Savchenko Problem 14.4.9

The key is recognizing that relativistic mass increases with velocity, and this affects how much a given force can accelerate a particle. When the proton moves perpendicular to the field, the electric force acts entirely perpendicular to its motion, so it only changes direction—but the relativistic mass factor $\gamma = 1/\sqrt{1-\beta^2}$ still applies. When moving at angle $\alpha$ to the field, the component of force along the velocity direction (parallel acceleration) experiences a larger effective mass than the perpendicular component. By comparing $F/m_{\text{eff}}$ in each case, you'll find the ratio depends on how the parallel and perpendicular components of acceleration combine differently.

The key is recognizing that relativistic mass increases with velocity, and this affects how much a given force can accelerate a particle. When the proton moves perpendicular to the field, the electric force acts entirely perpendicular to its motion, so it only changes direction—but the relativistic mass factor $\gamma = 1/\sqrt{1-\beta^2}$ still applies. When moving at angle $\alpha$ to the field, the component of force along the velocity direction (parallel acceleration) experiences a larger effective mass than the perpendicular component. By comparing $F/m_{\text{eff}}$ in each case, you'll find the ratio depends on how the parallel and perpendicular components of acceleration combine differently.