I can see I have confused you good folks. I apologize.
The first attachment, Z.docx, shows how I form the vector between the two objects...spheres or in some cases disks. I then take the magnitude of this vector and combine like terms. Mathematica does not do it automatically in a form that is efficient...when I combine terms I get Cos[t2-t1} and thereby lose other trigonometric functions.
This vector length connects two volume elements which can contain charge. I am looking for the mutual interaction between these elemental forces integrated over the whole volumes....I don't want to use the point-charge model except to model the point-charge interactions between the elemental volumes. In cylindrical coordinates, I want to find the mutual attraction between the plates of a capacitor exactly by using areal charge densities...or I can find the mutual attraction or repulsion for volume charges, too.
Since I am using force, I also want to find the Cosine of each elemental mutual force, which by symmetry is the only net non-zero mutual force, so I multiply the inverse-square function by Cos of each incremental vector with the z-axis....this is just the Z component divide by the total vector length. Therefore, when the parameter d...the separations of the centers of each volume or areal distribution goes to infinity, the total integral goes to unity, since I am dividing by the traditional inverse square model that only considers the value of inverse d square and does go to zero.
I believe what may be happening is that the numerator and denominator integrals get so small that I am approaching dividing zero by zero...or at least dividing two very small numbers by one another. The working hypothesis is that at large separations the two integrals approach the same value. In cylindrical coordinates, things seem to work out well, but the same problem set up in spherical coordinates does not "behave," which is what I am grappling with.
I have attached a new worksheet with both the spherical and the cylindrical models for comparison.
Attachments: