 Solution

For a Hertzian dipole of length $\ell$ driven at peak electric current $I_0$, the electric dipole moment amplitude is $p_0=I_0\ell /\omega$. ConsequentlyThe Hertzian dipole fixed-current power scaling is thus $\overline{P}\propto I_0^2{\ell }^2/{\lambda }^2$, givingThe inverse-fourth-power wavelength scaling would hold at fixed electric dipole moment, not at the fixed electric current specified here.