We consider the magnetophoresis problem within the Rubinstein-Duke model, i.e., a reptating polymer pulled by a constant field applied to a single end of a chain. Extensive density matrix renormalization calculations are presented of the drift velocity and the profile of the chain for various strengths of the driving field and chain lengths. We show that the velocities and the average densities of the stored length are well described by simple interpolating crossover formulas, derived under the assumption that the difference between the drift and curvilinear velocities vanishes for sufficiently long chains. The profiles, which describe the average shape of the reptating chain, also show such interesting features as some nonmonotonic behavior of the link densities for sufficiently strong pulling fields. We develop a description in which a distinction is made between links entering at the pulled head and at the unpulled tail. At weak fields the separation between the head zone and the tail zone meanders through the whole chain, while the probability of finding it close to the edges drops off. At strong fields the tail zone is confined to a small region close to the unpulled edge of the polymer.