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[sigma](r) = Q/(2*[pi]*R^2*sqrt(1 - (r/R)^2))
where r ranges over the interior of the disk, i.e. 0 <= r <= R . Note
the singularly divergent charge density on the outer edge of the disk.
I know that many people are able to read physics from
your equations. Unfortunately, I am not one of them.
I'm sorry about the cumbersome ASCII notation. That
makes the reading and comprehending job much more
difficult than with real mathematical symbols.
What you call S(r/R) I called [sigma](r) above (i.e. it's
the surface charge density per actual area, not per reduced
area in units of R^2).