How Do You Use Einstein's Photoelectric Equation (Emax = hf - Φ)?

Use Einstein's photoelectric equation by calculating the incident photon's energy, subtracting the metal's work function, and interpreting the remainder as the maximum kinetic energy of an emitted electron. This is HL-only content in IB Physics E.2 Quantum physics.

The Reasoning and Method

The equation is

$$E_{\max}=hf-\Phi$$

It expresses conservation of energy for one photon interacting with one electron.

Each photon transfers its energy to one electron. The work function is the least energy needed for escape; any remaining energy becomes kinetic energy. Electrons losing energy inside the metal emerge with less than $E_{\max}$, which is why the equation gives a maximum rather than every electron's energy.

Follow these steps:

1. Calculate the photon energy using $E=hf$.
2. Convert the work function into the same unit as the photon energy.
3. Subtract: $E_{\max}=hf-\Phi$.

For example, radiation of frequency $8.00\times10^{14}\,\mathrm{Hz}$ strikes a metal with $\Phi=2.20\,\mathrm{eV}$:

$$hf=(6.63\times10^{-34})(8.00\times10^{14})=5.30\times10^{-19}\,\mathrm{J}$$

Converting to electronvolts:

$$hf=\frac{5.30\times10^{-19}}{1.60\times10^{-19}}=3.31\,\mathrm{eV}$$

Therefore,

$$E_{\max}=3.31-2.20=1.11\,\mathrm{eV}$$

The corresponding stopping potential is $V_s=1.11\,\mathrm{V}$.

The equation also predicts a straight-line graph of $E_{\max}$ against $f$: gradient $h$, vertical intercept $-\Phi$, and horizontal intercept ${}_{}$. This graph is experimental evidence that photon energy is proportional to frequency rather than to light intensity.

A common misconception is that greater light intensity increases $E_{\max}$. It does not: intensity normally increases the number of emitted photoelectrons, while frequency determines their maximum kinetic energy. If $hf<\Phi$, no photoelectrons are emitted, regardless of intensity; the threshold frequency is $f_0=\Phi/h$.

Exam Technique

For a Calculate question, show the equation, substitution, unit conversion, and final value. Keep joules and electronvolts consistent, and never report a negative kinetic energy: a negative result means the frequency is below the threshold and no emission occurs.