here is the link for this page thank to w0btu

Optimum Ground Radial Wire Length
vs. Number of Ground Radials

ARRL ANTENNA BOOK 20th Edition, Chapter 3
(The Effects of the Earth), p. 3-10, Table 1

Ground radials under an end-fed antenna (such as a vertical monopole, inverted-L, or a "long wire") are important!

However, as you can see, there is little to be gained by using over 60 radials.

What is NOT shown in the above chart is how seriously the loss increases as we use fewer than 16 radials. Even with only 16 radials, we lose half of our transmitter power (3 dB) in ground losses.

Look at what happens with only a couple of long radials:

Ground Radial length vs. number of radials

If radials are elevated (rather than stapled to the earth), we can greatly reduce the number required. But they are still needed!

Relevant links:
newEvidence that just FOUR elevated radials (properly isolated from the earth) can equal the efficiency of 120 buried radials.
160 Meters
forums.qrz.com ... Ground-Radials-for-an-Inverted-L-Question

Verticals Without Grounds?

No one has ever designed an end-fed antenna (such as a bottom-fed vertical or "long-wire") that works good without some form of counterpoise. It needs a good RF ground, and a ground rod or water pipe does NOT meet that criteria.

The exceptions might be a properly designed coaxial dipole or center-fed 1/2λ dipole on its side.
Of all the physical laws that there are, none are more well-established than the fact that a bottom-fed vertical (vertical monopole) or an inverted-L needs something to "push against" to be effective. Call that 'something' a counterpoise, radials, or whatever you want to; but to be effective, it's got to be there. Period.
K5UJ put it nicely when he said "The reason for radials is to collect RF currents and return them to the feedpoint since you don't have the other half of the antenna to do that, the part that would make it a dipole. If you didn't have the radials you'd have earth resistance trying to do it and you'd have a nice wide 50 ohm antenna and it would stink because ground is a lousy conductor."

Case in point: someone recently shared his experience after he added a bunch of ground radials to his inverted-L. It was a WOW! experience for him. Sure, he made some contacts without the radials. But what a difference it made in the amount of RF that radiated into space and the amount of DX he worked afterward.

If we make the statement that adding a counterpoise to a λ/4 vertical (for example) is not really necessary, then we might as well say that opening the blinds won't make the room any brighter. Sure, maybe we can see our way around, but when we let the sun in, life is so much better. Or someone could say that we don't really need tires on a car. Sure, with enough ground clearance we can drive around on the rims, but isn't the car much more fun to drive with the rubber attached? Likewise, a proper RF ground makes the radiated signal stronger while preventing unwanted common-mode currents on the outside of the coax shield!
Some verticals have an RF ground built-in, like a 1/4λ ground plane, and they don't need a external ground. But take the radials off your ground plane and let us know what happens. :-)

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