The problem of estimating the entry length over which the laminar motion of an electrically conducting, viscous fluid in a pipe under a transverse field adjusts to a steady state from some initial state is linearized by a Rayleigh approximation adapted for cases where there is a pressure gradient. Solutions are found which express the difference between the initial and ultimate velocity profiles as a series of orthogonal, exponentially decaying terms. Approximate entry lengths for the most persistent terms are calculated for flow between two infinite parallel planes and in rectangular tubes when magnetic effects are present. Entry lengths in circular and rectangular pipes and between infinite planes are calculated in the case where magnetic effects are negligible to permit a comparison of the results with those derived by other methods. Pressure losses at entry are considered in the simpler cases.