The best answer I've seen, based on current research is:

N=(SQRT(2*PI*L))/A

N equals the square root of the quantity 2*PI*L divided by A, whe

N = optimum number of radials for high efficiency
L = amount of wire available in meters
A = distance between wire tips at the far end (a measure of radial density)

A = 1.3 for 95% or greater efficiency (approximation) read article for more
precise values.
A = 2.6 for 85% efficiency (approximation)

Example 1:

You have 500 meters of wire available. How many and how long for the two
values of A?

N = (SQR(2*pi*500))/1.3 = 43 radials. Length = 500/43 = 11.6 meters

N= (SQR(2*pie*500))/2.6 = 22 radials. Length = 500/22 = 18 meters

Example 2: (you have space constraints and the max radial length available
is 15 meters)

How many radials are required and how much wire is required?

The circumference of a circle with a radius of 15 meters is 2*PI*15 or 94.2
meters. With the tips of the radials seperated by 1.3 meters we have
94.2/1.3 = 72 radials. If we go for slightly more loss, we have 94.2/2.6 or
36 radials, 15 meters long.

Now you can plug in your own limitations for radial length, and get a feel
for how many of them you will need for 95% efficiency (A=1.3 meters) and 85%
(A=2.6 meters).

This info was presented in some ARRL publication, as I recall, and is also
presented in the 4th Edition of "Low-Band DXing" by John Devodere.

....hasan, N0AN

"Walter Maxwell" wrote in message
...
On Sun, 4 Sep 2005 22:24:24 -0500, (Richard
Harrison) wrote:

Reg, G4FGQ wrote:
"The rods are both wasted effort and wasted copper (or aluminum)."

Likely so.

Radials are placed to capture displacement current to and from the
vertical radiator, to prevent its travel at a high densitY in the earth
where it would cause high loss.

B, L, & E found that more radials were better than longer radials. More
radials put the displacement current capture closer to the vertical
radiator where its density is higher. Radials need extend outward only
as far as there is any current or until a point of diminishing returns
is reached. At great distance from the vertical radiator, the earth`s
crust, which may be affected, has a great cross section, so current
density is low and so are losses. There is much less displacement
current to deal with near the ends of radials. Displacement current is
low near the ends of the radials and the earth out there has a large
cross section and a low resistance. Due to skin effect, the closer to
the surface, the higher the current. This is especially true at high
frequencies.

Hi Richard,

You've just presented the best abstract of BL&E I've seen ever seen.
It should be must reading for anyone who asks questions concerning the
purpose of radials, how many, and how long.

Walt, W2DU