Foundations: every measurement combines a numerical magnitude and a unit. Mastery of SI units, dimensional analysis, errors, and uncertainties underpins every other topic.
1.1 SI Base Units
All physical quantities are derived from seven SI base quantities. In 9702 you need to know six (luminous intensity not required):
Base Quantity
Symbol
Base Unit
Symbol
Mass
m
kilogram
kg
Length
l
metre
m
Time
t
second
s
Electric current
I
ampere
A
Thermodynamic temp.
T
kelvin
K
Amount of substance
n
mole
mol
1.2 Derived Units & Dimensional Analysis
Every other quantity has a unit built from products and quotients of base units. The dimension of a quantity shows which base units it involves, written as [M]a[L]b[T]c[A]d.
Quantity
Derived Unit
In Base Units
Dimensions
Force
newton (N)
kg m s-2
[M][L][T]-2
Energy / Work
joule (J)
kg m2 s-2
[M][L]2[T]-2
Power
watt (W)
kg m2 s-3
[M][L]2[T]-3
Pressure / Stress
pascal (Pa)
kg m-1 s-2
[M][L]-1[T]-2
Charge
coulomb (C)
A s
[A][T]
Potential diff.
volt (V)
kg m2 s-3 A-1
[M][L]2[T]-3[A]-1
Resistance
ohm (Ω)
kg m2 s-3 A-2
[M][L]2[T]-3[A]-2
Homogeneous equation: the units (or dimensions) on both sides must be the same. Dimensional analysis detects algebraic errors but cannot verify dimensionless constants or sign correctness.
Example — checking homogeneity
Verify T = 2π√(l/g): Left side [T]. Right side: √([L] / [L][T]-2) = √([T]2) = [T]. ✓ Homogeneous.
1.3 SI Prefixes
Name
Symbol
Factor
Name
Symbol
Factor
pico
p
10-12
kilo
k
103
nano
n
10-9
mega
M
106
micro
μ
10-6
giga
G
109
milli
m
10-3
tera
T
1012
centi
c
10-2
1.4 Scalars and Vectors
Scalar: a quantity with magnitude only (e.g. mass, speed, distance, time, energy, temperature, density). Vector: a quantity with magnitude and direction (e.g. displacement, velocity, acceleration, force, momentum, field strength).
Vector addition: tip-to-tail (head-to-tail) method or parallelogram rule. Resolution splits a vector into perpendicular components.
Component resolutionFx = F cos θ, Fy = F sin θ
Resultant magnitude and directionR = √(Fx2 + Fy2), tan θ = Fy / Fx
Choose θ from the horizontal axis consistently. The angle of the resultant is measured anticlockwise from the positive x-axis.
1.5 Errors and Uncertainties
Absolute uncertainty (Δx): half the range of repeated readings: Δx = ½(xmax - xmin). Same unit as x. Fractional uncertainty: Δx / x (unit-free). Percentage uncertainty: (Δx / x) × 100%. Systematic error: consistent bias (zero error, parallax error, calibration errors). Not reduced by repeating; accuracy is affected. Random error: unpredictable scatter (reaction time, electronic noise). Reduced by repeating and averaging; precision is affected. Precision: how close repeated readings are to each other (indicated by % uncertainty and sig. figs.). Accuracy: how close the mean is to the true value.
Propagation rules — must know for Paper 3:
• Addition / subtraction: add absolute uncertainties: ΔS = ΔA + ΔB (for S = A + B or A - B)
• Multiplication / division: add percentage (or fractional) uncertainties.
• Power n: multiply the percentage uncertainty by |n|.
• Multiplication by a constant: % uncertainty is unchanged.
• General: for Q = AlBmCn …, ΔQ/Q = |l|(ΔA/A) + |m|(ΔB/B) + |n|(ΔC/C) + …
Worked example — propagation
L = (1.250 ± 0.005) m → % unc. = 0.4%. d = (0.40 ± 0.01) mm → % unc. = 2.5%.
Area A = πd2/4. % unc. in A = 2 × 2.5 = 5.0 %. (π and 4 are constants — ignore.) g = 4π2L/T2. % unc. in g = % unc. in L + 2 × % unc. in T.
Concept check: measurements are models of reality
A unit states the scale used to compare a measurement; a dimension states how a quantity is built from base quantities. Dimensional analysis can reject an equation with inconsistent units, but it cannot prove an equation is physically correct because a missing numerical factor or wrong sign may still have the right dimensions.
Random uncertainty causes readings to scatter around a mean and is reduced by repeats. Systematic error shifts readings consistently and requires calibration, a zero correction or a better method. Quote uncertainty honestly: more decimal places do not make a measurement more accurate.
1.6 Checking Homogeneity and Making Estimates
Every physical quantity is a magnitude and a unit. Use base units to test an equation: both sides must have the same combination of kg, m, s, A and K. Prefixes from pico (10−12) to tera (1012) must be converted before substituting.
You must be able to make a reasonable order-of-magnitude estimate (mass of a person ~70 kg, walking speed ~1.5 m s−1, atmospheric pressure ~105 Pa, visible wavelength ~5×10−7 m).
1.7 Combining Uncertainties
Sums and differences: add absolute uncertainties.
Products, quotients and powers: add percentage (or fractional) uncertainties. For xn multiply the % uncertainty by |n|.
A zero error is systematic. Repeating and averaging reduces random error, not systematic error. Precise = small scatter; accurate = close to the true value.
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 + 1/2at2 v2 = u2 + 2as s = 1/2(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 a 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.
Concept check: choose the quantity the graph represents
Distance is total path length and is scalar; displacement is the directed change in position. Likewise speed is scalar while velocity is vector. A negative velocity does not mean an object is slowing down - it means it is moving in the chosen negative direction.
On a displacement-time graph, gradient is velocity. On a velocity-time graph, gradient is acceleration and signed area is displacement. SUVAT equations are only a compact form of constant-acceleration motion; use components independently for projectile motion and do not apply one SUVAT equation across a stage where acceleration changes.
2.7 Deriving the SUVAT Equations
From the definitions a = dv/dt and v = ds/dt with constant a:
v = u + at follows at once by integrating constant acceleration.
Displacement is the area under the v–t graph: a trapezium of sides u and v gives s = ½(u+v)t.
Eliminate t or v to obtain s = ut + ½at2 and v2 = u2 + 2as.
2.8 Experiment: Acceleration of Free Fall
Drop an object through a measured height from rest and time the fall (light gates or a trapdoor–electromagnet). From s = ½gt2, a graph of s against t2 is a straight line of gradient g/2. Repeat, use a small dense object to reduce drag, and include the uncertainty in t (which is squared).
1st Law (Inertia): a body remains at rest or moves with constant velocity unless acted on by a resultant force. If ΣF = 0, the body is in equilibrium. 2nd Law: the resultant force on a body equals its rate of change of momentum: F = Δp/Δt. For constant mass: F = ma. 1 N = 1 kg m s-2. 3rd Law (Action-Reaction): when body A exerts a force on body B, body B exerts an equal and opposite force on body A. The forces act on different bodies, are of the same type, and are equal in magnitude and opposite in direction.
N3 pitfall: the reaction to a book's weight (mg downwards) is the book's gravitational pull upwards on the Earth — NOT the normal contact force from the table. Weight and normal force act on the same body so cannot be an action-reaction pair.
3.2 Mass and Weight
Mass (m): scalar measure of inertia — resistance to change in motion. Unit: kg. Constant everywhere. Weight (W): gravitational force on a body: W = mg. Vector, unit: N. Varies with g. On Earth, g ≈ 9.81 N kg-1.
W = mg
3.3 Free-Body Diagrams (FBD)
Reduce an object to a point mass and draw all forces as labelled arrows from that point. Include: weight (downwards), normal contact (perpendicular to surface), friction (parallel to surface, opposing motion), tension (along string, away from body), drag (opposite to velocity).
Always draw forces on the body you are analysing. Do not include forces exerted by that body on other objects.
3.4 Drag Force
Drag opposes motion through a fluid. It depends on speed, shape, cross-sectional area and fluid density.
Falling through a fluid: Fnet = mg - f. Initially a = g; as speed increases, f increases, acceleration decreases. When f = mg, a = 0 and the body falls at terminal velocity.
3.5 Normal Contact Force and Friction
Normal contact force (R): perpendicular component of contact force between two surfaces. Arises from electrostatic repulsion between surface atoms.
Friction (f): component of contact force parallel to the surface, opposing relative motion.
• Static friction (fS): self-adjusting up to a limit; opposes tendency of motion.
• Limiting friction (flim): maximum static friction: flim = μR, where μ = coefficient of friction.
• Kinetic friction (fK): opposes actual sliding. For most surfaces fK ≈ flim = μR.
3.6 Inclined Slope
For a body of mass m on a slope inclined at angle θ to the horizontal:
Component parallel to slope: W∥ = mg sinθ
Component perpendicular to slope: W⊥ = mg cosθ
For a smooth (frictionless) slope: Fnet = mg sinθ = ma, so a = g sinθ. For a rough slope: Fnet = mg sinθ - μmg cosθ.
The normal reaction R = mg cosθ (not mg) on an inclined slope. Students often forget this.
3.7 Connected Bodies
For two masses mA and mB connected by a light inextensible string over a smooth pulley:
Atwood machine (both hanging)a = (mA - mB)g / (mA + mB)
TensionT = 2mAmBg / (mA + mB)
Mass on table + hanging massa = mg / (M + m)
Method: (1) Draw separate FBDs for each body. (2) Apply Fnet = ma to each. (3) Use the same a and T for both (light inextensible string).
3.8 Momentum
Momentum (p): product of mass and velocity: p = mv. Vector. Unit: kg m s-1 = N s.
The area under an F-t graph equals the change in momentum (= impulse). The gradient of a p-t graph gives the net force.
3.9 Conservation of Momentum
For a closed system with zero net external force, total momentum remains constant: m1u1 + m2u2 = m1v1 + m2v2.
For 2-D collisions, apply conservation independently to x and y components.
3.10 Elastic vs Inelastic Collisions
Type
Momentum
K.E.
Relative speed
Perfectly elastic
Conserved
Conserved
v2-v1 = u1-u2
Inelastic
Conserved
Not conserved
Reduced
Perfectly inelastic
Conserved
Max K.E. lost
Bodies stick together
Momentum is always conserved in collisions (no external force). Kinetic energy is only conserved in perfectly elastic collisions.
Concept check: always identify the system first
A force is an interaction, not a property carried by an object. Draw only forces acting on the chosen body. Newton's third-law pair acts on two different bodies, so it never cancels on one free-body diagram.
Friction and drag oppose relative motion or attempted relative motion, while a normal contact force is perpendicular to the surface. Terminal speed occurs when drag plus upthrust balances weight, so resultant force and acceleration become zero even though the object keeps moving at constant speed.
3.11 Elastic Collision Extra Condition
Momentum of an isolated system is always conserved. For a perfectly elastic collision, kinetic energy is also conserved and the relative speed of approach equals the relative speed of separation along the line of centres.
3.12 Two-Dimensional Collisions
Resolve into two perpendicular components. Momentum is conserved separately in each direction. Do not apply the 1-D relative-speed rule unless the collision is head-on. After the collision, recombine components to find speed and direction.
04Forces, Density and Pressure
Turning effects of forces, equilibrium conditions, centre of gravity, density, fluid pressure and Archimedes' upthrust.
4.1 Moment of a Force
Moment (torque, τ): the product of a force and the perpendicular distance from the pivot to its line of action: τ = F d⊥. Unit: N m. Vector direction: clockwise or anticlockwise.
τ = F d sin θ, where d sinθ = lever arm
If the force is not perpendicular to the lever, resolve it into components or find the perpendicular distance from the pivot to the force's line of action.
4.2 Couple and Torque
Couple: a pair of equal and opposite parallel forces whose lines of action do not coincide. Produces pure rotation with no net translational force.
τcouple = F L⊥
where L⊥ is the perpendicular distance between the two force lines. The torque of a couple is independent of the choice of pivot.
4.3 Mechanical Equilibrium
A rigid body is in equilibrium when both conditions are satisfied:
1. Resultant force = 0 in any direction: ΣF = 0 (translational equilibrium).
2. Resultant moment = 0 about any point: Στ = 0 (rotational equilibrium).
Principle of momentsΣ clockwise moments = Σ anticlockwise moments (about any pivot)
When solving equilibrium problems, choose a pivot that eliminates unknown forces (where lines of action intersect).
4.4 Centre of Gravity / Centre of Mass
Centre of gravity: the point at which the entire weight of the body may be considered to act. Centre of mass: the average position of all the mass in an object.
Near Earth's surface, centre of mass = centre of gravity. For uniform bodies: at the geometrical centre. Found experimentally by suspending from multiple points and tracing plumb-lines.
4.5 Forces in Equilibrium
Two forces: equal, opposite, same line of action.
Three forces: their vector sum forms a closed force triangle, and their lines of action must be concurrent (pass through a single point). Use the Law of Sines to find unknown magnitudes.
n forces: general condition: the force vectors form a closed polygon.
4.6 Density and Pressure
ρ = m/V (kg m-3); p = F/A (Pa = N m-2)
4.7 Hydrostatic Pressure
At depth h in a fluid of uniform density ρ, the pressure due to the fluid alone is p = ρgh. Total pressure = ρgh + Patm.
Δp = ρgΔh
Derivation: column of cross-section A, height h: weight = ρAhg, pressure on base = ρAhg/A = ρgh.
Manometer: U-tube with liquid (often mercury, ρHg = 1.36 × 104 kg m-3): P1 - P2 = ρgΔh.
Barometer: inverted mercury-filled tube in basin; Patm = ρgh.
4.8 Upthrust (Archimedes' Principle)
Upthrust (FU): the upward buoyant force on a body immersed (partially or fully) in a fluid. It equals the weight of the fluid displaced.
FU = ρfluidgVdisplaced
Origin: the pressure on the bottom face (at greater depth) exceeds the pressure on the top face. Net force is upward.
Floating/sinking:Fnet = W - FU = (ρbody - ρfluid)gV. Body sinks if ρbody > ρfluid; rises if ρbody < ρfluid; floats partially submerged when FU = W.
Concept check: equilibrium needs both force and turning balance
A body in equilibrium has zero resultant force and zero resultant moment. A moment is a turning effect, so its perpendicular distance is measured from the pivot to the line of action of the force - not simply to the point where the force is drawn.
Pressure is normal force per unit area; in a stationary liquid, pressure difference depends on vertical depth because lower fluid supports the weight of liquid above. Upthrust is the resultant of the larger pressure on the bottom surface than on the top, and equals the weight of fluid displaced.
05Work, Energy and Power
Mechanical work, kinetic and potential energy, energy conservation, efficiency and power.
5.1 Work Done by a Force
Work (W): the product of a force and the displacement of its point of application in the direction of the force. W = F s cos θ. Scalar. Unit: joule (J); 1 J = 1 N m.
W = F s cos θ
When the force is perpendicular to the motion (θ = 90°), no work is done. Example: centripetal force does zero work. For a varying force, work = area under an F-s graph. For a gas expanding at constant pressure: W = pΔV.
5.2 Types of Energy
Kinetic energy: energy due to motion: Ek = ½mv2.
Derivation: W = F s = ma s. With v2 = u2 + 2as, W = ½m(v2 - u2) = ΔEk. Gravitational potential energy: energy due to position in a gravitational field: ΔEp = mgΔh. GPE is relative — only changes matter. Elastic potential energy (strain energy): energy stored in a deformed elastic body: Ep = ½kx2 = ½Fx.
5.3 Work–Energy Theorem
Work–energy theorem: the net work done on a body equals its change in kinetic energy: Wtotal = ΔEk = ½mv2 - ½mu2.
5.4 Conservative and Non-Conservative Forces
Conservative force: the work done is independent of the path taken and depends only on the initial and final positions. Examples: gravity, electric force, spring force. For a conservative force, ΔEp = -Wc. Non-conservative force: the work done depends on the path taken. Examples: friction, drag, air resistance. The work done by non-conservative forces equals the change in total mechanical energy: Wnc = ΔEm.
5.5 Conservation of Energy
Energy cannot be created or destroyed; it can only be transferred between different forms. In the absence of non-conservative forces, mechanical energy is conserved: Ek + Ep = constant.
5.6 Power
Power (P): the rate of doing work: P = ΔW/Δt. Unit: watt (W); 1 W = 1 J s-1.
P = W/t = F v (for constant force and velocity along same line)
P = F v is used for vehicles at constant speed against resistive forces. At maximum speed, driving force = resistive force.
5.7 Efficiency
Efficiency (η): the ratio of useful output to total input, often expressed as a percentage.
η = (useful energy output / total energy input) = (useful power output / total power input)
η is always < 1 (or < 100 %) due to energy dissipated as heat, sound, etc.
Concept check: energy is transferred, not used up
Work is energy transferred when a force has a component along a displacement. The work-energy theorem connects resultant work to change in kinetic energy. Conservation of energy still holds when a system slows due to friction; organised kinetic energy is transferred mainly to less useful thermal energy in the object and surroundings.
Power is the rate of energy transfer. Efficiency compares useful output with total input and has no unit; state clearly what counts as useful in the context. A force perpendicular to motion does no work, even if the force itself is large.
5.8 Useful Derivations
Power:P = W/t = F × (s/t) = Fv when the force is in the direction of the velocity.
Gravitational PE near Earth: work done lifting at constant speed is W = mg × Δh, so ΔEP = mgΔh.
Kinetic energy: a resultant force F = ma does work Fs. Using v2 = u2 + 2as with u = 0 gives EK = ½mv2.
06Deformation of Solids
Hooke's law, spring combinations, stress, strain, Young modulus, stress–strain curves, elastic limit and elastic potential energy.
6.1 Elasticity
Elastic material: returns to its original shape when the deforming force is removed. Plastic (inelastic) material: retains permanent deformation after the force is removed.
6.2 Hooke's Law
Hooke's law: the extension of an ideal spring is proportional to the applied force, up to the limit of proportionality. F = kx. The spring constant k (force constant) measures stiffness: larger k = stiffer spring. Unit: N m-1.
F = kx
On a F-x graph: the straight-line region obeys Hooke's law. Beyond the limit of proportionality, the graph curves. Beyond the elastic limit, the spring does not return to its original length on unloading.
6.3 Spring Combinations
Parallel springsk = k1 + k2 + … (same extension for each spring)
Series springs1/k = 1/k1 + 1/k2 + … (same force through each spring)
6.4 Elastic Potential Energy (Strain Energy)
Ep = ½Fx = ½kx2
The area under a F-x graph up to extension x equals the elastic potential energy stored. For Hookean springs: triangular area = ½ × base × height = ½kx2.
6.5 Stress, Strain, Young Modulus
Tensile stress (σ): force per unit cross-sectional area: σ = F/A. Unit: Pa (N m-2). Tensile strain (ε): extension per unit original length: ε = x/L (unit-free). Young modulus (E): ratio of tensile stress to tensile strain within the limit of proportionality: E = σ/ε. Unit: Pa. It is a property of the material — independent of the dimensions of the sample.
E = σ / ε = (F/A) / (x/L) = FL/(Ax)
For metals: Emetal ~ 1011 Pa. Related to spring constant: k = EA/L.
6.6 Stress–Strain Curve for a Metal Wire
Key points on a typical stress–strain curve for a ductile metal (e.g. copper):
Limit of proportionality (LP): end of the straight-line (linear) region. Beyond LP, Hooke's law (E = σ/ε) no longer holds. Elastic limit (LE): up to this point the wire returns to its original length when unloaded. Beyond LE, permanent deformation occurs (plastic). Usually LP ≈ LE for metals. Yield point: sudden extension without increased load. The wire yields. Ultimate tensile stress (UTS): the maximum stress the material can withstand before it fractures. Corresponds to the highest point on the curve. Breaking stress (fracture): the stress at which the material breaks. Occurs after necking.
Loading/unloading hysteresis: after plastic deformation, the unloading path is parallel to the original elastic line. The area enclosed between loading and unloading curves = energy dissipated as heat.
6.7 Ductile vs Brittle Materials
Ductile (e.g. copper, mild steel): large plastic region, significant necking before fracture. Brittle (e.g. glass, cast iron): no plastic region — fractures at the elastic limit.
Polymeric materials (e.g. rubber): very different stress–strain behaviour. Large strain for small stress; does not obey Hooke's law except at very small extensions. Shows hysteresis when loaded and unloaded.
6.8 Experiment — Measuring Young Modulus
Use a long thin wire (≥ 2 m), clamped securely at the top.
Measure the original length L with a metre rule.
Measure the diameter d with a micrometer screw gauge at several points along the wire → average → \( A = \dfrac{\pi d^2}{4} \).
Apply increasing loads, measure extension x each time (use a vernier scale or travelling microscope).
Plot F vs x; gradient = \( \dfrac{EA}{L} \) → \( E = \dfrac{\text{gradient} \times L}{A} \).
Safety: wear goggles; place a soft landing pad beneath the load; do not exceed the elastic limit.
Concept check: elastic does not mean perfectly proportional
An object is elastic if it returns to its original shape when the load is removed. Hooke's law is stricter: extension must be proportional to force, and applies only up to the limit of proportionality. The area under a force-extension graph is work done and hence elastic potential energy.
Stress compares force with cross-sectional area; strain compares extension with original length, so strain has no unit. Young modulus is a material property: a large value means a material is stiff, not necessarily strong or tough.
07Waves
Wave basics, progressive waves, intensity, transverse/longitudinal, Doppler, EM spectrum and polarisation.
7.1 Key Quantities
Displacement = distance from equilibrium position. Amplitude = maximum displacement. PeriodT = time for one complete oscillation. Frequencyf = 1/T = number of oscillations per second. Wavelength = distance travelled by a wave in one period, or distance between neighbouring points in phase.
7.2 Wave Equation
v = fλ = λ/T
Crossing media: v and λ change; f stays the same.
7.3 Intensity
I = P/A; I ∝ A2; point source: I ∝ 1/r2
7.4 Transverse vs Longitudinal
Transverse
Longitudinal
Oscillation ⊥ propagation
Oscillation ∥ propagation
Crests & troughs
Compressions & rarefactions
Can be polarised
Cannot be polarised
Mechanical waves need a medium, e.g. sound and waves on a string. Electromagnetic waves are oscillations of electric and magnetic fields and can travel through vacuum.
1. Transverse Wave
Displacement is perpendicular to the direction of wave travel. Particles oscillate up and down; energy moves horizontally.
Wave particles
Tracked particle
Amplitude
Wavelength
0.50 Hz
20 px
100 px/s
λ = 200 pxT = 2.00 s
Key idea: The medium moves up and down; the wave energy moves left to right. No net displacement of particles.
2. Longitudinal Wave — Sound
Displacement is parallel to the direction of wave travel. Compressions (high pressure) and rarefactions (low pressure) alternate.
Air particles
Tracked particle
Compression (high p)
Rarefaction (low p)
0.60 Hz
15 px
96 px/s
λ = 160 pxT = 1.67 s
Key idea: In a sound wave, particles oscillate back and forth about their equilibrium positions. The pressure variations are what our ears detect.
7.5 Doppler Effect (moving source)
fobs = fsv/(v ± vs)
Relative motion changes the observed frequency and wavelength. Use minus in the denominator when the source approaches; use plus when it moves away.
The effect applies to sound and electromagnetic waves. Approaching source or observer: observed frequency increases and wavelength decreases; separating source or observer: observed frequency decreases and wavelength increases.
7.6 EM Spectrum
All electromagnetic waves are transverse and travel at c = 3.00 × 108 m s-1 in vacuum. Order: Radio → Microwave → Infrared → Visible (red about 700 nm, violet about 400 nm) → UV → X-ray → γ.
7.7 Polarisation
Only transverse waves can be polarised. A polaroid transmits vibrations parallel to its transmission axis.
Unpolarised light contains vibrations in many transverse directions. After one polaroid, the emergent light is plane-polarised in the direction of the transmission axis.
Unpolarised through one polaroidI = I0/2
Malus's LawI = I0 cos2θ
7.8 CRO
Time-base × horizontal divisions per cycle = period. Y-gain × vertical divisions to peak = amplitude. For stationary-wave measurements, adjacent nodes are separated by λ/2.
Concept check: a wave transports a disturbance and energy
Particles in a medium oscillate about equilibrium; they do not travel with a progressive wave overall. Frequency is fixed by the source, while wave speed depends on the medium, so wavelength changes when a wave enters a medium with a different speed.
Intensity is power per unit area and is proportional to amplitude squared. Only transverse waves can be polarised because polarisation restricts oscillations to one plane; longitudinal sound waves cannot be polarised.
7.9 Doppler Formula (moving source)
fo = fsv / (v ± vs)
Use + in the denominator when the source moves away (lower observed frequency) and − when it moves towards the observer. The syllabus does not require a moving observer.
7.10 Malus's Law
Polarisation occurs only for transverse waves. After a polariser, intensity through an analyser at angle θ is
I = I0 cos2θ
Do not apply this formula to unpolarised incident light (that case is not required).
08Superposition
Principle of superposition, stationary waves, diffraction, interference, double-slit and diffraction grating.
8.1 Principle of Superposition
When two or more waves meet, the resultant displacement = vector sum of their individual displacements.
3. Superposition Principle
When two waves meet, the resultant displacement is the vector sum of the individual displacements.
Formed by two progressive waves of same frequency travelling in opposite directions.
Nodes: zero displacement. Antinodes: maximum displacement. Adjacent nodes = λ/2 apart. No net energy transfer.
For sound-speed experiments, adjacent nodes can be located with a CRO and microphone, or by sand / foam piling up at nodes in a tube. Measure the node-to-node separation, use λ = 2 × separation, then calculate v = fλ.
Fixed or closed end → node.
Free or open end → antinode.
Two fixed ends or two open ends: allowed frequencies are f1, 2f1, 3f1, ...
One closed end and one open end: allowed frequencies are f1, 3f1, 5f1, ...
4. Standing (Stationary) Wave
Two progressive waves of the same frequency travelling in opposite directions superpose. Nodes (zero displacement) and antinodes (maximum displacement) form.
Particles
Antinode (max amplitude)
Node (zero displacement)
0.50 Hz
35 px
Nodes: 5Antinodes: 4
Nodes (N) are points of zero displacement. Antinodes (A) are points of maximum amplitude. For a string of length L fixed at both ends: L = n·λ/2.
8.3 Diffraction
Spreading of a wave through a gap or around an obstacle. Most pronounced when gap ≈ λ.
Diffraction increases when wavelength increases, or when the gap / obstacle size becomes smaller.
8.4 Interference
Coherence: same frequency, constant phase difference.
Stable interference requires coherent sources. In practice, water-wave dippers may be driven by the same vibrator, sound sources by the same signal generator, and light sources by the same slit / double-slit system; two separate lamps are not coherent.
Path diff = nλ → constructive (max).
Path diff = (n+1/2)λ → destructive (min).
Time diff = nT or phase diff = 2nπ → constructive.
Time diff = (n+1/2)T or phase diff = (2n+1)π → destructive.
Δt/T = ΔL/λ = Δφ/(2π)
8.5 Young's Double Slit
x = λD/a or λ = ax/D (D ≫ a)
Bright fringes form where path difference = nλ. Changing slit width changes brightness, but not fringe spacing.
5. Double-Slit Interference
Two coherent sources produce an interference pattern. Bright fringes (constructive) and dark fringes (destructive) appear on a screen.
Wavefronts from slit 1
Wavefronts from slit 2
Constructive (bright)
Destructive (dark)
30 px
90 px
400 px
a/λ = 3.0Fringe spacing w = λD/a
Fringe spacing:w = λD/a. Bright fringes occur where path difference = nλ; dark fringes where path difference = (n + 1/2)λ.
8.6 Diffraction Grating
d sinθ = nλ
If there are N slits per metre, then d = 1/N. Highest possible order is nmax = floor(d/λ), so total number of principal maxima is 2nmax + 1.
With white light, the zero-order maximum is white, while first and higher orders are dispersed into colours; red appears at larger angles than violet.
Higher orders are more spread out. Maxima from different orders can overlap when different wavelengths satisfy n1λ1 = n2λ2 at the same angle.
6. Diffraction Grating
Multiple equally spaced slits produce sharp principal maxima. The grating equation: d sin θ = nλ. More slits → narrower, brighter peaks.
Principal maxima
Secondary maxima
Minima
30 px
120 px
5
Max orders n: 4Angular resolution ∝ 1/N
Grating equation:d sin θ = nλ. Principal maxima become sharper as N increases. Missing orders occur when a diffraction minimum coincides with an interference maximum.
Concept check: superposition is addition of displacement
When waves overlap, the resultant displacement is the algebraic sum of their individual displacements. Constructive interference occurs when waves arrive in phase; destructive interference occurs when they arrive in antiphase. Coherent sources must have the same frequency and a constant phase difference.
A stationary wave is made by two equal progressive waves travelling in opposite directions. It transfers no net energy along the medium and has fixed nodes and antinodes. Diffraction becomes significant when a gap or obstacle is comparable in size with the wavelength.
8.7 Stationary-Wave Experiments
Stretched string (sonometer / vibration generator): nodes at the fixed ends. λ = 2L/n. Speed from v = fλ.
Air column (resonance tube): closed end is a node, open end an antinode. First resonance at L = λ/4 (end correction is neglected).
Microwaves: a metal plate reflects the wave; a probe finds nodes and antinodes. Node-to-node distance is λ/2.
8.8 Conditions for Two-Source Interference
The sources must be coherent (constant phase difference, same frequency) and of comparable amplitude. For light this usually means one source split into two paths. Fringe spacing x = λD/a.
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
NTC thermistor
R falls as T rises
9.6 Power
P = VI = I2R = V2/R
9.7 Resistivity
R = ρL/A
9.8 LDR and Thermistor
LDR: R ↓ as light ↑. NTC thermistor: R ↓ as T ↑.
Concept check: current is charge flow, not energy flow
Current is the rate at which charge passes a point; conventional current is defined as the direction positive charge would move. In metals, electrons drift slowly opposite to conventional current while the electric field and energy transfer are established through the circuit much more quickly.
Potential difference is energy transferred per unit charge between two points. Emf is energy supplied per unit charge by a source. Resistance is the ratio of p.d. to current only for an ohmic conductor at constant temperature; a non-ohmic I-V curve needs a microscopic explanation involving temperature, carriers or a potential barrier.
9.9 Charge is Quantised
Charge comes in integer multiples of e = 1.60×10−19 C, so Q = Ne. Current is a flow of these carriers: I = Anvq.
9.10 Why a Filament Lamp is Non-Ohmic
Larger current heats the filament. Higher temperature increases the amplitude of ion vibrations, so the drift of electrons is hindered and resistance rises. The I–V graph therefore curves towards the V axis.
Vterminal vs I: y-intercept = ε, gradient = -r. When R → 0, the short-circuit current is Imax = ε/r.
10.2 Kirchhoff's Laws
1st (junction): ΣI in = ΣI out (charge conservation). 2nd (loop): Σε = ΣIR (energy conservation).
10.3 Series and Parallel
SeriesR = R1 + R2 + ...
Parallel1/R = 1/R1 + 1/R2 + ...
10.4 Potential Divider
V1/V2 = R1/R2
V1 = R1Vin/(R1 + R2), V2 = R2Vin/(R1 + R2)
Potential at a point is measured relative to a chosen zero-potential reference, often earth. Sensor circuits use this because changing one resistance changes the output pd.
10.5 Bridge Circuits
For the two divider branches, potential at the midpoints are VX = R2ε/(R1+R2) and VY = R4ε/(R3+R4).
VXY = VX - VY
A bridge is balanced when VXY = 0, so VX = VY and R1/R2 = R3/R4. This is useful for finding unknown resistances accurately.
10.6 Potentiometer
Long uniform wire; null method — no current at balance, so internal resistance of the test cell does not affect the reading.
Etest/Edrive = AP/AB
The emf or pd being tested is proportional to the balancing length along the wire.
The driver cell should have a known emf larger than the test emf, and its internal resistance should be negligible compared with the resistance wire. At null balance the galvanometer reads zero, so no current is drawn from the test cell.
10.7 Electrical Power in Practical Circuits
Ptotal = Iε = I2(R+r)
Pload = IVR = I2R
Pinternal = IVr = I2r
Pout = ε2R/(R+r)2
Power delivered to the external load is maximum when R = r, giving Pmax = ε2/(4r).
Concept check: Kirchhoff's laws are conservation laws
Kirchhoff's first law is conservation of charge: charge cannot accumulate at a junction in a steady circuit. Kirchhoff's second law is conservation of energy: the total energy supplied per unit charge around a complete loop equals the total energy transferred per unit charge.
Internal resistance represents energy transferred inside a source. When current increases, the lost volts Ir increase, so terminal p.d. falls. A potential divider is useful because the output p.d. is a fraction of the supply determined by the resistance ratio; changing a sensor resistance changes that fraction.
10.8 Terminal p.d. and Internal Resistance
e.m.f. is energy transferred per unit charge around the whole circuit. Terminal p.d. is energy per unit charge across the external load only. V = ε − Ir, so V falls as current rises. A graph of V against I has intercept ε and gradient −r.
10.9 Null Methods
A potentiometer compares p.d.s. At the balance (null) length the galvanometer reads zero, so no current is drawn from the unknown — the comparison is made at infinite resistance. Thermistors and LDRs in a potential divider make the output p.d. depend on temperature or light.
11Particle Physics
Nuclear atom, radioactivity, beta decay, quarks, leptons, antiparticles and the Standard Model basics required for AS.
11.1 Nuclear Atom
Tiny dense positive nucleus (protons + neutrons) surrounded by electrons. Evidence: Rutherford α-scattering (most pass through; few deflected; very few bounce back).
11.2 Notation
AZX: A = nucleon number, Z = proton number.
Isotopes: same Z, different A.
11.3 Fundamental Particles
Fundamental particles have no known internal structure. In the AS course, the required matter particles are quarks and leptons.
Every particle has an antiparticle with the same mass and opposite charge, e.g. electron / positron, proton / antiproton.
For 9702 AS, focus on matter and antimatter particles, quarks, leptons and hadrons. Detailed force-carrier particles and the Higgs boson are not required.
11.4 Radioactive Emissions
Type
Identity
Charge
Mass (u)
Penetration
α
42He nucleus
+2e
≈ 4
Few cm air, stopped by paper
β-
Fast electron
-e
≈ 0
~1 m air, few mm Al
β+
Positron
+e
≈ 0
Annihilates with electron → 2γ
γ
EM photon
0
0
Reduced by Pb/concrete
11.5 Beta Decay
β- decay: a neutron changes into a proton, electron and electron antineutrino.
10n → 11p + 0-1e + 00νe
d → u + e- + νe
β+ decay: a proton changes into a neutron, positron and electron neutrino.
11p → 10n + 0+1e + 00νe
u → d + e+ + νe
11.6 Quark Model
Six quark flavours: u, d, s, c, t, b. Charges: u, c, t have +2/3e; d, s, b have -1/3e.
Quarks are affected by the strong force and cannot be isolated individually.
Leptons come in six types: e, μ, τ, νe, νμ, ντ. Charged leptons have charge -e; neutrinos are neutral and interact only through the weak interaction (and gravity).
Strong force acts on quarks and hadrons, not leptons; it is short-ranged but very strong. Weak force acts on both quarks and leptons and is responsible for beta decay.
The strong interaction binds quarks into hadrons and holds hadrons close together inside nuclei; its range is about 10-15 m. The weak interaction has a still shorter range, about 10-17 m, and can change quark flavour during beta decay.
11.8 Useful Constants
1 u = 1.66 × 10-27 kg ≈ proton mass; e = 1.60 × 10-19 C; 1 eV = 1.60 × 10-19 J.
In every nuclear or particle equation, conserve nucleon number, charge, lepton number and energy. An emitted beta particle alone cannot account for the observed energy and momentum distribution, which is why a neutrino or antineutrino is included.
Do not confuse a nucleus with an atom: an atom includes electrons, whereas nuclear notation counts protons and neutrons only. Radioactive emission is random for one nucleus; the predictable behaviour of a sample appears only when very many nuclei are considered.
11.9 Discrete α Energies and Continuous β Spectra
α-particles from a given decay have discrete kinetic energies (two-body decay to a definite daughter). β-particles share energy with an (anti)neutrino, so their spectrum is continuous up to a maximum. β− produces an electron antineutrino; β+ produces an electron neutrino.
11.10 Writing Decay Equations
Nucleon number and charge are conserved. Example: 238U → 234Th + 4He. Use the unified atomic mass unit u when masses are given.
These notes are AI-assisted study material. Always cross-check against the official 9702 syllabus or your teacher before relying on them in an exam.
9702 A2 Level Physics
Complete A2 Notes · Cambridge International A Level (2025–2027)
Uniform circular motion: angular speed, centripetal acceleration and the net force that maintains it.
12.1 The Radian
Physical meaning: a radian is a dimensionless angle measure defined by arc length divided by radius. The relation s = rθ works only when θ is in radians, which is why radians are essential rather than optional in circular-motion equations.
Radian: the angle subtended at the centre of a circle by an arc equal in length to the radius. 2π rad = 360°.
Arc length s = rθ (where θ is in radians).
12.2 Angular Speed
Link to linear motion: all points on a rigid rotating object have the same angular speed, but a point farther from the axis travels a greater distance in the same time and therefore has a greater linear speed. One rotation is 2π rad, so period and frequency connect directly to angular speed.
ω = θ/t = 2π/T = 2πf (rad s-1)
Linear speedv = rω
For uniform circular motion, the speed is constant but the velocity is constantly changing direction — hence there is an acceleration.
12.3 Centripetal Acceleration and Force
Direction first: draw the instantaneous tangent velocity and then the inward acceleration before writing equations. The inward acceleration follows from the vector change in velocity, not from a drop in speed. If the real inward force disappears, the object travels tangentially, not radially outward.
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.
Concept check: why an object accelerates at constant speed
Velocity includes direction. In uniform circular motion, the object covers equal arc lengths in equal times, so its speed is constant; however, the direction of its velocity changes continuously. The change in velocity points towards the centre, so both acceleration and resultant force point towards the centre.
At any instant, velocity is tangent to the circle. The centripetal force is perpendicular to that velocity, so in ideal uniform circular motion it changes direction but does no work; it does not increase the kinetic energy. If a question names several real forces, first resolve them and identify their inward resultant - only that resultant equals mv2/r.
12.4 Using the Circular-Motion Equations
A force of constant magnitude, always perpendicular to the velocity, changes direction but not speed. Then a = v2/r = rω2 and F = mv2/r = mrω2, with ω = 2π/T and v = rω. The resultant towards the centre may be a single force or the difference of two (e.g. tension and weight at the top of a vertical circle).
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.
For a uniform spherical shell, the external field is as if all mass were at the centre; inside the shell, the resultant gravitational field is zero.
Newton's law of gravitation: every mass attracts every other mass. The force on each mass has the same magnitude, acts along the line joining their centres, and the two forces are an action-reaction pair acting on different objects.
What every symbol means
Symbol
Meaning
Exam condition
m1, m2
The two interacting masses, in kg
Use the mass of the source and the mass experiencing the force; do not replace either with weight.
r
Separation of the two mass centres, in m
For a satellite, r is planet radius + altitude, not altitude alone.
G
Universal gravitational constant
It is not g. G is the same everywhere; g changes with location.
Inverse-square meaning: if the centre-to-centre separation doubles, the force becomes one quarter; if it triples, the force becomes one ninth. This is why a planet's pull becomes weaker with altitude but never suddenly disappears.
Model boundary: use this equation directly for point masses and for spherically symmetric objects when the point is outside them. It does not say that all gravity inside an arbitrary solid sphere is zero; the zero-result statement is specifically for a uniform spherical shell.
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.
A gravitational field is the region in which another mass experiences gravitational force. Field strength describes the field created by the source mass: it is not a property of the test mass placed there. Since force and mass are both involved in F = mg, the test mass cancels when calculating g.
g is a vector: at every point it points towards the source mass. The unit N kg-1 is exactly equivalent to m s-2; a freely falling object has acceleration a = g because its resultant force is mg, so ma = mg.
Field-line language
Arrows show the force direction on a small positive test mass (towards the mass for gravity). Lines closer together mean a stronger field. Far from a spherical planet the field is radial; near a small patch of its surface the lines are nearly parallel, so the field is approximately uniform.
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.
Potential and potential energy: the physical meaning
Potential is a scalar energy quantity per kilogram. It is useful because potentials from several source masses add algebraically: φtotal = φ1 + φ2 + ... . Field strengths must instead be added as vectors, so their directions matter.
The zero at infinity is a chosen reference. At any finite distance, the potential is negative because an external agent must supply energy to take a mass from that point all the way to infinity against an attractive field. Moving outward therefore makes φ increase (become less negative).
Do not confuse the two equations:ΔEp = mgΔh works only where g is approximately constant. For a large change in distance from a planet or star, use Ep = -GMm/r, then calculate the change.
Why the gradient has a minus sign
The negative sign in g = -dφ/dr is physical, not cosmetic. Gravitational potential increases as you move away from the mass, whereas the gravitational field points back towards the mass. A steep potential-distance graph therefore represents a strong field; as the graph flattens far away, the field becomes weak.
Equipotential surfaces join points with the same potential. Moving a mass along an equipotential requires no work by or against gravity because there is no change in potential energy. Field lines cross equipotentials at right angles.
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
An orbit is not a place with no gravity. At orbital altitude, gravity is still the resultant inward force. The satellite continually falls towards the planet, but its sideways speed means the planet's surface curves away beneath it. The word centripetal describes the direction of the resultant force; it is not an extra force to add to gravity.
For the same central mass, a larger orbit has a lower orbital speed but a longer period. The satellite's mass cancels from the equations: changing satellite mass does not change the radius, speed or period of a specified circular orbit.
Energy extension: in a circular orbit, Ek = GMm/(2r), Ep = -GMm/r, so total energy is negative. Removing orbital energy by drag makes the orbit shrink and the orbital speed increase.
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.
All three conditions are required: the orbital plane must be the equatorial plane, the direction must match Earth's rotation, and the period must equal Earth's rotational period. A satellite with a 24-hour period but an inclined orbit is geosynchronous, not geostationary, because it moves north and south in the sky for a ground observer.
One fixed ground aerial can continuously communicate with a geostationary satellite, which is the key advantage for broadcasting. The trade-offs are the large signal delay caused by the long path, weak coverage near the poles, and the need for a high orbit and powerful launch.
14Temperature
The thermodynamic temperature scale, thermometric properties, and the energy transferred during heating and phase changes.
14.1 Thermal Equilibrium
Meaning: thermal equilibrium is a dynamic condition: microscopic energy transfers still occur in both directions, but their rates are equal so there is no net energy transfer. Temperature is the property that determines the direction of spontaneous thermal transfer.
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
Why calibration matters: a thermometer does not measure temperature directly; it measures a chosen property and maps it to a temperature scale. A useful thermometric property must vary reproducibly, measurably and preferably almost linearly over the required range.
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
Why use kelvin: the Kelvin scale starts at absolute zero and is proportional to microscopic kinetic-energy measures. This makes it the only valid temperature scale in gas laws and thermodynamic formulae; a temperature change of 1 K equals a change of 1 degree Celsius.
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
Interpretation: a high specific heat capacity means a substance requires a large energy transfer for a given mass and temperature rise. In a practical experiment, energy supplied electrically is not always equal to energy gained by the sample because heating the container and losses to surroundings must be considered.
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
During melting or boiling: temperature remains constant for a pure substance at constant pressure even while energy is supplied. The energy changes intermolecular separation and therefore potential energy; it does not raise the average random kinetic energy.
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 is needed 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.
Concept check: heating is not the same as raising temperature
Temperature is linked to the average random kinetic energy of particles; internal energy is the total random kinetic and potential energy of all particles. Heating transfers energy because of a temperature difference, but it may raise temperature, change state, or do both in stages.
Specific heat capacity tells how much energy is needed for a temperature rise with no state change. Specific latent heat tells how much energy is needed to change state with no temperature change: energy is used to separate or form intermolecular bonds, changing potential energy rather than average kinetic energy.
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
Scale bridge: the mole links an uncountably large number of particles to a measurable macroscopic mass. One mole contains exactly the Avogadro constant of specified entities, so always state whether the entities are atoms, molecules, ions or electrons.
One mole contains NA = 6.02 × 1023 particles. n = N/NA = mass/molar mass.
15.2 Equation of State for an Ideal Gas
State variables: pressure, volume and thermodynamic temperature specify the equilibrium state of a fixed amount of ideal gas. The equation does not describe how quickly a gas reaches that state; convert pressure to Pa, volume to m3 and temperature to K before substitution.
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
Model limitation: the ideal model becomes less accurate at high pressure or low temperature, where molecular volume and attractive forces can no longer be ignored. It is still powerful because its simplified assumptions predict the observed gas laws in ordinary conditions.
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
Microscopic interpretation: pressure depends on molecular number density and mean-square speed, not on one molecule's speed. Mean-square speed is used because molecular speeds vary; the root-mean-square speed is a useful single speed related to the gas temperature.
pV = 1/3Nm<c2>
where <c2> is the mean-square speed of the molecules. Comparing with pV = NkT:
Mean translational KE per molecule1/2m<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).
Concept check: what makes a gas ideal
The ideal-gas model treats molecules as particles of negligible volume, moving randomly, with negligible intermolecular forces except during perfectly elastic collisions. Pressure is the rate of momentum transfer when molecules collide with container walls.
At a fixed temperature, all gases have the same mean kinetic energy per molecule. Raising temperature increases molecular speed and collision momentum; at fixed volume this increases pressure. Use kelvin because gas-law relationships are proportional to absolute, not Celsius, temperature.
15.5 Kinetic-Theory Pressure and Mean Energy
pV = ⅓ Nm<c2>
One-dimensional collisions with a wall give pA = Nm<cx2>/L; extending to three equal dimensions replaces <cx2> by ⅓<c2>. The r.m.s. speed is √<c2>.
Compare with pV = NkT to get the mean translational KE of a molecule:
½ m<c2> = 3⁄2kT and k = R/NA
16Thermodynamics
Internal energy, the first law of thermodynamics, work done by/on a gas, and the distinction between temperature, heat and internal energy.
16.1 Temperature Scales and Thermal Equilibrium
Core distinction: temperature predicts the direction of net thermal energy transfer, whereas thermal energy/internal energy depends also on how much substance is present. A bathtub of warm water can contain more internal energy than a hot cup of water.
Thermal equilibrium: two bodies are in thermal equilibrium when there is no net flow of thermal energy between them — their temperatures are equal. Zeroth law of thermodynamics: if A and B are each in thermal equilibrium with C, then A and B are in thermal equilibrium with each other. Kelvin scale:T(K) = θ(°C) + 273.15. 0 K = -273 °C = absolute zero — the lowest possible temperature where particles have minimum kinetic energy.
The thermodynamic (Kelvin) scale is independent of the properties of any particular substance. It is defined using the triple point of water.
16.2 Specific Heat Capacity and Specific Latent Heat
Energy-accounting method: first identify whether the substance changes temperature, changes state, or does both in separate stages. Use mcΔT only on the sloping parts of a heating curve and mL only on the flat phase-change parts.
Specific heat capacity (c): the energy required per unit mass to raise the temperature by 1 K (or 1 °C), with no change of state. Unit: J kg-1 K-1. Specific latent heat (L): the energy required per unit mass to change the state of a substance at constant temperature. Unit: J kg-1. Lf = fusion (melting/freezing); Lv = vaporisation (boiling/condensing).
Q = mcΔT (heating/cooling); Q = mL (phase change at constant temp.)
During a phase change, the temperature stays constant because all energy supplied goes into breaking intermolecular bonds (increasing PE), not into increasing KE. On a heating curve, this appears as a horizontal plateau.
Worked example
Convert 0.50 kg of ice at -10 °C to water at 25 °C: cice = 2100, Lf = 3.34 × 105, cwater = 4200.
Warm ice: Q1 = 0.50 × 2100 × 10 = 1.05 × 104 J.
Melt: Q2 = 0.50 × 3.34 × 105 = 1.67 × 105 J.
Warm water: Q3 = 0.50 × 4200 × 25 = 5.25 × 104 J.
Total ≈ 2.30 × 105 J.
16.3 Internal Energy
Particle model: internal energy is the sum of random kinetic energy and intermolecular potential energy. For an ideal gas, intermolecular forces are negligible, so internal energy depends only on temperature; for real substances it also changes when the spacing between particles changes.
Internal energy (U): the sum of the random kinetic energy of all molecules (due to their motion) and the potential energy between molecules (due to intermolecular forces). U is a state function — it depends only on the current state of the system, not on how it got there.
Ideal gas: intermolecular PE = 0 (no forces between molecules) → U = total molecular KE = (3/2)NkT = (3/2)nRT. Depends on temperature only.
Real substance:U = Σ(random KE) + Σ(molecular PE). KE rises with T; PE changes during phase changes and when intermolecular separations change.
Temperature ≠ internal energy: Temperature measures the average KE per molecule, not the total internal energy. A swimming pool at 30 °C has far more internal energy than a cup of boiling water at 100 °C, because the pool has vastly more molecules.
16.4 The First Law of Thermodynamics
Sign discipline: state the convention being used before calculating. With the convention ΔU = Q + W, Q is positive when energy is transferred to the gas by heating and W is positive when work is done on the gas.
First law: the increase in internal energy of a system equals the sum of the heat supplied to the system and the work done on the system: ΔU = Q + W. This is a statement of energy conservation for thermodynamic systems.
ΔU = Q + W
Symbol
Meaning
+ (positive)
- (negative)
ΔU
Change in internal energy
U increases
U decreases
Q
Heat transferred
Supplied to system
Lost from system
W
Work done
Done on system (compression)
Done by system (expansion)
16.5 Work Done by a Gas
Area means work: on a p-V graph, work done by a gas in expansion is the area under its path because the gas exerts force over a changing distance. Compression reverses the sign: work is done on the gas, increasing its internal energy unless energy leaves by heating.
When a gas expands by ΔV against a constant external pressure p:
Work done BY gasWby gas = pΔV
So work done on the gas: W = -pΔV. On a p–V diagram, the area under the curve equals the magnitude of work done by (or on) the gas.
16.6 Special Processes
Read the constraint: isothermal means constant temperature, isobaric means constant pressure, isochoric means constant volume, and adiabatic means no heating. The same pressure-volume endpoints can be connected by different paths with different work and heat transfers.
Process
Condition
ΔU
Key point
Isothermal
ΔT = 0
ΔU = 0
Q = -W; all heat in/out = work done by/on gas
Adiabatic
Q = 0
ΔU = W
Compression → heats up; expansion → cools down
Isovolumetric (isochoric)
ΔV = 0
ΔU = Q
No work done; all heat changes U
Isobaric
Δp = 0
ΔU = Q - pΔV
Work done = pΔV at constant pressure
Isothermal vs adiabatic compression
Isothermal (slow, good thermal contact): ΔT = 0, ΔU = 0. Work done ON gas = heat OUT: Q = -W. Pressure rises less steeply (p ∝ 1/V). Adiabatic (rapid, insulated):Q = 0. Work done ON gas → ΔU = W > 0 → temperature rises. Pressure rises more steeply than isothermal because the gas heats up as it is compressed.
Concept check: the first law is energy conservation
Internal energy changes when energy is transferred by heating or when work is done. On a pressure-volume graph, the area under a path is the work done by the gas; the path matters, not just the end states.
For an isothermal ideal-gas change, temperature and internal energy stay constant, so energy transferred by heating equals work done by the gas. In an adiabatic change, no heating occurs; expansion cools the gas because work is done at the expense of internal energy.
17Oscillations
Simple harmonic motion (SHM): defining equation, kinematic solutions, energy exchange, damping and resonance.
17.1 Definitions
Language precision: displacement is measured from equilibrium and has a sign; amplitude is the maximum magnitude of that displacement. Period is the time for one complete cycle, while phase states where in a cycle the oscillator is. These are not interchangeable graph labels.
Displacement (x): distance from the equilibrium position. Amplitude (x0): maximum displacement from equilibrium. Period (T): time for one complete oscillation. T = 1/f. Frequency (f): number of oscillations per unit time. Unit: Hz. Angular frequency:ω = 2πf = 2π/T. Unit: rad s-1. Phase angle (φ): the point an oscillator has reached within a complete cycle: Δφ = (Δt/T) × 2π. Restoring force: the resultant force always acting to return the system to equilibrium.
17.2 SHM — Defining Equation
Defining test: an oscillation is SHM only when acceleration is proportional to displacement and opposite in direction. The minus sign carries the restoring direction; a periodic motion with a non-linear restoring acceleration is oscillatory but not necessarily SHM.
SHM: motion in which the acceleration is proportional to the displacement from a fixed point and is always directed toward that point. The acceleration acts opposite to the displacement.
Defining equationa = -ω2x → a ∝ -x
This is the critical test: if a = -ω2x holds, the motion is SHM with angular frequency ω.
17.3 Kinematic Equations for SHM
Phase relationship: displacement, velocity and acceleration are not maximum together. Velocity is a quarter cycle out of phase with displacement, and acceleration is in antiphase with displacement. Use this before choosing signs on a graph.
General solution (starting from equilibrium, t = 0 at x = 0):
x = x0 sin(ωt) v = ωx0 cos(ωt) = ±ω√(x02 - x2) a = -ω2x0 sin(ωt) = -ω2x
If starting from amplitude (t = 0 at x = x0): x = x0 cos(ωt).
Maximum speed:vmax = ωx0 (at equilibrium, x = 0). Maximum acceleration:amax = ω2x0 (at amplitude).
On displacement graphs: the v–x graph is an ellipse; the a–x graph is a straight line through origin with gradient -ω2.
17.4 Energy in SHM
Continuous exchange: in an ideal oscillator total energy is constant while energy changes between kinetic and potential stores. At equilibrium all the energy is kinetic; at either turning point all is potential. Damping makes the total energy decrease because energy is transferred to surroundings.
Total energyEtotal = ½mω2x02 = ½mvmax2 (constant throughout)
Kinetic energyEk = ½mv2 = ½mω2(x02 - x2)
Potential energyEp = ½mω2x2
At equilibrium (x = 0): Ek = max, Ep = 0. At amplitude (x = ±x0): Ek = 0, Ep = max.
KE and PE both oscillate at 2f (twice the frequency of the displacement), because both depend on x2.
17.5 SHM Systems
System conditions matter: a mass-spring system is close to SHM only within the proportional region of the spring. A small-angle pendulum approximates SHM because the tangential restoring force is approximately proportional to angular displacement; large amplitudes spoil this approximation.
Mass-spring systemω = √(k/m), T = 2π√(m/k)
Simple pendulum (small angles)ω = √(g/L), T = 2π√(L/g)
For the simple pendulum, the period is independent of mass and amplitude (for small angles < ~10°). This is isochronism.
17.6 Damping
Compare regimes: light damping lets the system continue oscillating with decreasing amplitude; heavy damping prevents oscillation but returns slowly; critical damping returns in the shortest time without oscillating. The damping force removes mechanical energy.
Damping: the gradual loss of energy from an oscillating system due to dissipative forces (e.g. friction, air resistance). The amplitude decreases over time.
Type
Behaviour
Amplitude decay
Light damping
Oscillations continue for many cycles; amplitude decreases gradually
Exponential: x0 ∝ e-λt
Critical damping
System returns to equilibrium in the shortest possible time without oscillating
Aperiodic (no overshoot)
Heavy (over) damping
System returns to equilibrium slowly without oscillation
Slower than critical
Applications of critical damping: car suspension systems, door closers, galvanometer needles.
17.7 Forced Oscillations and Resonance
Resonance is an energy-transfer effect: a periodic driving force transfers energy most efficiently when its frequency matches the natural frequency. Damping lowers and broadens the resonance peak, which can be desirable for safety but undesirable for a tuned receiver.
Free oscillation: the system oscillates at its natural frequency after a single disturbance, with no external driving force. Forced oscillation: the system is driven by an external periodic force at the driving frequency. Resonance: the large-amplitude oscillation that occurs when the driving frequency equals the natural frequency of the system. At resonance, the driving force is in phase with the velocity, so energy transfer to the oscillator is maximum.
Increased damping → broader resonance peak (wider range of frequencies gives large amplitude).
Increased damping → resonant frequency shifts slightly below the natural frequency.
Resonance hazards: Tacoma Narrows Bridge collapse, soldiers breaking step on bridges, machinery vibrating at natural frequency. Useful resonance: musical instruments, microwave ovens (water molecules), MRI scanners, radio tuning circuits.
Concept check: what makes motion SHM
SHM is not simply any repeating motion. Its defining condition is that acceleration, and therefore resultant restoring force, is proportional to displacement from equilibrium and directed towards equilibrium. At equilibrium speed and kinetic energy are maximum; at an extreme displacement speed is zero and potential energy is maximum.
Damping transfers energy from the oscillator to its surroundings. A forced oscillator follows the driving frequency; resonance occurs when the driving frequency is close to the natural frequency, producing the largest energy transfer per cycle and hence the largest amplitude.
18Electric Fields
Coulomb's law, uniform and radial fields, electric potential and PE.
18.1 Electric Field Strength
Test-charge convention: electric field direction is defined as the force direction on a small positive test charge. It points away from positive sources and towards negative sources; a negative particle feels a force opposite to the arrows.
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)
Uniform means constant vector: between large, close, oppositely charged plates, field lines are parallel and equally spaced away from the edges. A charged particle entering sideways experiences constant acceleration perpendicular to its initial velocity, so its path is parabolic.
E = V/d
Between parallel plates separated by distance d with pd V, the field is uniform from the + plate toward the - 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)
Interaction law: Coulomb's law gives the magnitude between stationary point charges in vacuum. The two forces are equal and opposite; signs decide attraction or repulsion, while the equation's magnitude uses positive values of charge.
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
Geometry: a point charge produces a radial field, so its strength decreases with inverse square of distance because the same field spreads over the surface area of ever-larger spheres. Field direction changes from point to point even when field strength is the same at equal radii.
E = Q / (4πε0r2)
18.5 Electric Potential
Energy viewpoint: electric potential is potential energy per unit positive charge. A positive charge released from rest moves from higher to lower potential; a negative charge moves from lower to higher potential because its potential energy is qV.
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
Graph interpretation: field strength is the negative potential gradient. The field points in the direction of decreasing potential. In a uniform field, potential changes linearly with distance; in a radial field the potential-distance graph curves and becomes flatter far away.
E = -dV/dr
Conductors in electrostatic equilibrium are equipotential. The electric field inside a conductor is zero, and electric field lines meet a conducting surface at right angles.
18.7 Gravitational vs Electric Fields
Use the analogy carefully: both are inverse-square fields with potential and potential energy, but gravity only attracts and acts on mass, while electric force can attract or repel and is enormously stronger at particle scale. Gravitational potential is conventionally negative; electric potential may be positive or negative.
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 negative)
V = Q/(4πε0r) (sign of Q)
Concept check: field, force, potential and energy
Electric field strength is force per unit positive charge, so its direction is the direction a positive test charge would move. Unlike gravity, electric forces may attract or repel because charges have signs. The force direction on a negative charge is opposite to the field direction.
Electric potential is potential energy per unit positive charge and is a scalar. In a uniform field between parallel plates, potential decreases steadily in the field direction. Field lines are perpendicular to equipotentials; moving along an equipotential needs no work.
19Capacitance
Charge storage, energy stored in a capacitor, combinations, and exponential discharge.
19.1 Definition
Capacitance is not charge: capacitance is the charge stored per volt of potential difference and depends on geometry and dielectric, not on the particular charge currently stored. A larger capacitance stores more charge for the same p.d.
Capacitance C: the charge stored per unit potential difference across a capacitor.
C = Q/V (F = C V-1)
19.2 Combinations of Capacitors
Decide what is shared: capacitors in parallel share the same p.d. and their charges add. Capacitors in series carry the same magnitude of charge and their p.d.s add. This is the reverse pattern from resistors in series/parallel.
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
Where the energy is: work is required to separate charge while charging, and that work is stored as energy in the electric field between the plates. The factor one half appears because the p.d. rises from zero to its final value during charging.
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 = 1/2QV = 1/2CV2 = 1/2Q2/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
Exponential behaviour: as a capacitor discharges, its p.d. falls; this reduces current through the resistor, so equal time intervals remove equal fractions rather than equal amounts of charge. The time constant RC sets the timescale of the decay.
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: t1/2 = 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.
Concept check: why capacitors store energy
Charging moves charge from one plate to the other, creating equal and opposite charges and an electric field between the plates. The increasing potential difference opposes further transfer, so current falls as the capacitor charges.
During discharge through a resistor, charge, potential difference and current all fall exponentially because the rate of flow is proportional to the amount still stored. One time constant does not mean the capacitor is empty; it is the time for charge, current or potential difference to fall to about 37% of its initial value.
19.5 Time Constant and Exponential Discharge
τ = RCQ = Q0 e−t/RC (same form for V and I)
After one time constant, Q has fallen to 37% of Q0. Graphs of Q, V and I against time are decaying exponentials. A graph of ln Q against t is a straight line of gradient −1/RC. Energy stored is the area under the V–Q graph: W = ½QV = ½CV2.
20Magnetic Fields
Force on current-carrying wires and moving charges, the Hall effect, and electromagnetic induction.
20.1 Magnetic Field
Field description: magnetic field lines show the direction a north pole would point, conventionally from north to south outside a magnet. Unlike electric field lines, magnetic field lines form continuous loops because isolated magnetic poles have not been observed.
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
Microscopic origin: a current-carrying wire contains moving charge carriers. Each carrier experiences a magnetic force; the sum is a force on the wire. Reverse either current or field and the force reverses; reverse both and its direction stays the same.
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
Path rule: the magnetic force is always perpendicular to both velocity and field. It therefore changes momentum direction without changing kinetic energy. With velocity perpendicular to a uniform field the path is circular; a velocity component parallel to the field makes a helix.
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
Charge separation: a magnetic force pushes moving carriers to one side of a conductor, producing a transverse p.d. The electric field created by that separation grows until its electric force balances the magnetic force; the Hall voltage reveals carrier sign and motion.
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
Orientation matters: flux is greatest when the field is perpendicular to the area and zero when it is parallel to the plane. Flux linkage multiplies flux by number of turns, provided each turn links the same flux.
Magnetic fluxΦ = BA cosθ (Wb)
Flux linkage of N turnsNΦ
20.6 Faraday's and Lenz's Laws
Two-law method: first use Lenz's law to determine the induced current direction by identifying the change it opposes. Then use Faraday's law for emf magnitude from the rate of change of flux linkage. Lenz's law follows from conservation of energy.
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
Applications include generators, magnetic braking and transformers. For power transmission, stepping up the voltage reduces the current for the same power, so the cable loss I2R is reduced.
20.7 Motional e.m.f.
A conducting rod of length L moving at speed v perpendicular to a uniform field B sweeps area Lv each second, so the flux through the circuit changes at rate BLv.
ε = BLv
If the rod, rails and a resistor R form a closed loop, I = BLv/R. Lenz's law requires the magnetic force on the induced current to oppose the motion, so a constant speed needs an equal external force F = BIL = B2L2v/R.
Worked example
Rails 0.45 m apart, B = 0.80 T, v = 3.0 m s−1, R = 6.0 Ω. ε = 0.80 × 0.45 × 3.0 = 1.08 V; I = 1.08/6.0 = 0.18 A; F = 0.80 × 0.18 × 0.45 = 0.065 N.
20.8 Eddy Currents and Magnetic Braking
A conductor moving in a non-uniform field, or sitting in a changing field, has different flux changes in different parts. Circulating eddy currents heat the metal (I2R) and experience a magnetic force that opposes the motion.
A conducting disc swinging between electromagnet poles stops quickly when the field is on.
A magnet falling through a copper tube is delayed; the same magnet in a plastic tube is not.
Used in train and roller-coaster brakes. Laminating an iron core cuts the eddy-current path and reduces transformer heating.
20.9 Simple Generator
A coil of N turns and area A rotating at angular speed ω in a uniform field has flux linkage NBA sinωt (or cos, depending on the zero of time). Faraday's law then gives a sinusoidal e.m.f.
ε = NBAω cosωt → ε0 = NBAω
Peak e.m.f. occurs when the coil sides cut field lines fastest (plane of the coil parallel to B). Doubling rotation rate doubles both peak e.m.f. and frequency.
Hall probe vs search coil: a Hall voltage is proportional to B itself and is non-zero in a steady field. A coil reads only dΦ/dt, so a steady field gives zero e.m.f. unless the coil is moved, rotated or the field is changed.
Concept check: magnetic force changes direction, not speed
A magnetic field exerts a force only on a moving charge or current with a component perpendicular to the field. The force is perpendicular to the motion, so it can bend a charged-particle path into a circle while doing no work and leaving speed unchanged.
Magnetic flux is a measure of how much field passes normally through an area. Flux linkage changes when field strength, area or orientation changes. Faraday's law gives the magnitude of induced emf; Lenz's law gives the direction, which must oppose the change in flux linkage.
20.10 Field Patterns due to Currents
Long straight wire: concentric circles; direction by the right-hand grip rule.
Flat circular coil: loops through the coil, similar to a short dipole.
Long solenoid: uniform and strong inside, weak outside; a ferrous core increases B.
Two parallel currents attract if they are in the same direction (each wire sits in the other's field). Opposite currents repel.
20.11 Velocity Selector
Perpendicular E and B fields: undeflected particles satisfy qE = Bqv, so v = E/B. Faster particles are bent one way, slower the other. Combined with r = mv/BQ this is the idea of a mass spectrometer.
21Alternating Currents
Sinusoidal currents, RMS values, mean power, and rectification using diodes and smoothing capacitors.
21.1 Sinusoidal AC
Waveform meaning: sinusoidal AC is produced when a coil rotates uniformly in a magnetic field, so the rate of flux-linkage change varies sinusoidally. Polarity reverses every half-cycle; frequency is the number of full cycles each second.
I = I0 sin(ωt); V = V0 sin(ωt)
where I0, V0 are peak values, ω = 2πf.
21.2 RMS Values
Not an arithmetic average: rms is obtained by squaring instantaneous values, averaging, then taking a square root. It is defined through equal heating effect in a resistor, which makes it the meaningful quoted value for mains supplies.
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
Power transfer: for a purely resistive AC circuit, instantaneous p.d. and current are in phase and mean power is VrmsIrms. Do not use peak values in this expression unless the required conversion has been made.
<P> = VrmsIrms = 1/2V0I0 = Irms2R
The "230 V mains" specification is an RMS value; peak is 230√2 ≈ 325 V.
21.4 Rectification
Purpose: a diode conducts mainly in one direction, so a rectifier converts alternating current into a unidirectional but still varying current. A bridge rectifier uses both halves of the input waveform and is more effective than half-wave 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
Capacitor role: the smoothing capacitor charges rapidly near peaks and discharges through the load between peaks, reducing ripple. A larger capacitance, smaller load current or higher supply frequency gives a smoother output.
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.
21.6 Transformers
An alternating current in the primary produces a changing flux in a soft-iron core. That flux links the secondary, so Faraday's law induces an alternating e.m.f. of the same frequency. A steady d.c. primary current produces no continuous secondary e.m.f.
Losses and how they are reduced: eddy currents in the core (laminate the core); I2R heating in the windings (thick copper); flux leakage (closed iron loop); hysteresis (soft iron that is easily remagnetised).
Power stations generate at a few kilovolts. A step-up transformer raises the grid voltage so the same power is sent at a much smaller current. Cable loss is I2R, so a smaller current cuts heating. Local step-down transformers then give a safer domestic voltage (230 V r.m.s. in the UK).
Concept check: RMS is an equivalent heating value
An alternating supply repeatedly reverses direction. Its instantaneous voltage and current vary continuously, so an ordinary average over a complete cycle is zero. RMS values are the steady DC values that would produce the same mean power dissipation in a resistor.
Rectification makes current one-directional; smoothing uses a capacitor that charges near voltage peaks and discharges between them. High-voltage transmission reduces current for the same power, so it reduces energy loss in cables because loss is proportional to I2R.
21.8 Measuring a.c. on a CRO
Y-gain (V/div) × peak-to-peak divisions gives 2V0. Time-base (s/div) × divisions per cycle gives the period. For a sinusoid, mean power in a resistor is half the peak power, which is why Irms = I0/√2.
22Quantum Physics
Photons, the photoelectric effect, wave–particle duality, electron energy levels and atomic spectra.
22.1 The Photon
Quantised radiation: a photon is a discrete packet of electromagnetic energy, with energy proportional to frequency. Higher-frequency radiation has more energetic photons, even if the total intensity is low.
A photon is a discrete packet (quantum) of electromagnetic energy.
E = hf = hc/λ
Although photons have zero rest mass, they carry energy and momentum. Classical p = mv is not used for a photon.
Photon momentump = h/λ
Photon energy-momentum relationE = pc
Evidence for the photon model includes the photoelectric effect and radiation pressure. Higher-level discussions may also use Compton scattering as evidence that photons carry momentum.
22.2 The Photoelectric Effect
Evidence for photons: emission is effectively instantaneous and has a threshold frequency because one electron absorbs one photon. Increasing intensity increases the number of emitted electrons only when each photon already has enough energy to overcome the work function.
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.
where Φ = hf0 is the work function — the minimum energy needed to release an electron from the metal surface.
Emitted electrons have a range of kinetic energies because electrons deeper in the metal lose different amounts of energy before escaping. The maximum kinetic energy is for electrons released from the surface with no extra energy loss.
Change
Effect
Increase frequency above f0
Increases Ek,max and stopping potential.
Increase intensity at fixed frequency
Increases emission rate / photocurrent, not Ek,max.
Use frequency below f0
No emission, no matter how large the intensity.
On a graph of Ek,max against frequency, the gradient is h, the x-intercept is f0, and the y-intercept is -Φ.
Pitfall: Increasing intensity does NOT increase max KE. It only increases the rate of electron emission (photocurrent).
22.3 Wave–Particle Duality
Complementary models: light shows interference and diffraction like a wave, but exchanges energy and momentum in localised photons. Matter particles have de Broglie wavelength and can diffract; the useful model depends on which observation is being explained.
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)
The de Broglie relation links a particle property, momentum, to a wave property, wavelength. Electron diffraction is observable because fast electrons can have wavelengths comparable with atomic spacing, about 10-10 m. For everyday macroscopic objects, the de Broglie wavelength is too small to detect.
Wave-particle duality does not mean "sometimes fake wave, sometimes fake particle". It means classical wave-only and particle-only models are each incomplete.
22.4 Discrete Atomic Energy Levels
Allowed states: electrons in an atom cannot have arbitrary energies. Absorption promotes an electron only if photon energy exactly matches an energy gap; emission occurs when it falls to a lower level and releases a photon with that gap's energy.
Electrons in an isolated atom can occupy only certain discrete energy levels. No intermediate energy values are allowed.
Ground state: the lowest energy level available to the electron.
Excited state: any allowed level above the ground state.
Energy levels are usually negative because zero energy is defined when the electron is free at infinity. Energy must be supplied to remove the electron from the atom.
Ionisation means raising the electron to E = 0, so it is no longer bound to the atom.
When an electron changes level, the photon energy equals the energy difference between the two levels:
hf = hc/λ = Ehigh - Elow
Transition to a lower level: energy is lost by the electron, so a photon is emitted.
Transition to a higher level: energy is gained by the electron, so a photon is absorbed.
Only specific energy differences are possible, so only specific photon frequencies and wavelengths appear.
22.5 Line Spectra
Fingerprint idea: each element has unique energy-level separations, so its emission and absorption lines occur at characteristic wavelengths. Continuous spectra come from hot dense sources; line spectra reveal transitions in isolated atoms or gases.
Emission spectrum: bright lines on a dark background. Hot atoms drop to lower energy levels and emit photons with specific wavelengths.
Absorption spectrum: dark lines on a continuous bright background. Cool gas absorbs only photons with energies matching its level gaps; re-emitted photons leave in random directions, so those wavelengths are missing from the transmitted beam.
Each element has a unique set of energy levels, so it has a unique line spectrum. Spectra can identify elements in stars, gases and unknown samples.
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).
Concept check: photons explain the threshold
Light arrives in photons, each with energy hf. One electron absorbs one photon. Emission occurs only if that individual photon provides at least the work function, which explains why raising intensity below threshold frequency cannot eject electrons.
Increasing intensity above threshold sends more photons each second, so it increases photocurrent; increasing frequency increases each photon's energy and therefore the maximum kinetic energy of emitted electrons. Atomic line spectra arise because electrons can change only between discrete energy levels.
22.6 Photoelectric Equation in Full
hf = Φ + ½m vmax2 and p = E/c = h/λ
Threshold frequency f0 = Φ/h. Below f0 no emission, however intense the light. Intensity changes the number of photoelectrons (current) but not KEmax. Use the electronvolt as an energy unit: 1 eV = 1.60×10−19 J.
22.7 Electron Diffraction and de Broglie
Interference of light is wave evidence; the photoelectric effect is particle evidence. Electrons produce diffraction rings, so particles have a wavelength λ = h/p.
23Nuclear Physics
Mass–energy equivalence, binding energy, fission/fusion, and the laws of radioactive decay.
23.1 Mass–Energy Equivalence
Mass is energy: rest mass is one form of energy, so a change in total rest mass can release or absorb energy. In nuclear processes the mass change is tiny but multiplying by c2 produces a very large energy.
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
Bound systems have less mass: energy released when separate nucleons form a nucleus leaves the system, so the final nucleus has smaller mass than its separated parts. The binding energy is the energy required to reverse that process and separate the nucleus completely.
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
Stability comparison: binding energy per nucleon compares nuclei of different sizes. Fusion of light nuclei and fission of very heavy nuclei release energy because their products move towards the peak of the binding-energy-per-nucleon curve near iron.
The most useful measure for comparing nuclear stability:
Peaks at 5626Fe (~8.8 MeV/nucleon) — the most tightly bound nucleus.
Fission of a heavy nucleus (e.g. 23592U) splits it into two medium-mass nuclei of higher BE/nucleon → energy released.
Fusion of light nuclei (e.g. 21H + 31H → 42He + 10n) produces a nucleus higher up the graph → energy released. Larger energy per unit mass than fission.
23.4 Radioactive Decay
Probability, not a clock: no one can predict when an individual nucleus will decay. A constant fraction of a large sample decays in each equal interval, producing an exponential curve. Correct count rates for background before using half-life or activity data.
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-lifet1/2 = ln 2 / λ ≈ 0.693/λ
Worked example — dating
A bone has activity 1/8 of a modern bone. 146C has t1/2 = 5730 yr. How old is it?
1/8 = (1/2)3 → 3 half-lives → age ≈ 3 × 5730 = 17 200 years.
Concept check: binding energy measures stability
Mass defect is the difference between the total mass of separated nucleons and the mass of the nucleus. The missing mass has been released as binding energy. A high binding energy per nucleon means nucleons are more tightly bound and the nucleus is generally more stable.
In radioactive decay, each nucleus has a constant probability of decaying per unit time; decay is random for an individual nucleus but predictable for a very large sample. Half-life is the time for the number of undecayed nuclei, activity, or count rate after background correction to halve.
23.5 Fusion, Fission and the Binding-Energy Curve
Binding energy per nucleon has a maximum near 56Fe. Fusion of light nuclei and fission of heavy nuclei both move towards that peak and release energy ΔE = c2Δm. Write the nuclear equation first, then convert the mass defect in u to kg or use 1 u = 931 MeV/c2.
23.6 Activity and Exponential Decay
Decay is spontaneous (cannot be forced by T or P) and random (fluctuations in count rate). Activity A is the number of decays per unit time.
A = λNλ = ln 2 / t1/2N = N0 e−λt
The same exponential applies to activity and to received count rate.
24Medical Physics
Ultrasound imaging, X-ray production and CT scanning, and PET positron-emission tomography.
24.1 Ultrasound
Image mechanism: a transducer converts electrical pulses into ultrasound and receives returning echoes. Echo time gives depth only after dividing the round-trip distance by two; echo strength depends on the acoustic-impedance difference at a boundary.
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
Contrast mechanism: different tissues attenuate X-rays by different amounts, so bone appears more absorbing than soft tissue. X-rays are ionising, so an imaging answer should balance diagnostic benefit against dose, shielding and exposure time.
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
Why CT improves on a single radiograph: many X-ray measurements from different angles are computer-processed to reconstruct a slice, reducing the overlap of structures. The trade-off is a greater radiation dose than a simple X-ray image.
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)
Functional imaging: a positron-emitting tracer follows a biochemical process such as glucose uptake. Positron-electron annihilation produces two gamma photons in opposite directions; coincident detection locates the decay line and maps activity rather than anatomy alone.
A β+-emitting tracer (e.g. 189F-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.
24.5 MRI
Soft-tissue contrast: MRI uses the magnetic behaviour of hydrogen nuclei and radio-frequency pulses to produce signals that vary with local tissue environment. It is non-ionising, but expensive, slower and unsuitable for some patients with certain implants or ferromagnetic objects.
Magnetic resonance imaging uses a strong magnetic field and radio-frequency pulses to make hydrogen nuclei absorb and re-emit energy at resonance. It is non-ionising and gives strong soft-tissue contrast, but it is unsuitable for some patients with ferromagnetic implants or pacemakers.
Concept check: every scan balances information and risk
Ultrasound uses reflected sound pulses and is non-ionising; a boundary gives a strong echo when acoustic impedance changes greatly. X-rays are ionising and are attenuated differently by tissue, so image quality, dose and contrast media must be considered together.
CT combines many X-ray projections to reconstruct a slice. PET detects two gamma photons from positron-electron annihilation and maps tracer activity, while MRI uses nuclear magnetic resonance and gives excellent soft-tissue contrast without ionising radiation. State both a benefit and a limitation when comparing methods.
24.6 Ultrasound in Full
A piezoelectric crystal changes shape when a p.d. is applied (transmitter) and generates an e.m.f. when it is deformed by a returning pulse (receiver). Specific acoustic impedance Z = ρc. The intensity reflection coefficient at a boundary is
IR/I0 = (Z1 − Z2)2 / (Z1 + Z2)2
A large Z mismatch (tissue/air or tissue/bone) reflects almost everything — hence coupling gel. Attenuation: I = I0 e−μx.
24.7 X-ray Minimum Wavelength
Electrons accelerated through p.d. V have energy eV. The shortest X-ray wavelength (all KE in one photon) is λmin = hc/eV. Contrast comes from different attenuation in bone and soft tissue. The same exponential I = I0 e−μx applies.
24.8 PET Annihilation Energy
A β+ tracer annihilates with an electron. Mass–energy and momentum are conserved, so two 511 keV γ photons travel in opposite directions. Arrival-time differences reconstruct the tracer map.
25Astronomy and Cosmology
Luminosity and flux, Wien's and Stefan's laws, redshift and Hubble's law.
25.0 Distance Units
Choose a practical scale: AU suits Solar-System distances, light-year is a distance rather than a time, and parsec is defined geometrically through parallax. Convert units carefully before using inverse-square relationships.
Useful astronomical distance units: 1 AU = mean Earth-Sun distance; 1 light-year = distance travelled by light in one year; 1 parsec = distance at which 1 AU subtends 1 arcsecond.
25.1 Luminosity and Radiant Flux
Source versus observer: luminosity belongs to the star and is total emitted power; radiant flux is measured at the observer and falls with distance. A dim-looking star can be intrinsically luminous but far away, so standard candles are needed to infer distance.
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
Colour and temperature: Wien's law concerns the wavelength where a black-body spectrum peaks. A smaller peak wavelength means a higher surface temperature; use metres and kelvin in calculations, and do not confuse peak wavelength with total luminosity.
λmaxT = 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
Two effects on luminosity: a star becomes more luminous if its surface is larger or hotter. Temperature has a fourth-power effect, while radius enters through surface area, so temperature changes can have a much larger effect than intuition suggests.
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
Observed shift: redshift means spectral lines are measured at longer wavelengths than their laboratory values. For small recession speeds, the fractional wavelength shift gives speed divided by light speed; use the rest wavelength in the denominator.
Light from distant galaxies arrives at longer wavelengths than emitted — they are receding.
z = Δλ/λ ≈ v/c (non-relativistic)
For redshift, wavelength increases and frequency decreases. The fractional frequency change has the same magnitude as the fractional wavelength change for small speeds.
25.5 Hubble's Law
Trend not a local speed limit: Hubble's law is an empirical relationship for distant galaxies showing that recession speed is proportional to distance. The gradient of a recession-speed versus distance graph is the Hubble constant, and its reciprocal estimates the age scale of the Universe.
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 must be linked: galactic redshifts support expansion, the CMB is relic thermal radiation from an early hot Universe, and light-element abundances agree with early nucleosynthesis predictions. These independent observations together support the model more strongly than any one alone.
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.
Concept check: what observations actually show
Luminosity is a star's total power output; radiant flux is the power per unit area received by an observer, so flux falls with the square of distance even when luminosity stays constant. A standard candle is useful only because its luminosity can be estimated independently.
Redshift is an observed increase in wavelength. Hubble's law shows that more distant galaxies generally have larger recession speeds, supporting expansion of space. It is evidence for the Big Bang when combined with the CMB and light-element abundances; no single observation alone proves every feature of the model.
25.7 Standard Candles
Luminosity L is the total power radiated by the star. Radiant flux intensity at distance d is F = L/(4πd2). An object of known L is a standard candle: measure F, calculate d. This is how distances to galaxies are found.
25.8 Estimating a Star's Radius
Wien: λmaxT is constant, so the peak wavelength gives surface temperature. Stefan–Boltzmann: L = 4πr2σT4. Combine to estimate r.
25.9 Redshift and Hubble
Δλ/λ ≈ Δf/f ≈ v/c and v ≈ H0d
Spectral lines from distant galaxies are shifted to longer wavelength. That recession, with Hubble's law in SI units, is the evidence that the Universe is expanding and began in a hot dense state (Big Bang).
These reference cards summarise the 9702 data sheet for quick lookup. Always cross-check with the official Cambridge data & formulae booklet.
9702 Data & Formulae
Reference cards for AS & A2 examinations
ConstantsPaper structureLinearisationExam tips
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
Acceleration of free fall
g
9.81 m s-2
Avogadro constant
NA
6.02 × 1023 mol-1
Electron rest mass
me
9.11 × 10-31 kg
Proton rest mass
mp
1.67 × 10-27 kg
Unified atomic mass unit
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 %
Complete Formula Map - AS Mechanics, Matter, Waves and Electricity
Area
Formulae
Condition / meaning
Motion
v = s/t; a = Δv/Δt; v = u + at; s = ut + ½at2; v2 = u2 + 2as
SUVAT only for constant acceleration. The area under a velocity-time graph is displacement; its gradient is acceleration.
Forces and momentum
F = ma; W = mg; p = mv; F = Δp/Δt; moment = F×perpendicular distance
Resolve vectors before applying Newton's second law. Momentum is conserved in an isolated system.
Energy and solids
W = Fs cosθ; Ek = ½mv2; ΔEp = mgΔh; P = W/t = Fv; efficiency = useful/input; F = kx; Eelastic = ½kx2
For P = Fv, use the velocity component in the force direction. Hooke's law applies only up to the limit of proportionality.
Fluids and materials
ρ = m/V; p = F/A; Δp = ρgΔh; upthrust = weight of displaced fluid; stress = F/A; strain = ΔL/L; E = stress/strain
Use vertical depth for hydrostatic pressure. Strain has no unit; Young modulus is in Pa.
Waves
v = fλ; I = P/A; I ∝ A2; ax = Dλ; dsinθ = nλ
ax = Dλ is for small angles in Young double slit; grating equation requires the path difference to be an integer number of wavelengths.
Circuits
Q = It; V = W/Q; R = V/I; P = IV = I2R = V2/R; V = ε - Ir
At a junction, current in = current out; around a loop, total emf = total p.d. For a potential divider, use the resistor across which the output is measured.
Concept Mastery - How the Physics Fits Together
Core idea
Precise explanation
What an examiner expects
Models, data and uncertainty
A measurement is an estimate: random uncertainty causes scatter, while a systematic error shifts every value in one direction. Accuracy is closeness to the accepted value; precision is small spread between repeats.
Quote an uncertainty with units, identify a genuine cause, and give a matching improvement - repeats for random uncertainty, calibration/zero correction for systematic error.
Force is not motion
A resultant force changes velocity, not velocity itself. If the resultant force is zero, an object is at rest or continues with constant velocity. A free-body diagram contains only forces acting on the chosen object.
Resolve forces parallel/perpendicular to motion, then apply ΣF = ma in each direction. Never draw a reaction force on the same body as its action partner.
Energy, work and power
Work transfers energy when a force has a component along a displacement. Energy is conserved, but useful energy may become spread out thermally, so efficiency can be below 100%.
State the energy stores before and after, then explain the transfer pathway. Include resistive work/thermal transfer when mechanical energy is not conserved.
Fields
A field assigns a force per unit mass or charge at each point. Gravitational and electric potentials are energy per unit mass/charge; because they are scalars, potentials add algebraically.
Separate field strength from potential. A negative gravitational potential means work must be done to move a mass from that point to infinity.
Waves and superposition
Waves transfer energy without net transfer of matter. Superposition is the algebraic addition of displacements; a stable interference pattern needs coherent sources with a fixed phase relationship.
Use path difference for maxima/minima and distinguish a stationary-wave node (always zero displacement) from an antinode (maximum amplitude).
Electric circuits
Charge is conserved at a junction and energy is conserved around a closed loop. Emf is energy supplied per unit charge by a source; terminal p.d. falls when current produces energy loss in internal resistance.
Use current conservation and loop equations before selecting a resistance formula. Explain an I-V curve by linking temperature, lattice vibration and carrier motion.
Thermal physics
Temperature measures the average random kinetic energy of particles; internal energy is the sum of random kinetic and potential energies. During a change of state, supplied energy changes separation rather than temperature.
Use particle spacing, attractions and random motion in explanations. Convert Celsius to kelvin before any gas calculation.
Oscillations and resonance
In SHM the restoring acceleration is proportional to displacement and directed toward equilibrium. Damping removes energy; driven oscillations have maximum amplitude when driving and natural frequencies match.
Identify equilibrium, amplitude and phase. Explain resonance in terms of energy transfer per cycle, not merely "vibration gets bigger".
Electromagnetic induction
An emf is induced when flux linkage changes. Lenz's law says the induced effect opposes the change that produced it, which is required by conservation of energy.
Give a direction first using Lenz's law, then calculate magnitude from rate of change of flux linkage.
Quantum and nuclear physics
Photons have discrete energy; the photoelectric effect requires one photon with sufficient energy, rather than energy being accumulated from many weak photons. Nuclear transformations conserve charge and nucleon number.
Distinguish intensity from frequency in photoelectric questions; show both conservation laws in every nuclear equation.
Complete Formula Map - A2 Physics
Area
Formulae
Condition / meaning
Circular motion and gravitation
ω = 2πf; v = rω; a = v2/r = rω2; F = GmM/r2; g = GM/r2; φ = -GM/r; T2 = 4π2r3/GM
Centripetal force is the resultant toward the centre, not an additional force. Use centre-to-centre distance r.
Temperature must be in K and volume in m3. In the first law, state the sign convention before using ΔU = Q + W.
Oscillations
a = -ω2x; vmax = Aω; amax = Aω2; E = ½mω2A2; T = 2π√(m/k)
Displacement is measured from equilibrium. Maximum speed is at equilibrium; maximum acceleration and elastic energy are at amplitude.
Fields, capacitors and magnetism
E = F/Q = V/d; F = QE; F = Q1Q2/(4πε0r2); C = Q/V; W = ½QV = ½CV2; F = BILsinθ = BQvsinθ; Φ = BAcosθ; ε = -NΔΦ/Δt
Capacitor discharge: Q, V and I fall exponentially with time constant RC. Apply Lenz's law to determine direction before calculating magnitude.
AC, quantum and nuclear
Vrms = V0/√2; Irms = I0/√2; P = VrmsIrms; E = hf = hc/λ; hf = φ + Ek,max; λ = h/p; E = mc2; N = N0(1/2)t/T1/2
For photoelectric effect, frequency controls maximum kinetic energy; intensity controls photon rate. In decay equations, conserve nucleon number and charge.
Medical physics and cosmology
z = Δλ/λ ≈ v/c; v = H0d; F = L/(4πd2); λmaxT = 2.9×10-3 m K; L = 4πr2σT4
Use the non-relativistic redshift approximation only when specified/appropriate. Ultrasound depth uses distance = ct/2 because the pulse makes a return journey.
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.