Working through Purcell. In particular, doing my best to put everything on a firm footing, mathematically, so that I can understand it. But, I have struggled all week in establishing from first principles that the potential/field/distribution for a configuration of two capacitive disks of radius 1 and separation s along the same cylindrical axis is "nearly" uniform. In the book, like all the other books at this level, it seems, it is merely evinced by a field line diagram drawn according to a numerically estimated solution to the boundary value problem--none of which is developed in the text or even the exercises.
I don't care to get the exact distribution, or even a sharp estimate; I just want an estimate good enough to establish the claim that this setup, with finite disks at finite separation, does exhibit the sort of behavior of the easy-thought-experiment-examples with infinite plates or infinitesimal separation or where the plates are surfaces of nested spheres so large that the curvature is trivial.
I have attempted to get estimates many ways: using the harmonic relationship between partial derivatives and the knowledge that the field is linear at the origin; using the averaging properties of the laplacian and also the fact that the laplacian of the potential is proportional to charge density; I even successfully got (by hand!) estimates for the charge distribution and electric field on an isolated disk. The only problem in the book that seems related is that of verifying that an infinitesimally thin capacitive rod of finite length will have a uniform distribution (this is trivial when you've done the disk already; I also did it the way suggested in the book: where you divide the rod into N uniform segments and estimate the "correction" needed at location i+1 (of the order of N/2) to account for the field at i due to all other segments... but I also struggled to get this method to work for the case of two disks).
Of course, since I have a crude estimate for the electric field from a single disk, I can see quite evidently that the field lines will have a tenancy to straighten out in the region between two disks. But, I am struggling to put any quantitative measure to this notion. Any ideas?
One thing: from looking at more advanced books, it seems that you can get estimates on the solution by appealing to PDE theory and estimating the special functions involved in the solution you get. I don't care about getting a sharp estimate; is there any first principles way to do this? Feel free to apply extra assumptions--an answer that uses the words "uniform continuity" would not be off putting, if you get my drift.