Here is a very good article written by Rudy Severns, N6LF.
http://www.antennasbyn6lf.com/files/radiation_resistance_variation_with_radial_system_design.pdf
It answers many questions hams have about vertical antennas and how the number of and length of radials influence the radation resistance of vertical antennas made for the 40 M band.
Ground resistance soil conditions are also taken into account and discussed.
NEC models are used.
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