Topics

How the marks work

9702 mark schemes label every mark with a code. The mark schemes on this site use the same codes, so you learn what examiners are looking for.

Data you are given in the exam

These constants are printed at the start of Papers 1, 2 and 4, so you don't need to memorise their values.

Each topic lists its key equations. Those marked On formulae page are printed on the paper; those marked Learn it are not, so you must memorise them.

Every measurement in physics is a number with a unit and an uncertainty. This topic sets up the tools used everywhere else: SI base units, checking equations, estimating, combining uncertainties, and working with vectors.

Physical quantities and estimates

A physical quantity has a numerical magnitude and a unit, such as 9.81 m s⁻². Paper 1 often asks you to pick a sensible estimate, so learn some reference values:

SI units

All units are built from six SI base units used in this course:

Every other unit is a derived unit. To express one in base units, start from a defining equation:

Homogeneity: an equation is homogeneous if every term has the same base units. A homogeneous equation is not necessarily correct, because pure numbers such as ½ have no units, but an equation that is not homogeneous must be wrong.

Prefixes

Errors and uncertainties

Systematic error

Makes every reading too high or too low by the same amount or proportion, such as a zero error on a meter or a stretched tape. Repeating readings does not reduce it; it is removed by correcting the instrument or the method.

Random error

Makes readings scatter unpredictably either side of the true value, such as reaction time when using a stopwatch. Reduced by repeating readings and averaging.

Accuracy

How close a measurement is to the true value. Affected by systematic error.

Precision

How close repeated measurements are to each other (small scatter). Affected by random error.

If a length is x = 25.0 ± 0.5 cm, the absolute uncertainty is 0.5 cm, the fractional uncertainty is 0.5/25.0 = 0.02, and the percentage uncertainty is 2%.

Scalars and vectors

A scalar has magnitude only (mass, speed, distance, energy, time, temperature). A vector has magnitude and direction (displacement, velocity, acceleration, force, momentum, weight).

To add vectors, draw them tip to tail; the resultant goes from the start of the first to the end of the last. For two perpendicular vectors, use Pythagoras and trigonometry. To resolve a vector F at angle θ to the horizontal into two perpendicular components:

Exam-style questions

Kinematics describes motion without asking what causes it. You need precise definitions, the meaning of the gradient and area of motion graphs, the four equations of uniformly accelerated motion, and projectile motion.

Definitions

Distance

The total length of path travelled. A scalar.

Displacement

The distance in a straight line from the starting point, in a stated direction. A vector.

Speed

Distance travelled per unit time. A scalar.

Velocity

The rate of change of displacement. A vector.

Acceleration

The rate of change of velocity. A vector, in m s⁻².

Motion graphs

• The gradient of a displacement–time graph is the velocity.
• The gradient of a velocity–time graph is the acceleration.
• The area under a velocity–time graph is the displacement.

Equations of uniformly accelerated motion

These apply only when the acceleration is constant and the motion is in a straight line. Choose a positive direction and keep to it: a deceleration is a negative acceleration.

Free fall and measuring g

Near the Earth's surface, with no air resistance, every object falls with the same acceleration g = 9.81 m s⁻².

Experiment: drop a steel ball from rest through a measured height h, timing the fall with an electromagnet release and a trapdoor (or with light gates) connected to an electronic timer. Repeat for several heights. Since h = 12gt², plot h against t²: it is a straight line through the origin with gradient g/2, so g = 2 × gradient. Using a graph also reveals a systematic delay in the release (an intercept).

Projectile motion

Without air resistance, the horizontal and vertical motions are independent:

• Horizontally: no force, so constant velocity ucos θ. Use s = vt.
• Vertically: constant acceleration g downwards, starting at usin θ. Use the equations of motion.
• The time is the same for both, and links them.

With air resistance, the projectile reaches a lower maximum height and a shorter range, and its path is no longer symmetrical: the descent is steeper than the ascent.

Exam-style questions

Dynamics explains why motion changes. The central idea is momentum: a resultant force changes it, and in a collision it is conserved. Air resistance adds the idea of terminal velocity.

Mass, weight and momentum

Mass

The property of an object that resists changes in its motion (its inertia). Measured in kg, the same everywhere.

Weight

The gravitational force on an object: W = mg. It depends on the gravitational field strength.

Linear momentum

The product of mass and velocity, p = mv. A vector, in kg m s⁻¹ or N s.

Newton's laws of motion

1. First law: an object stays at rest or moves with constant velocity unless a resultant force acts on it.
2. Second law: the resultant force on an object is proportional to its rate of change of momentum, and acts in the same direction: F = ΔpΔt. For constant mass this becomes F = ma.
3. Third law: if body A exerts a force on body B, then B exerts a force on A that is equal in magnitude and opposite in direction. The two forces are of the same type and act on different bodies.

Non-uniform motion: drag and terminal velocity

Drag (air resistance or a viscous force in a liquid) acts opposite to the velocity and increases with speed. For an object falling from rest:

1. At first the speed is zero, so there is no drag; the resultant force is the weight and the acceleration is g.
2. As speed increases, drag increases, so the resultant force (weight − drag) and the acceleration decrease.
3. Eventually drag equals weight. The resultant force is zero, so the object moves at a constant terminal velocity.

Conservation of momentum

Principle of conservation of momentum: for a system of interacting objects, the total momentum in any direction stays constant, provided no resultant external force acts on the system.

• In every collision, momentum is conserved (and total energy is conserved).
• In a perfectly elastic collision, kinetic energy is also conserved. In this case the relative speed of approach equals the relative speed of separation.
• In an inelastic collision, some kinetic energy becomes other forms (thermal energy, sound, deformation).
• In two dimensions, momentum is conserved separately in each of two perpendicular directions.

Exam-style questions

Forces can make things turn as well as accelerate. This topic covers moments and couples, the two conditions for equilibrium, and pressure in fluids, including why things float.

Types of force

• Weight: acts through the centre of gravity, the point where the whole weight of an object appears to act.
• Normal contact (reaction) force: perpendicular to a surface.
• Friction: acts along a surface, opposing relative motion.
• Drag (viscous force, air resistance): opposes motion through a fluid and increases with speed.
• Upthrust: the upward force a fluid exerts on an object in it.

Turning effects

Moment of a force

The product of the force and the perpendicular distance from the pivot to the line of action of the force. Unit: N m.

Couple

A pair of equal and opposite forces whose lines of action are parallel but do not coincide. A couple produces rotation only, with no resultant force.

Torque of a couple

The product of one of the forces and the perpendicular distance between their lines of action.

Principle of moments

For an object in equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about that point.

Equilibrium

An object is in equilibrium when both conditions hold:

1. the resultant force is zero (in every direction), and
2. the resultant moment (torque) about any point is zero.

If three forces keep an object in equilibrium, they can be drawn tip to tail to form a closed triangle (a triangle of forces).

Density and pressure

Pressure is the normal force per unit area, in pascals (Pa = N m⁻²). In a fluid, pressure increases with depth: Δp = ρgΔh, where ρ is the density of the fluid. The total pressure at a depth also includes the atmospheric pressure on the surface.

Upthrust arises because the pressure on the bottom surface of a submerged object is greater than on its top surface (the bottom is deeper). The resulting upward force equals the weight of fluid displaced, F = ρgV, where ρ is the fluid's density and V the volume submerged (Archimedes' principle). A floating object displaces its own weight of fluid.

Exam-style questions

Energy is never created or destroyed, only transferred. Work is how a force transfers energy, power is how fast it happens, and efficiency tells you how much of it is useful.

Work done

Work done by a force is the product of the force and the displacement in the direction of the force. One joule is the work done when a force of 1 N moves its point of application 1 m in the direction of the force. If the force is at angle θ to the displacement, W = Fscos θ.

A gas expanding against a constant external pressure p does work W = pΔV, where ΔV is the increase in volume.

Energy conservation and efficiency

Principle of conservation of energy: energy cannot be created or destroyed; it can only be transferred from one form to another. The total energy of a closed system is constant.

Power is the rate of doing work (or transferring energy), in watts: 1 W = 1 J s⁻¹. When a force F moves an object at constant velocity v, P = Fs/t = Fv.

Potential and kinetic energy

Gravitational potential energy is the energy an object has because of its position in a gravitational field. Kinetic energy is the energy it has because of its motion. You may be asked to derive both:

• ΔE_p = mgΔh: lifting a mass at constant speed needs a force equal to its weight, mg. The work done is force × distance = mgΔh, and this equals the gain in potential energy. It applies only in a uniform field, near the Earth's surface.
• E_k = 12mv²: a constant force F accelerates a mass from rest through distance s. Work done W = Fs = mas. From v² = u² + 2as with u = 0, as = 12v², so W = 12mv², the kinetic energy gained.

Exam-style questions

When forces stretch or squash a solid, it changes shape. Stress and strain describe this independently of the sample's size, and the Young modulus is the property of the material that links them.

Hooke's law

Tensile forces stretch an object; compressive forces squash it. Hooke's law: the extension of a spring (or wire) is proportional to the applied force, provided the limit of proportionality is not exceeded.

Stress, strain and the Young modulus

Stress

The force per unit cross-sectional area, σ = F/A. Unit: Pa.

Strain

The extension per unit original length, ε = x/L. No unit.

Young modulus

Stress divided by strain, for a material obeying Hooke's law. Unit: Pa. A large Young modulus means a stiff material.

Measuring the Young modulus of a metal wire

1. Use a long, thin wire (so the extension is measurable), clamped at one end and passing over a pulley, with a marker attached near the loaded end against a fixed millimetre scale. (Alternatively, use a vernier on a Searle's apparatus with a reference wire.)
2. Measure the original length L from the clamp to the marker with a metre rule.
3. Measure the diameter at several places with a micrometer screw gauge and take the mean; calculate A = πd²/4.
4. Add masses in steps, recording the extension x for each load F. Check by removing the loads that the wire returns to its original length (it stayed elastic).
5. Plot F against x. In the straight-line region, E = gradient × LA.

Elastic and plastic behaviour

Elastic deformation

The object returns to its original shape and size when the load is removed.

Plastic deformation

The object is permanently deformed; it does not return to its original shape when the load is removed.

Limit of proportionality

The point beyond which force is no longer proportional to extension.

Elastic limit

The point beyond which the deformation becomes plastic.

Elastic potential energy (strain energy) is the energy stored in a deformed object. It equals the area under the force–extension graph. In the Hooke's law region this area is a triangle: E_p = 12Fx = 12kx².

Exam-style questions

A progressive wave transfers energy from one place to another without transferring matter. This topic covers the vocabulary of waves, the wave equation, the Doppler effect, the electromagnetic spectrum and polarisation.

Describing a wave

Displacement

The distance of a point on the wave from its equilibrium position.

Amplitude

The maximum displacement from the equilibrium position.

Wavelength

The distance between two adjacent points on the wave that are in phase (such as two crests).

Period

The time for one complete oscillation of a point on the wave.

Frequency

The number of oscillations per unit time. Unit: hertz (Hz).

Phase difference

How far one oscillation is ahead of another, as a fraction of a cycle, in degrees or radians. Points one wavelength apart are in phase; points half a wavelength apart are in antiphase (180°).

Wave speed

The distance travelled by the wave energy per unit time.

Intensity

The power transmitted per unit area at right angles to the wave direction. Intensity ∝ amplitude².

Deriving v = fλ: in one period T the wave moves forward one wavelength, so v = λ/T = fλ.

Using an oscilloscope: the time-base setting (such as 0.5 ms per division) converts horizontal divisions into time. Measure the number of divisions for one cycle to find T, then f = 1/T. The vertical (Y-gain) setting converts divisions into voltage, giving the amplitude.

Transverse and longitudinal waves

The Doppler effect

When a source of sound moves relative to a stationary observer, the observed frequency differs from the emitted frequency. As the source approaches, the wavefronts in front of it are squashed together (shorter wavelength), so the observer hears a higher frequency; as it recedes, the wavefronts are spread out and the frequency is lower.

The electromagnetic spectrum

All electromagnetic waves are transverse, can travel through a vacuum, and travel at the same speed in a vacuum, c = 3.00 × 10⁸ m s⁻¹. They differ in wavelength and frequency.

Polarisation

In plane-polarised light, the oscillations are in one plane only. Only transverse waves can be polarised. When plane-polarised light of intensity I₀ passes through a polarising filter whose transmission axis is at angle θ to the plane of polarisation, the transmitted intensity is given by Malus's law:

Exam-style questions

When two waves meet, their displacements add. This principle of superposition explains stationary waves, interference fringes and the sharp bright lines of a diffraction grating.

The principle of superposition

When two or more waves meet at a point, the resultant displacement is the sum of the individual displacements at that point.

Stationary waves

A stationary (standing) wave forms when two progressive waves of the same frequency (and similar amplitude), travelling at the same speed in opposite directions, superpose. Usually one wave is a reflection of the other.

• Nodes are points of zero amplitude; antinodes are points of maximum amplitude.
• The distance between adjacent nodes (or adjacent antinodes) is λ/2.
• All points between two adjacent nodes oscillate in phase; points either side of a node are in antiphase.
• A stationary wave stores energy but does not transfer it.

Experiments

• Stretched string: a string is driven by a vibration generator and passes over a pulley to a hanging mass. Adjust the frequency until a stationary wave forms. Measure the node-to-node distance with a metre rule, so λ = 2 × node separation and v = fλ.
• Air column: a tuning fork of known frequency is held over a tube closed at one end (for example by water). Change the length of the air column until the sound is loudest (resonance). The first two resonant lengths differ by λ/2, so λ = 2(L₂ − L₁) and the speed of sound v = fλ. Using the difference removes the end correction.
• Microwaves: a microwave transmitter faces a metal reflector. A detector moved along the line between them finds maxima and minima; adjacent minima (nodes) are λ/2 apart.

Diffraction

Diffraction is the spreading of a wave as it passes through a gap or around the edge of an obstacle. It is most noticeable when the gap width is similar to the wavelength. The wavelength and speed do not change.

Interference

Two sources are coherent if they have a constant phase difference (which requires the same frequency). Interference fringes can only be observed with coherent sources. For two sources in phase:

• Constructive interference (maximum) where the path difference is a whole number of wavelengths, nλ.
• Destructive interference (minimum) where the path difference is (n + 12)λ.

Two-source interference can be shown with water waves in a ripple tank, with microwaves from one transmitter passing through two slits, and with light through a double slit (Young's experiment).

The diffraction grating

A grating has a very large number of equally spaced parallel slits. Light diffracts through each slit, and the waves interfere constructively only at precise angles where the path difference between adjacent slits is a whole number of wavelengths, giving sharp, bright maxima. If the grating has N lines per metre, d = 1/N. The highest order seen is the largest integer n for which nλ/d ≤ 1.

Exam-style questions

Electric current is a flow of charge, potential difference measures the energy each coulomb transfers, and resistance links the two. Learn the precise definitions: they are asked for again and again.

Electric current and charge

Electric current is the rate of flow of charge. In metals the charge carriers are free electrons; in electrolytes they are ions. Charge is quantised: any charge is a whole-number multiple of the elementary charge e = 1.60 × 10⁻¹⁹ C. One coulomb is the charge passing a point when a current of 1 A flows for 1 s.

Potential difference and power

The potential difference across a component is the energy transferred from electrical to other forms per unit charge passing through it. One volt is one joule per coulomb.

Resistance and I–V characteristics

Resistance is the potential difference across a component divided by the current through it. One ohm is one volt per ampere. Ohm's law: for a metallic conductor at constant temperature, the current is proportional to the potential difference.

Why a filament's resistance rises: a larger current heats the metal, so its ions vibrate with greater amplitude and the electrons collide with them more often.

• The resistance of a light-dependent resistor (LDR) decreases as light intensity increases.
• The resistance of an NTC thermistor decreases as its temperature increases.

Resistivity ρ is a property of the material: the resistance of a 1 m length with 1 m² cross-section. Unit: Ω m. A wire's resistance increases with length and decreases with cross-sectional area.

Exam-style questions

Real sources have internal resistance, Kirchhoff's two laws let you solve any network, and the potential divider is the basis of every sensor circuit.

E.m.f. and internal resistance

Electromotive force (e.m.f.)

The energy transferred from other forms (such as chemical) to electrical energy per unit charge, by the source, when charge passes round a complete circuit.

Potential difference

The energy transferred from electrical to other forms per unit charge, in a component.

Internal resistance

The resistance of the source itself. Some energy is transferred to thermal energy inside the source.

The terminal p.d. V is less than the e.m.f. by the "lost volts" Ir. The more current drawn, the lower the terminal p.d. A graph of V against I is a straight line with y-intercept E and gradient −r.

Kirchhoff's laws

First law

The sum of the currents entering a junction equals the sum of the currents leaving it. This is a consequence of conservation of charge.

Second law

Around any closed loop in a circuit, the sum of the e.m.f.s equals the sum of the p.d.s. This is a consequence of conservation of energy.

The series and parallel formulas follow from these laws. For series resistors the current is the same, so V = IR₁ + IR₂. For parallel resistors the p.d. is the same, so I = V/R₁ + V/R₂.

Potential dividers

Two resistors in series across a supply share its p.d. in the ratio of their resistances:

Replacing one resistor with a thermistor or LDR makes a sensor: for example, with an LDR as R₂, more light means a lower R₂ and so a lower output p.d.

The potentiometer

A potentiometer is a uniform resistance wire with a steady current from a driver cell, so the p.d. along it is proportional to length. To compare two p.d.s, connect each in turn (through a galvanometer) to a sliding contact and find the balance (null) point where the galvanometer reads zero. Then V₁/V₂ = L₁/L₂. At balance no current is drawn from the cell being measured, so its e.m.f. is measured without any lost volts.

Exam-style questions

The atom has a tiny, dense nucleus; nuclei can be unstable and emit radiation; and protons and neutrons are themselves made of quarks. This topic links all three.

The nuclear atom

In the α-particle scattering experiment, a thin gold foil in a vacuum was bombarded with α-particles:

A nuclide is written AZX, where A is the nucleon number (protons + neutrons) and Z the proton number. Isotopes are atoms of the same element with the same number of protons but different numbers of neutrons. A nucleus has a diameter of about 10⁻¹⁵ to 10⁻¹⁴ m, compared with about 10⁻¹⁰ m for the atom.

Radioactive decay

In every nuclear process, nucleon number and charge are conserved.

β particles have a continuous range of energies because the energy released in each decay is shared with a second particle: an antineutrino in β⁻ decay and a neutrino in β⁺ decay. These particles have almost no mass and no charge.

Example decay equations:

• α: 23892U → 23490Th + 42α
• β⁻: 146C → 147N + 0−1e + ν
• β⁺: 116C → 115B + 0+1e + ν

Fundamental particles

Quarks are fundamental particles. You need three of the six flavours. Each quark has an antiquark with the opposite charge.

The other three are charm, top and bottom.

Hadrons

Particles made of quarks, which feel the strong force. Baryons are three quarks (proton = uud, neutron = udd); mesons are a quark and an antiquark.

Leptons

Fundamental particles that are not made of quarks, such as the electron and the neutrino.

Antiparticles

Every particle has an antiparticle with the same mass and opposite charge (the positron is the antiparticle of the electron).

Quark changes in β decay: in β⁻ decay a down quark changes into an up quark (a neutron becomes a proton): d → u + e⁻ + ν. In β⁺ decay an up quark changes into a down quark (a proton becomes a neutron): u → d + e⁺ + ν. Both are caused by the weak interaction.

Exam-style questions

The AS papers make up the other half of the A Level: Paper 1 is 15.5%, Paper 2 is 23% and Paper 3 is 11.5%.

Topics

Revising A Level physics

• Learn definitions word for word. Gravitational potential, electric field strength, the tesla, simple harmonic motion and binding energy are asked for again and again, and each phrase earns a mark.
• Spot the shared maths. Capacitor discharge and radioactive decay are both exponential decays; gravitational and electric fields follow the same inverse-square pattern.
• Practise with logarithms. Paper 5 expects you to linearise relationships such as x = x₀e^−kt by plotting ln x against t.

An object moving in a circle at constant speed is accelerating, because its direction keeps changing. The acceleration, and the resultant force that causes it, point towards the centre.

Angles and angular speed

Radian

The angle subtended at the centre of a circle by an arc equal in length to the radius. A full circle is 2π rad.

Angular displacement

The angle through which an object has moved around a circle, in radians.

Angular speed ω

The rate of change of angular displacement, in rad s⁻¹.

Centripetal acceleration and force

Velocity is a vector. Moving in a circle at constant speed, the direction of the velocity changes continuously, so the object is accelerating. This centripetal acceleration is directed towards the centre of the circle, perpendicular to the velocity, so it changes the direction but not the speed.

By Newton's second law, a resultant force towards the centre is needed: the centripetal force. It is not a new kind of force; it is provided by an existing one:

• a car on a flat bend: friction between the tyres and the road;
• a satellite in orbit: gravitational force;
• a ball whirled on a string: tension (plus a component of the weight in a vertical circle);
• an electron orbiting in a magnetic field: the magnetic force.

Because the force is perpendicular to the velocity, it does no work, and the kinetic energy stays constant.

Vertical circles

For a mass on a string in a vertical circle, at the top both tension and weight point to the centre: T + mg = mv²/r. At the bottom the tension points up and the weight down: T − mg = mv²/r. So the tension is greatest at the bottom, and the string is most likely to go slack at the top.

Exam-style questions

Every mass attracts every other mass. This topic describes that attraction as a field, uses it to explain orbits, and introduces gravitational potential, which is always negative.

The gravitational field

A gravitational field is a region of space where a mass experiences a force. Gravitational field strength g at a point is the gravitational force per unit mass at that point, in N kg⁻¹. Field lines show the direction of the force on a mass. Close to the Earth's surface the field is approximately uniform: parallel, equally spaced lines, with g = 9.81 N kg⁻¹.

Newton's law of gravitation

The gravitational force between two point masses is proportional to the product of their masses and inversely proportional to the square of their separation. For a uniform sphere, outside the sphere it acts as if all its mass were concentrated at its centre, so r is measured from the centre.

Orbits

For a satellite in a circular orbit, gravity provides the centripetal force:

GMmr² = mv²r, so v² = GMr. Using v = 2πr/T gives T² = 4π²GMr³, so T² ∝ r³ (Kepler's third law).

A geostationary satellite stays above the same point on the Earth. It must have a period of 24 hours, orbit above the equator, and move in the same direction as the Earth rotates (west to east). This fixes its orbital radius at about 4.2 × 10⁷ m. Ground aerials do not need to track it.

Gravitational potential

The gravitational potential at a point is the work done per unit mass in bringing a small test mass from infinity to that point. It is zero at infinity and negative everywhere else, because gravity is attractive: work is done by the field as the mass moves in. The potential energy of a mass m at that point is E_p = mφ.

When a mass moves from radius r₁ to r₂, the change in potential energy is ΔE_p = GMm(1r₁ − 1r₂). Near the Earth's surface this reduces to mgΔh.

The field strength is the negative of the potential gradient: g = −ΔφΔr.

Exam-style questions

Temperature decides which way thermal energy flows. This topic covers the absolute temperature scale and how much energy it takes to warm a substance or change its state.

Thermal equilibrium

Thermal energy is transferred from a region of higher temperature to a region of lower temperature. Objects in thermal equilibrium are at the same temperature, and there is no net transfer of thermal energy between them.

Temperature scales

A practical thermometer relies on a physical property that varies with temperature, such as the resistance of a thermistor, the e.m.f. of a thermocouple, or the expansion of a liquid. The thermodynamic (Kelvin) scale does not depend on the property of any particular substance. Its zero is absolute zero, the temperature at which a substance has minimum internal energy.

A temperature change of 1 K is the same as a change of 1 °C.

Specific heat capacity and specific latent heat

Specific heat capacity

The energy needed per unit mass to raise the temperature of a substance by one degree (1 K). Unit: J kg⁻¹ K⁻¹.

Specific latent heat

The energy needed per unit mass to change the state of a substance without any change in temperature. Unit: J kg⁻¹. Of fusion for melting, of vaporisation for boiling.

During a change of state the temperature stays constant: the energy supplied increases the potential energy of the molecules (breaking bonds), not their kinetic energy. The specific latent heat of vaporisation is much larger than that of fusion because, on boiling, the molecules must be separated completely and work is done pushing back the atmosphere as the volume increases greatly.

Measuring specific heat capacity: the electrical method

1. Place an electrical heater and a thermometer in holes in a metal block (or in a liquid), and insulate it.
2. Measure the mass with a balance. Record the starting temperature.
3. Switch on the heater for a measured time t, recording the p.d. V and current I (or use a joulemeter).
4. Record the highest temperature reached. Energy supplied = VIt = mcΔθ, so c = VItmΔθ.
5. Energy losses to the surroundings make the value of c too high. To reduce them: insulate, or start below room temperature and finish the same amount above it.

Exam-style questions

A gas is a huge number of molecules moving at random. The ideal gas equation describes its large-scale behaviour, and kinetic theory explains that behaviour from the motion of the molecules.

The mole

The mole is the SI unit of amount of substance. One mole contains N_A = 6.02 × 10²³ particles (the Avogadro constant). For n moles, the number of molecules is N = nN_A.

The equation of state

An ideal gas is one that obeys pV ∝ T exactly, at all pressures, volumes and temperatures, where T is the thermodynamic temperature in kelvin. Real gases behave nearly ideally at low pressure and high temperature.

The Boltzmann constant is k = R/N_A. The root-mean-square speed is c_rms = √⟨c²⟩.

Kinetic theory

Assumptions

• A gas contains a very large number of molecules moving randomly.
• The volume of the molecules is negligible compared with the volume of the container.
• There are no forces between molecules except during collisions.
• Collisions between molecules and with the walls are perfectly elastic.
• The time spent in collisions is negligible compared with the time between collisions.

Deriving the pressure

1. A molecule of mass m moving with velocity component c_x hits a wall and rebounds: its change of momentum is 2mc_x.
2. In a cube of side l, it returns to the same wall every 2l/c_x, so the force on the wall is 2mc_x2l/c_x = mc_x²l.
3. For N molecules, total force = Nm⟨c_x²⟩l, and pressure = forcel² = Nm⟨c_x²⟩V.
4. Motion is random, so ⟨c_x²⟩ = ⟨c_y²⟩ = ⟨c_z²⟩ = 13⟨c²⟩, giving pV = 13Nm⟨c²⟩.

Linking to temperature: comparing pV = 13Nm⟨c²⟩ with pV = NkT gives 13m⟨c²⟩ = kT, so the mean kinetic energy of a molecule is 12m⟨c²⟩ = 32kT. The thermodynamic temperature is proportional to the mean kinetic energy of the molecules.

Exam-style questions

Internal energy is the total energy of the molecules in a system. The first law of thermodynamics is conservation of energy applied to it: heat it or do work on it, and its internal energy rises.

Internal energy

The internal energy of a system is the sum of a random distribution of the kinetic and potential energies of its molecules. A rise in temperature increases the kinetic energy of the molecules, so it increases the internal energy.

In an ideal gas there are no intermolecular forces, so the molecules have no potential energy. The internal energy is purely kinetic, U = N × 32kT, and so it is proportional to the thermodynamic temperature.

The first law of thermodynamics

ΔU

The increase in internal energy of the system.

q

The energy transferred to the system by heating (negative if the system loses thermal energy).

W

The work done on the system. When a gas is compressed, work is done on it and W is positive; when it expands, it does work on the surroundings and W is negative.

Exam-style questions

Simple harmonic motion describes pendulums, masses on springs and vibrating molecules. Its defining equation leads to sinusoidal graphs, a constant total energy, and the dramatic effect of resonance.

Describing oscillations

Displacement x

The distance from the equilibrium position in a given direction.

Amplitude x₀

The maximum displacement.

Period T

The time for one complete oscillation.

Frequency f

The number of oscillations per unit time.

Angular frequency ω

ω = 2πf = 2π/T, in rad s⁻¹.

Phase difference

The fraction of a cycle by which one oscillation leads another, in radians or degrees.

Simple harmonic motion

Simple harmonic motion is motion in which the acceleration is proportional to the displacement from a fixed point and is always directed towards that point.

• The velocity is maximum at the equilibrium position and zero at the extremes.
• The acceleration is zero at equilibrium and maximum (magnitude ω²x₀) at the extremes.
• Velocity leads displacement by a quarter cycle (π/2); acceleration is in antiphase with displacement (π).
• A graph of a against x is a straight line through the origin with gradient −ω².

Energy in s.h.m.

Energy is continuously exchanged between kinetic and potential energy, while the total stays constant (if there is no damping). Kinetic energy = 12mω²(x₀² − x²); potential energy = 12mω²x². Both graphs against displacement are parabolas; their sum is a horizontal line.

Damping and resonance

Damping is the loss of energy from an oscillating system, usually by resistive forces, so the amplitude decreases.

• Light damping: the system oscillates with amplitude decreasing gradually (exponentially) over many cycles; the period stays almost constant.
• Critical damping: the system returns to equilibrium in the shortest possible time without oscillating (as in car suspension).
• Heavy damping: the system returns to equilibrium very slowly, without oscillating.

In forced oscillations, a periodic driving force makes a system oscillate at the driving frequency. Resonance is when the amplitude of the forced oscillation is a maximum, which happens when the driving frequency equals the natural frequency of the system. At resonance, energy is transferred most efficiently from the driver.

Resonance can be useful (tuning a radio circuit, magnetic resonance imaging, musical instruments) or destructive (bridges and buildings driven by wind or earthquakes, rattling vehicle parts), which is why structures are damped.

Exam-style questions

Electric fields follow the same inverse-square pattern as gravitational fields, with one big difference: charges can repel as well as attract. This topic covers uniform fields between plates, Coulomb's law and electric potential.

Electric fields and field lines

An electric field is a region of space where a charge experiences a force. Electric field strength is the force per unit positive charge, in N C⁻¹ (equivalently V m⁻¹). Field lines show the direction of the force on a positive charge: away from positive charges, towards negative charges.

Uniform fields

Between two parallel plates with a p.d. across them, the field is uniform (except near the edges): field lines are parallel, equally spaced and perpendicular to the plates. A charged particle entering the field at right angles moves in a parabola, like a projectile: constant velocity parallel to the plates, and constant acceleration a = qE/m towards one plate.

Point charges

Coulomb's law: the force between two point charges is proportional to the product of the charges and inversely proportional to the square of their separation. For a charged conducting sphere, outside the sphere the charge acts as if concentrated at its centre.

Electric potential

The electric potential at a point is the work done per unit positive charge in bringing a small test charge from infinity to that point. It is positive near a positive charge and negative near a negative charge. The field strength equals the negative of the potential gradient: E = −ΔVΔr.

Exam-style questions

A capacitor stores charge, and with it energy. When it discharges through a resistor the charge, current and p.d. all fall exponentially, governed by one number: the time constant RC.

Capacitance

Capacitance is the charge stored per unit potential difference. A capacitor stores equal and opposite charges on two plates separated by an insulator (dielectric); the charge Q means the charge on one plate. One farad is one coulomb per volt. The idea also applies to an isolated conducting sphere.

The series and parallel formulas are the reverse of those for resistors. In parallel, the p.d. is the same and the charges add. In series, the charge on each is the same and the p.d.s add.

Energy stored

On a graph of p.d. V against charge Q for a capacitor, the line is straight through the origin (since V = Q/C). The work done to charge it is the area under this graph, a triangle: W = 12QV. Substituting Q = CV gives W = 12CV². When a capacitor is charged from a battery, the battery supplies energy QV; only half is stored, and half is dissipated in the resistance of the circuit.

Discharging a capacitor

When a charged capacitor discharges through a resistor R, the current is proportional to the p.d., which is proportional to the remaining charge. So the rate of loss of charge is proportional to the charge: an exponential decay. The time constant τ = RC is the time for the charge (or p.d., or current) to fall to 1/e (about 37%) of its initial value. It takes 0.69RC to halve.

Taking natural logs: ln Q = ln Q₀ − tRC, so a graph of ln Q (or ln V) against t is a straight line of gradient −1RC.

Exam-style questions

Magnetic fields push on moving charges and on currents, and changing magnetic flux induces an e.m.f. Together these explain motors, particle beams, Hall probes and generators.

Magnetic fields

A magnetic field is a region of space where a magnet, a current-carrying conductor or a moving charge experiences a force. Field lines run from north to south outside a magnet; closer lines mean a stronger field.

Fields due to currents

• Long straight wire: concentric circles around the wire, getting further apart with distance. Direction given by the right-hand grip rule (thumb along the current, fingers curl with the field).
• Flat circular coil: like that of a short bar magnet, with field lines through the centre of the coil.
• Long solenoid: a strong, uniform field inside, like a bar magnet outside. Inserting a ferrous (iron) core greatly increases the field strength.

Two parallel wires carrying currents in the same direction attract; in opposite directions they repel. Each wire sits in the other's field.

Force on a current-carrying conductor

The direction of the force is given by Fleming's left-hand rule: First finger = Field, seCond finger = Current, thuMb = Motion (force). There is no force when the current is parallel to the field.

Magnetic flux density B is the force per unit length per unit current on a conductor placed at right angles to the field. One tesla is the flux density when a wire carrying 1 A at right angles to the field experiences a force of 1 N per metre of its length (1 T = 1 N A⁻¹ m⁻¹).

Force on a moving charge

A charge moving at right angles to a uniform magnetic field feels a force perpendicular to its velocity, so it moves in a circle: BQv = mv²r, giving r = mvBQ. Faster or heavier particles make bigger circles; a stronger field makes smaller ones. The force does no work, so the speed is constant.

Velocity selection: crossed electric and magnetic fields exert opposite forces. Only particles with EQ = BQv, that is v = E/B, pass through undeflected.

The Hall effect: when a current flows through a thin slice of conductor in a magnetic field perpendicular to it, the magnetic force pushes the charge carriers towards one side. Charge builds up until the electric force from it balances the magnetic force, producing a steady Hall voltage across the slice. Since V_H ∝ B, a Hall probe measures flux density. Semiconductors give a larger Hall voltage than metals because their number density of charge carriers n is much smaller.

Electromagnetic induction

Magnetic flux

The product of the flux density and the area perpendicular to the field, Φ = BA. Unit: weber (Wb).

Magnetic flux linkage

The product of the flux through a coil and its number of turns, NΦ.

Faraday's law

The induced e.m.f. is proportional to the rate of change of magnetic flux linkage.

Lenz's law

The direction of the induced e.m.f. is such that it tends to oppose the change producing it. This is a consequence of conservation of energy: work must be done to produce the induced current.

An e.m.f. is induced whenever the flux linking a circuit changes: moving a magnet into a coil, moving a wire across a field, rotating a coil in a field, or changing the current in a nearby coil. Experiments show the e.m.f. is larger if the change is faster, the field stronger, or the coil has more turns.

Exam-style questions

Mains electricity is alternating: the current reverses direction many times a second. This topic covers how to describe an a.c. supply, why we quote r.m.s. values, and how diodes and a capacitor turn a.c. into steady d.c.

Describing alternating current

Period

The time for one complete cycle.

Frequency

The number of cycles per unit time.

Peak value

The maximum value of the current or p.d., I₀ or V₀.

Root-mean-square (r.m.s.) value

The value of the direct current (or p.d.) that would produce the same mean power in a resistor as the alternating current (or p.d.).

For a sinusoidal supply, the mean power in a resistor is half the peak power, because the power I²R varies between 0 and I₀²R and the mean of sin ² is ½. Mains voltages are quoted as r.m.s. values: a 230 V supply has a peak value of 230√2 = 325 V.

Rectification and smoothing

• Half-wave rectification uses a single diode in series with the load. The current flows only during the half-cycles when the diode is forward-biased.
• Full-wave rectification uses a bridge rectifier of four diodes. In each half-cycle a different pair of diodes conducts, so the current always passes through the load in the same direction.
• Smoothing: a capacitor connected in parallel with the load charges up as the p.d. rises, then, as the supply p.d. falls, discharges through the load, keeping the output p.d. fairly steady. The remaining variation is the ripple. A larger time constant RC (larger capacitor or larger load resistance) gives less ripple.

Exam-style questions

Light behaves as particles called photons, and electrons behave as waves. The photoelectric effect, electron diffraction and atomic line spectra are the three pieces of evidence you need to know.

Photons

Electromagnetic radiation has a particulate nature: it is emitted and absorbed in discrete packets of energy called photons. A photon's energy depends only on its frequency.

The electronvolt is the energy gained by an electron accelerated through a p.d. of 1 V: 1 eV = 1.60 × 10⁻¹⁹ J. A photon of wavelength 500 nm has energy hc/λ = 3.98 × 10⁻¹⁹ J = 2.49 eV.

The photoelectric effect

When electromagnetic radiation of high enough frequency falls on a metal surface, electrons are emitted. The observations:

• No electrons are emitted below a threshold frequency, however intense the radiation.
• Above it, emission is instantaneous, even for very weak radiation.
• The maximum kinetic energy of the electrons depends on the frequency, not on the intensity.
• The number of electrons emitted per second (the photoelectric current) is proportional to the intensity.

Explanation: each electron absorbs the energy of one photon. It needs at least the work function Φ (the minimum energy needed to release an electron from the surface) to escape, so the photon energy hf must be at least Φ: the threshold frequency is f₀ = Φ/h. Any extra energy becomes kinetic energy; electrons from deeper in the metal lose some on the way out, which is why the equation gives the maximum kinetic energy. Greater intensity means more photons per second, so more electrons, but not more energy per electron.

The wave theory fails: a wave's energy depends on its intensity, so it predicts that intense light of any frequency would eventually release electrons, with energy increasing with intensity, and with a time delay at low intensity. None of these happen.

Wave-particle duality

Electron diffraction is evidence that particles have a wave nature. A beam of electrons passed through a thin film of polycrystalline graphite produces a pattern of concentric rings on a fluorescent screen, which is a diffraction pattern. Increasing the accelerating p.d. increases the electrons' momentum, so their wavelength decreases and the rings get smaller.

Any particle of momentum p has a de Broglie wavelength λ = h/p. For an electron accelerated from rest through p.d. V: 12mv² = eV, so p = √2meV.

Energy levels and line spectra

Electrons in an isolated atom can only have certain discrete energy levels. When an electron falls from a higher level E₁ to a lower level E₂, a photon of energy hf = E₁ − E₂ is emitted. Because only certain differences are possible, only certain frequencies appear: an emission line spectrum of bright lines. When white light passes through a cool gas, photons with exactly these energies are absorbed to excite electrons, giving an absorption line spectrum: dark lines on a continuous spectrum.

Exam-style questions

A nucleus has less mass than the protons and neutrons it is made from; the missing mass is the binding energy. This explains why fission and fusion release energy. The second half of the topic treats radioactive decay mathematically.

Mass and energy

Mass defect

The difference between the total mass of the separate nucleons and the mass of the nucleus.

Binding energy

The minimum energy needed to separate a nucleus into its individual protons and neutrons. Binding energy = mass defect × c².

Binding energy per nucleon

The binding energy divided by the nucleon number. A higher value means a more stable nucleus.

Fission is the splitting of a large nucleus (such as uranium-235, after absorbing a neutron) into two smaller nuclei and a few neutrons. Fusion is the joining of two light nuclei (such as hydrogen isotopes) to form a heavier one; it needs very high temperatures to overcome the electrostatic repulsion. In both, the total mass of the products is less than the total mass before, and the mass difference appears as energy, ΔE = Δmc².

Radioactive decay

Radioactive decay is spontaneous (not affected by external factors such as temperature or pressure) and random (it is impossible to predict which nucleus will decay next, or when; each nucleus has a constant probability of decaying per unit time). Evidence for randomness: the count rate from a source fluctuates.

Activity

The number of nuclei decaying per unit time, in becquerels (1 Bq = 1 decay per second).

Decay constant λ

The probability that a nucleus decays per unit time.

Half-life

The mean time taken for half of the active nuclei in a sample to decay.

Because A = λN, the number of nuclei, the activity and the count rate all decay exponentially with the same half-life. After n half-lives, the fraction remaining is (12)ⁿ. Plotting ln A against t gives a straight line of gradient −λ.

Exam-style questions

Three ways of seeing inside the body: ultrasound reflects from boundaries between tissues, X-rays are absorbed differently by bone and soft tissue, and PET detects gamma photons from a radioactive tracer.

Ultrasound

Ultrasound is sound with a frequency above the range of human hearing (above 20 kHz); medical scans use about 1 to 10 MHz.

The piezoelectric transducer

• Generating: an alternating p.d. applied across a piezoelectric crystal makes it change shape (expand and contract) at the same frequency, so it vibrates and emits ultrasound. The crystal's thickness is chosen so it resonates at the required frequency.
• Detecting: when reflected ultrasound hits the crystal, it is made to vibrate, and the changing shape produces an alternating p.d. across it, which is processed.

Reflection at boundaries

The specific acoustic impedance Z of a medium is its density × the speed of sound in it, unit kg m⁻² s⁻¹. When ultrasound meets a boundary, the fraction of intensity reflected depends on the difference in Z. A large difference means strong reflection.

Why gel is used: air and skin have very different impedances, so almost all ultrasound would be reflected at the skin. A gel with an impedance close to skin's fills the gap and removes the air layer, so most of the ultrasound enters the body.

In an A-scan, the time between sending a pulse and receiving each echo gives the depth of each boundary: depth = 12ct. A B-scan combines many A-scans taken at different angles into a two-dimensional image.

X-rays

In an X-ray tube, electrons emitted from a heated cathode are accelerated through a large p.d. (tens of kilovolts) and strike a metal target (such as tungsten). Most of their energy becomes thermal energy, so the anode is cooled, often by rotating it. About 1% becomes X-rays:

• Braking radiation: electrons decelerating in the target emit a continuous spectrum of X-ray photons. The highest-energy photon carries all of one electron's kinetic energy eV, which sets a minimum wavelength λ_min = hc/eV.
• Characteristic lines: sharp peaks at wavelengths set by electron energy levels in the target atoms.

A higher accelerating p.d. gives "harder" X-rays (higher photon energies, more penetrating). Increasing the tube current gives more photons, so a more intense beam, but does not change the photon energies.

Image quality. Sharpness improves with a smaller target area (a near point source) and with a lead grid that stops scattered X-rays. Contrast depends on the difference in attenuation between tissues: bone attenuates far more than soft tissue. Contrast media such as barium (high attenuation) are used to show soft tissues such as the digestive tract.

CT scanning: an X-ray tube and detectors rotate around the patient, taking images of a thin slice from many angles. A computer combines them to find the attenuation in each small volume element (voxel) of the slice. Repeating for many slices builds a three-dimensional image that can be viewed from any angle.

PET scanning

1. A tracer is injected: a substance used by the body (such as a glucose analogue, FDG) labelled with a β⁺-emitting isotope such as fluorine-18, with a short half-life.
2. The tracer collects where it is used most, for example in active tumours.
3. Each emitted positron travels a very short distance before meeting an electron. They annihilate, turning their mass into two gamma photons.
4. To conserve momentum, the two photons travel in opposite directions, each with energy m_ec² = 0.51 MeV (8.2 × 10⁻¹⁴ J).
5. A ring of detectors around the patient detects both photons. The line between the two detectors, and the small difference in arrival times, locate where the annihilation happened. Many events build up an image of tracer concentration.

Exam-style questions

How do we measure the size of a star we can't resolve, or the distance to a galaxy? Luminosity, the inverse-square law, the laws of black-body radiation and redshift together answer both questions, and point to an expanding universe.

Luminosity and standard candles

Luminosity

The total power of the electromagnetic radiation emitted by a star, in watts.

Radiant flux intensity

The radiant power passing normally through unit area, in W m⁻². This is what we measure from Earth.

Standard candle

An astronomical object of known luminosity. Examples: Cepheid variable stars (whose period of brightness variation is linked to their luminosity) and type 1a supernovae (which all reach about the same peak luminosity).

Radiation spreads out equally in all directions over the surface of a sphere of area 4πd². If we know a standard candle's luminosity and measure its flux, we can find its distance: d = √L/4πF.

Stellar radii

A star behaves approximately as a black body. Wien's displacement law: the wavelength at which the intensity of radiation is greatest is inversely proportional to the thermodynamic temperature, λ_maxT = constant (about 2.9 × 10⁻³ m K). So a star's colour gives its surface temperature: hotter stars are bluer.

The Stefan–Boltzmann law links luminosity to surface temperature and radius. Knowing L (from a standard candle or flux and distance) and T (from Wien's law), the radius is r = √L4πσT⁴.

Hubble's law and the Big Bang

Spectral lines in light from distant galaxies are shifted to longer wavelengths: a redshift. This is the Doppler effect for light from a source moving away. The fractional shift gives the recession speed: v ≈ cΔλ/λ (for speeds much less than c).

Hubble's law: the speed at which a galaxy is receding is proportional to its distance from us, v ≈ H₀d, where H₀ is the Hubble constant (about 2.2 × 10⁻¹⁸ s⁻¹).

Since more distant galaxies recede faster, and in every direction, the universe is expanding. Running the expansion backwards suggests it began from an extremely small, hot, dense state: the Big Bang. A rough age for the universe is 1/H₀.

Exam-style questions