Hertzian dipole

Example

A Hertzian dipole is a current element in vacuum, with time factor $e^{-i\omega t}$. Its current has constant complex amplitude $I_0$ and flows along the $z$-axis over a length $\ell$ much smaller than the wavelength $\lambda$. Let $k = \omega/c = 2\pi/\lambda$ and $\eta_0 = \sqrt{\mu_0/\varepsilon_0}$. Its retarded vector potential is $\mathbf{A} = \dfrac{\mu_0 I_0 \ell}{4\pi} \dfrac{e^{ikr}}{r} \hat{\mathbf{z}}$, and $\mathbf{H} = \mu_0^{-1} \nabla \times \mathbf{A}$ gives, in spherical coordinates $(r, \theta, \varphi)$, $$H_\varphi = \frac{I_0 \ell \sin\theta}{4\pi} \, e^{ikr} \left( \frac{1}{r^2} - \frac{ik}{r} \right).$$ Check: $\nabla \times (g \hat{\mathbf{z}}) = -g'(r) \sin\theta \, \hat{\boldsymbol{\varphi}}$ for a radial function $g$, and $\dfrac{d}{dr} \dfrac{e^{ikr}}{r} = e^{ikr}\left(\dfrac{ik}{r} - \dfrac{1}{r^2}\right)$. For $kr \ll 1$ the $1/r^2$ term dominates: this is the near field, which looks like the static Biot–Savart field of the current element. For $kr \gg 1$ the $1/r$ term dominates, and the radiated field is $E_\theta = \eta_0 H_\varphi \approx -\dfrac{i \eta_0 k I_0 \ell \sin\theta}{4\pi r} e^{ikr}$. The time-averaged radiated power is $$P = \frac{\eta_0 k^2 \abs{I_0}^2 \ell^2}{12\pi},$$ obtained by integrating $\abs{E_\theta}^2/(2\eta_0)$ over a sphere with $\int \sin^2\theta \, d\Omega = 8\pi/3$. Writing $P = \frac{1}{2} R \abs{I_0}^2$ gives the radiation resistance $R = \dfrac{2\pi \eta_0}{3} \left(\dfrac{\ell}{\lambda}\right)^2 \approx 789 \left(\dfrac{\ell}{\lambda}\right)^2 \Omega$.