KE_max = hf − Φ — Photoelectric Equation

Einstein's photoelectric equation — photon energy minus work function — with a worked sodium example.

KE_{max} = hf - \Phi

Maximum electron kinetic energy equals photon energy minus the metal's work function.

What it means

A photon's energy ( $hf$ ) is spent in two parts: the work function $\Phi$ — the energy cost of escaping the metal — and whatever remains becomes the electron's kinetic energy. Below the threshold frequency no electrons escape at all, no matter how bright the light: energy arrives in indivisible packets.

Variables

Symbol	Name	Unit	Meaning
$KE_max$	Max kinetic energy	eV or J	Fastest ejected electrons' energy
$h$	Planck constant	J·s	Planck's constant
$f$	Frequency	Hz	Light frequency
$Φ$	Work function	eV or J	Energy to free an electron from the metal

Assumptions

Single photon ejects a single electron
Photon energy exceeds the work function (hf > Φ)
Electron escapes without extra collisions

Worked example

Problem

Violet light of wavelength 400 nm strikes sodium (work function 2.75 eV). What is the ejected electrons' maximum kinetic energy?

Given:

λ = 400 nm (violet light)
Φ = 2.75 eV (sodium work function)

Photon energy: $hf = hc/\lambda = 1240/400 = 3.10\ \mathrm{eV}$
Write the equation: $KE_{max} = hf - \Phi$
Substitute: $KE_{max} = 3.10 - 2.75 = 0.35\ \mathrm{eV}$

0.35 eV

Each violet photon carries 3.10 eV; 2.75 eV pays the 'exit fee' and 0.35 eV remains as the electron's kinetic energy. Redder light (lower hf) below the threshold ejects nothing — proof light comes in quanta.