Turning Points in Physics

The discovery of the electron, wave-particle duality, and special relativity.

Spec Points Covered
  • Explain how cathode raysStreams of electrons emitted from the cathode of a discharge tube at very low pressures. Originally observed as fluorescence on the glass walls. are produced in a discharge tube and describe the key observations at different pressures.
  • Describe thermionic emissionThe release of electrons from a heated metal surface. The thermal energy of the metal gives electrons enough energy to escape the surface. and calculate the work done on an electron accelerated through a potential difference using $W = eV$.
  • Determine the specific chargeThe charge-to-mass ratio of a particle, $e/m_e$, measured in C kg$^{-1}$. of the electron using deflection in electric and magnetic fields.
  • Explain Thomson's experiment and why the electron's specific charge being 1800 times larger than hydrogen's was significant.
  • Describe Millikan's oil drop experiment and explain how it demonstrated the quantisation of chargeElectric charge exists only in integer multiples of the elementary charge $e = 1.60 \times 10^{-19}$ C..
  • Compare Newton's corpuscular theory and Huygens' wave theory of light and explain how each accounts for reflection, refraction, and diffraction.
  • Explain how Young's double-slit experiment provided evidence for the wave nature of light.
  • Describe how electromagnetic waves were predicted by Maxwell ($c = 1/\sqrt{\mu_0 \varepsilon_0}$) and confirmed by Hertz.
  • Explain the ultraviolet catastropheThe failure of classical wave theory to predict the observed spectrum of black-body radiation at short wavelengths, where it predicted infinite intensity. and how Planck resolved it with quantised energy $E = nhf$.
  • Apply de Broglie's hypothesis $\lambda = h/p$ and describe evidence for electron diffraction.
  • Describe the Michelson-Morley experiment and explain how its null result led to Einstein's postulates of special relativity.
  • Apply time dilation $\Delta t = \gamma \Delta t_0$ and length contraction $L = L_0 / \gamma$ using the Lorentz factor $\gamma = 1/\sqrt{1 - v^2/c^2}$.
  • Describe the muon lifetime experiment as evidence for time dilation and length contraction.
  • Apply mass-energy equivalence $E = mc^2$ and calculate relativistic kinetic energy $E_k = (\gamma - 1)m_0 c^2$.
  • Describe Bertozzi's experiment and explain how it confirmed that no particle can reach the speed of light.
Notes
12.1: The Discovery of the Electron
01 Cathode rays are produced in discharge tubes at very low pressures Cathode rays 3.12.1.1 → 02 Thermionic emission releases electrons from a heated filament Thermionic emission 3.12.1.2 → 03 Measuring the specific charge of the electron using electric and magnetic fields Specific charge 3.12.1.3 → 04 Thomson's experiment revealed the electron as a universal subatomic particle Thomson's experiment 3.12.1.4 → 05 Millikan's oil drop experiment proved charge is quantised Millikan's experiment 3.12.1.5 → 06 Stokes' law and terminal velocity in Millikan's experiment $F = 6\pi \eta r v$ 3.12.1.5 →
12.2: Wave-Particle Duality
07 Newton's corpuscular theory treated light as a stream of particles Corpuscular theory 3.12.2.1 → 08 Huygens' wave theory explains light using wavefronts and wavelets Huygens' principle 3.12.2.1 → 09 Young's double-slit experiment and the discovery of electromagnetic waves $c = 1/\sqrt{\mu_0 \varepsilon_0}$ 3.12.2.2 → 10 The UV catastrophe, Planck's quantisation, and the photoelectric effect $E = nhf$ 3.12.2.3 → 11 De Broglie's hypothesis and electron diffraction $\lambda = h/p$ 3.12.2.4 → 12 Electron microscopes: the TEM and the scanning tunnelling microscope TEM & STM 3.12.2.5 →
12.3: Special Relativity
13 The Michelson-Morley experiment and the death of the luminiferous aether Michelson-Morley 3.12.3.1 → 14 Einstein's two postulates of special relativity Special relativity 3.12.3.2 → 15 Time dilation: moving clocks run slower $\Delta t = \gamma \Delta t_0$ 3.12.3.3 → 16 Length contraction and the muon lifetime experiment $L = L_0 / \gamma$ 3.12.3.4 → 17 Mass-energy equivalence and Bertozzi's experiment $E = mc^2$ 3.12.3.5 →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
Work done on an electron
$$W = eV$$
  • Where:
    • $W$ = work done / kinetic energy gained (J)
    • $e$ = charge of the electron (C)
    • $V$ = accelerating potential difference (V)
Used in thermionic emission and in Bertozzi's experiment.
Specific charge (magnetic field only)
$$\frac{e}{m_e} = \frac{2V_a}{B^2 r^2}$$
  • Where:
    • $V_a$ = accelerating p.d. (V)
    • $B$ = magnetic flux density (T)
    • $r$ = radius of circular path (m)
Derived by equating kinetic energy to work done and centripetal force to magnetic force.
De Broglie wavelength
$$\lambda = \frac{h}{p} = \frac{h}{mv}$$
  • Where:
    • $\lambda$ = de Broglie wavelength (m)
    • $h$ = Planck constant (J s)
    • $p$ = momentum (kg m s$^{-1}$)
All particles have a wavelength. Wavelength decreases with increasing momentum.
Thomson's crossed-fields method
$$\frac{e}{m_e} = \frac{V}{B^2 r d}$$
  • Where:
    • $V$ = p.d. across plates (V)
    • $B$ = magnetic flux density (T)
    • $r$ = radius of curvature in B field alone (m)
    • $d$ = plate separation (m)
Derived by balancing electric and magnetic forces to find $v$, then using magnetic deflection.
Millikan's stationary drop condition
$$QE = mg \quad \Rightarrow \quad Q = \frac{mgd}{V}$$
  • Where:
    • $Q$ = charge on the drop (C)
    • $E$ = electric field strength (V m$^{-1}$)
    • $d$ = plate separation (m)
    • $V$ = p.d. across plates (V)
When the drop is stationary, the electric force balances gravity.
Relativistic kinetic energy
$$E_k = (\gamma - 1)\,m_0 c^2$$
  • Where:
    • $E_k$ = relativistic kinetic energy (J)
    • $\gamma$ = Lorentz factor
    • $m_0$ = rest mass (kg)
    • $c$ = speed of light (m s$^{-1}$)
At low speeds, this reduces to the Newtonian $\frac{1}{2}m_0 v^2$.
Stokes' law
$$F = 6\pi \eta r v$$
  • Where:
    • $F$ = viscous drag force (N)
    • $\eta$ = viscosity of the fluid (Pa s)
    • $r$ = radius of the sphere (m)
    • $v$ = speed of the sphere (m s$^{-1}$)
Used to find the radius and mass of the oil drop at terminal velocity.
Speed of light from Maxwell's equation
$$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$$
  • Where:
    • $c$ = speed of light (m s$^{-1}$)
    • $\mu_0$ = permeability of free space (H m$^{-1}$)
    • $\varepsilon_0$ = permittivity of free space (F m$^{-1}$)
Maxwell showed light is an electromagnetic wave from this calculation alone.
Lorentz factor
$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$
  • Where:
    • $\gamma$ = Lorentz factor (dimensionless)
    • $v$ = speed of the object (m s$^{-1}$)
    • $c$ = speed of light (m s$^{-1}$)
$\gamma$ is always $\geq 1$. At everyday speeds it is effectively 1.
Time dilation
$$\Delta t = \gamma \Delta t_0$$
  • Where:
    • $\Delta t$ = dilated time measured by external observer (s)
    • $\Delta t_0$ = proper time in the moving frame (s)
Moving clocks run slower. $\Delta t_0$ is always the shorter time.
Length contraction
$$L = \frac{L_0}{\gamma}$$
  • Where:
    • $L$ = contracted length measured by external observer (m)
    • $L_0$ = proper length in the rest frame (m)
Objects moving relative to an observer appear shorter in the direction of motion.
Mass-energy equivalence
$$E = mc^2$$
  • Where:
    • $E$ = total energy (J)
    • $m$ = relativistic mass (kg)
    • $c$ = speed of light (m s$^{-1}$)
The total energy of a particle includes its rest energy and its kinetic energy.
Q Retrieval Practice
Q1. What are cathode raysStreams of electrons emitted from the cathode of a discharge tube at very low pressures. and how are they produced?
Cathode rays are streams of electrons. They are produced in a discharge tube when a high voltage is applied across a gas at very low pressure. Electrons from the cathode are accelerated towards the anode, striking the glass and causing fluorescence.
Q2. State the equation for the work done on an electron accelerated through a potential difference, and define each term.
  • $W = eV$.
  • $W$ is the work done on the electron (equal to the kinetic energy gained), $e$ is the electron charge, and $V$ is the accelerating potential difference.
  • This applies whenever an electron is accelerated from rest through a p.d.
Q3. Why was the electron's specific chargeThe charge-to-mass ratio of a particle, $e/m_e$, measured in C kg$^{-1}$. being 1800 times that of the hydrogen ion so significant?
  • The hydrogen ion was the smallest known charged particle at the time.
  • Since the electron had the same magnitude of charge but a much higher specific charge, its mass had to be roughly 1800 times smaller than hydrogen.
  • This proved the electron was a new subatomic particle, far smaller than any atom.
Q4. What is the ultraviolet catastropheThe failure of classical wave theory to predict the observed spectrum of black-body radiation at short wavelengths, where it predicted infinite intensity. and how did Planck resolve it?
  • Classical wave theory predicted that the intensity of radiation emitted by a black body should increase without limit at short wavelengths. This disagreed with experimental observations.
  • Planck proposed that energy is emitted in discrete packets (quanta), with $E = nhf$.
  • This quantisation naturally limits the energy at high frequencies, producing the correct spectrum.
Q5. State Einstein's two postulates of special relativityEinstein's theory applying to objects in uniform relative motion. Built on two postulates: (1) the laws of physics are the same in all inertial frames; (2) the speed of light in a vacuum is the same for all observers..
  • First postulate: The laws of physics are the same in all inertial frames of reference.
  • Second postulate: The speed of light in a vacuum is invariant, meaning it is the same for all observers regardless of their relative motion or the motion of the source.