  The expression for the dynamical mass of fermions in QED in a magnetic field
is obtained for a large number of the fermion flavor N in the framework of 1/N
expansion. The existence of a threshold value N_{thr}, dividing the theories
with essentially different dynamics, is established. For the number of flavors
N << N_{thr}, the dynamical mass is very sensitive to the value of the coupling
constant \alpha_b, related to the magnetic scale \mu = |eB|. For N of order
N_{thr} or larger, a dynamics similar to that in the Nambu-Jona-Lasinio model
with cutoff of order |eB| and the dimensional coupling constant G \sim
1/(N|eB|) takes place. In this case, the value of the dynamical mass is
essentially \alpha_b independent (the dynamics with an infrared stable fixed
point). The value of N_{thr} separates a weak coupling dynamics (with
\tilde{\alpha}_b \equiv N\alpha_b << 1) from a strong coupling one (with
\tilde{\alpha}_b \gtrsim 1) and is of order 1/\alpha_b.
