According to Maxwell wave theory what would be the exact relationship between the light beam intensity and the stopping potential?

Question

According to Maxwell wave theory , what would be the quantitative relationship between the light beam intensity and the stopping potential in a typical photoelectric experiment (Millikan one for example)? I ask this question because all books simply say that increasing the intensity $V_{stop}$ should increase, but don't explain the exact function that connects the two quantities.

Answer 1

We can't really say without a more detailed model of the ejection process, which is horrendously complicated in the classical model.

Classically, one could model a metal as a "sea" of negatively charged point particles (conduction electrons) all moving around in a complicated potential formed by the nuclei (and other more tightly bound electrons). An incident electromagnetic wave would then exert forces on these electrons, which would alter their trajectories and possibly be ejected from the metal.

To calculate the kinetic energy of the ejected electrons from first principles, you'd have to figure out how a given electron would move in response to the potential from the nuclei, the electric forces from the other electrons, and the electric and magnetic fields from the wave. In particular, you'd need to know what the electrostatic potential from the nuclei would be. It would depend on the exact structure of the crystal lattice of the atoms, and it probably wouldn't be expressible as a "nice" function amenable to calculations in closed form. This is why your textbooks don't tell you what this dependence is; it would vary from metal to metal, and it wouldn't be expressible as a simple formula. In fact, I suspect that it wouldn't be expressible as a closed-form formula at all.

The important point, though, is that it would make sense qualitatively that applying a larger intensity of the wave (at a fixed frequency) would result in a greater kinetic energy of the ejected electrons; after all, you're applying more force to them. But experimentally, increasing the intensity of the wave at a fixed frequency does not alter the kinetic energy of the electrons. This is A Problem™ for the classical model.