 Solution

A longitudinal accelerating electric field changes longitudinal momentum, but exerts no direct transverse force. With a constant guiding magnetic field and adiabatic transport, the transverse gyro-energy is consequently retained from the cathode:More generally the magnetic moment of the orbit gives ${\Theta }_{\perp }/B\simeq \text{constant}$, so the equality assumes the same relevant field strength, not merely that a field is present all along the line.

In the strong-magnetization regime, many electron gyro-orbits occur during an important long-range electron-ion interaction and the gyroradius is small. The ion then interacts predominantly with a guiding centre; rapid transverse cyclotron motion does not act like an unmagnetized bath at the full transverse electron temperature. This allows a proton equilibrium scale far below $T_c$ despite the hot transverse gyro-motion.^[ref]

That conclusion is conditional on magnetization and collision scales. A finite gyroradius, nonadiabatic fields or transverse beam errors can restore heating and degrade cooling; it is not generally true that transverse temperature can never affect the cooled beam.