photoelectric laws and Einstein equation

Laws of the photoelectric effect (Stoletov): 1) at fixed frequency the saturation photocurrent is proportional to the light intensity; 2) every substance has a red threshold $\nu_0$ — for $\nu<\nu_0$ there is no emission, however high the intensity; 3) emission begins almost instantaneously ($t\approx10^{-9}$ s).

Wave theory cannot explain the second and third laws, whereas the photon model explains all three: light is a stream of photons of energy $h\nu$, and one photon interacts with one electron (photon energy and momentum: $E=h\nu$, $p=h/\lambda=h\nu/c$). Energy conservation yields the Einstein equation

$$h\nu = A_{\text{out}} + K_{\max},$$

where $A_{\text{out}}$ is the work function and $K_{\max}$ the maximum kinetic energy of an emitted electron. The stopping voltage $U_{s}$ at which the current vanishes measures that energy: $K_{\max}=eU_{s}$, hence

$$U_{s} = \frac{h}{e}\nu - \frac{A_{\text{out}}}{e}.$$

The graph of $U_{s}$ against $\nu$ is a straight line of slope $h/e$, meeting the frequency axis at $\nu_0=A_{\text{out}}/h$.

A metal with work function $A_{\text{out}}=2.2\ \text{eV}$ is illuminated by light of wavelength $\lambda=400\ \text{nm}$. Find the maximum kinetic energy of the photoelectrons and the stopping voltage.

The red threshold of a metal is $\lambda_0=600\ \text{nm}$. The metal is illuminated with light of $\lambda=400\ \text{nm}$. Find the work function, the stopping voltage, and the maximum speed of the photoelectrons ($m_e=9.11\times10^{-31}\ \text{kg}$).

A laser of power $P=2.0\ \text{mW}$ sends light of $\lambda=500\ \text{nm}$ onto a cathode with work function $A_{\text{out}}=1.5\ \text{eV}$. (a) How many photons hit the cathode per second? (b) How do the saturation current and the stopping voltage change if the laser power is doubled at the same wavelength?

A simulation instead of a hazardous experiment. Do not use real ultraviolet lamps: work in the PhET "Photoelectric Effect" simulation (or an equivalent one).

1. Choose a metal (for example, sodium) and record its work function.
2. Set the wavelength $\lambda=400\ \text{nm}$ and a moderate intensity. Slowly increase the stopping voltage until the photocurrent becomes zero; record $U_{s}$.
3. Repeat for 5–6 wavelengths (for example, 350, 400, 450, 500, 550, 600 nm).
4. Plot $U_{s}$ against $\nu=c/\lambda$ and draw a straight line. The slope gives $h/e$ — estimate $h$; the intercept with the $\nu$ axis gives the red threshold $\nu_0$.
5. Changing only the intensity, check that the saturation current grows in proportion to intensity while $U_{s}$ stays the same.
6. Lower the frequency below $\nu_0$ and check that the current is zero at any intensity.

Show that the stopping voltage depends linearly on frequency, and link the slope and the intercept of the line to Planck's constant and the work function.

Graph: frequency $\nu$ on the horizontal axis, stopping voltage $U_{s}$ on the vertical axis. The experimental points fall on a straight line $U_{s}=\frac{h}{e}\nu-\frac{A_{\text{out}}}{e}.$ The graph marks the slope $h/e$, the intercept with the frequency axis $\nu_0=A_{\text{out}}/h$ (the red threshold), and the negative intercept on the $U_{s}$ axis equal to $-A_{\text{out}}/e$. For different metals the lines are parallel and shifted vertically by the difference of their work functions divided by $e$.

At a school 100 m race, a blue light beam with wavelength $\lambda = 450\ \text{nm}$ falls on a cesium photocathode with work function $W = 2.14\ \text{eV}$. The light flux produces a photocurrent; when a runner blocks the beam, the current drops and the timer records the time. The maximum kinetic energy of emitted electrons is $E_{k\max}=h\nu - W=\frac{hc}{\lambda}-W \approx 2.76-2.14=0.62\ \text{eV}$, and the stopping voltage is $U_s=E_{k\max}/e \approx 0.62\ \text{V}$.

The scenario exercises the first photoelectric law: the photocurrent is proportional to the number of photons per second, that is, to light intensity. Einstein's equation $h\nu = W + E_{k\max}$ shows that the maximum kinetic energy and stopping voltage are set by light frequency and work function, not by beam brightness.

A smartphone screen automatically changes brightness from the signal of a silicon photodiode. Green light from a window has $\lambda = 550\ \text{nm}$, so the photon energy is $E=\frac{1240}{550}\approx 2.25\ \text{eV}$. For silicon the threshold energy is about $W=1.12\ \text{eV}$, so the maximum photoelectron energy is $E_{k\max}\approx 1.13\ \text{eV}$, and the red threshold is $\lambda_{\max}=\frac{hc}{W}\approx 1107\ \text{nm}$. If the window brightness doubles, the photocurrent doubles, but $E_{k\max}$ does not change.

The scenario tests the photoelectric red threshold and the independence of $E_{k\max}$ from intensity. If the wavelength exceeds $\lambda_{\max}$, a single photon cannot eject an electron, so the sensor does not respond. Einstein's equation gives $E_{k\max}=h\nu-W$ for each absorbed photon.

A kitchen faucet turns on when a hand reflects an infrared beam with $\lambda=940\ \text{nm}$ back to a photodetector. The photon energy is $E=\frac{1240}{940}\approx 1.32\ \text{eV}$. For a silicon detector with $W=1.12\ \text{eV}$, the maximum kinetic energy is $E_{k\max}=1.32-1.12=0.20\ \text{eV}$, and the stopping voltage is $U_s=0.20\ \text{V}$. Brighter infrared light only increases the number of electrons, not their maximum energy.

The scenario shows the instantaneous nature of the photoelectric effect: current appears immediately upon illumination, and its magnitude is proportional to intensity. The threshold frequency $\nu_{\min}=W/h$ explains why the detector is chosen for a particular wavelength, while Einstein's equation links photon energy to work function and electron kinetic energy.