The Nature of Light

Module 7 is worth about a quarter of the paper. Special relativity is the most heavily examined part (about 7 marks a year), with the quantum model, the electromagnetic spectrum and the wave model each worth about 5.

• Electromagnetic spectrum: Maxwell's contribution, the speed of light, spectra and what stellar spectra tell us, Wien's law.
• Light: wave model: Newton's and Huygens' models, interference, diffraction, polarisation and Malus's law.
• Light: quantum model: black-body radiation, Planck's hypothesis, the photoelectric effect and Einstein's photon explanation.
• Light and special relativity: frames of reference, the aether and Michelson–Morley, Einstein's postulates, time dilation, length contraction, relativistic momentum, $E=m{c}^2$ and the evidence for each.

A big theme across the module is how evidence changes models. Many extended-response questions ask you to explain which observation supported which model, so keep a clear list.

Maxwell's unification

James Clerk Maxwell (1860s) combined the laws of electricity and magnetism into one theory. His equations predicted that a changing electric field produces a magnetic field and a changing magnetic field produces an electric field, so the two can sustain each other as a wave travelling through empty space. The predicted speed was

which matched the measured speed of light. Maxwell concluded that light is an electromagnetic wave and predicted waves at other frequencies. Hertz produced and detected radio waves in 1887, confirming the prediction. In an electromagnetic wave the electric and magnetic fields oscillate at right angles to each other and to the direction of travel, so the wave is transverse.

The speed of light and the spectrum

Early measurements included Fizeau's toothed wheel (1849) and Foucault's rotating mirror. Modern methods time laser pulses over known distances or measure $f$ and $\lambda$ of microwaves in a resonant cavity; since 1983 $c$ is fixed and defines the metre. All electromagnetic waves travel at $c$ in a vacuum, and

with $f$ in Hz and $\lambda$ in m. In order of increasing frequency (decreasing wavelength): radio, microwave, infrared, visible, ultraviolet, X-ray, gamma.

Spectra and stars

• Continuous spectrum: all wavelengths, from a hot dense object (a black body).
• Emission spectrum: bright lines from a hot low-density gas. Each element has its own pattern.
• Absorption spectrum: dark lines where a cooler gas in front of a hot source absorbs particular wavelengths.

From a star's spectrum astronomers can find its surface temperature (peak wavelength, using Wien's law), chemical composition (absorption lines), radial (translational) velocity (Doppler shift of the lines towards red or blue), rotational velocity (line broadening from one side approaching and the other receding) and density (pressure broadening of lines).

Wien's displacement law (data sheet)

${\lambda }_{max}$ is the peak wavelength (m), $T$ is the absolute temperature (K) and $b=2.898\times 10^{-3}$ m K. Hotter objects peak at shorter wavelengths.

Example. A star's spectrum peaks at 410 nm:

It is hotter than the Sun and appears white.

Competing models

Newton proposed a particle (corpuscular) model: light is made of tiny particles, which explained reflection and straight-line travel, and predicted refraction because light speeds up in denser media. Huygens proposed a wave model: every point on a wavefront acts as a source of secondary wavelets, which explained reflection and refraction with light slowing down in denser media. Newton's reputation kept the particle model dominant for over a century.

The evidence that swung opinion to waves:

• Interference: Young's double-slit experiment (about 1801) produced bright and dark fringes, which particles could not explain.
• Diffraction: light spreads around edges and through narrow slits, most noticeably when the slit width is comparable to the wavelength.
• Speed in water: Foucault (1850) showed light travels more slowly in water than in air, as the wave model predicted.
• Polarisation: light can be polarised, showing it is a transverse wave.

Interference (data sheet)

$d$ is the slit separation (m), $\theta$ is the angle from the central maximum to the bright fringe, and $m=0,1,2,\dots$ is the order. Bright fringes occur where the path difference is a whole number of wavelengths (constructive interference); dark fringes occur where the path difference is $(m+\frac{1}{2})\lambda$ (destructive interference). A diffraction grating has many slits, giving sharper, brighter maxima at the same angles.

Worked example: diffraction grating

A green laser (532 nm) shines through a grating with 600 lines per mm.

The slit spacing is

For the first order ($m=1$):

so $\theta =18.6^{\circ }$. For the second order, $\sin \theta =0.638$, so $\theta =39.7^{\circ }$.

Highest order: $\sin \theta \le 1$ needs $m\le d/\lambda =3.13$, so the third order (at $73.3^{\circ }$) is the last one visible.

Polarisation and Malus's law (data sheet)

$I_{max}$ is the intensity of polarised light reaching a polariser, $I$ is the transmitted intensity and $\theta$ is the angle between the light's plane of polarisation and the polariser's axis. Unpolarised light passing through one polariser loses half its intensity (the ${\cos }^2$ averages to one half).

Example. Unpolarised light of intensity $60{\text{W m}}^{-2}$ passes through two polarisers whose axes are $35^{\circ }$ apart. After the first: $30{\text{W m}}^{-2}$. After the second: $I=30{\cos }^{2}35^{\circ }=20{\text{W m}}^{-2}$.

Practicals

Expect questions on using a laser with double slits or a grating to measure wavelength (measure fringe spacing or angle, use a long screen distance to reduce percentage uncertainty), and on testing Malus's law with a light sensor and rotating polariser (plot $I$ against ${\cos }^2\theta$ for a straight line).

Black-body radiation

A black body absorbs all radiation that falls on it and emits a continuous spectrum that depends only on its temperature. Classical wave theory predicted that emitted intensity should keep increasing as wavelength decreases, giving infinite energy in the ultraviolet. This failure is called the ultraviolet catastrophe. In 1900 Max Planck matched the measured curves by assuming that energy is emitted or absorbed only in discrete packets (quanta):

with $h=6.626\times 10^{-34}$ J s. Planck saw this as a mathematical device; Einstein later treated it as a real property of light.

The photoelectric effect

When light of high enough frequency shines on a metal, electrons (photoelectrons) are emitted. The observations:

• No electrons are emitted below a threshold frequency $f_0$, however bright the light.
• Above $f_0$, emission is instant, even for dim light.
• The maximum kinetic energy of the electrons depends on frequency, not intensity.
• Increasing intensity increases the number of electrons (the current), not their energy.

The wave model predicts that any frequency should work if the light is bright enough, that a delay would be needed for dim light, and that brighter light would give more energetic electrons. All three predictions fail. In 1905 Einstein explained the effect with photons: light arrives as particles of energy $hf$, and one photon gives all its energy to one electron.

$\varphi$ is the work function, the minimum energy needed to free an electron from the metal (J or eV, $1\text{eV}=1.602\times 10^{-19}$ J). At threshold, $h{f}_0=\varphi$. The maximum kinetic energy is measured with a stopping voltage $V_0$: $K_{max}=q{V}_0$.

A graph of $K_{max}$ against $f$ is a straight line with gradient $h$ (the same for every metal), $f$-intercept $f_0$ and vertical intercept $-\varphi$.

Worked example: finding a work function

Light of wavelength 400 nm falls on a metal surface and the stopping voltage is 0.75 V.

Photon energy:

so $E=4.97\times 10^{-19}$ J (3.10 eV). Then the maximum kinetic energy and the work function are

which is 2.35 eV. The threshold frequency is $f_0=\varphi /h=5.69\times 10^{14}$ Hz (threshold wavelength 528 nm).

Practical

The photoelectric investigation (real or simulated) measures stopping voltage for several frequencies and plots $K_{max}$ against $f$ to find $h$ and $\varphi$. Know why intensity is varied separately and why only the frequency changes the stopping voltage.

Frames of reference and the aether

A frame of reference is a viewpoint from which measurements are made. An inertial frame is one that is not accelerating; the laws of motion work the same in all of them, and no experiment inside a closed inertial frame can tell whether it is moving. Nineteenth-century physicists assumed light waves needed a medium, the aether, which would give a single absolute frame.

The Michelson–Morley experiment (1887) used an interferometer to look for a change in the speed of light as Earth moved through the aether. No shift in the interference pattern was found (a null result). This was evidence against the aether.

Einstein's postulates (1905)

1. The laws of physics are the same in all inertial frames of reference.
2. The speed of light in a vacuum is the same for all observers, regardless of the motion of the source or the observer.

If the speed of light is fixed, time and space must change instead. Einstein used thought experiments to show how.

Thought experiments

Light clock. A light clock is a pulse of light bouncing between two mirrors a distance $L$ apart. In the clock's own frame each tick (there and back) takes $t_0=2L/c$. To an observer who sees the clock moving sideways at speed $v$, the light travels a longer diagonal path. In time $t$ for one tick, the light covers $ct$ while the clock moves $vt$, so by Pythagoras

Because the light still travels at $c$ for this observer, the longer path takes longer. Solving for $t$ gives the time dilation formula below: the moving clock ticks slower.

Train and lightning (simultaneity). Lightning strikes both ends of a moving train. An observer on the platform, standing midway between the strikes, sees both flashes at the same moment and says the strikes were simultaneous. An observer at the middle of the train is moving towards the front flash, so its light reaches her first. Light travels at $c$ in both directions in her frame and she is midway between the ends, so she concludes the front strike happened first. Events that are simultaneous in one frame need not be simultaneous in another.

Formulas (data sheet)

• $t_0$ is proper time: measured in the frame where the two events happen at the same place (the clock moves with the object). Any other observer measures a longer time $t$ (time dilation).
• $l_0$ is proper length: measured in the frame where the object is at rest. Observers moving relative to it measure a shorter length $l$ along the direction of motion only (length contraction).
• $p_v$ is relativistic momentum; $m_0$ is rest mass. As $v\to c$, momentum (and the energy needed) grows without limit, so no object with mass can reach $c$.
• $E=m{c}^2$: mass and energy are equivalent; a mass $m$ (kg) corresponds to energy $E$ (J).

The factor $\gamma =1/\sqrt{1-v^2/c^2}$ (the Lorentz factor) is always at least 1.

Worked example: a fast pion

A charged pion has a proper mean lifetime of $2.6\times 10^{-8}$ s and is produced moving at $0.990c$ in a laboratory.

This is the lifetime measured in the lab. The distance travelled in the lab is

compared with only 7.7 m if time dilation did not happen.

Momentum example. An electron at $0.90c$ has $\gamma =2.29$, so

This is more than twice the classical value of $2.5\times 10^{-22}{\text{kg m s}}^{-1}$.

Rest energy example. A proton ($1.673\times 10^{-27}$ kg) has rest energy $m{c}^2=1.51\times 10^{-10}$ J, about 940 MeV.

Evidence

• Muons made by cosmic rays high in the atmosphere reach the ground in far greater numbers than their short lifetime would allow without time dilation (or, in the muon's frame, length contraction of the atmosphere).
• Atomic clocks flown around the world (Hafele and Keating, 1971) disagreed with ground clocks by the amounts relativity predicts.
• Particle accelerators show particles approaching but never reaching $c$, and momentum and lifetimes match the relativistic formulas.
• GPS satellites must correct their clocks for relativistic effects or positions would drift by kilometres a day.
• Nuclear reactions release energy matching the mass lost, as $E=m{c}^2$ predicts.

• Mixing up proper time and dilated time. Ask: in which frame does the clock stay with the events? That one measures $t_0$.
• Contracting lengths perpendicular to the motion. Only the length along the motion contracts.
• Saying brighter light gives faster photoelectrons. Intensity changes the number of photons, not the energy of each.
• Forgetting to halve the intensity of unpolarised light at the first polariser.
• Leaving wavelengths in nm or energies in eV when the formula needs SI units.
• Writing that Michelson–Morley "proved relativity". It gave a null result for the aether; Einstein's postulates explained it.
• Calling Planck's quantisation a property of light rather than of the oscillators emitting it (that was Einstein's step).

• Explain how X supports a model: describe the observation, state what each model predicts, and say which prediction matches. For example, for the photoelectric effect name at least two observations the wave model cannot explain.
• Evaluate evidence for relativity: describe the experiment, the result, how it matches the prediction, and any limitation (such as uncertainty in early muon or clock experiments).
• Assess the contribution of a scientist (Maxwell, Planck, Einstein): what they proposed, what problem it solved, what evidence followed.
• Relativity calculations: write $\gamma$ first, then decide whether to multiply or divide. Check that moving clocks run slow and moving rulers are short.
• Photoelectric graphs are common: label gradient, intercepts and units, and note that different metals give parallel lines.
• Round to the data's significant figures. Convert eV and J carefully, and include units such as ${\text{W m}}^{-2}$, Hz and K.

• Module 6: Maxwell's unification builds on electromagnetic induction; the photoelectric effect uses stopping voltage, which is work done by an electric field ($W=qV$).
• Module 8: spectra and Wien's law are used to classify stars and study the Big Bang; $E=hf$ links to the Bohr model and line spectra; $E=m{c}^2$ is the basis of binding energy, fission and fusion; de Broglie extended wave–particle duality to electrons.
• Year 11 Waves and Thermodynamics: wave properties, superposition and the wave equation are assumed knowledge.