The radiation resistance of a Hertzian (elementary, uniform-current) dipole is

\(R_{r}=80\pi^{2}\left(\dfrac{dl}{\lambda}\right)^{2}\ \Omega\)

Substitute \(dl=\dfrac{\lambda}{80}\), so the ratio is 1/80 :

\(R_{r}=80\pi^{2}\left(\dfrac{1}{80}\right)^{2}=\dfrac{80\times9.8696}{6400}=\dfrac{789.6}{6400}=0.1234\ \Omega\)

which is option 4.

The squared dependence is the whole physics of the result. Radiation resistance falls as the square of the length in wavelengths, so a short antenna is a very poor radiator:

Why such a small value is a serious problem. Efficiency is

\(\eta=\dfrac{R_{r}}{R_{r}+R_{loss}}\)

and a real antenna's conductor and ground losses easily amount to an ohm or more. With \(R_{r}=0.123\ \Omega\) against, say, 1 Ω of loss, the efficiency would be only 11 % — almost all the power delivered would heat the wire rather than radiate. This is exactly the difficulty of AM broadcast antennas at long wavelengths and of the short whips used on hand-held VHF radios.

The second problem is matching. A fraction of an ohm of resistance in series with a large capacitive reactance is very hard to match to a 50 Ω feeder, and any matching network's own losses are then comparable with the radiation resistance itself.

Note the contrast with the half-wave dipole, whose radiation resistance is about 73 Ω — six hundred times larger. That is why resonant-length antennas are used wherever there is room for them, and why the Hertzian dipole survives mainly as the theoretical building block from which longer antennas are constructed by integration.

Hence, the radiation resistance is 0.123 Ω.