Turning points in physics (A-level only)

Explanation

• Cathode rays are produced in a discharge tube containing gas at low pressure and a large potential difference between electrodes. The field ionises gas atoms.
• Positive ions accelerate towards the negative cathode and release electrons on impact; these electrons accelerate away from the cathode towards the anode.
• A hole in the anode can form a narrow beam, which may produce fluorescence when it reaches the glass or a screen.
• Electric deflection towards a positive plate identifies negative charge, while magnetic deflection confirms moving charged particles.
• The independence of the ray's behaviour from gas and electrode material supported the conclusion that electrons are constituents of all atoms.

Explanation

• Thermionic emission occurs when heating a metal gives some conduction electrons enough energy to escape from its surface. A heated cathode therefore supplies electrons continuously, and a positive anode can accelerate them into a beam.
• When an electron starts from rest and moves through potential difference $V$, electrical work $eV$ becomes kinetic energy: $eV=\frac12m_{\mathrm e}v^2$.
• This relation assumes negligible energy loss and non-relativistic speed.
• Potential difference must be in volts, $e$ in coulombs and $m_{\mathrm e}$ in kilograms.
• The examiner may ask for the emission principle, an energy-transfer explanation, or a speed calculated by rearranging the energy equation.

Explanation

• Specific charge is charge divided by mass; the electron's magnitude is $e/m_{\mathrm e}$ in $\text{C kg}^{-1}$. In one determination, crossed electric and magnetic fields are adjusted so the beam is undeflected.
• Balancing $eE=evB$ gives the selected speed $v=E/B$.
• The electrons then enter a perpendicular magnetic field and follow a circular path because $evB=m_{\mathrm e}v^2/r$, giving $e/m_{\mathrm e}=v/(Br)$.
• Thomson found a magnitude far greater than the hydrogen-ion specific charge.
• Since charge magnitudes are comparable, this showed that the electron has far smaller mass and is a subatomic constituent rather than a whole atom.

Explanation

• In Millikan's method, a charged oil droplet lies between oppositely charged parallel plates.
• When it is stationary, upward electric force balances weight: $QE=mg$, with $E=V/d$, so $QV/d=mg$.
• With the field removed, a falling droplet reaches terminal speed when Stokes' viscous force $6\pi\eta rv$ balances its effective weight; together with the droplet density, this determines radius and mass.
• Motion may also be observed with the field applied to find the charge.
• Repeating the measurement produced charges that were integer multiples of a smallest value, $e$, demonstrating that electric charge is quantised rather than continuous.

Explanation

• Newton's corpuscular theory treated light as tiny particles emitted by sources and travelling in straight lines. This readily explained sharp shadows, and forces acting on corpuscles at boundaries were invoked to account for reflection and refraction.
• Huygens instead treated every point on a wavefront as a source of secondary wavelets.
• Newton's theory was preferred because his mechanical ideas were highly successful and authoritative, ray-like propagation was familiar, and diffraction was not obvious in everyday conditions.
• The models made different predictions: Newton's corpuscles should speed up in a denser optical medium, whereas Huygens' waves should slow.
• Later speed measurements and interference evidence supported the wave account.

Explanation

• Young's double-slit experiment uses one monochromatic source to illuminate two narrow slits, producing waves with a stable phase relationship. Diffraction at each slit allows the two waves to overlap on a screen.
• Where they arrive in phase, constructive superposition forms a bright fringe; where they arrive in antiphase, destructive superposition forms a dark fringe.
• Only a qualitative account is required here, not fringe-spacing calculations.
• The repeated bright and dark pattern was powerful evidence for wave behaviour because independent classical corpuscles could add intensity but could not explain cancellation to darkness.
• Acceptance of Huygens' wave theory was nevertheless delayed by Newton's authority and the established corpuscular view.

Explanation

• An electromagnetic wave consists of oscillating electric and magnetic fields that are perpendicular to each other and to the propagation direction. Maxwell predicted the vacuum speed $c=1/\sqrt{\mu_0\varepsilon_0}$.
• The permittivity $\varepsilon_0$ relates to electric-field strength produced by charge in free space, while permeability $\mu_0$ relates to magnetic flux density produced by current.
• Agreement between Maxwell's value and the measured speed of light identified light as electromagnetic.
• Hertz generated and detected radio waves, measured a speed close to $c$ and observed wave behaviour, supporting Maxwell.
• Fizeau's toothed-wheel experiment independently measured the finite speed of light on Earth, strengthening the connection.

Explanation

• Classical theory predicted the ultraviolet catastrophe: black-body intensity should rise without limit at high frequency, contrary to observed spectra. Planck resolved this by proposing that energy is exchanged in quanta $E=hf$, making high-frequency quanta less likely at a fixed temperature.
• Classical wave theory also failed for photoelectricity: it could not explain threshold frequency, immediate emission, or maximum electron kinetic energy depending on frequency rather than intensity.
• Einstein treated each quantum as a photon absorbed by one electron, giving $hf=\phi+K_{\max}$.
• Above threshold, greater intensity supplies more photons and raises the emission rate, but at fixed frequency it does not raise $K_{\max}$.
• This established a particle aspect of electromagnetic radiation.

Explanation

• de Broglie's hypothesis assigns wavelength $\lambda=h/p$ to a particle of momentum $p$. For a non-relativistic electron accelerated from rest through potential difference $V$, $eV=p^2/(2m_{\mathrm e})$, so $\lambda=h/\sqrt{2m_{\mathrm e}eV}$.
• Low-energy electrons diffracted by a crystal produce rings or maxima, demonstrating wave superposition, while localised detection retains particle behaviour.
• Increasing electron speed or accelerating voltage raises momentum and reduces wavelength.
• With fixed crystal spacing, a smaller wavelength produces diffraction maxima at smaller angles.
• Calculations require potential difference in volts and energy $eV$ in joules; the voltage itself is not an energy.

Explanation

• Electron microscopes exploit electron de Broglie wavelengths that can be comparable with atomic dimensions. For non-relativistic electrons accelerated through anode voltage $V$, $\lambda=h/\sqrt{2m_{\mathrm e}eV}$, so the voltage required for an atomic-scale wavelength can be estimated.
• In a transmission electron microscope, electrons are accelerated, pass through a thin specimen, and electromagnetic lenses form an image from transmitted and scattered electrons.
• A scanning tunnelling microscope instead moves a sharp conducting tip very close to a conducting surface.
• Quantum tunnelling current changes steeply with tip–surface separation, allowing surface height or electronic structure to be mapped.
• The STM's atomic sensitivity is not produced by a TEM-style imaging beam.

Explanation

• A Michelson interferometer sends coherent light to a half-silvered beam splitter, producing two beams that travel along perpendicular arms. Mirrors return the beams to the splitter, where they recombine and form interference fringes.
• If Earth moved through a stationary ether, an 'ether wind' would make the round-trip light times depend on arm orientation.
• Rotating the apparatus should then change the phase difference and shift the fringes.
• Michelson and Morley found no significant periodic shift.
• This null result did not prove Earth stationary; it failed to detect absolute motion, undermined the stationary-ether model and supported the invariance of the speed of light.

Explanation

• An inertial frame is non-accelerating: an object with no resultant force moves at constant velocity within it.
• Einstein's first postulate states that physical laws have the same form in all inertial frames, so no inertial frame is privileged and no experiment within one reveals absolute uniform motion.
• The second postulate states that the speed of light in free space is invariant: every inertial observer measures $c$, independent of the source's or observer's motion.
• Keeping both postulates requires measured time intervals, lengths and simultaneity to depend on reference frame.
• Galilean addition cannot be applied to make a measured light speed greater or smaller than $c$.

Explanation

• Proper time $t_0$ is measured by one clock present at both events, so the events occur at the same position in that clock's frame.
• An observer for whom the clock moves measures the dilated interval $t=t_0/\sqrt{1-v^2/c^2}=\gamma t_0$, which is longer than $t_0$.
• The effect is a consequence of special relativity, not a mechanical fault in the clock.
• Atmospheric muons provide evidence: their proper mean lifetime is short, but in Earth's frame their moving lifetime is dilated, allowing many more to reach the surface than classical timing predicts.
• Calculations must first identify which interval is proper and which belongs to the laboratory frame.

Explanation

• Proper length $l_0$ is measured in the object's rest frame, where its endpoints have fixed positions. An observer who sees the object moving parallel to its length measures $l=l_0\sqrt{1-v^2/c^2}=l_0/\gamma$, so the moving length is shorter.
• Only the component parallel to relative motion contracts; transverse dimensions do not.
• Measuring a moving length requires recording both endpoint positions simultaneously in the observer's frame, linking contraction to the relativity of simultaneity.
• A laboratory track is therefore its own proper length in the laboratory frame, while a passing particle describes that track as contracted.
• Examiners expect correct identification of the rest frame before substitution.

Explanation

• Mass and energy are equivalent: $E=mc^2$.
• In the specification's relativistic-mass convention, $m=m_0/\sqrt{1-v²/c²}=\gamma m_0$, so total energy is $E=\gamma m_0c^2$ and kinetic energy is $K=(\gamma-1)m_0c^2$.
• At low speed this approaches $\frac12m_0v^2$.
• Graphs of relativistic mass and kinetic energy against speed rise increasingly steeply and tend towards infinity as $v$ approaches $c$, so a massive particle cannot reach $c$.
• Bertozzi independently measured electron kinetic energy and speed: added energy produced ever smaller speed increases near $c$, directly supporting the relativistic relation and rejecting the classical prediction of speeds above $c$.