According to IRCA reprint A009 a matched condition exists when when the ratio of length of the ferrite rod to its diameter equals the square root of the rod's permeability.

So for

a mu of 125 and a diameter of .5" the matched length is 5.6"

a mu of 800 and diameter of .5" the matched length is 14.1"

a mu of 2000 and diameter of 1" the matched length is 44.7"

The author says a matched condition occurs when the "Available Flux Field" (AFF) - that is, the flux flowing into an air loop of effectively the same size as the ferrite loop - is equally divided between the rod and the air surrounding the rod.

While trying to understand more how this works, I found out magnetism has more or less an ohm's law. Instead of resistance you use reluctance. The reluctance of a ferrite rod is 1/(mu*pi/4*D^2) while the reluctance of the air within the AFF but not in the rod is 1/(pi/4*(L^2-D^2). The flux
(which is analogous to current) flowing into the AFF is the flux density of air times pi/4*L^2.

Using this information and solving some equations, I calculated a few plots. The first two plots below show the ratio of the magnetic power - I think this exists - flowing into the rod of given length (flux_into_rod^2 * reluctance_of_rod) to the magnetic power flowing into an air loop of effectively the same size as the rod (flux_into_aff^2 * reluctance_of_aff). A matched condition in effect gives you the best magnetic power ratio. Once past this length you hit a point of diminishing returns.

http://img178.imageshack.us/img178/5495/screenshot1oe.png http://img411.imageshack.us/img411/235/screenshot2djm.pngFlux however continues to increase and doesn't flatten out till later. Another plot shows this.

http://img192.imageshack.us/img192/5138/screenshotaz.pngGeorge