I am working through the design of a ferrite core antenna with a
necked down center section. I am trying to calculate how much
permeability is needed to get most of the available signal out of the
antenna, compared to that needed for a cylindrical rod.

I think that for a full cylinder of material to be a lower reluctance
path than the same volume of space, the relative permeability just has
to be higher than 1. If it is much more, then considerable additional
flux will be gathered around the ends of the cylinder (that would not
have passed through the cylinder's volume of free space).

If the rod consists of large diameter ends and a thinner central
section, then the permeability of the long center section must be
higher by the ratio of the (end area)/(center area), for the same
assumptions to hold.

For instance, the present design has end diameters of 2.4 inches and a
center diameter of 0.5 inches. So the center section needs a
permeability (2.4^2)/(.5^2)=23 times higher than would be needed for a
solid 2.4 inch diameter cylinder the same overall size to pass about
the same flux in a given field.

Is this reasonable?