Turning points in physics: A-level Physics AQA revision notes

3.12.1

The discovery of the electron

Cathode rays and thermionic emission 3.12.1.1–3.12.1.2

Definitions

Cathode rays: a stream of fast-moving electrons emitted from the cathode of a discharge tube and travelling towards the anode.
Thermionic emission: the release of electrons from the surface of a heated metal, once they gain enough thermal energy to overcome the metal's own attractive forces and escape.

Key results

Energy gained accelerating through a potential difference: $e V_{a} = \frac{1}{2} m_{e} v^{2}$ , giving $v = \frac{2 e V _{a}}{m _{e}}$ (valid only for $v ≪ c$ , since it uses non-relativistic kinetic energy).

Notes

A discharge tube is a sealed glass tube containing gas at very low pressure, with a metal cathode (negative electrode) and anode (positive electrode) at either end. Applying a large potential difference across the electrodes ionises some of the residual gas; the resulting positive ions accelerate towards the cathode and, on impact, release electrons from its surface. These electrons then accelerate away from the negative cathode towards the anode, and some pass beyond it to strike a fluorescent screen, producing a visible glowing spot or line.
Three observations established that cathode rays are negatively charged particles, not a form of electromagnetic radiation: they travel in straight lines in a field-free region (casting a sharp shadow of an object placed in their path); they are deflected towards a positive plate and away from a negative one when passed between charged parallel plates, showing they carry negative charge; and they are deflected by a magnetic field in exactly the way a moving negative charge should be, by the motor effect.
A cold-cathode discharge tube (as just described) relies on gas ionisation to supply electrons and needs no heater; it is a different arrangement from the heated electron gun in the next section, though both eventually produce a controlled beam of electrons.
A heated-cathode electron gun replaces the cold cathode with a filament heated by its own separate low-voltage supply. Heating gives a much larger, more controllable fraction of the free electrons in the metal enough energy to escape its surface — this is thermionic emission, the electron analogue of evaporation. A separate, larger potential difference then accelerates the emitted electrons towards an anode; the work done on each electron by this accelerating field is converted entirely into kinetic energy, since the electron's initial thermal kinetic energy on leaving the cathode is negligible by comparison.
Because the accelerating field does a known, fixed amount of work on every electron ( $e V_{a}$ ), every electron reaching the anode aperture has (very nearly) the same, calculable speed — this controllability is exactly what makes a thermionic electron gun useful as a source of a well-defined beam, unlike the more haphazard cathode rays of a simple discharge tube. e here is taken as the positive magnitude of the elementary charge; the electron's own charge is $- e$ , but the accelerating work done on it, $e V_{a}$ , is a positive quantity regardless.

Specific charge of the electron 3.12.1.3

Definitions

Specific charge of a particle: the ratio of its charge to its mass, $e / m$ — a quantity that can be measured directly from a particle's motion in electric and magnetic fields, without first knowing $e$ or $m$ individually.

Key results

Magnetic force provides centripetal force for a charge moving perpendicular to a uniform field: $B e v = \frac{m _{e} v ^{2}}{r}$ .
Specific charge from the circular-beam method: $\frac{e}{m _{e}} = \frac{2 V _{a}}{B ^{2} r ^{2}}$ (combining $e V_{a} = \frac{1}{2} m_{e} v^{2}$ with $B e v = \frac{m _{e} v ^{2}}{r}$ to eliminate $v$ ).
Modern value: $e / m_{e} \approx 1.76 \times 1 0^{11} C kg^{- 1}$ — roughly 1836 times the hydrogen ion's specific charge, $e / m_{p} \approx 9.58 \times 1 0^{7} C kg^{- 1}$ .

Notes

A uniform magnetic field directed perpendicular to an electron's velocity deflects it into a circular path, since the magnetic force always acts at right angles to the electron's motion — it changes the direction of the velocity continuously but never its magnitude, so it does no work on the electron. This magnetic force supplies exactly the centripetal force needed to maintain the circular path, giving $B e v = m_{e} v^{2} / r$ .
Combining this with the accelerating-field relation $e V_{a} = \frac{1}{2} m_{e} v^{2}$ eliminates the unknown speed $v$ algebraically, leaving specific charge expressed entirely in terms of quantities that can be set or measured directly: the accelerating p.d. $V_{a}$ , the magnetic flux density $B$ , and the orbit radius $r$ (found from the visible circular path's diameter, $r = diameter /2$ ).
J.J. Thomson's 1897 measurement of a consistent specific charge — the same value regardless of the cathode material or the residual gas used — was the key evidence that cathode rays are a single kind of particle, present inside every kind of atom, rather than an effect specific to one particular metal or gas. The size of the ratio, about 1836 times the hydrogen ion's own specific charge, pointed to a particle very much lighter than a whole atom: the electron.
This school-laboratory circular-beam method is a modern teaching arrangement, not a reconstruction of Thomson's own 1897 apparatus, which instead balanced electric and magnetic deflections of a straight cathode-ray beam.

Millikan's oil-drop experiment: holding a droplet stationary 3.12.1.4

Definitions

Terminal velocity: the constant velocity reached once the resultant force on a falling (or rising) object is zero, so it no longer accelerates.

Key results

Holding a charged droplet stationary between parallel plates: $QE = m g$ , so $Q = \frac{m g d}{V}$ (using $E = V / d$ ).

Notes

Millikan sprayed a fine mist of oil droplets above two horizontal, charged parallel plates; a few droplets, having picked up a small charge by friction as they left the atomiser, fell through a small hole in the upper plate into the space between them, where they could be illuminated and observed side-on through a microscope. Adjusting the potential difference between the plates until a chosen droplet hung exactly stationary balances the upward electric force on it against its own weight, $QE = m g$ , giving its charge directly once $E = V / d$ and its mass are known.
Oil was chosen specifically because it evaporates extremely slowly, so a droplet's mass stays effectively constant for as long as it takes to complete a careful observation — a fast-evaporating liquid would make the whole measurement drift under the experimenter's own eyes.
This simple treatment neglects air buoyancy on the droplet; a more careful analysis replaces the droplet's weight with its weight minus the buoyant force in every force balance.

Millikan's oil-drop experiment: finding a droplet's mass 3.12.1.4

Definitions

Stokes' law: the viscous drag force on a small sphere moving slowly through a fluid, $F_{D} = 6 π η a u$ , where $η$ is the fluid's viscosity, $a$ is the sphere's radius and $u$ is its speed.

Key results

Field off, falling at terminal velocity $u_{0}$ : $m g = 6 π η a u_{0}$ , giving the droplet's radius $a = \frac{9 η u _{0}}{2 ρ g}$ and hence its mass $m = \frac{4}{3} π a^{3} ρ$ .
Field on, terminal velocity $u_{d}$ (falling, weaker field) or $u_{r}$ (rising, stronger field): $m g = QE + 6 π η a u_{d}$ , or $QE = m g + 6 π η a u_{r}$ .

Notes

A droplet's mass is found separately, with the field switched off, from how fast it falls: at terminal velocity its weight exactly balances the upward viscous drag given by Stokes' law, letting its radius (and hence its mass, from the density of the oil) be calculated purely from its measured falling speed, timed over a calibrated distance in the microscope's own graticule.
With the field switched back on, the SAME droplet — now with a known mass from the field-off measurement — can be timed falling more slowly (field weaker than needed for equilibrium) or rising (field stronger), giving a second route to its charge that cross-checks the equilibrium method: the electric force no longer has to exactly balance gravity, so the resultant of gravity, the electric force and drag together determines a new, different terminal velocity.
The same droplet, not a fresh one, must be used across every stage of one measurement — its radius and charge are both assumed constant throughout, and a second droplet would carry an unrelated, unknown charge.

Charge quantisation and the electron's mass 3.12.1.4

Key results

Charge quantisation: $Q = n e$ , where $n = 1, 2, 3, \dots$ and $e \approx 1.602 \times 1 0^{- 19} C$ is the elementary charge.
Electron mass from specific charge: $m_{e} = \frac{e}{e / m _{e}} \approx 9.11 \times 1 0^{- 31} kg$ .

Notes

Repeating the charge measurement across many different droplets, Millikan found that every measured charge was a whole-number multiple of the same smallest value — never a fraction of it. This was the first direct evidence that electric charge is quantised: it exists only in discrete lumps of size $e$ , never as a continuously variable quantity. Agreement with integer multiples must always be judged within the uncertainty of the individual charge measurements, not by demanding an exact match — real data never lands on integers with zero scatter.
Combining this elementary charge with Thomson's specific charge closes the loop begun in the previous two sections: the electron's own mass follows directly, $m_{e} = e \div (e / m_{e}) \approx 9.11 \times 1 0^{- 31} kg$ — a mass roughly 1836 times smaller than a hydrogen atom's, confirming Thomson's own conclusion that the electron is a genuinely sub-atomic constituent of every element.

Worked examples

Worked example 3.12.1.3 · 5 marks

In a circular-beam apparatus, electrons are accelerated through a potential difference of 240 V and enter a region of uniform magnetic flux density $2.4 \times 1 0^{- 3} T$ , perpendicular to their velocity, where they follow a circular path of radius 0.075 m.

Calculate the specific charge of the electron from this data.

Show worked solution

\frac{e}{m _{e}} = \frac{2 V _{a}}{B ^{2} r ^{2}} = \frac{2 \times 240}{( 2.4 \times 1 0 ^{- 3} ) ^{2} \times ( 0.075 ) ^{2}} = \frac{480}{( 5.76 \times 1 0 ^{- 6} ) \times ( 5.625 \times 1 0 ^{- 3} )} \approx 1.48 \times 1 0^{10} C kg^{- 1}

(This is lower than the accepted:

1.76 \times 1 0^{11} C kg^{- 1}

purely because these illustrative values were chosen for round numbers, not to reproduce the real experimental result.)

Mark scheme · 5 marks

Correctly states or applies $e / m_{e} = 2 V_{a} / (B^{2} r^{2})$ 2 marks
Correctly squares B and r before substituting 2 marks
Reaches a final value with correct units, C kg⁻¹, from correct arithmetic on the given (illustrative) data 1 mark

Do not count matching words alone — ask whether your answer actually makes the same claim.

Worked example 3.12.1.4 · 6 marks

In a Millikan-type experiment, a charged oil droplet of mass $9.8 \times 1 0^{- 15} kg$ is held stationary between two horizontal plates 6.0 mm apart with a potential difference of 360 V across them.

(a) Calculate the charge on the droplet.

(b) State how many elementary charges this corresponds to, given:

e = 1.60 \times 1 0^{- 19} C

Show worked solution

(a) At equilibrium, $QE = m g$ , so:

Q = \frac{m g d}{V} = \frac{9.8 \times 1 0 ^{- 15} \times 9.81 \times 6.0 \times 1 0 ^{- 3}}{360} \approx 1.60 \times 1 0^{- 18} C

(b):

n = \frac{Q}{e} = \frac{1.60 \times 1 0 ^{- 18}}{1.60 \times 1 0 ^{- 19}} = 10

the droplet carries 10 elementary charges.

Mark scheme · 6 marks

Correctly identifies the equilibrium condition $QE = m g$ 2 marks
Correctly substitutes $E = V / d$ to reach $Q = m g d / V$ 2 marks
Correctly calculates the charge, ≈1.60×10⁻¹⁸ C 1 mark
Correctly divides by e to find n = 10, and states this is a whole number consistent with charge quantisation 1 mark

Do not count matching words alone — ask whether your answer actually makes the same claim.

3.12.2

Wave–particle duality

Newton, Huygens and Young: competing models of light 3.12.2–3.12.2.1–3.12.2.2

Notes

The second turning point runs in the opposite direction to the first: having just established that cathode rays (matter) are made of discrete particles, physics spent the next thirty years discovering that light — long assumed to be a pure wave — sometimes behaves as discrete particles too, and that matter itself can show wave behaviour in return. Neither picture, wave or particle, turns out to be the whole story for either light or matter.
In the seventeenth century, Newton proposed that light consists of a stream of fast-moving particles ("corpuscles") travelling in straight lines, while Huygens proposed instead that light is a wave, spreading out from a source in the same way sound or water waves do. Newton's corpuscular theory could explain reflection and the straight-line travel of light easily, and — because of Newton's own enormous scientific authority — it remained the dominant view for over a century, even though it struggled to explain phenomena like diffraction (light bending slightly around obstacles) that a wave theory predicts naturally.
A genuine experimental test between the two theories turned on the speed of light in a dense medium such as water: Newton's corpuscular theory predicted light should travel faster in water than in air (since it treated refraction as the medium pulling the particles inward, speeding them up), while Huygens' wave theory predicted the opposite, that light should travel slower in water than in air. Armand Fizeau's 1849 experiment supplied the tool this test needed: a beam of light passes through a rotating toothed wheel, out to a distant mirror and back; at certain rotation speeds, the returning light is blocked by the next tooth rather than passing back through the same gap it left through, and timing this "first extinction" against the wheel's known rotation speed and the known distance $D$ gives the speed of light directly, $c = 4 D n f_{w}$ (equivalently $2 D / c = 1/ (2 n f_{w})$ , where $n$ is the number of teeth and $f_{w}$ the wheel's rotation frequency). Fizeau's own result, close to $3 \times 1 0^{8} m s^{- 1}$ , measured light's speed in air; it was slightly later, comparative measurements finding light slower in water than in air that directly contradicted Newton's prediction and decisively favoured the wave theory — evidence that took over 150 years to arrive after the two theories were first proposed.
Thomas Young's 1801 double-slit experiment passed light through two closely spaced, narrow slits and observed the resulting pattern on a screen: not two bright bands (as two independent particle streams would produce) but a regular series of alternating bright and dark fringes. This interference pattern — bright fringes where light from the two slits arrives in phase and reinforces, dark fringes where it arrives out of phase and cancels — is exactly the behaviour expected of overlapping waves, and cannot be explained by picturing light as a stream of independent particles travelling in straight lines.
Despite this compelling evidence, wave theory took decades to become fully accepted, partly because Newton's corpuscular view still carried great authority and partly because a satisfactory answer to what, physically, was supposed to be waving was still missing. That gap would not be closed convincingly until Maxwell's electromagnetic theory, decades later.

Electromagnetic waves 3.12.2.3

Key results

Maxwell's prediction of the speed of an electromagnetic wave in a vacuum: $c = \frac{1}{μ _{0} ε _{0}}$ , where $μ_{0}$ is the permeability and $ε_{0}$ the permittivity of free space.

Notes

James Clerk Maxwell's mid-nineteenth-century theory of electromagnetism unified electricity, magnetism and light into a single framework: a changing electric field generates a magnetic field, and a changing magnetic field generates an electric field, so the two can sustain each other and propagate together as a self-supporting electromagnetic wave through empty space, needing no material medium to carry it. Maxwell's equations predicted the speed of this wave purely from two previously separate, independently measurable electrical and magnetic constants, $μ_{0}$ and $ε_{0}$ — and the predicted value matched the already-measured speed of light so closely that Maxwell concluded light itself must be an electromagnetic wave.
Heinrich Hertz confirmed this experimentally in 1887, generating and detecting radio waves — electromagnetic waves at a much longer wavelength than visible light — using oscillating electric circuits, and showing they travelled at the predicted speed and displayed the same wave behaviours (reflection, refraction, interference, polarisation) that visible light does. This closed the case in the wave theory's favour: light was now understood as one member of a much broader electromagnetic family, all governed by the same equations.

The photoelectric effect 3.12.2.4

Definitions

Work function, $ϕ$ : the minimum energy needed to remove an electron from the surface of a given metal.
Threshold frequency, $f_{0}$ : the minimum frequency of incident light that can just release a photoelectron from a given metal, where $h f_{0} = ϕ$ .

Key results

Photon energy: $E = h f$ .
Einstein's photoelectric equation: $h f = ϕ + E_{k (m a x)}$ , where $E_{k (m a x)} = \frac{1}{2} m_{e} v_{m a x}^{2}$ is the maximum kinetic energy of an emitted photoelectron.

Notes

Shining light on a clean metal surface can eject electrons from it — the photoelectric effect. Classically (treating light as a continuous wave), a dimmer light should simply take longer to build up enough energy at the surface to release an electron, and this should happen at any frequency given enough time and intensity. Experiment showed the opposite on both counts: below a sharp threshold frequency specific to the metal, no electrons are emitted at all, however intense the light and however long it shines; above that threshold, emission begins the instant the light is switched on, with no measurable delay, and a brighter light increases the *number* of photoelectrons emitted per second but never their maximum kinetic energy — only raising the light's frequency does that.
This flat contradiction with wave theory's predictions is closely related to a separate classical puzzle Max Planck had already resolved a few years earlier: the ultraviolet catastrophe, where classical wave theory wrongly predicted a hot object should radiate an infinite amount of energy at ever-shorter wavelengths. Planck's fix — initially treated as a mathematical device rather than a physical claim — was to propose that energy is emitted and absorbed only in discrete packets, or quanta, of size $h f$ , each oscillator in the hot object restricted to a fixed ladder of energy levels rather than a continuous range.
Einstein took Planck's quantum hypothesis literally, in 1905, proposing that light itself, not just its emission and absorption, consists of discrete packets of energy — photons — each carrying energy $E = h f$ . A single photon can transfer its entire energy to a single electron in one interaction; if that energy exceeds the work function $ϕ$ needed to free the electron, the electron escapes immediately (explaining the lack of any delay) with the surplus energy $h f - ϕ$ as kinetic energy. Below the threshold frequency $f_{0} = ϕ / h$ , an individual photon simply never carries enough energy to free an electron at all, however many additional photons (however intense the light) arrive per second — since each interaction is one photon with one electron, more photons only means more emissions per second, never a higher energy per emission.
The maximum kinetic energy, not the average, appears in Einstein's equation: electrons freed from just below the metal's surface lose some energy escaping through the material itself, so only electrons freed from the very surface, with no such loss, reach $E_{k (m a x)}$ .

Wave-particle duality 3.12.2.5

Key results

De Broglie's hypothesis: every moving particle has an associated wavelength, $λ = \frac{h}{p} = \frac{h}{m v}$ .

Notes

The photoelectric effect showed that light — traditionally understood as a wave, on the strength of Young's interference fringes — sometimes behaves as discrete particles. In 1924 Louis de Broglie proposed the converse and, at the time, far stranger idea: that matter, traditionally understood as particles, should likewise show wave-like behaviour, with a wavelength inversely proportional to its momentum. For an everyday object this wavelength is absurdly small (a wavelength depends on $h / p$ , and $h$ is tiny) and utterly undetectable; but for a single electron, accelerated to a modest speed, the associated wavelength comes out comparable to the spacing between atomic planes in a crystal.
This is exactly what makes electron diffraction the decisive test of de Broglie's hypothesis. Davisson and Germer's 1927 experiment fired a beam of electrons at a nickel crystal and found a pattern of intensity peaks at specific angles — not the smooth scattering a particle picture would predict, but a diffraction pattern directly analogous to X-ray diffraction from the same crystal lattice. G.P. Thomson (J.J. Thomson's own son) independently confirmed the effect by firing electrons through a thin polycrystalline graphite film, producing the concentric diffraction rings characteristic of transmission diffraction through many randomly oriented tiny crystals. In both cases, the measured diffraction geometry matched the wavelength de Broglie's equation predicted for the electrons' known momentum, closing the loop: light shows particle behaviour (the photoelectric effect) and matter shows wave behaviour (electron diffraction) — wave and particle are not two separate categories of thing, but two complementary aspects every quantum object shows depending on the situation.

Electron microscopes 3.12.2.6

Notes

An electron's de Broglie wavelength at typical microscope accelerating voltages is thousands of times shorter than visible light's wavelength, and since the finest detail an imaging system can resolve is fundamentally limited by the wavelength it uses, an electron beam can resolve far finer structure than any light microscope ever could, however good its lenses. A transmission electron microscope (TEM) fires a high-energy electron beam through an extremely thin sample; electrons scatter by different amounts passing through denser or less dense regions of the sample, and magnetic lenses focus the transmitted beam onto a detector to form a high-resolution image of the sample's internal structure.
A scanning tunnelling microscope (STM) works on a completely different principle, requiring no lenses at all: an extremely fine conducting probe is scanned just above a conducting sample's surface, close enough that electrons can quantum-mechanically tunnel across the tiny gap between probe and surface even though they classically lack the energy to cross it. This tunnelling current is exquisitely sensitive to the gap's exact width, so keeping the current constant as the probe scans traces out the surface's height profile with atomic-scale resolution, imaging individual atoms directly.
A TEM images internal structure via the transmitted beam and needs a very thin sample; an STM images only the outer surface, via a tunnelling current, and needs a conducting sample.

Worked examples

Worked example 3.12.2.4 · 6 marks

The work function of sodium is $3.78 \times 1 0^{- 19} J$ .

Light of wavelength 350 nm is shone onto a clean sodium surface.

(a) Show that this light can cause photoelectric emission.

(b) Calculate the maximum kinetic energy of an emitted photoelectron, in eV. (:

h = 6.63 \times 1 0^{- 34} Js

c = 3.00 \times 1 0^{8} m s^{- 1}

e = 1.60 \times 1 0^{- 19} C

)

Show worked solution

(a):

f = \frac{c}{λ} = \frac{3.00 \times 1 0 ^{8}}{350 \times 1 0 ^{- 9}} \approx 8.57 \times 1 0^{14} Hz

Photon energy:

= h f = 6.63 \times 1 0^{- 34} \times 8.57 \times 1 0^{14} \approx 5.68 \times 1 0^{- 19} J

which exceeds the work function ( $3.78 \times 1 0^{- 19} J$ ), so photoelectric emission occurs. (b):

E_{k (m a x)} = h f - ϕ = 5.68 \times 1 0^{- 19} - 3.78 \times 1 0^{- 19} = 1.90 \times 1 0^{- 19} J

In eV:

\frac{1.90 \times 1 0 ^{- 19}}{1.60 \times 1 0 ^{- 19}} \approx 1.19 eV

Mark scheme · 6 marks

Correctly calculates the frequency from $c = f λ$ 2 marks
Correctly calculates the photon energy hf and compares it with the work function to justify emission 2 marks
Correctly applies $E_{k (m a x)} = h f - ϕ$ 1 mark
Correctly converts the final answer to eV, ≈1.19 eV 1 mark

Do not count matching words alone — ask whether your answer actually makes the same claim.

Worked example 3.12.2.5 · 4 marks

Calculate the de Broglie wavelength of an electron accelerated from rest through a potential difference of 3.0 kV. (:

m_{e} = 9.11 \times 1 0^{- 31} kg

e = 1.60 \times 1 0^{- 19} C

h = 6.63 \times 1 0^{- 34} Js

)

Show worked solution

e V_{a} = \frac{1}{2} m_{e} v^{2} \Rightarrow v = \frac{2 e V _{a}}{m _{e}} = \frac{2 \times 1.60 \times 1 0 ^{- 19} \times 3000}{9.11 \times 1 0 ^{- 31}} \approx 3.25 \times 1 0^{7} m s^{- 1}

λ = \frac{h}{m _{e} v} = \frac{6.63 \times 1 0 ^{- 34}}{9.11 \times 1 0 ^{- 31} \times 3.25 \times 1 0 ^{7}} \approx 2.24 \times 1 0^{- 11} m

(about 22 pm — comparable to typical atomic spacings, which is exactly why fast electrons diffract usefully off crystals).

Mark scheme · 4 marks

Correctly finds the electron's speed from $e V_{a} = \frac{1}{2} m_{e} v^{2}$ 1 mark
Correctly applies $λ = h / (m_{e} v)$ 1 mark
Reaches the correct final wavelength, ≈2.2×10⁻¹¹ m 1 mark
Notes this wavelength is comparable to atomic/crystal-plane spacing, explaining why electron diffraction is observable 1 mark

Do not count matching words alone — ask whether your answer actually makes the same claim.

3.12.3

Special relativity

The Michelson–Morley experiment and Einstein's postulates 3.12.3.1–3.12.3.2

Definitions

Luminiferous aether: the hypothetical medium nineteenth-century physicists assumed light waves needed to propagate through, filling all of space, including a vacuum.
Inertial reference frame: a frame of reference moving at constant velocity (not accelerating), in which Newton's first law holds.

Notes

If light were a wave in a physical aether filling space, the Earth's motion through that aether (orbiting the Sun at around 30 km/s) should be detectable as an "aether wind" — light travelling parallel to the Earth's motion through the aether should take a measurably different time than light travelling perpendicular to it. In 1887, Michelson and Morley built an extremely sensitive interferometer to test exactly this: it split a single beam of light into two perpendicular paths of equal length, reflected each back, and recombined them, so that any difference in travel time along the two paths would show up as a shift in the resulting interference fringe pattern when the whole apparatus was rotated.
The experiment found no fringe shift of the size predicted for any assumed aether wind, however the apparatus was oriented or however the measurement was repeated across the year as the Earth's orbital velocity changed direction — a genuinely surprising null result, since a wave needing a medium to travel through was the settled assumption of the time. This forced physicists to confront the possibility that light does not need a medium at all, and that the speed of light might simply be the same in every direction, regardless of any observer's own motion — exactly the assumption Einstein would build special relativity on eighteen years later.
A null result alone does not, by itself, prove every conceivable aether model impossible, nor does it single-handedly establish special relativity — it is one strong piece of evidence among several that motivated Einstein's theory, not a self-contained proof of it.
Einstein's 1905 theory of special relativity rests on two postulates. First, the principle of relativity: the laws of physics are the same in every inertial reference frame — no experiment performed entirely inside a closed, uniformly moving laboratory can tell you how fast, or even whether, that laboratory is moving, since there is no experiment whose outcome would differ from one inertial frame to another. Second, the constancy of the speed of light: the speed of light in a vacuum, $c$ , has the same value in every inertial reference frame, regardless of the motion of the light source or the observer measuring it.
The second postulate is the genuinely radical one: it directly explains the Michelson-Morley null result (light travels at exactly $c$ in every direction, in every inertial frame, with no aether wind to detect), but it also breaks a deep, previously unquestioned assumption of Newtonian mechanics — that velocities simply add (if you run forward on a moving train, your speed relative to the ground is the train's speed plus your running speed). If light's speed must come out as exactly $c$ regardless of the emitting source's own motion, then something else that Newtonian mechanics treated as absolute and universal — time itself, or length itself — has to give way instead.

Time dilation 3.12.3.3

Definitions

Proper time, $t_{0}$ : the time interval between two events measured by an observer for whom both events happen at the same place — i.e. in the frame moving with whatever is being timed.

Key results

Time dilation: $t = \frac{t _{0}}{1 - v ^{2} / c ^{2}} = γ t_{0}$ , where the Lorentz factor $γ = \frac{1}{1 - v ^{2} / c ^{2}} \geq 1$ .

Notes

A direct consequence of the constancy of the speed of light is that a moving clock runs slow, as measured by a stationary observer. This can be derived from a simple thought experiment, the light clock: imagine a clock that ticks by bouncing a single pulse of light back and forth between two mirrors a fixed distance apart. Viewed in the frame where the clock is at rest, the light simply travels straight up and down between the mirrors, taking a proper time $t_{0}$ per round trip. Viewed from a frame in which the whole clock is moving sideways at speed $v$ , the light must instead travel along a longer, diagonal zig-zag path to cover the same vertical mirror separation while the clock itself moves sideways — and since the speed of light must still be exactly $c$ in this frame too (Einstein's second postulate), covering a longer path at the same speed necessarily takes a longer time, $t = γ t_{0}$ .
Time dilation is not a trick of perception or a delay in signals reaching an observer — it is a genuine difference in how much time elapses between two events, depending on the observer's own state of motion. It becomes significant only as $v$ approaches $c$ : at everyday speeds $γ$ is indistinguishable from 1, which is exactly why the effect went unnoticed until physics had reason to examine particles moving at a substantial fraction of the speed of light.
The clearest experimental confirmation comes from muons, unstable particles created high in the atmosphere by cosmic rays, which decay with a short proper half-life measured at rest in a laboratory. Far more muons survive the journey down to sea-level detectors than that proper half-life alone would predict for particles travelling even at very nearly the speed of light — because, in the Earth's reference frame, each muon's own internal decay 'clock' is time-dilated by its high speed, so a much longer time elapses on the ground before an equivalent amount of proper time has passed for the muon and it decays.

Length contraction 3.12.3.4

Definitions

Proper length, $l_{0}$ : the length of an object measured in the frame in which the object is at rest.

Key results

Length contraction: $l = l_{0} 1 - v^{2} / c^{2} = \frac{l _{0}}{γ} \leq l_{0}$ — contraction applies only to the length component parallel to the direction of relative motion.

Notes

Just as a moving clock runs slow, a moving object is measured as shorter, along its direction of travel, than its own proper (rest-frame) length — length contraction, the companion effect to time dilation, and equally a genuine consequence of the same two postulates rather than an optical illusion. A length measurement requires locating both of an object's endpoints at the same instant in whichever frame is doing the measuring; because different inertial frames disagree about which distant events count as simultaneous, they disagree correspondingly about the resulting measured length, with the object's own rest frame always measuring the largest (proper) length.
The muon example illustrates length contraction just as well as time dilation, viewed from the opposite frame: in the muon's own rest frame, its short proper lifetime is not stretched at all — instead, the depth of the Earth's atmosphere it must cross is length-contracted, so a much shorter distance needs to be covered before reaching the ground. Both descriptions — a dilated muon lifetime in the Earth's frame, or a contracted atmosphere in the muon's frame — predict exactly the same observable outcome: more muons reach the ground than a naive, non-relativistic calculation would suggest.

Mass-energy equivalence 3.12.3.5

Key results

Mass-energy equivalence: $E = m c^{2}$ .
Relativistic total energy: $E = γ m_{0} c^{2}$ , so relativistic kinetic energy is $E_{k} = (γ - 1) m_{0} c^{2}$ — not the familiar $\frac{1}{2} m v^{2}$ , which is only the low-speed approximation of this exact result.

Notes

Special relativity's best-known result, $E = m c^{2}$ , states that mass and energy are equivalent and interconvertible: a given amount of mass corresponds to an enormous amount of energy, since $c^{2}$ is such a large number, and conversely a system that gains energy (for example, a nucleus absorbing energy) genuinely gains mass, however immeasurably small that mass increase usually is at everyday energies. A key practical consequence is that a particle's kinetic energy no longer obeys the simple Newtonian formula $\frac{1}{2} m v^{2}$ as its speed approaches $c$ : instead, its total energy is $γ m_{0} c^{2}$ (rest energy $m_{0} c^{2}$ plus kinetic energy), and as $v \to c$ , $γ \to \infty$ , so an unbounded amount of energy would be needed to accelerate any object with mass all the way to the speed of light — this is why $c$ acts as an ultimate speed limit for anything with mass, not merely a very large but achievable speed.
William Bertozzi's 1964 experiment gave direct experimental confirmation of this, accelerating electrons through a range of known, increasingly large potential differences (giving each a precisely known kinetic energy) and independently measuring their resulting speed by timing their flight over a fixed distance. As the electrons' kinetic energy increased, their measured speed rose ever more slowly, approaching but never reaching $c$ , exactly as the relativistic formula predicts — while a simple Newtonian calculation using $E_{k} = \frac{1}{2} m v^{2}$ would have predicted speeds exceeding $c$ at the higher energies used, a physically impossible result the Newtonian formula does not itself forbid.

Worked examples

Worked example 3.12.3.3 · 5 marks

A muon is created in the upper atmosphere travelling at $0.995 c$ towards the ground.

Its proper half-life at rest is $1.56 μ s$ .

Calculate the half-life of this muon as measured by an observer on the ground.

Show worked solution

γ = \frac{1}{1 - 0.99 5 ^{2}} = \frac{1}{1 - 0.990025} = \frac{1}{0.009975} \approx 10.0

t = γ t_{0} = 10.0 \times 1.56 μ s = 15.6 μ s

Mark scheme · 5 marks

Correctly calculates the Lorentz factor γ ≈ 10.0 2 marks
Correctly identifies 1.56 μs as the proper time $t_{0}$ 1 mark
Correctly applies $t = γ t_{0}$ to reach 15.6 μs 1 mark
Recognises the ground-frame half-life is longer than the proper (rest-frame) half-life, consistent with time dilation 1 mark

Do not count matching words alone — ask whether your answer actually makes the same claim.

Worked example 3.12.3.5 · 4 marks

Calculate the energy released, in MeV, when 1.0 mg of mass is entirely converted into energy. (:

c = 3.00 \times 1 0^{8} m s^{- 1}

e = 1.60 \times 1 0^{- 19} C

)

Show worked solution

E = m c^{2} = 1.0 \times 1 0^{- 6} \times (3.00 \times 1 0^{8})^{2} = 9.0 \times 1 0^{10} J

In MeV:

\frac{9.0 \times 1 0 ^{10}}{1.60 \times 1 0 ^{- 19} \times 1 0 ^{6}} \approx 5.6 \times 1 0^{23} MeV

Mark scheme · 4 marks

Correctly converts 1.0 mg to kg (1.0×10⁻⁶ kg) 1 mark
Correctly applies $E = m c^{2}$ to reach 9.0×10¹⁰ J 1 mark
Correctly converts joules to eV and then to MeV 1 mark
Reaches the correct final answer, ≈5.6×10²³ MeV, and recognises this is an enormous energy from a tiny mass 1 mark

Do not count matching words alone — ask whether your answer actually makes the same claim.