Art wrote:
"The phrase within or on the surface is somewhat inaccurate."

At risk of boring some to tears, please consult the 1955 opus of F.E.
Terman, pages 865 and 903 for radiation patterns of elemental and
1/2-wave doublets when currents flow in an axial direction on a
conductor. Recall that the conductor can be ever so thin and that
reciprocity rules in antennas.

Best regards, Richard Harrison, KB5WZI