= Solution
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> $[\dot m]/(cr^2)$ term and a transverse radiation $[\ddot m]/(c^2r)$ 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 $E_\phi=-cB_\theta$. The radial Poynting <flux> is therefore
$$
S_r=\frac{cB_\theta^2}{\mu_0}
=\frac{\mu_0[\ddot m]^2}{16\pi^2c^3r^2}\sin^2\theta.
$$
Using $\int\sin^2\theta\,d\Omega=8\pi/3$ yields
$$
\boxed{P(t)=\frac{\mu_0[\ddot m]^2}{6\pi c^3}
=\frac{[\ddot m]^2}{6\pi\epsilon_0c^5}.}
$$
For 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 $I(t)=I_0\cos\omega t$, $m=AI$, so
$$
\overline P=\frac{A^2I_0^2\omega^4}{12\pi\epsilon_0c^5}
=\frac12I_0^2R_{\rm rad},\qquad
\boxed{R_{\rm rad}(\omega)=\frac{A^2\omega^4}{6\pi\epsilon_0c^5}.}
$$
This uses peak current $I_0$; with rms current the <power> is $I_{\rm rms}^2R_{\rm rad}$.
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 gives
$$
\Phi=BA\cos\omega t,\quad\mathcal E=BA\omega\sin\omega t,\quad
I(t)=\frac{BA\omega}{R}\sin\omega t.
$$
Write $m_0=BA^2\omega/R$. The full magnetic moment is
$$
\mathbf m(t)=m_0\sin\omega t\,\mathbf n(t)
=\frac{m_0}{2}\left[\sin(2\omega t)\widehat{\mathbf x}
+(1-\cos(2\omega t))\widehat{\mathbf y}\right].
$$
It has a static component and a rotating component at $2\omega$. Consequently $|\ddot{\mathbf m}|^2=4m_0^2\omega^4$, and
$$
\boxed{\overline P_{\rm loop}
=\frac{2B^2A^4\omega^6}{3\pi\epsilon_0c^5R^2}
=\frac{2\pi^3B^2b^8\omega^6}{3\epsilon_0c^5R^2}.}
$$
Holding the dipole axis fixed while inserting the sinusoidal current would instead give $B^2A^4\omega^6/(12\pi\epsilon_0c^5R^2)$, 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.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2013/paper-1-oe2013-rotating-moment.png]
{title=Original calculated rotation-induced current and vector magnetic-moment components, illustrating the static plus twice-spin-frequency decomposition}
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> $I_0=BA\omega/\sqrt{R^2+\omega^2L^2}$. Its <phase> changes the static moment direction but not the twice-frequency component's magnitude, so the <power> becomes
$$
\boxed{\overline P_{\rm loop}(L)=
\frac{2B^2A^4\omega^6}{3\pi\epsilon_0c^5(R^2+\omega^2L^2)}.}
$$
The 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 $B=50\,\mathrm{\mu T}$, a representative near-Earth field rather than a supplied orbital measurement. The terrestrial surface range reported by NOAA is roughly $25$–$65\,\mathrm{\mu T}$; the actual orbital value depends on position.<Geomagnetism Frequently Asked Questions by National Oceanic and Atmospheric Administration>[What are the magnetic elements? Total field intensity range]{ref} With $b=1\,\mathrm m$ and $\omega=2\pi(500\,\mathrm{s^{-1}})$, the resistance-only model gives
$$
\boxed{\overline P\simeq2.3\times10^{-10}\,\mathrm W
\left(\frac{B}{50\,\mathrm{\mu T}}\right)^2,
\qquad f_{\rm rad}=1000\,\mathrm{Hz},\qquad\lambda_{\rm rad}\simeq300\,\mathrm{km}.}
$$
<Inductance> 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.<NASA Rocket Mission Studying Escaping Radio Waves by National Aeronautics and Space Administration>[VIPER description: absorption/reflection layers and condition-dependent trans-ionospheric propagation]{ref}
At a separation $r=100\,\mathrm{km}$, $k_{\rm rad}r\simeq2.1$, so the observer is not securely in the far field. The radiation term alone has at most the characteristic magnetic <amplitude>
$$
B_{\rm rad}\sim\frac{\mu_0(2m_0\omega^2)}{4\pi c^2r}
\simeq3.4\times10^{-18}\,\mathrm T
\left(\frac{B}{50\,\mathrm{\mu T}}\right)
$$
in 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. \b[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.]
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