We study the dynamical generation of masses for fundamental fermions in quenched quantum electrodynamics, in the presence of magnetics fields of arbitrary strength, by solving the Schwinger-Dyson equation for the fermion self-energy in the rainbow approximation. We employ the Ritus eigenfunction formalism which provides a neat solution to the technical problem of summing over all Landau levels. It is well known that magnetic fields catalyze the generation of fermion mass $m$ for arbitrarily small values of electromagnetic coupling $\ensuremath{\alpha}$. For intense fields it is also well known that $m\ensuremath{\propto}\sqrt{eB}$. Our approach allows us to span all regimes of parameters $\ensuremath{\alpha}$ and $eB$. We find that $m\ensuremath{\propto}\sqrt{eB}$ provided $\ensuremath{\alpha}$ is small. However, when $\ensuremath{\alpha}$ increases beyond the critical value ${\ensuremath{\alpha}}_{c}$ which marks the onslaught of dynamical fermion masses in vacuum, we find $m\ensuremath{\propto}\ensuremath{\Lambda}$, the cutoff required to regularize the ultraviolet divergences. Our method permits us to verify the results available in literature for the limiting cases of $eB$ and $\ensuremath{\alpha}$. We also point out the relevance of our work for possible physical applications.