As you mentioned, Thevenin does apply, and your effective area formula
is correct (gain w.r.t. isotropic radiator). So a thin halve wave
dipole antenna has an internal resistance of about 70 Ohms, and half
the power is lost, but not as heat in the antenna.

If I may add just a simple observation to the exhaustive Wim's explanation, I
would say that we generally tend to consider resistors as things that
necessarily dissipate power, turning it into heat. That is not always the case.



As a matter of fact, having current in phase with voltage (what we usually call
resistance) signify the transformation of electrical power into any form of
power, not just heat. For instance:



- an ideal electrical engine (no ohmic loss, no friction) fully
transforming the applied electrical power into mechanical power. It looks like a
resistor, but no heat is generated anywhere

- an ideal antenna (no ohmic loss) fully transforming the applied
electrical power into electromagnetic power. Again it looks like a resistor, but
no heat is generated anywhere

..

73



Tony I0JX