Motion of a Charge in a Uniform Field

A uniform field has the same magnitude and direction everywhere between, say, two charged plates. A charge q sitting in that field feels a force F = qE that never changes as it moves through the region. Constant force means constant acceleration, a = qE/m, by Newton's second law. Nothing about the geometry or the speed of the particle changes that — as long as the field stays uniform, the acceleration is locked in.

Send an electron in sideways, with some velocity v0 along one axis, while the field points along a perpendicular axis. The electron's motion splits cleanly into two independent pieces. Along its original direction, nothing pushes or pulls it, so it moves at constant velocity. Perpendicular to that, the field accelerates it from rest, exactly like gravity accelerating a thrown ball downward. Constant velocity in one direction plus constant acceleration in the perpendicular direction is the exact recipe for a parabola — the field problem and the projectile problem are the same mathematics wearing different costumes, with qE/m standing in for g.

This is why the topic sits right after Gauss's law: once you know a configuration produces a uniform field, you can drop straight into kinematics and forget about charge distributions entirely. The deflection, the time of flight, the exit angle — all of it comes from the same three equations you already know from projectile motion, just relabeled.