  The dichotomy between radio-loud (RL) and radio-quiet (RQ) active galactic
nuclei (AGN) is thought to be intrinsically related to radio jet production.
This difference may be explained by the presence of a strong magnetic field
(B-field) that enhances, or is the cause of, the accretion activity and the jet
power. Here, we report the first evidence of an intrinsic difference in the
dust polarized emission cores of four RL and five RQ obscured AGN using 89
$\mu$m polarization with HAWC+/SOFIA. We find that the thermal polarized
emission increases with the nuclear radio-loudness, $R_{20} = L_{\rm 5GHz}/
L_{\rm 20\mu m}$. The dust emission cores of RL AGN are measured to be
polarized, $\sim5-11$%, while RQ AGN are unpolarized, $<1$%. For RQ AGN, our
results are consistent with the observed region being filled with an
unmagnetized or highly turbulent, disk and/or expanding outflow at scales of
$5-130$ pc from the AGN. For RL AGN, the measured $89$ $\mu$m polarization
arises primarily from magnetically aligned dust grains associated with a
$5-130$ pc-scale dusty obscuring structure with a toroidal B-field orientation
highly offset, $65\pm22^{\circ}$, with respect to the jet axis. Our results
indicate that the size and strength of the B-fields surrounding the AGN are
intrinsically related to the strength of the jet power -- the stronger the jet
power is, the larger and stronger the toroidal B-field is. The detection of a
$\le130$ pc-scale ordered toroidal B-field suggests that a) the infalling gas
that fuels RL AGN is magnetized, b) there is a magnetohydrodynamic wind that
collimates the jet, and/or c) the jet is able to magnetize its surroundings.
