Charged Particles Accelerated in a Field

An old television tube fires electrons from a hot filament, then uses a strong electric field between two plates to yank them forward. Thousands of volts means a big potential difference, and a big potential difference means each electron picks up a lot of kinetic energy: W = qDV. That energy shows up as speed by the time the electron slams into the phosphor screen and makes a dot of light.

An electron gun has two jobs done by two different fields. First, an accelerating field, pointing along the electron's path, speeds it up in a straight line, same as a mass falling in gravity. Then the electron flies into a second region between deflecting plates, where the field points sideways, perpendicular to its velocity. This is projectile motion with electricity standing in for gravity: constant velocity along the original direction, constant acceleration a = qE/m sideways. The time spent between the plates is t = L/v0, and the sideways deflection is y = (1/2)(qE/m)t^2. Change the field strength and you change how far the beam bends, which is exactly how old televisions aimed electrons at different pixels and how an ink-jet printer steers charged ink droplets onto the exact right spot on a page.

The deeper idea tying this together is energy conservation, and it is the same bookkeeping as a circuit. A charge crossing a potential difference DV gains energy qDV, whether that charge is an electron between deflecting plates in a vacuum tube or a charge pushed by a battery through a wire. Voltage is voltage: it tells you energy per charge, and the charge spends or gains that energy the same way no matter what device it is in.