Particle nature of light: Einstein's explanation

What this figure shows. Two small panels each show a metal surface being struck by an incoming photon of energy $E=h\nu$, with an electron shown leaving the surface in panel (a) carrying maximum kinetic energy $K_{max}=h\nu-h\nu_0$, corresponding to light above the threshold frequency, while panel (b) shows the special case at exactly the threshold frequency $\nu_0$, where the photon's energy $E=h\nu_0$ is just enough to free the electron and it leaves the surface with essentially zero kinetic energy, $K=0$. The two panels together are the pictorial version of Einstein's photoelectric equation: as the incident frequency is reduced toward $\nu_0$, the emitted electron's kin …

What this figure shows. A graph with frequency $\nu$ of the incident light along the horizontal axis and maximum kinetic energy $K_{max}$ of the photoelectrons along the vertical axis, drawn as a single straight line that crosses the frequency axis at the threshold frequency $\nu_0$ and has a y-intercept of $-\phi_0$ when extended back to zero frequency. The line's slope is explicitly labelled $h$ -- Planck's constant -- making this graph the direct experimental route to measuring $h$: it is exactly the straight-line form $K_{max}=h\nu-\phi_0$ that Einstei …

What this figure shows. The same kind of $K_{max}$-versus-$\nu$ straight-line graph as Figure 7.14, but now drawn simultaneously for four different metals -- caesium, potassium, sodium and calcium -- as four parallel lines, each starting from its own y-intercept marked with that metal's own negative work function value (-2.14 eV, -2.30 eV, -2.75 eV and -3.20 eV respectively) but all sharing the identical slope, explicitly labelled 'Slope = h'. This is exactly the graph Robert Millikan produced to experimentally verify Einstein's equation: the four lines' common slope, independent of which metal is used, gave Millikan his precise measurement of Planck's constant, $h=6.626\times10^{-34}$ Js, wh …

Worked out. A radiation of wavelength 300 nm is incident on a silver surface, and the task is to determine whether photoelectrons will be observed at all. Computing the incident photon's energy using $E=hc/\lambda$ (expressed in eV) with the standard constants gives $E\approx4.14$ eV. Comparing this against silver's work function, listed in Table 7.1 as 4.7 eV, shows the photon energy (4.14 eV) is less than the work function (4.7 eV) required to free even the most weakly bound surface electron -- so no photoelectrons are observed for this radiation on silver, however long or brightly it is shone, illustrating the strict quantum threshold-energy condition that a …

Worked out. When light of wavelength 2200 Å falls on copper (work function $\phi_0=4.65$ eV), photoelectrons are emitted, and the task is to find both the threshold wavelength and the stopping potential. The threshold wavelength follows directly from $\lambda_0=hc/\phi_0$, giving $\lambda_0\approx2672$ Å -- the longest wavelength (lowest frequency) copper can still respond to. The incident photon's own energy at 2200 Å works out to $E=hc/\lambda\approx9.035\times10^{-19}$ J $\approx5.65$ eV, so applying Einstein's equation gives the photoelectrons' maximum kinetic energy as $K_{max}=h\nu-\phi_0=5.65-4.65=1$ eV. Since $K_{max}=eV_0$, the corresponding stopping potential is simply $V_0=1$ V -- a clean numerical illustration of how a single incident wavelength, together with a metal's tabulated work function, pins down both the thresho …

Worked out. UV light of wavelength 3000 Å and intensity $2\ \text{W m}^{-2}$ is incident on a potassium surface (work function 2.30 eV) of area $2\ \text{cm}^2$, and the task has two parts. First, the incident photon's energy works out to $E=hc/\lambda\approx6.626\times10^{-19}$ J $\approx4.14$ eV, so by Einstein's equation the maximum kinetic energy of the ejected photoelectrons is $K_{max}=h\nu-\phi_0=4.14-2.30=1.84$ eV. Second, assuming 40% of the incident photons actually succeed in producing a photoelectron, the number of photons striking the surface per second is found from the incident power divided by the single-photon energy, $n_p=IA/E\approx6.04\times10^{14}$ photons per second, so the rate of photoelectron emission is $0.40\times n_p\approx2.415\times10^{14}$ photoelectrons per second -- showing how a photon-counting argument …

Worked out. Light of wavelength 390 nm directed at a metal electrode is found to be completely stopped by an opposing potential difference of 1.10 V, and the task is to find both the metal's work function and its threshold wavelength for ejecting electrons. Since $K_{max}=eV_0$ at the stopping potential, Einstein's equation $\phi_0=h\nu-K_{max}$ can be rewritten as $\phi_0=\dfrac{hc}{\lambda}-eV_0$; substituting the given wavelength and stopping potential gives $\phi_0\approx3.34\times10^{-19}$ J $\approx2.09$ eV. The threshold wavelength then follows from $\lambda_0=hc/\phi_0$, giving $\lambda_0\approx5969$ Å (about 5963-5969 Å depending on rounding) -- demonstrating that a single stopping-potential measurement at one known wavelength is, by itself, enough to fully characterise …