Assuming the averaged (background) electron distribution in a crossed-field device, such as a magnetron or crossed-field amplifier, to be slowly varying between the cathode and the anode, we develop a WKB approximation for the cold-fluid plasma equations of a planar magnetron. In this approximation, we can give general expressions for the solution of linearized, high-frequency oscillations in such a device in terms of two integrals. Assuming also that the high-frequency wave in the slow-wave structure drives the response in the electron plasma, we are then able to show that the current drawn by a crossed-field device will be proportional to the power propagating in the slow-wave structure. Thus the device will operate as a linear amplifier. We also show, in the same approximation, that the averaged electron sheath that forms when the device is operating is independent of the current being drawn. Thus the current will not be limited by the sheath, but only by the ability of the cathode to emit electrons. In the process, we also obtain expressions for the linear growth rate and the value of the quasilinear diffusion coefficient at the diocotron resonance, in terms of parameters of the background electron sheath.