Photoclectrons are liberated from a plate by ultraviolet radiation, their initial velocity being negligible. A magnetic field \(\mathbf{B}\) is maintained parallel to the plate and an E-field perpendicular to it. The E-field is produced by a second plate parallel to the first, a distance \(d\) from it and at a positive potential \(V\) with respect to it. Show that the value of \(d\) for which the current just fails to pass between the plates is \(\left(2 m, V / e B^{2}\right)^{1 / 2}\), where e and \(m_{e}\) are the charge and mass of the clectron.

Imagine you're in a dimly lit room and suddenly the lights flicker on; electrons, much like the room's darkness, can also be displaced by a sudden surge of energy. This phenomenon, known as the photoelectric effect, occurs when light, particularly ultraviolet radiation, hits a material causing electrons to be ejected from its surface. This ejection takes place because the incoming photons' energy outstrips the electrons' 'binding energy' - the glue holding them to the atoms. Once liberated, these electrons, called photoelectrons, can exhibit versatile movement, which can be manipulated by external magnetic and electric fields, giving us intriguing insights into their behavior.

When these free electrons meander through a space with magnetic fields, they experience what's called the Lorentz force. It's like the wind that alters the course of a flying kite. Specifically, as electrons move, the Lorentz force acts perpendicular to their velocity and the magnetic field, curvedly altering their trajectories. Capturing the essence of electron kinetics, this force, denoted by \( \mathbf{F_{B}} = e(\mathbf{v} \times \mathbf{B}) \), is instrumental in understanding how electrons weave through circuits and devices, and in our case, how they might traverse or fail to make it between the plates described in the exercise.

In addition to magnetic fields, photoelectrons can also be influenced by electric fields. Harking back to our exercise, imagine a tug-of-war where one team is the electric force, pulling in a direction perpendicular to the magnetic tug. A force felt by charged particles when they are within the vicinity of other charges, the electric force expresses itself through electric fields. These fields are quantified as force per charge and can be envisaged as lines that show the trajectory a positive test charge would follow. They dictate why the electrons don’t simply follow the straight path but curve, and the stronger the field, the tighter the curvature, altering the space between the inherent 'tracks' electrons follow.

The story of electron movement, or electron kinetics, involves understanding how these tiny particles zip and zoom when influenced by forces. Electron kinetics marries the concepts of energy, mass, charge, and force to explain the electrons' motions. Whether we're looking at the photoelectric effect launching electrons or the mingling of forces from electric and magnetic fields shaping their pathways, electron kinetics is essentially about predicting where an electron will go under a set of circumstances. It’s akin to guessing the exact spot a freestyle skier will land, taking into consideration speed, wind, and other forces at play. In our context, this understanding helps us pinpoint where and why an electron might not finish its seemingly simple journey from one plate to another due to these compelling forces of nature.