  We have measured the surface magnetic flux on four accreting young brown
dwarfs and one non-accreting young very low-mass star utilizing high resolution
spectra of absorption lines of the FeH molecule. A magnetic field of 1-2 kG had
been proposed for one of the brown dwarfs, 2MASS J1207334$-$393254, because of
its similarities to higher mass T Tauri stars as manifested in accretion and
the presence of a jet. We do not find clear evidence for a kilo-Gauss field in
any of our young brown dwarfs but do find a 2 kG field on the young VLM star.
Our 3-$\sigma$ upper limit for the magnetic flux in 2MASS J1207334$-$393254
just reaches 1 kG. We estimate the magnetic field required for accretion in
young brown dwarfs given the observed rotations, and find that fields of only a
few hundred Gauss are sufficient for magnetospheric accretion. This predicted
value is less than our observed upper limit. We conclude that magnetic fields
in young brown dwarfs are a factor of five or more lower than in young stars of
about one solar mass, and in older stars with spectral types similar to our
young brown dwarfs. It is interesting that, during the first few million years,
the fields scale down with mass in line with what is needed for magnetospheric
accretion, yet no such scaling is observed at later ages within the same
effective temperature range. This scaling is opposite to the trend in rotation,
with shorter rotation periods for very young accreting brown dwarfs compared
with accreting solar-mass objects (and very low Rossby numbers in all cases).
We speculate that in young objects a deeper intrinsic connection may exist
between magnetospheric accretion and magnetic field strength, or that magnetic
field generation in brown dwarfs may be less efficient than in stars. Neither
of these currently have an easy physical explanation.
