3.12 Turning points in physics (A-level only) — AQA 7408 spec coverage

3.12.1.1

Cathode rays

A discharge tube contains gas at low pressure between electrodes at a high potential difference; positive ions striking the cathode release electrons that form a cathode ray.
Cathode rays travel from the negative cathode towards the anode and can produce fluorescence or heating when they strike a target.
Deflection towards a positive plate and the direction of magnetic deflection show that cathode rays are negatively charged particles.
Their behaviour did not depend on the gas or cathode material, supporting Thomson's conclusion that electrons are constituents of all atoms.
A common error is to describe cathode rays as electromagnetic radiation; they are streams of electrons and are deflected by electric and magnetic fields.

Tier 1 · Easy

2 marks

ORIGINAL

State the direction in which a cathode ray travels inside a discharge tube and identify the particles in the ray.

Tier 2 · Standard

3 marks

ORIGINAL

A large potential difference is connected across a discharge tube containing low-pressure gas. Describe the sequence that creates the electron beam.

Tier 3 · Hard

5 marks

ORIGINAL

A beam from the cathode bends towards a positive plate. Reversing a transverse magnetic field reverses its magnetic deflection, and the results are unchanged when the gas and cathode metal are replaced. Explain the conclusions drawn from these observations and their importance for the atomic model.

3.12.1.2

Thermionic emission of electrons

Thermionic emission occurs when heating a metal gives some conduction electrons enough energy to escape from its surface.
A heated cathode supplies electrons continuously; a positively charged anode accelerates and focuses them into an electron beam.
For an electron accelerated from rest through potential difference $V$ , the electrical work $eV$ becomes kinetic energy: $eV=\frac{1}{2}m_{\mathrm e}v^2$ .
Use $e=1.60\times10^{-19}\,\text{C}$ and $m_{\mathrm e}=9.11\times10^{-31}\,\text{kg}$ , and convert kilovolts to volts before substitution.
The non-relativistic equation becomes inaccurate when the calculated speed is an appreciable fraction of $c$ ; do not assume that increasing $V$ can make $v>c$ .

Tier 1 · Easy

1 mark

ORIGINAL

Explain why heating a metal cathode can cause electrons to leave its surface.

Tier 2 · Standard

3 marks

ORIGINAL

Electrons emitted from a heated cathode start from rest and are accelerated through $2.50\,\text{kV}$ . Calculate their speed using a non-relativistic model.

Tier 3 · Hard

5 marks

ORIGINAL

An electron gun accelerates a steady current of $3.00\,\text{mA}$ through $7.50\,\text{kV}$ . Calculate the electron speed, the number of electrons emitted each second, and the power transferred to the beam. Use a non-relativistic model.

3.12.1.3

Specific charge of the electron

Specific charge is charge divided by mass; for the electron its magnitude is $e/m_{\mathrm e}$ in $\text{C kg}^{-1}$ .
In crossed electric and magnetic fields adjusted for no deflection, $eE=evB$ , so the selected electron speed is $v=E/B$ .
In a magnetic field perpendicular to the velocity, $evB=m_{\mathrm e}v^2/r$ , giving $e/m_{\mathrm e}=v/(Br)$ .
Thomson's much larger specific charge for electrons than for hydrogen ions implied either a much larger charge or, correctly, a far smaller mass.
Keep the selector field and deflecting field distinct. A common error is to omit $r$ or invert $e/m_{\mathrm e}=v/(Br)$ .

Tier 1 · Easy

2 marks

ORIGINAL

An electron beam passes undeflected through crossed fields of strength $E=2.40\times10^4\,\text{V m}^{-1}$ and $B=8.00\times10^{-3}\,\text{T}$ . Calculate the electron speed.

Tier 2 · Standard

3 marks

ORIGINAL

Electrons selected at $3.00\times10^6\,\text{m s}^{-1}$ enter a perpendicular magnetic field of $1.80\times10^{-4}\,\text{T}$ and follow a circular path of radius $9.50\,\text{cm}$ . Determine the magnitude of their specific charge.

Tier 3 · Hard

5 marks

ORIGINAL

In a Thomson-style experiment, electrons pass between plates separated by $15.0\,\text{mm}$ with $360\,\text{V}$ across them. A crossed field of $8.00\,\text{mT}$ makes the beam undeflected. The selected beam then follows a circular path of radius $10.7\,\text{mm}$ in a separate $1.60\,\text{mT}$ field. Determine $e/m_{\mathrm e}$ and compare it with the hydrogen-ion specific charge $9.58\times10^7\,\text{C kg}^{-1}$ .

3.12.1.4

Principle of Millikan's determination of the electronic charge, e

For a stationary charged droplet between parallel plates, the electric force balances its weight: $QE=mg$ , with $E=V/d$ .
With the field off and air buoyancy neglected, a droplet at terminal speed satisfies $6\pi\eta rv=mg$ ; using $m=\frac{4}{3}\pi r^3\rho$ gives its radius and mass.
The field polarity determines whether the electric force is upward or downward; use force directions before taking magnitudes.
Millikan found that measured droplet charges were integer multiples of a smallest value, $e=1.60\times10^{-19}\,\text{C}$ , demonstrating charge quantisation.
A common error is to use $Q=mg/V$ rather than $Q=mg/E=mgd/V$ , or to apply Stokes' law before terminal speed is reached.

Tier 1 · Easy

2 marks

ORIGINAL

A negatively charged oil droplet is held stationary between horizontal plates. State the relationship between the magnitudes of the electric force and the droplet's weight.

Tier 2 · Standard

3 marks

ORIGINAL

An oil droplet of mass $6.52\times10^{-15}\,\text{kg}$ is held stationary between plates $3.00\,\text{mm}$ apart with $600\,\text{V}$ across them. Calculate the magnitude of its charge and express the result as a multiple of $e$ .

Tier 3 · Hard

5 marks

ORIGINAL

With the electric field off, an oil droplet of density $850\,\text{kg m}^{-3}$ falls through air at terminal speed $8.00\times10^{-5}\,\text{m s}^{-1}$ . The air viscosity is $1.80\times10^{-5}\,\text{Pa s}$ . Neglect air buoyancy. The droplet is then held stationary by a field of $7.50\times10^4\,\text{V m}^{-1}$ . Use Stokes' law to determine the droplet charge and show how the result supports charge quantisation.

3.12.2.1

Newton's corpuscular theory of light

Newton proposed that light consists of tiny corpuscles emitted by sources and travelling in straight lines.
Straight-line travel accounted naturally for sharp shadows, while forces at a boundary were used to explain reflection and refraction.
Newton's authority, the apparent success of his mechanics and the lack of obvious everyday diffraction helped the corpuscular theory dominate over Huygens' wave theory.
The corpuscular model predicted that light should move faster in a more optically dense medium, whereas wave theory predicted a lower speed; later measurements supported the wave prediction.
A strong historical answer links evidence to theory change: interference and diffraction could not be explained by independent classical corpuscles.

Tier 1 · Easy

1 mark

ORIGINAL

State one observation that made Newton's corpuscular model of light appear plausible.

Tier 2 · Standard

3 marks

ORIGINAL

Newton modelled light as corpuscles, whereas Huygens used wavefronts. Compare their speed predictions for light entering glass from air and state why the later measurement mattered.

Tier 3 · Hard

5 marks

ORIGINAL

Discuss why Newton's particle account of light was accepted for so long and why later optical evidence forced physicists to replace it with a wave account.

3.12.2.2

Significance of Young's double slits experiment

Young's arrangement uses one source to illuminate two narrow slits so that the emerging waves have a stable phase relationship.
The two diffracted waves overlap: constructive superposition gives bright fringes and destructive superposition gives dark fringes.
A path difference of a whole number of wavelengths gives waves in phase; a half-integer number gives antiphase waves. No fringe-spacing calculation is required for this option.
Alternating bright and dark fringes were compelling evidence for wave behaviour because a classical corpuscle model predicted addition, not cancellation.
Acceptance of Huygens' wave theory was delayed by Newton's influence; exam answers must state why the evidence discriminated between the theories.

Tier 1 · Easy

1 mark

ORIGINAL

State the feature of Young's double-slit pattern that provides evidence for interference.

Tier 2 · Standard

3 marks

ORIGINAL

Explain qualitatively how bright and dark fringes are formed when monochromatic light passes through Young's two slits.

Tier 3 · Hard

5 marks

ORIGINAL

A screen behind two illuminated narrow slits shows many regularly spaced dark bands between bright bands rather than two bright images. Explain why this result was a turning point in the debate between Newton's and Huygens' models of light.

3.12.2.3

Electromagnetic waves

An electromagnetic wave consists of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation.
Maxwell predicted the vacuum speed $c=1/\sqrt{\mu_0\varepsilon_0}$ ; $\varepsilon_0$ sets electric-field strength from charge and $\mu_0$ sets magnetic flux density from current.
The agreement of Maxwell's calculated speed with measured light speed led to the identification of light as an electromagnetic wave.
Hertz generated and detected radio waves and demonstrated reflection, refraction, diffraction and a speed close to $c$ , providing experimental support for Maxwell.
Fizeau's terrestrial measurement showed that light has a finite speed; in toothed-wheel questions, connect the wheel's rotation during the round trip to the blocked return beam.

Tier 1 · Easy

2 marks

ORIGINAL

Describe the relative directions of the electric field, magnetic field and travel direction in a plane electromagnetic wave.

Tier 2 · Standard

3 marks

ORIGINAL

Use $\mu_0=4\pi\times10^{-7}\,\text{H m}^{-1}$ and $\varepsilon_0=8.85\times10^{-12}\,\text{F m}^{-1}$ to calculate the speed predicted by Maxwell for an electromagnetic wave in a vacuum.

Tier 3 · Hard

5 marks

ORIGINAL

In a Fizeau-type experiment, light travels to a mirror $8.63\,\text{km}$ away and back through a wheel with $720$ teeth. The first extinction occurs at $12.1\,\text{Hz}$ because the wheel turns from a gap to the adjacent tooth during the round trip. Calculate the speed of light and explain how Fizeau's and Hertz's results supported Maxwell's theory.

3.12.2.4

The discovery of photoelectricity

Classical theory predicted an ultraviolet catastrophe because continuously shared energy gave black bodies unlimited high-frequency emission; observed spectra instead fall at high frequency.
Planck resolved the spectrum by proposing quantised exchanges of energy $E=hf$ .
Photoelectric observations include a threshold frequency, immediate emission, maximum electron kinetic energy depending on frequency, and emission rate depending on intensity above threshold.
Einstein treated each quantum as a photon: $hf=\phi+K_{\max}$ . One electron absorbs one photon, explaining the threshold and lack of delay.
The photoelectric effect restored a particle aspect to electromagnetic radiation; a common error is to say that greater intensity increases $K_{\max}$ at fixed frequency.

Tier 1 · Easy

1 mark

ORIGINAL

State one photoelectric observation that classical wave theory could not explain.

Tier 2 · Standard

3 marks

ORIGINAL

Explain how Planck's quantum hypothesis avoided the ultraviolet catastrophe in the black-body spectrum.

Tier 3 · Hard

5 marks

ORIGINAL

Light of wavelength $350\,\text{nm}$ illuminates a metal with work function $2.20\,\text{eV}$ . Calculate the maximum speed of an emitted electron and explain why doubling the light intensity does not change this speed.

3.12.2.5

Wave-particle duality

de Broglie proposed that a particle of momentum $p$ has wavelength $\lambda=h/p$ .
For a non-relativistic electron accelerated from rest through $V$ , $eV=p^2/(2m_{\mathrm e})$ , so $\lambda=h/\sqrt{2m_{\mathrm e}eV}$ .
Electron diffraction from a crystal demonstrates wave behaviour; localised electron detection and momentum transfer demonstrate particle behaviour.
Increasing electron speed or accelerating voltage increases momentum, reduces wavelength and moves diffraction maxima to smaller angles.
Use SI units throughout. A common error is to use $V$ as an energy without multiplying by $e$ .

Tier 1 · Easy

2 marks

ORIGINAL

Calculate the de Broglie wavelength of an electron with momentum $3.00\times10^{-24}\,\text{kg m s}^{-1}$ .

Tier 2 · Standard

3 marks

ORIGINAL

An electron is accelerated from rest through $150\,\text{V}$ . Calculate its de Broglie wavelength using a non-relativistic model.

Tier 3 · Hard

5 marks

ORIGINAL

Electrons accelerated through $100\,\text{V}$ produce a first diffraction maximum from a crystal plane at angle $37.8^\circ$ . The accelerating voltage is raised to $400\,\text{V}$ . Calculate the new de Broglie wavelength and, using $d\sin\theta=\lambda$ , the new angle. Explain the turning-point significance of electron diffraction.

3.12.2.6

Electron microscopes

Electron wavelengths can be much smaller than visible-light wavelengths, so electron microscopes can resolve structures on the atomic scale.
For non-relativistic electrons, $\lambda=h/\sqrt{2m_{\mathrm e}eV}$ ; increasing a TEM anode voltage reduces wavelength and improves the diffraction-limited resolution.
A TEM accelerates electrons through a thin specimen and uses electromagnetic lenses to form an image from transmitted and scattered electrons.
An STM scans a sharp conducting tip close to a conducting surface; the tunnelling current changes very rapidly with gap width and maps surface height or electronic structure.
Do not explain STM resolution using an anode voltage: its atomic sensitivity comes from quantum tunnelling, while excessive TEM voltage can also damage specimens despite the shorter wavelength.

Tier 1 · Easy

2 marks

ORIGINAL

Explain why increasing the anode voltage of a transmission electron microscope can improve its resolution.

Tier 2 · Standard

3 marks

ORIGINAL

Estimate the anode voltage required to give non-relativistic electrons a wavelength of $0.100\,\text{nm}$ .

Tier 3 · Hard

5 marks

ORIGINAL

A TEM is designed for an electron wavelength of $5.00\,\text{pm}$ . Estimate the non-relativistic anode voltage and compare how a TEM and an STM obtain atomic-scale information.

3.12.3.1

The Michelson-Morley experiment

The Michelson interferometer splits coherent light along perpendicular arms, reflects the beams and recombines them to form interference fringes.
If Earth moved through a stationary luminiferous ether, the two arms would have different round-trip light times; rotating the apparatus should therefore shift the fringes.
Michelson and Morley found no significant periodic fringe shift, so the experiment failed to detect absolute motion through an ether.
The null result undermined the ether model and supported the invariance of the speed of light, preparing the way for special relativity.
A common error is to claim that the experiment showed Earth is stationary; it showed that this optical experiment could not reveal an absolute state of motion.

Tier 1 · Easy

1 mark

ORIGINAL

State the effect that Michelson and Morley expected to observe if Earth moved through a stationary ether.

Tier 2 · Standard

3 marks

ORIGINAL

Explain why a Michelson interferometer was rotated through $90^\circ$ and state the significance of the null result.

Tier 3 · Hard

5 marks

ORIGINAL

For an interferometer with equal arm length $L=11.0\,\text{m}$ moving at $v=3.00\times10^4\,\text{m s}^{-1}$ through a proposed ether, use $\Delta t\approx Lv^2/c^3$ for the initial difference in round-trip times. Calculate the fringe shift predicted on rotating the apparatus through $90^\circ$ for light of wavelength $550\,\text{nm}$ , using $n=2c\Delta t/\lambda$ . Explain why observing no such shift was decisive.

3.12.3.2

Einstein's theory of special relativity

An inertial frame is one that is not accelerating: a free object moves at constant velocity in it unless a resultant force acts.
Einstein's first postulate states that the laws of physics have the same form in every inertial frame, so no inertial experiment identifies absolute uniform motion.
The second postulate states that the speed of light in free space is the same for every inertial observer, independent of source or observer motion.
Retaining both postulates requires space and time intervals to depend on the observer's frame, leading to relativity of simultaneity, time dilation and length contraction.
Do not use Galilean velocity addition for light; the invariant quantity is $c$ , not the measured frequency or wavelength.

Tier 1 · Easy

1 mark

ORIGINAL

Define an inertial frame of reference.

Tier 2 · Standard

3 marks

ORIGINAL

State Einstein's two postulates of special relativity and explain how they account for the Michelson-Morley null result.

Tier 3 · Hard

5 marks

ORIGINAL

Two flashes occur simultaneously at the front and rear of a platform according to an observer at its midpoint. A train moves towards the front flash, and a passenger is at the train's midpoint as the flashes occur. Explain why the passenger does not judge the flashes to be simultaneous and why this follows from Einstein's postulates rather than from light travelling faster from one end.

3.12.3.3

Time dilation

Proper time $t_0$ is measured by a single clock present at both events; it is the interval in the frame where the events occur at the same position.
A frame in which that clock moves measures the longer interval $t=\gamma t_0$ , where $\gamma=1/\sqrt{1-v^2/c^2}$ .
Time dilation is reciprocal between inertial frames because each observer compares a moving clock with clocks synchronised in their own frame.
Atmospheric muons survive to sea level in much larger numbers because their dilated lifetime in Earth's frame permits greater travel distances.
Use the proper lifetime in the particle's rest frame. A common error is to divide by $\gamma$ when finding the lifetime measured in the laboratory.

Tier 1 · Easy

1 mark

ORIGINAL

Define proper time.

Tier 2 · Standard

3 marks

ORIGINAL

A muon has proper mean lifetime $2.20\,\mu\text{s}$ and moves through a laboratory at $0.800c$ . Calculate its mean lifetime in the laboratory and the mean distance it travels there.

Tier 3 · Hard

5 marks

ORIGINAL

Muons are created $10.0\,\text{km}$ above sea level with speed $0.995c$ and proper mean lifetime $2.20\,\mu\text{s}$ . Assuming exponential decay, calculate the fraction that survive to sea level. Compare this with the prediction if time dilation were ignored.

3.12.3.4

Length contraction

Proper length $l_0$ is the length measured in the object's rest frame, using the positions of its ends in that frame.
An observer who sees the object moving parallel to its length measures $l=l_0/\gamma=l_0\sqrt{1-v^2/c^2}$ .
Only the dimension parallel to the relative motion contracts; transverse dimensions are unchanged.
The moving frame measures the endpoints simultaneously in its own frame, so length contraction is linked to relativity of simultaneity.
A common error is to treat the laboratory distance as contracted in the laboratory frame; it is the proper length when its endpoints are fixed there.

Tier 1 · Easy

1 mark

ORIGINAL

Define the proper length of an object.

Tier 2 · Standard

3 marks

ORIGINAL

A spacecraft has proper length $120\,\text{m}$ and passes an observer parallel to its length at $0.600c$ . Calculate the length measured by the observer.

Tier 3 · Hard

5 marks

ORIGINAL

A particle moves at $0.980c$ along a straight accelerator of proper length $3.00\,\text{km}$ in the laboratory. Calculate the accelerator length and the transit time in the particle's frame. Show that the result is consistent with the laboratory transit time and time dilation.

3.12.3.5

Mass and energy

Mass and energy are equivalent: rest energy is $E_0=m_0c^2$ , and a change of rest mass $\Delta m$ corresponds to energy $\Delta E=\Delta mc^2$ .
Using the specification's relativistic-mass convention, $m=\gamma m_0$ and total energy is $E=mc^2=\gamma m_0c^2$ .
Relativistic kinetic energy is $K=(\gamma-1)m_0c^2$ ; it approaches $\frac{1}{2}m_0v^2$ only when $v\ll c$ .
Graphs of relativistic mass and kinetic energy against speed rise increasingly steeply and tend to infinity as $v$ approaches $c$ , so a massive particle cannot be accelerated to $c$ .
Bertozzi measured electron speed and kinetic energy directly: additional energy produced diminishing speed increases near $c$ , agreeing with relativity and contradicting the classical prediction.

Tier 1 · Easy

2 marks

ORIGINAL

A reaction reduces the total rest mass of a system by $3.00\times10^{-11}\,\text{kg}$ . Calculate the energy released.

Tier 2 · Standard

3 marks

ORIGINAL

An electron moves at $0.800c$ . Calculate its total energy and kinetic energy. Use $m_0=9.11\times10^{-31}\,\text{kg}$ .

Tier 3 · Hard

5 marks

ORIGINAL

In a Bertozzi-type accelerator experiment, an electron has kinetic energy $4.00\,\text{MeV}$ . Its rest energy is $0.511\,\text{MeV}$ . Calculate its relativistic speed and the speed predicted by $K=\frac{1}{2}m_0v^2$ . Explain why the comparison supports special relativity.