Effect of Frequency of Incident Radiation on Stopping Potential

Fig. 11.4 is a graph of photoelectric current (vertical axis) against collector plate potential (horizontal axis). The horizontal axis is marked with negative (retarding) potentials to the left of the origin and positive (accelerating) potentials to the right. Three separate curves are drawn, one for each of three frequencies $\nu_3 > \nu_2 > \nu_1$, all at the same intensity of incident light.

On the right side of the graph, all three curves rise to exactly the same saturation current. This tells you that the total number of photoelectrons emitted per second is the same for all three frequencies — because the intensity (number of photons per second) is kept fixed. On the left side, each curve cuts the horizontal axis at a different negative potential: $-\!V_{03}$ (most negative) for $\nu_3$, then $-\!V_{02}$, then $-\!V_{01}$ (closest to the origin) for $\nu_1$. These are the stopping potentials. The higher the frequency, the more negative the stopping potential.

The physical idea is straightforward. The stopping potential $V_0$ is the retarding voltage that just barely stops the fastest photoelectrons from reaching the collector. A more negative $V_0$ means those electrons had a larger maximum kinetic energy when they left the metal. So the graph shows that higher-frequency light gives photoelectrons more kinetic energy, even though the light intensity (and therefore the number of electrons) is unchanged.

From this figure, the textbook develops the central linear relation between stopping potential and frequency. For a given photosensitive material, the stopping potential $V_0$ varies linearly with the frequency $\nu$ of the incident radiation:

Here $h$ is Planck's constant, $e$ is the magnitude of the electron charge, and $\nu_0$ is the threshold frequency — the minimum frequency below which no photoelectrons are emitted, no matter how intense the light. The maximum kinetic energy of the emitted photoelectrons is $K_{\text{max}} = eV_0$, so the same relation can be written as:

This is Einstein's photoelectric equation. The figure directly demonstrates that $K_{\text{max}}$ depends only on frequency, not on intensity — because the three curves have the same saturation current (same number of electrons) but different stopping potentials (different maximum energies). The threshold frequency $\nu_0$ is the intercept on the frequency axis when $V_0 = 0$; for frequencies below $\nu_0$, the stopping potential would be negative of a positive number that doesn't exist — meaning no emission occurs. …

Fig. 11.5 is a graph that captures the most important quantitative relationship in the photoelectric effect. The horizontal axis is the frequency $\nu$ of the incident light. The vertical axis is the stopping potential $V_0$, which is the retarding voltage needed to just stop the most energetic photoelectrons from reaching the collector.

For a given photosensitive material, the data points fall on a straight line. The line does not start at the origin. It meets the frequency axis at a positive value labelled $\nu_0$, the threshold frequency for that material. To the left of $\nu_0$, no photoelectrons are emitted at all, so there is no data and no line. The line itself has a positive slope, and the textbook explicitly annotates that slope as $h/e$, where $h$ is Planck's constant and $e$ is the elementary charge.

The figure actually shows two such straight lines, one for metal A and one for metal B. Both lines have exactly the same slope $h/e$ — they are parallel. The only difference is where each line cuts the frequency axis. The metal with the higher work function has its threshold frequency $\nu_0$ further to the right. This is a direct visual demonstration that the threshold frequency is a material property, while the slope of the $V_0$ vs $\nu$ graph is a universal constant.

The physical idea is straightforward: the stopping potential measures the maximum kinetic energy $K_{\text{max}}$ of the photoelectrons, because $K_{\text{max}} = e V_0$. The graph shows that $K_{\text{max}}$ increases linearly with frequency, not with intensity. The line's equation, read directly from the graph, is the Einstein photoelectric equation in its most useful form:

or equivalently

Here $e$ is the charge of the electron ($1.6 \times 10^{-19}$ C), $h$ is Planck's constant ($6.63 \times 10^{-34}$ J s), $\nu$ is the frequency of the incident light, and $\nu_0$ is the threshold frequency for that metal. The term $h \nu_0$ is the work function $\phi$ of the metal — the minimum energy needed to free an electron from its surface.