A general treatment of the input boundary value problem for two‐dimensional space‐charge flow is developed using a theory of electrostatics (Thomson's) which notes that in an electrostatic problem the electric field distribution is such that the electrostatic‐field energy is a minimum. This principle is used to determine the average energy density mismatch and the starting conditions for finite thickness beams inM‐type backward‐wave oscillators. It is shown that the mismatch is greatest for strong space‐charge fields, high‐circuit attenuation, and signal frequencies below the cyclotron frequency. The starting length and beam velocity are greater as calculated by the variational procedure than obtained by the approximate method. Under certain conditions the average energy density mismatch is greater than 20% of the electrostatic field energy in the absence of the beam.