A low‐&bgr; plasma can stream across a “slab” of magnetic field provided the spatial extentLof the field along the streaming direction is small compared with the geometric mean cyclotron radius and provided the magnetic field is not too strong(&ohgr;ce≲ &ohgr;pc). The behavior of the plasma which emerges from the magnetic field region is discussed here. On a scale large compared withL, the effect of the magnetic field is simply to impart an impulsive transverse velocity to the electrons of the streaming plasma. If this velocity exceeds the stability limit for the two‐stream instability, long‐wavelength, longitudinal thermal fluctuations should grow to such an amplitude that the resulting weak turbulence reduces the transverse velocity to zero, the associated kinetic energy being converted to electron thermal energy. A solution to this problem within the quasilinear approximation for a “collisionless” plasma is given here, together with numerical calculations of the spatial decay of the current and the associated electron heating. As expected, the length of the turbulent transition region, in which the directed transverse energy is thermalized, is an increasing function of the initial plasma density, but a decreasing function of the initial electron temperature and transverse velocity. A small charge separation(ni− ni≲ 10−4ne)provides an electric field just sufficient to balance the pressure gradient,n dT / dx.