  The far infrared counterpart of hot spot D, the terminal hot spot of the
eastern jet hosted by the radio galaxy Cygnus A, is detected with
\textit{Herschel} Aperture photometery of the source performed in 5 photometric
bands covering the wavelength range of $70\unicode{x2013}350$ $\mathrm{\mu m}$.
After removing the contamination from another nearby hot spot, E, the
far-infrared intensity of hot spot D is derived as $83\pm13$ and $269\pm66$ mJy
at $160$ and $350$ $\mathrm{\mu m}$, respectively. Since the far-infrared
spectrum of the object smoothly connects to the radio one, the far-infrared
emission is attributed to the synchrotron radiation from the radio-emitting
electron population. The radio-to-near-infrared spectrum is confirmed to
exhibit a far-infrared break feature at the frequency of
$\nu_\mathrm{br}=2.0^{+1.2}_{-0.8} \times10^{12}$ Hz. The change in energy
index at the break ($\Delta\alpha=0.5$) is interpreted as the impact of
radiative cooling on an electron distribution sustained by continuous injection
from diffusive shock acceleration. By ascribing the derived break to this
cooling break, the magnetic field, $B$, in the hot spot is determined as a
function of its radius, $R$ within a uniform one-zone model combined with the
strong relativistic shock condition. An independent $B$-$R$ constraint is
obtained by assuming the X-ray spectrum is wholly due to
synchrotron-self-Compton emission. By combining these conditions, the two
parameters are tightly determined as $B=120\unicode{x2013}150$ $\mathrm{\mu G}$
and $R=1.3\unicode{x2013}1.6$ kpc. A further investigation into the two
conditions indicates the observed X-ray flux is highly dominated by the
synchrotron-self-Compton emission.
