  The exchange interaction is investigated theoretically for electrons confined
to a 2-D sample placed in a linearly varying magnetic field perpendicular to
the plane. Unusual and interesting behavior is predicted: starting from zero,
as one adds electrons to the system, its size will increase continuously but
will then collapse. However this collapse will be reversed as the number
crosses another critical value, which we estimate here. For electron parameters
typical for 2DEG's, the instability could be observable at sufficiently low
electron densities. A Hartree Fock equation is derived and investigated. We
also show that in an appropriate asymptotic limit an approximate local
potential can be derived. One key lesson is that the exchange interaction is
large and cannot be reasonably excluded from any valid theoretical
investigation.
