Vector addition by tip-to-tail or parallelogram rule. Resolution into perpendicular components Fcosθ, Fsinθ.
R = √(Fx² + Fy²); tanθ = Fy/Fx
1.5 Errors and Uncertainties
Systematic error: consistent bias (e.g. zero error). Not reduced by repeating. Random error: unpredictable scatter. Reduced by repeating and averaging. Precision: how close repeated readings are to each other. Accuracy: how close mean is to true value.
Common pitfall: Add ABSOLUTE uncertainties for + / −; add PERCENTAGE uncertainties for × / ÷ and powers.
Worked example
L = (1.250 ± 0.005) m, d = (0.40 ± 0.01) mm. % unc. in d = 2.5 %. A = πd²/4, so % unc. in A = 2 × 2.5 = 5.0 %.
Distance: total path length (scalar). Displacement: vector from start to finish. Speed: rate of change of distance. Velocity: rate of change of displacement. Acceleration: rate of change of velocity (m s⁻²).
2.2 Motion Graphs
Graph
Gradient
Area under
s–t
Velocity
—
v–t
Acceleration
Displacement
a–t
—
Change in velocity
2.3 SUVAT (Constant Acceleration)
v = u + at s = ut + ½at² v² = u² + 2as s = ½(u + v)t
2.4 Free Fall
g = 9.81 m s⁻² downward, neglecting air resistance.
2.5 Projectile Motion
Horizontal and vertical components are independent. Horizontal: constant velocity. Vertical: uniform acceleration g. Trajectory is a parabola.
Worked example
Ball launched horizontally at 15 m s⁻¹ from 20 m cliff. Time of flight: t = √(2×20/9.81) = 2.02 s. Range = 15 × 2.02 = 30.3 m.
2.6 Air Resistance and Terminal Velocity
Drag increases with speed. When drag = weight, resultant force = 0, body falls at terminal velocity.
03Dynamics
Newton's laws, mass, weight, momentum, conservation, and collisions.
3.1 Newton's Three Laws
1st Law: body remains at rest or constant velocity unless acted on by a resultant force. 2nd Law:F = Δp/Δt; for constant mass F = ma. 3rd Law: body A on body B = equal and opposite from B on A; forces act on DIFFERENT bodies and are of the same type.
3.2 Mass and Weight
Mass: inertia (kg). Weight:W = mg, at centre of gravity (N).
N3 pitfall: the reaction to a book's weight is the book's gravitational pull on Earth — NOT the normal force from the table.
3.3 Momentum
p = mv; impulse = FΔt = Δp
3.4 Conservation of Momentum
Total momentum of an isolated system is constant.
3.5 Elastic vs Inelastic
Type
Momentum
KE
Perfectly elastic
Conserved
Conserved (speed of approach = speed of separation)
A couple: two equal opposite parallel forces, lines of action don't coincide. Produces only rotation. Torque = force × perpendicular distance between them.
4.3 Equilibrium
Resultant force = 0 AND resultant torque = 0.
Principle of moments: sum of clockwise moments = sum of anticlockwise moments about any point.
4.4 Centre of Gravity
Point at which the entire weight may be considered to act. For uniform body: geometric centre.
4.5 Density and Pressure
ρ = m/V; p = F/A
4.6 Hydrostatic Pressure
Δp = ρgh
Derivation: column of cross-section A, height h: weight = ρAhg, pressure on base = ρgh.
4.7 Upthrust (Archimedes)
Upthrust on an immersed body = weight of fluid displaced.
Fupthrust = ρfluidgVdisplaced
05Work, Energy and Power
Energy conservation, mechanical work, efficiency and power.
5.1 Work Done
Work done by a force = force × displacement in the direction of the force.
W = F s cosθ (J)
Force ⊥ motion does ZERO work. Centripetal force does no work.
5.2 Conservation of Energy
Energy cannot be created or destroyed; only transferred between forms.
5.3 GPE and KE
ΔEp = mgΔh; Ek = ½mv²
Derive Ek: v² = 2as, work done = Fs = mas = ½mv².
5.4 Efficiency
η = (useful output / total input) × 100 %
5.5 Power
P = W/t = F v (W = J s⁻¹)
P = Fv is essential for cars at constant speed against drag.
06Deformation of Solids
Hooke's law, stress, strain, Young modulus and elastic PE.
6.1 Hooke's Law
Extension ∝ applied force, up to the limit of proportionality.
F = kx
6.2 Stress, Strain, Young Modulus
σ = F/A (Pa); ε = x/L; E = σ/ε = FL/(Ax)
Young modulus = gradient of linear region of stress–strain graph. Property of the MATERIAL.
Pd: energy per unit charge transferred from electrical to other forms (component). EMF: energy per unit charge transferred from other forms to electrical (source).
V = W/Q
9.4 Ohm's Law
For a metallic conductor at constant temperature, I ∝ V.
9.5 I–V Characteristics
Component
Behaviour
Resistor (ohmic)
Straight line through origin
Filament lamp
S-curve; R rises with T
Diode
Threshold ~0.6 V forward; almost no reverse current
Common error: "Centripetal force" is NOT a new kind of force. It is the name we give to the resultant force that happens to be directed toward the centre. In real problems it is gravity (orbits), tension (string), normal contact (banked track), friction (cars on flat curves), or the magnetic force (charged particles).
Worked example — conical pendulum
A 0.20 kg bob on a string of length 0.80 m moves in a horizontal circle, with the string making 30° to the vertical. Find the period.
Vertical: T cos30° = mg. Horizontal: T sin30° = m(2π/P)²r, where r = L sin30° = 0.40 m.
Dividing: tan30° = (4π²r)/(gP²) → P = 2π√(r/(g tan30°)) = 2π√(0.40/(9.81 × 0.577)) = 1.68 s.
13Gravitational Fields
Newton's law of gravitation, field strength, potential, and circular orbits including geostationary satellites.
13.1 Newton's Law of Gravitation
F = Gm₁m₂/r²
Always attractive, directed along the line joining the two point masses. G = 6.67 × 10⁻¹¹ N m² kg⁻². Treat extended spherical bodies as if all the mass is concentrated at the centre.
13.2 Gravitational Field Strength
Gravitational field strength g: the gravitational force per unit mass at a point in the field.
g = F/m = GM/r² (outside a uniform sphere)
Near Earth's surface (r ≈ RE), g ≈ 9.81 N kg⁻¹. For small changes in height, g is taken as constant.
13.3 Gravitational Potential
Gravitational potential φ: the work done per unit mass in bringing a small test mass from infinity to that point.
φ = −GM/r
Always negative because work is done by the gravitational field as a mass moves from infinity (where φ = 0) to a finite distance.
Gravitational PE of mass mEp = mφ = −GMm/r
13.4 Field–Potential Relationship
g = −dφ/dr
Field strength is the (negative) gradient of potential vs distance. On a φ–r graph, gradient = −g.
13.5 Circular Orbits
For a satellite of mass m in a circular orbit of radius r around a body of mass M, gravity provides the centripetal force:
GMm/r² = mv²/r ⟹ v = √(GM/r)
Using v = 2πr/T:
Kepler's third lawT² = (4π²/GM) r³ ⟹ T² ∝ r³
13.6 Geostationary Satellites
A geostationary satellite has period T = 24 h, orbits above the equator, and moves west to east (same direction as Earth's rotation). Its orbital radius is r ≈ 4.22 × 10⁷ m from Earth's centre (~36 000 km altitude). It appears stationary from the ground — useful for telecommunications and TV broadcasts.
14Temperature
The thermodynamic temperature scale, thermometric properties, and the energy transferred during heating and phase changes.
14.1 Thermal Equilibrium
Energy flows spontaneously from a region of higher temperature to one of lower temperature. Two bodies are in thermal equilibrium when there is no net flow of energy between them; their temperatures are then equal.
14.2 Thermometric Properties
Any physical property that varies measurably and predictably with temperature can be used to make a thermometer:
Density / volume of a liquid (mercury or alcohol in a glass tube).
Pressure of a fixed-volume gas (constant-volume gas thermometer).
Resistance of a metal wire (platinum resistance thermometer).
EMF generated at a junction of two dissimilar metals (thermocouple).
14.3 The Kelvin Scale
The thermodynamic (Kelvin) scale is the SI temperature scale. It does not depend on the properties of any particular substance.
T/K = θ/°C + 273.15
Absolute zero (0 K): the temperature at which a substance has the minimum possible internal energy. It is unreachable.
14.4 Specific Heat Capacity
Specific heat capacity c: the energy required to raise the temperature of 1 kg of a substance by 1 K (without phase change).
Q = mcΔT (J kg⁻¹ K⁻¹ for c)
14.5 Specific Latent Heat
Specific latent heat L: the energy required to change the state of 1 kg of a substance at constant temperature.
Q = mL
Lf = latent heat of fusion (solid ↔ liquid); Lv = latent heat of vaporisation (liquid ↔ gas). During a phase change, all input energy goes into breaking intermolecular bonds — no temperature change occurs.
Worked example
How much energy to convert 0.50 kg of ice at −10 °C to water at 25 °C? Use cice = 2100, Lf = 3.34 × 10⁵, cwater = 4200.
Heat ice: Q₁ = 0.50 × 2100 × 10 = 1.05 × 10⁴ J.
Melt: Q₂ = 0.50 × 3.34 × 10⁵ = 1.67 × 10⁵ J.
Heat water: Q₃ = 0.50 × 4200 × 25 = 5.25 × 10⁴ J.
Total ≈ 2.30 × 10⁵ J.
15Ideal Gases
The equation of state, the kinetic theory model and the link between temperature and molecular KE.
15.1 The Mole and Avogadro Constant
One mole contains NA = 6.02 × 10²³ particles. n = N/NA = mass/molar mass.
15.2 Equation of State for an Ideal Gas
pV = nRT = NkT
where R = 8.31 J mol⁻¹ K⁻¹ (molar gas constant) and k = R/NA = 1.38 × 10⁻²³ J K⁻¹ (Boltzmann constant).
An ideal gas is one that obeys pV ∝ T at all pressures, volumes and temperatures.
15.3 Kinetic Theory Assumptions
A gas contains a very large number of molecules in random motion.
The volume of the molecules is negligible compared with the volume of the gas.
Intermolecular forces are negligible except during collisions.
Collisions with each other and with the walls are perfectly elastic.
The time of each collision is negligible compared with the time between collisions.
15.4 The Kinetic-Theory Equation
pV = ⅓Nm<c²>
where <c²> is the mean-square speed of the molecules. Comparing with pV = NkT:
Mean translational KE per molecule½m<c²> = (3/2) kT
This is the central conclusion of kinetic theory: temperature is a direct measure of the average translational KE of molecules.
The root-mean-square speed is crms = √<c²> = √(3kT/m).
16Thermodynamics
Internal energy of a system, the first law of thermodynamics, and work done by/on a gas.
16.1 Internal Energy
Internal energy: the sum of the random kinetic and potential energies of all the molecules in a system.
For an ideal gas, intermolecular PE = 0, so internal energy = total molecular KE = (3/2)NkT.
For a real substance, internal energy includes both random KE (rises with T) and molecular PE (changes during phase changes).
16.2 The First Law of Thermodynamics
ΔU = q + W
ΔU: increase in internal energy.
q: heat supplied to the system (positive if heating).
W: work done on the system (positive if compressed).
Statement: ΔU = heat supplied to system + work done on system. Energy is conserved.
16.3 Work Done by a Gas
When a gas expands by ΔV against an external pressure p, it does work:
Work done BY gasWby gas = pΔV
So work done on the gas (used in the first law above) is W = −pΔV. On a p–V diagram, area under the curve = work done by the gas.
Worked example — isothermal vs adiabatic
Isothermal compression (slow): ΔT = 0 → ΔU = 0. Work done on gas = heat out of gas; q = −W. Adiabatic compression (rapid, insulated): q = 0. All the work goes into raising internal energy: ΔU = W > 0, so temperature rises.
17Oscillations
Simple harmonic motion (SHM), energy exchange between KE and PE, damping and resonance.
17.1 SHM Definition
SHM: motion in which the acceleration is proportional to the displacement from a fixed point and is always directed toward that point.
a = −ω²x
17.2 Kinematic Equations for SHM
For an oscillation starting at the equilibrium position:
x = x₀ sin(ωt) v = v₀ cos(ωt) = ±ω√(x₀² − x²) a = −ω²x₀ sin(ωt) = −ω²x
Maximum speed: vmax = ωx₀ (at equilibrium, x = 0).
Maximum acceleration: amax = ω²x₀ (at amplitude, x = ±x₀).
17.3 Energy in SHM
Total energyE = ½mω²x₀² (constant)
KE = ½m(v²) = ½mω²(x₀² − x²) (max at equilibrium).
PE = ½mω²x² (max at amplitude).
KE and PE oscillate at 2f (twice the frequency of the displacement).
17.4 Examples of SHM Systems
System
Period
Mass on a spring
T = 2π√(m/k)
Simple pendulum (small angles)
T = 2π√(L/g)
17.5 Damping
Type
Behaviour
Light damping
Amplitude decays exponentially; oscillations continue many cycles.
Critical damping
Returns to equilibrium in the shortest time without oscillating.
Heavy damping
Slow return to equilibrium, no oscillation.
17.6 Forced Oscillations and Resonance
Resonance: the large-amplitude oscillation that occurs when the driving frequency equals the system's natural frequency.
At resonance the driving force is in phase with the velocity, so energy transfer to the oscillator is maximum.
Effect of damping on resonance curves:
More damping → lower peak amplitude.
More damping → broader peak.
More damping → resonant frequency shifts slightly below the natural frequency.
18Electric Fields
Coulomb's law, uniform and radial fields, electric potential and PE.
18.1 Electric Field Strength
Electric field strength E: the electric force per unit positive charge at a point.
E = F/q (N C⁻¹ or V m⁻¹)
18.2 Uniform Fields (Parallel Plates)
E = V/d
Between parallel plates separated by distance d with pd V, the field is uniform from + plate toward − plate. A charged particle in this field experiences a constant force, so its motion is parabolic (analogous to projectile motion under gravity).
18.3 Coulomb's Law (Point Charges)
F = Q₁Q₂ / (4πε₀r²)
Where ε₀ = 8.85 × 10⁻¹² F m⁻¹ is the permittivity of free space. Like charges repel, unlike attract — by symmetry with gravity, the magnitude form is the same but charges may take either sign.
18.4 Radial Field of a Point Charge
E = Q / (4πε₀r²)
18.5 Electric Potential
Electric potential V: the work done per unit positive charge in bringing a small test charge from infinity to that point.
V = Q/(4πε₀r)
Sign of V follows sign of Q: positive near a positive charge, negative near a negative charge.
Electric PE between two point chargesEp = Qq/(4πε₀r)
18.6 Field–Potential Relationship
E = −dV/dr
18.7 Gravitational vs Electric Fields
Property
Gravitational
Electric
Source
Mass (always +)
Charge (+ or −)
Force
Always attractive
Attractive or repulsive
Law
F = GMm/r²
F = Qq/(4πε₀r²)
Field at distance r
g = GM/r²
E = Q/(4πε₀r²)
Potential
φ = −GM/r (always −)
V = Q/(4πε₀r) (sign of Q)
19Capacitance
Charge storage, energy stored in a capacitor, combinations, and exponential discharge.
19.1 Definition
Capacitance C: the charge stored per unit potential difference across a capacitor.
C = Q/V (F = C V⁻¹)
19.2 Combinations of Capacitors
Series1/Ctotal = 1/C₁ + 1/C₂ + …
ParallelCtotal = C₁ + C₂ + …
Note: opposite to resistors. In series the same charge sits on each capacitor; in parallel each plate is at the same pd.
19.3 Energy Stored
Building up charge on a capacitor takes work against the rising pd. The energy stored is the area under the Q–V graph (a triangle):
W = ½QV = ½CV² = ½Q²/C
When a capacitor is charged through a resistor by a battery, only half the energy provided by the battery is stored — the other half is dissipated as heat in the resistor, regardless of R.
19.4 Capacitor Discharge through a Resistor
Solving I = V/R with Q = CV and I = −dQ/dt gives an exponential:
Q = Q₀ e−t/RC V = V₀ e−t/RC I = I₀ e−t/RC
Time constant τ = RC: time for the quantity to fall to 1/e ≈ 37% of its initial value.
Half-life: t½ = RC ln 2 ≈ 0.693RC.
Common mistake: Charging up obeys V = V₀(1 − e−t/RC) — discharging obeys V = V₀ e−t/RC. Use the right one for the situation.
20Magnetic Fields
Force on current-carrying wires and moving charges, the Hall effect, and electromagnetic induction.
20.1 Magnetic Field
A magnetic field is a region in which a force is exerted on a moving charge or on a current-carrying conductor.
20.2 Force on a Current
F = BIL sinθ
where θ is the angle between the current and the field, and B is the magnetic flux density. Direction given by Fleming's left-hand rule: First finger = field, seCond finger = current, thuMb = motion.
Magnetic flux density (B): the force per unit current per unit length on a wire placed perpendicular to the field. Unit: tesla (T).
20.3 Force on a Moving Charge
F = BQv sinθ
For a charged particle moving perpendicular to a uniform B-field, this force is always perpendicular to the velocity → circular motion at constant speed:
BQv = mv²/r ⟹ r = mv/(BQ)
20.4 The Hall Effect
When a current flows in a thin slab in a perpendicular magnetic field, charge carriers experience a sideways force, accumulate on one face, and produce a small transverse Hall voltage:
VH = BI/(ntq)
where n is the number density of carriers and t the slab thickness. A Hall probe can be used to measure B.
20.5 Magnetic Flux and Flux Linkage
Magnetic fluxΦ = BA cosθ (Wb)
Flux linkage of N turnsNΦ
20.6 Faraday's and Lenz's Laws
Faraday's Law: the induced EMF is proportional to the rate of change of flux linkage. Lenz's Law: the induced current flows in a direction such that its effect opposes the change producing it (consequence of conservation of energy).
ε = −d(NΦ)/dt
21Alternating Currents
Sinusoidal currents, RMS values, mean power, and rectification using diodes and smoothing capacitors.
21.1 Sinusoidal AC
I = I₀ sin(ωt); V = V₀ sin(ωt)
where I₀, V₀ are peak values, ω = 2πf.
21.2 RMS Values
The root-mean-square value of an AC quantity equals the steady DC value that would dissipate the same average power in a resistive load.
Irms = I₀/√2; Vrms = V₀/√2
21.3 Mean Power
<P> = Vrms·Irms = ½V₀I₀ = I²rms·R
The "230 V mains" specification is an RMS value; peak is 230√2 ≈ 325 V.
21.4 Rectification
Half-wave: single diode in series with the load — passes one half of each cycle, blocks the other. Output is unidirectional but pulsating, and half the cycle is wasted.
Full-wave: four diodes in a bridge arrangement direct current the same way through the load on both halves of the cycle. More efficient and smoother.
21.5 Smoothing
A capacitor in parallel with the load fills in the gaps between peaks: it discharges slowly through R while the diode is reverse-biased. Smoothing improves as RC increases — but a very large capacitor stresses the diodes with high inrush currents.
22Quantum Physics
Photons, the photoelectric effect, wave–particle duality, electron energy levels and atomic spectra.
22.1 The Photon
A photon is a discrete packet (quantum) of electromagnetic energy.
E = hf = hc/λ
22.2 The Photoelectric Effect
When EM radiation above a threshold frequency strikes a metal, electrons are emitted instantly. Observations cannot be explained by the wave model:
Below threshold frequency f₀, no emission, however intense the light or however long the exposure.
Above threshold, emission is essentially instantaneous, even at very low intensities.
Maximum KE of emitted electrons depends on f, not on intensity.
Intensity controls the number of electrons emitted, not their energy.
Einstein's explanation: a single photon transfers all its energy to one electron.
Einstein's photoelectric equationhf = Φ + ½mv²max
where Φ = hf₀ is the work function — the minimum energy to release an electron from the metal surface.
Pitfall: Increasing intensity does NOT increase max KE. It only increases the rate of electron emission (photocurrent).
22.3 Wave–Particle Duality
Light shows wave behaviour (interference, diffraction) and particle behaviour (photoelectric effect). Conversely, electrons show wave behaviour (electron diffraction through thin polycrystalline foils).
de Broglie wavelengthλ = h/p = h/(mv)
22.4 Discrete Atomic Energy Levels
Electrons in an isolated atom can occupy only certain discrete energy levels. Transitions between levels produce or absorb photons of definite frequencies:
hf = Eupper − Elower
Emission spectrum: bright lines on dark background — atoms drop to lower levels.
Absorption spectrum: dark lines on continuous bright background — atoms absorb specific photons and re-emit in all directions.
Worked example
A photon is emitted when an electron drops from −1.5 eV to −3.4 eV. Find its wavelength.
ΔE = (−1.5) − (−3.4) = 1.9 eV = 1.9 × 1.60 × 10⁻¹⁹ = 3.04 × 10⁻¹⁹ J. λ = hc/ΔE = (6.63 × 10⁻³⁴ × 3.00 × 10⁸)/(3.04 × 10⁻¹⁹) = 6.54 × 10⁻⁷ m (visible red).
23Nuclear Physics
Mass–energy equivalence, binding energy, fission/fusion, and the laws of radioactive decay.
23.1 Mass–Energy Equivalence
E = mc²
Energy and mass are equivalent. A mass change of Δm corresponds to an energy change of Δm·c². The atomic mass unit u corresponds to 931 MeV.
23.2 Mass Defect and Binding Energy
Mass defect (Δm): the difference between the mass of the separated nucleons and the mass of the nucleus they form. Binding energy (BE): the energy released when nucleons combine to form a nucleus — equivalently, the energy needed to break a nucleus into its individual nucleons.
BE = Δm·c²
23.3 Binding Energy per Nucleon
The most useful measure for comparing nuclear stability:
Peaks at ⁵⁶Fe (~8.8 MeV/nucleon) — the most tightly bound nucleus.
Fission of a heavy nucleus (e.g. ²³⁵U) splits it into two medium-mass nuclei of higher BE/nucleon → energy released.
Fusion of light nuclei (e.g. ²H + ³H → ⁴He + n) produces a nucleus higher up the graph → energy released. Larger energy per unit mass than fission.
23.4 Radioactive Decay
Decay is spontaneous (cannot be triggered or affected by external conditions) and random (cannot predict which nucleus decays next). Activity is proportional to the number of undecayed nuclei:
A = λN
where λ = decay constant (probability of decay per unit time), A = activity (Bq), N = number of undecayed nuclei.
Exponential decayN = N₀ e−λt; A = A₀ e−λt
Half-lifet½ = ln 2 / λ ≈ 0.693/λ
Worked example — dating
A bone has activity 1/8 of a modern bone. ¹⁴C has t½ = 5730 yr. How old is it?
1/8 = (1/2)³ → 3 half-lives → age ≈ 3 × 5730 = 17 200 years.
24Medical Physics
Ultrasound imaging, X-ray production and CT scanning, and PET positron-emission tomography.
24.1 Ultrasound
Ultrasound = longitudinal sound waves above 20 kHz. Generated and detected by a piezoelectric crystal: an alternating pd at the crystal's resonant frequency causes mechanical oscillation; conversely, mechanical pressure on the crystal produces an alternating pd that can be amplified and displayed.
A large mismatch in Z (e.g. air-to-skin) reflects almost all the ultrasound. A coupling gel between the probe and the patient matches impedance and allows the pulse to enter the body.
AttenuationI = I₀ e−μx
24.2 X-rays
Production: electrons accelerated through a high pd (~100 kV) strike a heavy metal target (tungsten). Their KE converts to X-ray photons (a few %) and heat (mostly).
Minimum wavelength (KE all → 1 photon)λmin = hc/(eV)
X-rays attenuate exponentially through tissues:
I = I₀ e−μx
where μ is the linear attenuation coefficient (different for bone, soft tissue, etc.). The difference in μ gives image contrast; tube voltage and current control sharpness and brightness.
24.3 CT Scanning
A rotating X-ray tube takes many 2-D images of a slice from different angles. A computer reconstructs a 3-D map of attenuation. Compared with a simple X-ray, CT gives 3-D information and can distinguish soft tissues, but uses a much higher radiation dose.
24.4 PET (Positron Emission Tomography)
A β⁺-emitting tracer (e.g. ¹⁸F-FDG) is injected. When a positron is emitted, it travels a short distance, then annihilates with an electron, producing two γ-ray photons of 0.511 MeV travelling in opposite directions (back-to-back).
A ring of detectors records pairs of γ-photons arriving in coincidence; the line joining the two detector hits passes through the annihilation point. Many such lines reconstruct a 3-D map of tracer concentration → reveals high-metabolism regions such as tumours and active brain areas.
25Astronomy and Cosmology
Luminosity and flux, Wien's and Stefan's laws, redshift and Hubble's law.
25.1 Luminosity and Radiant Flux
Luminosity L: the total power radiated by a star, in all directions, across all wavelengths (W). Radiant flux intensity F: power received per unit area at a distance d.
F = L / (4πd²) (inverse-square law)
If L is known independently (a standard candle such as a Type Ia supernova or a Cepheid variable) and F is measured, distance d follows immediately.
25.2 Wien's Displacement Law
λmax·T = 2.9 × 10⁻³ m K
The wavelength at which a black-body spectrum peaks is inversely proportional to its temperature. Hotter stars peak in the blue/UV; cooler stars peak in the red/IR.
25.3 Stefan–Boltzmann Law
L = 4πr²σT⁴
where σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. Combining Wien's and Stefan's allows the radius of a star to be deduced from its peak wavelength and apparent flux.
Worked example — stellar radius
A star has λmax = 500 nm and L = 3.84 × 10²⁶ W (the Sun). T = (2.9 × 10⁻³)/(5.0 × 10⁻⁷) = 5800 K. r² = L/(4πσT⁴). Plugging in: r ≈ 6.96 × 10⁸ m.
25.4 Doppler Redshift of Galaxies
Light from distant galaxies arrives at longer wavelengths than emitted — they are receding.
Δλ/λ ≈ Δf/f ≈ v/c (non-relativistic)
25.5 Hubble's Law
v ≈ H₀d
where H₀ ≈ 2.2 × 10⁻¹⁸ s⁻¹. The further away a galaxy, the faster it recedes — implying that space itself is expanding.
25.6 The Big Bang Model
Evidence supporting the Big Bang:
Hubble's law — universal recession indicates expansion from a single hot dense state.
Cosmic microwave background (CMB) — almost perfectly uniform 2.73 K radiation, the redshifted afterglow.
Observed abundances of light elements (H, He, Li) match Big Bang nucleosynthesis predictions.
📋Data, Formulae and Exam Information
Reference cards for the A2 examination.
Fundamental Constants
Quantity
Symbol
Value
Gravitational constant
G
6.67 × 10⁻¹¹ N m² kg⁻²
Permittivity of free space
ε₀
8.85 × 10⁻¹² F m⁻¹
Molar gas constant
R
8.31 J mol⁻¹ K⁻¹
Boltzmann constant
k
1.38 × 10⁻²³ J K⁻¹
Stefan–Boltzmann constant
σ
5.67 × 10⁻⁸ W m⁻² K⁻⁴
Hubble constant
H₀
2.2 × 10⁻¹⁸ s⁻¹
Planck constant
h
6.63 × 10⁻³⁴ J s
Speed of light
c
3.00 × 10⁸ m s⁻¹
Elementary charge
e
1.60 × 10⁻¹⁹ C
1 u
—
1.66 × 10⁻²⁷ kg ≡ 931 MeV
A Level Paper Structure
Paper
Style
Time
Marks
% of A2
P4
A2 structured questions
2 h
100
38.5 %
P5
Planning, Analysis & Evaluation
1 h 15 min
30
11.5 %
Paper 5 — Linearisation Cheatsheet
Form
Plot
Gradient
Intercept
y = mx + c
y vs x
m
c
y = axn
lg y vs lg x
n
lg a
y = a ekx
ln y vs x
k
ln a
Exam Tips for A2
In gravitation and electric-field questions, watch the sign of φ and V. Potentials are scalars and add algebraically.
In SHM questions, when asked for max speed and max acceleration, mark which point (equilibrium vs amplitude) they occur at.
For capacitor discharge, identify what is constant (RC), what is exponential (Q, V, I), and what is asked (often a half-life or a time to fall to a given fraction).
Photoelectric effect — always state the work function in joules when used with hf; convert eV to J first.
For nuclear binding-energy calculations, always be explicit: mass defect in kg, then × c² in J, then ÷ 1.60 × 10⁻¹³ to convert to MeV.
9702 Physics AI Tutor
AI can make mistakes. Always check important answers against the official 9702 syllabus or your teacher.