AQA A-Level Physics: Turning points in physics (7408) – Revision Notes

These notes condense section 3.12 Turning points in physics of the AQA AS and A-level Physics specification (7407/7408, version 1.4, AS and A-level exams June 2016 onwards), sub-sections 3.12.1 to 3.12.3. Turning points is one of the options (sections 3.9 to 3.13); you study one option, assessed in Paper 3 Section B. It is A-level only. For full explanations and worked examples, read the Turning points study guide first.

Useful links: course hub, printable checklist, free diagnostics, and the Turning points practice questions. Core background on the photoelectric effect and de Broglie wavelength is in Particles and radiation.

Formula table

Formula	Use	Spec ref
½mv² = eV	electron accelerated from rest through pd V	3.12.1.2
e/m = 2V/(B²r²)	fine-beam tube (combine Bev = mv²/r with ½mv² = eV)	3.12.1.3
QV/d = mg	oil drop held stationary	3.12.1.4
F = 6πηrv	Stokes’ law, viscous drag on a drop	3.12.1.4
c = 1/√(μ₀ε₀)	speed of EM waves in a vacuum	3.12.2.3
p = h/λ	de Broglie	3.12.2.5
λ = h/√(2meV)	electron accelerated through V	3.12.2.5
t = t₀/√(1 − v²/c²)	time dilation	3.12.3.3
l = l₀√(1 − v²/c²)	length contraction	3.12.3.4
E = mc², E = m₀c²/√(1 − v²/c²)	mass-energy	3.12.3.5

Useful derived results: Millikan radius r = √(9ηv/(2ρg)); Fizeau c = 2d/t with t = 1/(2Nf); kinetic energy Ek = mc² − m₀c².

3.12.1 The discovery of the electron

• Cathode rays: low-pressure gas, high pd. Positive ions hit the cathode and free electrons; these accelerate to the anode, ionising and exciting gas atoms. De-excitation gives the glow.
• Thermionic emission: heating a metal gives conduction electrons enough energy to leave the surface. An anode in a vacuum accelerates them; work done = eV.
• e/mₑ (one method): fine-beam tube, or Thomson’s crossed fields (v = E/B when undeflected).
• Significance of Thomson’s e/m: same value for any gas or cathode, so electrons are in all matter. The electron’s e/m is about 1800 times the hydrogen ion’s specific charge, so (for equal charge) much less massive than any atom.
• Millikan: field off → terminal speed → radius from Stokes’ law → mass. Field on, drop stationary → Q = mgd/V. Drop rising at terminal speed → QV/d = mg + 6πηrv.
• Significance of Millikan: all charges were integer multiples of e. Charge is quantised.

Method: Millikan in steps

1. Field off: mg = 6πηrv with m = (4/3)πr³ρ → r = √(9ηv/(2ρg)).
2. m = (4/3)πr³ρ.
3. Field on, stationary: Q = mgd/V.
4. Divide by 1.60 × 10⁻¹⁹ C; the answer should be close to a whole number.

3.12.2 Wave-particle duality

• Newton (corpuscles): refraction by attraction into the denser medium, so light faster in water. Huygens (waves): secondary wavelets, light slower in water.
• Why Newton won: his reputation; it explained reflection, refraction and sharp shadows; the speed of light in water could not be measured then.
• Young: two coherent sources; in phase → bright, antiphase → dark. General explanation only. Wave theory still slow to be accepted because of Newton’s authority.
• EM wave: E and B fields oscillating in phase, perpendicular to each other and to the direction of travel; no medium needed.
• ε₀ relates to E near a charged object; μ₀ to B near a current-carrying wire.
• Hertz: spark-gap transmitter, loop detector; stationary waves from a metal reflector give λ (nodes λ/2 apart); with f known, c = fλ = speed of light.
• Fizeau: rotating toothed wheel and distant mirror. Implication: measured c equals Maxwell’s predicted speed, so light is an EM wave.
• Black body / UV catastrophe: classical theory predicts intensity → ∞ as λ → 0; observed curve peaks and falls to zero. Planck: energy in quanta E = hf.
• Photoelectricity — wave theory fails: threshold frequency; instant emission; Ek(max) depends on f not intensity. Einstein: one photon (hf) to one electron. Significance: EM radiation has a particle nature.
• de Broglie: p = h/λ. LEED: faster electrons → shorter λ → less diffraction → smaller rings.
• Anode pd for atom-sized λ: V = h²/(2meλ²), a few hundred volts for λ near 10⁻¹⁰ m.
• TEM: filament → anode → magnetic condenser lens → thin sample → objective and projector lenses → fluorescent screen. Limits: wavelength, sample thickness, lens aberrations.
• STM: conducting tip about 1 nm from a conducting surface; electrons tunnel; current very sensitive to gap; constant-current or constant-height mode.

3.12.3 Special relativity

• Michelson-Morley: semi-silvered plate splits light into perpendicular arms; mirrors return it; fringes form. Ether motion should shift fringes when rotated through 90°. No shift: absolute motion undetectable; speed of light invariant.
• Inertial frame: not accelerating; Newton’s first law holds.
• Postulates: (1) physical laws have the same form in all inertial frames; (2) the speed of light in free space is invariant.
• Proper time t₀: measured in the frame where the clock is at rest. Moving observers measure longer t.
• Muons: made high in the atmosphere; many more reach the ground than their proper half-life allows without dilation.
• Proper length l₀: measured at rest relative to the object. Moving lengths contract along the motion only.
• Graphs: m rises from m₀ towards ∞ as v → c; Ek ≈ ½m₀v² at low v, then → ∞ as v → c.
• Bertozzi: electrons of known energy timed over a known distance; energy checked by heating of the target. v approaches but never exceeds c.

Method: relativistic speed from kinetic energy

1. Rest energy m₀c² (electron: 8.20 × 10⁻¹⁴ J).
2. Total E = Ek + m₀c².
3. γ = E/(m₀c²).
4. v = c√(1 − 1/γ²).

Must-know distinctions

• Thermionic emission vs photoelectric emission: heating vs photons.
• Discharge tube (low-pressure gas) vs electron gun (vacuum, heated filament).
• Stationary drop vs terminal-speed drop: QV/d = mg vs mg = 6πηrv (field off).
• Planck vs Einstein: Planck quantised emission from a black body; Einstein said light itself travels as photons.
• TEM vs STM: TEM transmits electrons through a thin sample; STM uses tunnelling between tip and surface.
• t vs t₀, l vs l₀: the proper value is measured in the object’s own rest frame; t > t₀ but l < l₀.
• m vs m₀: rest mass is fixed; m = γm₀ grows with speed.

Quick self-test

1. An electron is accelerated from rest through 800 V. Calculate its speed.
2. A fine-beam tube runs at 300 V with B = 1.0 mT. Taking e/mₑ = 1.76 × 10¹¹ C kg⁻¹, calculate the radius of the beam.
3. A drop of mass 3.2 × 10⁻¹⁵ kg is held stationary between plates 5.0 mm apart at 490 V. Find its charge in terms of e.
4. Calculate the viscous force on a drop of radius 1.5 × 10⁻⁶ m falling at 2.0 × 10⁻⁴ m s⁻¹ in air of viscosity 1.8 × 10⁻⁵ N s m⁻².
5. Which constant, μ₀ or ε₀, relates to the magnetic flux density near a current-carrying wire?
6. Calculate the de Broglie wavelength of an electron accelerated from rest through 2.0 kV.
7. The anode pd in an electron diffraction tube is increased. State the effect on the rings.
8. A clock records a proper time of 4.0 μs. What time does an observer moving at 0.60c relative to it measure?
9. A rod of proper length 2.5 m moves along its length at 0.60c. Find its measured length.
10. A proton moves at 0.80c. Calculate its total energy and kinetic energy (m₀ = 1.67 × 10⁻²⁷ kg).
11. State Einstein’s two postulates.
12. Why must the sample in an STM conduct?

Answers

1. v = √(2eV/m) = √(2 × 1.60 × 10⁻¹⁹ × 800 / 9.11 × 10⁻³¹) = 1.68 × 10⁷ m s⁻¹.
2. r = (1/B)√(2V/(e/m)) = 1000 × √(600/1.76 × 10¹¹) = 5.84 × 10⁻² m (5.8 cm).
3. Q = mgd/V = 3.2 × 10⁻¹⁵ × 9.81 × 5.0 × 10⁻³ / 490 = 3.20 × 10⁻¹⁹ C = 2e.
4. F = 6π × 1.8 × 10⁻⁵ × 1.5 × 10⁻⁶ × 2.0 × 10⁻⁴ = 1.02 × 10⁻¹³ N.
5. μ₀ (ε₀ relates to the field of a charged object).
6. λ = 6.63 × 10⁻³⁴ / √(2 × 9.11 × 10⁻³¹ × 1.60 × 10⁻¹⁹ × 2000) = 2.75 × 10⁻¹¹ m.
7. Electrons go faster, λ is shorter, so the rings get smaller (less diffraction).
8. γ = 1/√(1 − 0.36) = 1.25, so t = 5.0 μs.
9. l = 2.5 × √(1 − 0.36) = 2.0 m.
10. γ = 1/√(1 − 0.64) = 5/3. m₀c² = 1.67 × 10⁻²⁷ × (3.00 × 10⁸)² = 1.503 × 10⁻¹⁰ J. E = γm₀c² = 2.51 × 10⁻¹⁰ J; Ek = E − m₀c² = 1.00 × 10⁻¹⁰ J.
11. Physical laws have the same form in all inertial frames; the speed of light in free space is invariant.
12. A tunnelling current must flow between the tip and the surface, so both must conduct.

Where marks are usually lost

• Using ½mv² = eV when the speed is a large fraction of c.
• Forgetting the square on B or r in e/m = 2V/(B²r²).
• Using the field-on reading to find a Millikan drop’s radius, or including the field in the terminal-speed balance with the field off.
• Not dividing a Millikan charge by e and commenting on quantisation when asked about significance.
• Giving the fringe formula when asked to explain Young’s experiment in general terms; the marks are for phase and path difference.
• Saying wave theory fails photoelectricity “because light is particles” instead of naming each observation it cannot explain.
• Mixing up the TEM (transmission through a thin sample) and the STM (tunnelling to a tip).
• Calling Michelson-Morley a failure of the apparatus; its significance is the null result.
• Swapping proper and dilated time, or contracting a length perpendicular to the motion.
• Forgetting to subtract m₀c² when asked for kinetic energy rather than total energy.

Official syllabus

AQA AS and A-level Physics specification (7407/7408), version 1.4, July 2026, AS and A-level exams June 2016 onwards: section 3.12 Turning points in physics (A-level only). Published by AQA.