38.2.2 A Prediction

Not only do Einstein’s hypotheses explain all of Lenard’s observations, they also make a new prediction. According to Equation 38.8, the stopping potential should be a linearly increasing function of the light’s frequency $f$ . We can rewrite Equation 38.8 in terms of the threshold frequency $f_{0} = E_{0} / h$ as

V_{stop} = \frac{h}{e} (f - f_{0})

(38.9)

A graph of the stopping potential $V_{stop}$ versus the light frequency $f$ should start from zero at $f = f_{0}$ , then rise linearly with a slope of $h / e$ . In fact, the slope of the graph provides a way to measure Planck’s constant $h$ .

Lenard had not measured the stopping potential for different frequencies, so Einstein offered this as an untested prediction of his postulates. Robert Millikan, known for his oil-drop experiment to measure $e$ , took up the challenge. Some of Millikan’s data for a cesium cathode are shown in FIGURE 38.9. As you can see, Einstein’s prediction of a linear relationship between $f$ and $V_{stop}$ was confirmed.

The figure illustrates a potential versus frequency graph.

FIGURE 38.9 Full Alternative Text

Millikan measured the slope of his graph and multiplied it by the value of $e$ (which he had measured a few years earlier in the oil-drop experiment) to find $h$ . His value agreed with the value that Planck had determined in 1900 from an entirely different experiment. Light quanta, whether physicists liked the idea or not, were real.

EXAMPLE 38.3 The photoelectric threshold frequency

What are the threshold frequencies and wavelengths for photo-emission from sodium and from aluminum?

SOLVE Table 38.1 gives the sodium work function as $E_{0} = 2.36 eV$ . Aluminum has $E_{0} = 4.28 eV$ . We can use Equation 38.6, with $h$ in units of $eV s$ , to calculate

f_{0} = \frac{E_{0}}{h} = {\begin{matrix} 5.70 \times 10^{14} Hz & sodium \\ 10.34 \times 10^{14} Hz & aluminum \end{matrix}

These frequencies are converted to wavelengths with $λ = c / f$ , giving

λ = {\begin{matrix} 526 nm & sodium \\ 290 nm & aluminum \end{matrix}

REVIEW The photoelectric effect can be observed with sodium for $λ < 526 nm$ . This includes blue and violet visible light but not red, orange, yellow, or green. Aluminum, with a larger work function, needs ultraviolet wavelengths $λ < 290 nm$ .

EXAMPLE 38.4 Maximum photoelectron speed

What is the maximum photoelectron speed if sodium is illuminated with light of 300 nm?

SOLVE The light frequency is $f = c / λ = 1.00 \times 10^{15} Hz,$ so each light quantum has energy $h f = 4.14 eV$ . The maximum kinetic energy of a photoelectron is

\begin{matrix} K_{max} & = & h f - E_{0} = 4.14 eV - 2.36 eV = 1.78 eV \\ = & 2.85 \times 10^{- 19} J \end{matrix}

Because $K = \frac{1}{2} m v^{2}$ , where $m$ is the electron’s mass, not the mass of the sodium atom, the maximum speed of a photoelectron leaving the cathode is

v_{max} = \sqrt{\frac{2 K_{max}}{m}} = 7.91 \times 10^{5} m / s

Note that we had to convert $K_{max}$ to SI units of J before calculating a speed in $m / s$ .