Hertzian dipole fixed-current power scaling ($\overline{P}\propto I_0^2{\ell }^2/{\lambda }^2$)

A uniform electrically short current element of length $\ell$ with sinusoidal peak current $I_0$ has electric dipole moment amplitude $p_0=I_0\ell /\omega$. The mean electric dipole radiation power is ${\omega }^4{p}_0^2/(12\pi {ϵ}_0{c}^3)$, so fixing current makes it proportional to ${\ell }^2{\omega }^2$, rather than ${\ell }^2{\omega }^4$. Current, charge excursion and dipole moment are different quantities to hold fixed when comparing antennas. The approximation requires $\ell \ll \lambda$ and does not impose the nonuniform standing-current profile of a long antenna.