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A Hertzian dipole is an ideal electrically short oscillating electric dipole used to model radiation from a localized current element. In vacuum its far-field electric dipole radiation has ${\sin }^2\theta$ angular power distribution and total power $P={\mu }_0|\ddot{\mathbf{p}}{|}^2/(6\pi c)$, evaluated at the retarded time. The electric-dipole approximation for radiation requires a source much smaller than its radiation wavelength.

For a lossless ideal Hertzian dipole driven by an incident wave, use its induced open-circuit voltage and its feed impedance $R_{\mathrm{rad}}+iX$. A conjugately matched load has impedance $R_{\mathrm{rad}}-iX$. The same induced current therefore delivers equal power to the load and to radiation resistance: $P_{\mathrm{sca}}=P_{\mathrm{load}}$. The scattered wave has the usual dipole angular pattern. In direction of gain $G$, its scattering cross-section is $G{A}_{\mathrm{eff}}$, since the load receives $S_{\mathrm{inc}}{A}_{\mathrm{eff}}$.

This ideal single-mode scattering is not the same as a signal that travels through an already matched antenna port, reflects from a mismatched terminating load, and returns. A zero port reflection coefficient removes that latter echo but does not make the physical dipole invisible to the incident field. Additional structural scattering, ohmic loss or a different load changes the result.

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.

A Hertzian dipole is an ideal localized oscillating source much smaller than its radiation wavelength. Its electric version is an infinitesimal current element; its magnetic version is a small current loop with moment $\mathbf{m}=IA\mathbf{n}$. Such elements model the far-field radiation and radiation resistance of electrically small antennas. The approximation requires source dimensions much smaller than $c/\omega$.

The supplied magnetic field has a quasistatic $[m]/r^3$ term, an induction $[\dot{m}]/(c{r}^2)$ term and a transverse radiation $[\ddot{m}]/(c^{2}r)$ term. The electric field has induction and radiation terms. Brackets mean evaluation at the retarded time $t_r=t-r/c$, not a time average. Near terms mainly exchange stored energy; only the $r^{-1}$ fields give a nonvanishing outgoing power integral as $r\to \infty$.

For a fixed dipole axis the radiation fields satisfy $E_{\varphi }=-c{B}_{\theta }$. The radial Poynting flux is thereforeUsing $\int {\sin }^2\theta d\Omega =8\pi /3$ yieldsFor a general changing dipole direction, replace the squared scalar acceleration by $|\ddot{\mathbf{m}}{|}^2$.

Radiation resistance is the equivalent input resistance whose mean power equals radiated power, not ohmic heating. For a fixed small loop carrying $I(t)=I_0\cos \omega t$, $m=AI$, soThis uses peak current $I_0$; with rms current the power is $I_{\mathrm{rms}}^2{R}_{\mathrm{rad}}$.

The satellite loop needs its changing orientation as well as its induced current. Put the spin axis along z, the external field along x, and let $\mathbf{n}(t)=(\cos \omega t,\sin \omega t,0)$. With $A=\pi b^2$ and resistance $R=10^{-4}\Omega$, the resistance-only approximation givesWrite $m_0=B{A}^2\omega /R$. The full magnetic moment isIt has a static component and a rotating component at $2\omega$. Consequently $|\ddot{\mathbf{m}}{|}^2=4m_0^2{\omega }^4$, andHolding the dipole axis fixed while inserting the sinusoidal current would instead give $B^2{A}^4{\omega }^6/(12\pi {ϵ}_0{c}^5{R}^2)$, eight times smaller. That is the fixed-axis-loop result; it does not describe the stated rotating loop. This is frequency doubling of an induction-driven rotating loop; the changing vector direction is essential.

The resistance-only model is conditional: loop inductance is not supplied. If self-inductance L matters, solve $L\dot{I}+RI=BA\omega \sin \omega t$. In the steady state $I_0=BA\omega /\sqrt{R^2+{\omega }^2{L}^2}$. Its phase changes the static moment direction but not the twice-frequency component's magnitude, so the power becomesThe first expression is the requested resistive idealization and an upper estimate within this passive RL correction. Radiation reaction is negligible compared with the stipulated ohmic resistance. The radiated energy comes from rotational kinetic energy: the induced magnetic torque brakes the loop. Most lost rotational energy is Joule heating; a small part escapes as radiation. Maintaining a constant spin would require external mechanical power.

For scale only, take $B=50\mu \mathrm{T}$, a representative near-Earth field rather than a supplied orbital measurement. The terrestrial surface range reported by NOAA is roughly $25$–$65\mu \mathrm{T}$; the actual orbital value depends on position.^[ref] With $b=1\mathrm{m}$ and $\omega =2\pi (500{\mathrm{s}}^{-1})$, the resistance-only model givesInductance would reduce this power. The kilohertz signal is exceptionally weak and a metre-scale receiving antenna is electrically tiny. Ionospheric absorption/reflection and strong natural or instrumental low-frequency backgrounds make ordinary ground reception impractical. This is not a theorem that the ionosphere blocks every low-frequency mode: field-guided propagation and condition-dependent leakage are real, as the VIPER mission's measurements were designed to test.^[ref]

At a separation $r=100\mathrm{km}$, $k_{\mathrm{rad}}r\simeq 2.1$, so the observer is not securely in the far field. The radiation term alone has at most the characteristic magnetic amplitudein this optimistic resistance-only vacuum model. Near and induction fields must be retained for a full prediction; a simple far-field inverse-square power estimate is not an exact received signal here. Another satellite avoids the whole ground ionospheric path, but plasma propagation, receiving sensitivity, bandwidth, orientation, integration time and spacecraft noise remain unspecified. Detection is not forbidden in principle, but routine practical detection is implausible at this scale; a definitive instrument-level yes/no does not follow from the supplied data.

For the oblique coupling of parallel Hertzian dipoles, a lossless Hertzian dipole has directional gain $G(\theta )=\frac{3}{2}{\sin }^2\theta$. The separation is $r=\sqrt{2}d$, and both transmitting and receiving patterns have ${\sin }^2\theta =1/2$, so $G_t=G_r=3/4$. Reciprocity gives matched receiving effective area $A_e={\lambda }^2{G}_r/(4\pi )$. The emitted polarization matches the transverse projection of the receiving dipole axis, so there is no additional polarization mismatch. ConsequentlyBoth the off-axis transmitting pattern and the off-axis receiving aperture matter. This assumes the stated far field, conjugate matching and negligible dissipative antenna losses.

Use axes adapted to the two orthogonal rotations in the diagram:For a linearly oscillating Hertzian dipole with real unit axis $\mathbf{d}$ and total time-averaged radiated power $P$, the angular distribution of electric dipole radiation isThe normalization follows from $\int {\sin }^2\vartheta d\Omega =8\pi /3$. Equal readings therefore implyHere $d_z\ne 0$, since otherwise all components would vanish. Reversing the axis is immaterial, so choose $d_z>0$. The complete family isDefine $a=\sqrt{2}-1$, $b=-\sqrt{2}-1$, and $D=1+(s_B\sqrt{2}-1{)}^2+(s_C\sqrt{2}-1{)}^2$. Then $d_z^2=1/D$, and the absolute intensity fixesFor the measured distance and intensity, the allowed axes and powers are:

The intensity data do not select a unique orientation or total power. This is the dipole-axis ambiguity from three intensity measurements: the readings constrain squared projections and lose their signs. For example, the first branch makes an angle $30.36^{\circ }$ with $A$, but the other branches are equally consistent with the three measurements. A further suitable observation direction or polarization measurement is needed to remove the ambiguity. The calculation uses radiation-zone intensities, as appropriate to the Hertzian dipole model.

Use the ideal electrically short Hertzian dipole, $d\ll \lambda$, with negligible ohmic loss. Far-field statements additionally require an observation distance large compared with the wavelength and source dimensions. These assumptions distinguish its radiation from near-field coupling and from scattering by a general extended antenna structure.

Only the $1/r$ fields transport a nonzero outgoing power to infinity. Retain those terms from the Hertzian dipole fields:The brackets mean evaluation at the retarded time,for a point source at the origin in vacuum. They are not a time average. The radial Poynting vector in the radiation zone isIntegrating over a sphere usesThereforeNear-field terms either give inverse powers of $r$ after the surface integral or describe reversible exchange with the near field. The emitted power labelled by source time $t_{\mathrm{src}}$ is ${\mu }_0\ddot{p}(t_{\mathrm{src}}{)}^2/(6\pi c)$; the enclosing surface receives it one light-travel time later. For a changing dipole orientation, the general form is $P={\mu }_0|\ddot{\mathbf{p}}{|}^2/(6\pi c)$.

Radiation resistance represents emitted electromagnetic power as a real input resistance of an antenna. It does not describe conversion to heat inside the conductor. If peak current amplitude is used, $P_{\mathrm{rad}}=|I_0{|}^2{R}_{\mathrm{rad}}/2$. Real ohmic loss adds a separate resistance and lowers radiation efficiency. For an ideal short uniform-current element of length $d$, the Hertzian dipole gives $R_{\mathrm{rad}}=80{\pi }^2(d/\lambda {)}^2$ in ohms, using the usual vacuum impedance approximation $120\pi \Omega$.

Two aligned lossless Hertzian dipoles in one another's broadside radiation zone have gain $G=3/2$ and one-way received-power fraction $\eta =G^2[\lambda /(4\pi r){]}^2$. If the second remains conjugately matched, its matched-load scattering of a Hertzian dipole reradiates the same power as its load absorbs. Applying the reciprocal far-field link on the return gives $P_{\mathrm{echo}}=P_t{\eta }^2$, proportional to $r^{-4}$. The statement concerns physical dipole reradiation; a pure antenna load-mismatch echo model omits that scattering and is not interchangeable with it.