  We study the sensitivity of $T_c$ and the pseudogap onset temperature,
  $T^*$, to low fields, $H$, for cuprate superconductors, using a BCS-based
approach extended to arbitrary coupling. We find that $T^*$ and $T_c$, which
are of the same superconducting origin, have very different $H$ dependences.
The small coherence length makes $T^*$ rather insensitive to the field.
However, the presence of the pseudogap at $T_c$ makes $T_c$ more sensitive to
$H$. Our results for the coherence length $\xi$ fit well with existing
experiments. We predict that very near the insulator $\xi$ will rapidly
increase.
