The Vlasov-Maxwell system models collisionless plasma. Solutions are considered that depend on one spatial variable,
x
x
, and two velocity variables,
v
1
v_1
and
v
2
v_2
. As
x
→
−
∞
x\rightarrow - \infty
it is required that the phase space densities of particles approach a prescribed function,
F
(
v
1
,
v
2
)
F\left (v_1,v_2\right )
, and all field components approach zero. It is assumed that
F
(
v
1
,
v
2
)
=
0
F\left (v_1,v_2\right ) = 0
if
v
1
≤
W
1
v_1 \leq W_1
, where
W
1
W_1
is a positive constant. An external magnetic field is prescribed and taken small enough so that no particle is reflected (
v
1
v_1
remains positive). The main issue is to identify the large-time behavior; is a steady state approached and, if so, can it be identified from the time independent Vlasov-Maxwell system? The time-dependent problem is solved numerically using a particle method, and it is observed that a steady state is approached (on a bounded
x
x
interval) for large time. For this steady state, one component of the electric field is zero at all points, the other oscillates without decay for
x
x
large; in contrast the magnetic field tends to zero for large
x
x
. Then it is proven analytically that if the external magnetic field is sufficiently small, then (a reformulation of) the steady problem has a unique solution with
B
→
0
B \rightarrow 0
as
x
→
+
∞
x \rightarrow +\infty
. Thus the “downstream” condition,
B
→
0
B \rightarrow 0
as
x
→
+
∞
x\rightarrow + \infty
, is used to identify the large time limit of the system.