Turning points: discovery of the electron and the nature of light

What the exam asks

• The electron: cathode rays, thermionic emission, $e/m$ by one method and its significance, and Millikan’s oil drops and the quantisation of charge.
• Light: Newton versus Huygens, the significance of Young’s fringes (no calculations), Maxwell’s $c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}$, Hertz and Fizeau.
• Quanta: the ultraviolet catastrophe, Planck, and why Einstein’s photons succeeded where wave theory failed.
• Matter waves: de Broglie, electron diffraction, the TEM and STM, and the anode voltage needed for atomic-scale wavelengths.

Core ideas

Cathode rays and thermionic emission

In a discharge tube, gas at low pressure has a pd of several kilovolts across it. The few ions present are accelerated. Their collisions ionise more atoms, and positive ions striking the cathode release electrons. These electrons stream away from the cathode as cathode rays. The tube glows because electrons and ions collide with gas atoms and excite them. The atoms emit photons when they de-excite, and also when ions recombine with electrons. The colour is characteristic of the gas.

In thermionic emission, a heated metal filament gives some conduction electrons enough kinetic energy to escape from the surface. An anode at pd $V$ accelerates them, so $\frac{1}{2}mv^2 = eV$ if they start from rest. The tube must be evacuated so the beam is not stopped by collisions.

Measuring the specific charge $e/m$

Fine beam tube. A magnetic field at right angles to the beam provides the centripetal force: $Bev = \frac{mv^2}{r}$, so $v = \frac{Ber}{m}$. Substituting into $\frac{1}{2}mv^2 = eV$ gives

A graph of $r^2$ against $V$ at constant $B$ is a straight line through the origin with gradient $\frac{2m}{eB^2}$.

Crossed fields (Thomson). Perpendicular electric and magnetic fields are adjusted until the beam is undeflected. Then $eE = Bev$, so $v = \frac{E}{B} = \frac{V_p}{dB}$. Then either $\frac{e}{m} = \frac{v^2}{2V_a}$ using the accelerating pd, or, with the electric field switched off, $\frac{e}{m} = \frac{v}{Br}$ from the radius of the path.

Significance. Thomson (1897) found $\frac{e}{m} \approx 1.76 \times 10^{11}\ \text{C kg}^{-1}$, the same whatever gas or cathode metal was used. So the particles are a common part of all atoms. The value is about 1800 times the specific charge of the hydrogen ion ($9.58 \times 10^{7}\ \text{C kg}^{-1}$). If the charges are the same size, the electron has about $\frac{1}{1800}$ of the mass of a hydrogen atom. It was the first subatomic particle, so atoms are not indivisible.

Millikan’s oil-drop experiment

• Stationary drop: the electric force balances the weight, $\frac{QV}{d} = mg$.
• Falling with no field: at terminal speed, $mg = 6\pi\eta rv$ (ignoring upthrust). With $m = \frac{4}{3}\pi r^3\rho$, this gives $r = \sqrt{\frac{9\eta v}{2\rho g}}$. A measured terminal speed therefore gives the radius, then the mass, and the balancing pd then gives $Q$.
• Moving with the field on: the drop reaches a new terminal speed $v'$ where $mg \pm \frac{QV}{d} = 6\pi\eta rv'$.

Significance. Every charge was a whole-number multiple of $1.6 \times 10^{-19}$ C, so charge is quantised and $e$ is its basic unit. Combined with Thomson’s $e/m$, this gave the electron’s mass, $9.1 \times 10^{-31}$ kg.

Newton versus Huygens

Newton’s theory was preferred for over a century because of Newton’s enormous authority, because it explained reflection, refraction and sharp shadows, and because diffraction of light (with its tiny wavelength) was hard to observe. The speed of light in water could not be measured to test the two predictions until 1850.

Young’s double slits (about 1801). Young saw bright and dark fringes. The wave theory explains them by superposition: bright where the waves arrive in phase, dark where they arrive in antiphase. The corpuscular theory predicts only two bright bands opposite the slits and cannot explain darkness where two beams overlap. Acceptance was delayed by Newton’s reputation. It came after Fresnel’s mathematical wave theory (transverse waves explaining polarisation) and Foucault’s 1850 measurement showing that light is slower in water.

Electromagnetic waves: Maxwell, Hertz and Fizeau

Maxwell predicted waves of oscillating electric and magnetic fields, perpendicular to each other and to the direction of travel, moving at

This matched the measured speed of light, so light is an electromagnetic wave. Here $\varepsilon_0$ (the permittivity of free space) relates to the electric field strength around a charged object, and $\mu_0$ (the permeability of free space) relates to the magnetic flux density around a current-carrying wire.

Hertz (1887) made radio waves with a spark gap driven by an induction coil and detected them with a wire loop containing a small gap. Reflecting the waves from a metal sheet set up stationary waves: adjacent nodes are $\frac{\lambda}{2}$ apart, and the frequency came from the oscillating circuit, so $v = f\lambda$ gave a speed close to $c$. Rotating the detector until the sparks stopped showed the waves were polarised, so they are transverse.

Fizeau (1849) sent light through a gap in a rotating toothed wheel to a mirror about 8.6 km away. At a certain rotation rate, the returning light was blocked by the next tooth. The light’s round-trip time $\frac{2d}{c}$ then equals the time for the wheel to turn half a tooth spacing, $\frac{1}{2Nf}$, where $N$ is the number of teeth and $f$ the rotation frequency. So $c = 4dNf$. Fizeau’s value, about $3.1 \times 10^8\ \text{m s}^{-1}$, agreed with Maxwell’s prediction.

Black-body radiation and the photoelectric effect

Classical physics predicted that a black body would radiate ever more intensity at shorter wavelengths, meaning infinite energy in the ultraviolet: the ultraviolet catastrophe. Real curves peak and fall to zero. Planck (1900) fixed this by assuming the energy of the oscillators in the body is quantised in units of $hf$, so high-frequency modes are rarely excited.

Photoelectric observations that wave theory could not explain:

• There is a threshold frequency. Wave theory predicted that any frequency would work if intense enough.
• Emission is instant, even at very low intensity. Wave theory predicted a delay while energy built up.
• The maximum kinetic energy depends on frequency, not intensity. Wave theory linked energy to intensity.

Einstein (1905) proposed that light itself is quantised into photons of energy $E = hf$, each absorbed by one electron: $hf = \phi + E_{k(\max)}$. More intensity means more photons, so more electrons, but not more energy per electron. The significance: electromagnetic radiation has a particle nature as well as a wave nature. Millikan’s experiments in 1916 confirmed the equation.

Matter waves and electron microscopes

De Broglie (1924): every particle has a wavelength $\lambda = \frac{h}{p}$. For an electron accelerated through a pd $V$, $\lambda = \frac{h}{\sqrt{2meV}}$. Electron diffraction by thin crystals confirmed this. At a higher speed, the wavelength is shorter, so there is less diffraction and the rings are smaller.

To get $\lambda \approx 0.1$ nm (the size of an atom), $V = \frac{h^2}{2me\lambda^2} \approx 150$ V.

TEM. An electron gun and an anode pd of about 100 kV produce the beam. Magnetic condenser lenses make it parallel onto a very thin sample, the objective lens forms a magnified image, and the projector lens puts it onto a fluorescent screen or detector. Resolution is limited by the wavelength and by lens aberrations. Electrons slowed inside a thick sample have a longer wavelength and are focused in a different place, blurring the image.

STM.

• A very fine conducting tip, ending in a single atom, is held about 1 nm above a conducting surface with a small pd between them.
• Because of their wave nature, electrons tunnel across the gap. The tunnelling current changes extremely rapidly (exponentially) with the gap width.
• Piezoelectric controls scan the tip. In constant-current mode, feedback moves the tip up and down to keep the current constant, and the recorded tip height maps the surface, resolving individual atoms. In constant-height mode, the changes in current are recorded instead.

Worked examples

Exam technique

• “Explain the significance” means what the result showed and which idea it changed, e.g. “the same $e/m$ for all cathode materials, so electrons are in all atoms”.
• Show-that derivations ($e/m = \frac{2V}{B^2r^2}$, $r = \sqrt{\frac{9\eta v}{2\rho g}}$) need every starting equation stated before you combine them.
• Millikan: find $r$, then $m$, then $Q$, and finish by dividing by $1.6 \times 10^{-19}$ C to show a whole number.
• 6-markers on light or photoelectricity: observation → failed prediction → new explanation → significance. Keep the story in order: Newton, Huygens, Young, Maxwell/Hertz/Fizeau, Planck/Einstein, de Broglie.

Common mistakes

Quick recap

• Thermionic emission plus $\frac{1}{2}mv^2 = eV$ gives the beam speed. The discharge-tube glow comes from atoms de-exciting.
• $\frac{e}{m} = \frac{2V}{B^2r^2}$ (fine beam) or $v = \frac{E}{B}$ then $\frac{v^2}{2V}$ (crossed fields). It is the same for all materials and about 1800 times the hydrogen ion’s value.
• Millikan: $\frac{QV}{d} = mg$ and $mg = 6\pi\eta rv$. Charges are whole-number multiples of , so charge is quantised.
• Newton predicted light is faster in water and Huygens slower. Young’s fringes and Foucault’s measurement settled it for waves.
• $c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}$. Hertz measured radio-wave speed from stationary waves, and Fizeau got .
• Planck quantised oscillator energy; Einstein quantised light itself, giving threshold frequency, instant emission and $E_k$ depending on $f$.
• $\lambda = \frac{h}{\sqrt{2meV}}$, so about 150 V gives atomic-size wavelengths. The TEM needs thin samples, and the STM relies on tunnelling current.