinates or write the spherical coordinates in terms of cylindrical coordinates. I figured that the second option turns the integral into an intractable mess so I want to pursue the first option(Then I can use Legendre polynomials and spherical harmonics to do the integral). I calculated the Jacobian of the transformation from cylindrical to spherical coordinates(its $\frac 1 r$ where $r=\sqrt{\rho^2+(\frac l 2)^2}$ is the radial component of the spherical coordinates of the point on the cylinder ). But I don't know how should I do the transformation. I'm confused here. Can anyone help?
Thanks

 

[zoki85]

If I wanted to know potentials of such charge distributions I would use software package like Ansoft Maxwell.