Notes on Relativistic Electron Beam Deflection in Cathode Ray Tubes

We present a rigorous relativistic treatment of electron beam deflection in cathode ray tubes (CRTs) under homogeneous electric and magnetic fields. A common approach to account for relativistic effects, namely replacing the rest mass m by the relativistic mass gamma m in classical equations, is shown to be inadequate and field dependent. Using a proper four-vector formulation, we derive exact trajectories and demonstrate that this naive substitution yields correct results for magnetic deflection but fails for electric deflection, where the increase in kinetic energy must be taken into account explicitly. In the appropriate approximation regime, both cases lead to a universal correction factor of 1/gamma for the deflection angle, although the underlying physical mechanisms differ fundamentally. Quantitative results for typical CRT operating voltages are provided.