Photoelectron in a magnetic field: quantum physics and magnetism

Problem

A sodium surface (work function $W_0 = 2{,}3$ eV) is illuminated by monochromatic light of wavelength $\lambda = 300$ nm. The emitted electrons enter perpendicularly into a uniform magnetic field $B = 1{,}0$ mT. Determine: (a) the maximum kinetic energy of the electrons, (b) the radius of the circular trajectory in the field.

Solution

(a) Maximum kinetic energy (photoelectric effect). Einstein’s equation for the photoelectric effect states that the photon’s energy is spent partly on extracting the electron ($W_0$) and partly becomes kinetic energy: $E_{k,\max} = hf - W_0 = \frac{hc}{\lambda} - W_0$ It is useful to remember the practical constant $hc \approx 1240\;\text{eV}\cdot\text{nm}$: $\frac{hc}{\lambda} = \frac{1240}{300} \approx 4{,}13\;\text{eV}$ $\ev{E_{k,\max} = 4{,}13 - 2{,}3 \approx 1{,}83\;\text{eV}}$

(b) Radius in the magnetic field (Lorentz force). First, the electron’s velocity (non-relativistic: $E_k\ll m_e c^2$). Converting $E_k = 1{,}83\;\text{eV} = 2{,}93\cdot 10^{-19}\;$J: $v = \sqrt{\frac{2E_k}{m_e}} = \sqrt{\frac{2\cdot 2{,}93\cdot 10^{-19}}{9{,}11\cdot 10^{-31}}} \approx 8{,}0\cdot 10^{5}\;\text{m/s}$ In the field, the Lorentz force $\pq{F}=q\pq{v}\times\pq{B}$ is centripetal and generates uniform circular motion. Equating the Lorentz force and the centripetal force, $evB = m_e v^2/r$, gives: $r = \frac{m_e v}{eB} = \frac{9{,}11\cdot 10^{-31}\cdot 8{,}0\cdot 10^{5}}{1{,}602\cdot 10^{-19}\cdot 1{,}0\cdot 10^{-3}}$ $\ev{r \approx 4{,}6\cdot 10^{-3}\;\text{m} \approx 4{,}6\;\text{mm}}$

The connections: the photoelectric effect (ch. 21) to convert light into kinetic energy, the Lorentz force (ch. 17) for the circular motion, with conservation of energy (ch. 4) linking the light’s frequency and the electron’s velocity. Note the bridge between the two worlds: a quantum phenomenon (the photon) fixes the velocity, while a classical phenomenon (motion in a magnetic field) determines its trajectory.

Links

Topics: Quantum physics · Magnetism Concepts: Photoelectric effect · Photon · Lorentz force · Uniform circular motion