GCSE Physics (AQA) formulas

Energy

Energy stores, energy changes and specific heat capacity

• kinetic energy = ½ × mass × speed² (E_k = ½mv²), mass in kg, speed in m/s, energy in J
• gravitational potential energy = mass × gravitational field strength × height (E_p = mgh), with g = 9.8 N/kg
• elastic potential energy = ½ × spring constant × extension² (E_e = ½ke²), k in N/m, e in m
• change in thermal energy = mass × specific heat capacity × temperature change (ΔE = mcΔθ), c in J/kg °C
• Doubling the speed makes the kinetic energy 4 times bigger, because speed is squared
• With no air resistance, E_p lost in a fall = E_k gained
• In the specific heat capacity practical, insulate the block so less energy escapes to the surroundings

Power, conservation and dissipation of energy, and efficiency

• power = energy transferred ÷ time (P = E/t) and power = work done ÷ time (P = W/t)
• 1 W = 1 J/s; time must be in seconds; 1 kW = 1000 W
• efficiency = useful output energy transfer ÷ total input energy transfer
• efficiency = useful power output ÷ total power input
• Efficiency is never more than 1 (100%)
• The total energy of a closed system stays the same
• Higher thermal conductivity means a faster rate of energy transfer by conduction; thicker walls mean a slower rate

National and global energy resources

• Non-renewable: coal, oil, gas, nuclear fuel. Renewable: bio-fuel, wind, hydro-electricity, geothermal, tides, the Sun, water waves
• Burning fossil fuels releases carbon dioxide, which contributes to global warming
• Nuclear power releases no carbon dioxide while generating, but leaves radioactive waste that must be stored safely for a very long time
• Wind, the Sun and waves depend on the weather, so they are not reliable
• Tides are not constant but they are predictable; hydro-electricity can be switched on quickly when needed
• Geothermal energy comes from hot rocks underground, heated by the decay of radioactive elements
• Most road transport uses petrol and diesel, which are made from oil

Electricity

Current, potential difference, resistance and I–V characteristics

• charge flow = current × time (Q = It), charge in coulombs (C), current in amperes (A), time in seconds
• potential difference = current × resistance (V = IR), resistance in ohms (Ω)
• Ohmic conductor at constant temperature: current is directly proportional to p.d. (straight line through the origin)
• Filament lamp: resistance increases as temperature increases, so the I–V graph curves and gets less steep
• Diode: very high resistance in the reverse direction, so current flows one way only
• Thermistor: resistance decreases as temperature increases (used in thermostats)
• LDR: resistance decreases as light intensity increases (used to switch lights on when it gets dark)

Series and parallel circuits

• Series: same current through every component
• Series: total p.d. of the supply = sum of the p.d.s across the components
• Series: total resistance = R₁ + R₂
• Parallel: same p.d. across every branch
• Parallel: total current = sum of the currents in the branches
• Parallel: total resistance is less than the smallest individual resistance
• Use V = IR for the whole circuit (total V, total R) or for one component (its own V and R)

Mains electricity, power and the National Grid

• Live wire: brown, about 230 V. Neutral wire: blue, at or close to 0 V. Earth wire: green and yellow stripes, 0 V
• power = potential difference × current (P = VI)
• power = current² × resistance (P = I²R)
• energy transferred = power × time (E = Pt), time in seconds
• energy transferred = charge flow × potential difference (E = QV)
• Step-up transformers increase the p.d. for transmission; step-down transformers decrease it to a safer value for homes
• The live wire is dangerous even when the switch is open, because it is still at about 230 V

Static electricity and electric fields

• Gaining electrons gives a negative charge; losing electrons gives an equal positive charge
• Like charges repel; unlike charges attract
• The electric force is a non-contact force, and it gets weaker as the distance increases
• Field lines show the direction of the force on a positive charge: away from a positive sphere, towards a negative sphere
• Around an isolated charged sphere the field is radial; lines closer together mean a stronger field
• A spark happens when the p.d. between a charged object and an earthed conductor is large enough for charge to flow through the air
• Connecting an object to earth lets the charge flow away, so no large p.d. builds up

Particle model of matter

Density and changes of state

• density = mass ÷ volume (ρ = m/V)
• Units: kg/m³ (mass in kg, volume in m³) or g/cm³ (mass in g, volume in cm³)
• 1 g/cm³ = 1000 kg/m³
• Regular solid: volume = length × width × height, mass from a balance
• Irregular solid: volume = volume of water it displaces (displacement can or the rise in a measuring cylinder)
• Liquid: mass = mass of cylinder with liquid − mass of empty cylinder; read the volume from the cylinder
• Mass is conserved in a change of state

Internal energy, specific heat capacity and specific latent heat

• change in thermal energy = mass × specific heat capacity × temperature change (ΔE = mcΔθ), c in J/kg °C
• energy for a change of state = mass × specific latent heat (E = mL), L in J/kg
• Specific latent heat: the energy needed to change the state of 1 kg of a substance with no change in temperature
• Fusion: solid ⇌ liquid. Vaporisation: liquid ⇌ vapour
• Sloping part of a heating graph: temperature changing, use mcΔθ. Flat part: change of state, use mL
• More mass at the same temperature means more internal energy
• When a substance condenses or freezes it releases the same latent heat

Particle model and pressure in gases

• Higher temperature means higher average kinetic energy and higher average speed of the molecules
• Constant volume: raising the temperature raises the pressure
• Fixed mass at constant temperature: pressure × volume = constant (pV = constant)
• So p₁ × V₁ = p₂ × V₂; pressure in Pa, volume in m³ (or the same units on both sides)
• Halving the volume doubles the pressure; doubling the volume halves the pressure
• Work done on a gas increases its internal energy, which can raise its temperature (a bicycle pump gets warm)

Atomic structure

Atoms, isotopes and the atomic model

• Radius of an atom is about 1 × 10⁻¹⁰ m; the radius of the nucleus is less than 1/10 000 of this.
• Atomic number = number of protons. Mass number = protons + neutrons.
• Number of neutrons = mass number − atomic number.
• A neutral atom has equal numbers of protons and electrons.
• Most alpha particles went straight through the gold foil: the atom is mostly empty space.
• A few alpha particles bounced back: the mass and positive charge are concentrated in a tiny nucleus.
• Bohr: electrons orbit at specific distances (energy levels). Chadwick: evidence for neutrons in the nucleus.

Radioactive decay, half-life and contamination

• Alpha: most ionising, least penetrating; stopped by paper or a few centimetres of air.
• Beta: stopped by a few millimetres of aluminium; range in air about a metre.
• Gamma: least ionising, most penetrating; reduced by thick lead or concrete.
• Alpha decay: mass number falls by 4, atomic number falls by 2.
• Beta decay: mass number stays the same, atomic number rises by 1 (a neutron becomes a proton).
• Gamma emission: no change to the mass or the charge of the nucleus.
• After n half-lives the fraction remaining is (½)ⁿ: ½, ¼, ⅛, 1/16 ...

Background radiation, uses, fission and fusion

• 1 sievert (Sv) = 1000 millisieverts (mSv).
• Medical tracer: a gamma emitter (it passes out of the body) with a half-life of a few hours.
• Radiotherapy: gamma beams are aimed from several directions so the tumour gets a high dose and healthy tissue a lower dose.
• Fission: a uranium-235 nucleus absorbs a neutron, then splits into two smaller nuclei of roughly equal size, plus two or three neutrons and gamma rays.
• Chain reaction: the neutrons released go on to cause more fissions.
• Control rods absorb neutrons to control the rate of fission in a reactor.
• Fusion: two light nuclei join to form a heavier nucleus; some mass is converted into energy.

Forces

Scalars, vectors, weight and resultant forces

• weight = mass × gravitational field strength (W = mg); W in N, m in kg, g in N/kg.
• On Earth, g = 9.8 N/kg. Mass stays the same everywhere; weight changes with g.
• Weight is measured with a newtonmeter (a calibrated spring balance).
• Forces in the same direction: add. Forces in opposite directions: subtract.
• Two forces at right angles: resultant² = F₁² + F₂², or use a scale drawing.
• A force can be resolved into two components at right angles that together have the same effect as the force.

Work done and elasticity

• work done = force × distance moved along the line of action of the force (W = Fs).
• 1 joule = 1 newton-metre: 1 N moving an object 1 m.
• force = spring constant × extension (F = ke); k in N/m, e in m.
• extension = stretched length − original length.
• elastic potential energy = ½ × spring constant × extension² (E_e = ½ke²).
• On a force–extension graph, the straight line through the origin shows F is proportional to e; it ends at the limit of proportionality.

Moments, levers and gears

• moment = force × perpendicular distance from the pivot to the line of action (M = Fd); unit N m.
• Balanced object: total clockwise moment = total anticlockwise moment.
• A longer distance from the pivot gives a bigger moment for the same force.
• A small gear driving a larger gear: the larger gear turns more slowly but with a larger moment.
• A large gear driving a smaller gear: the smaller gear turns faster but with a smaller moment.

Pressure in fluids and the atmosphere

• pressure = force normal to the surface ÷ area (p = F/A); unit pascal, 1 Pa = 1 N/m².
• pressure due to a column of liquid = height × density × gravitational field strength (p = hρg).
• h in m, ρ in kg/m³, g = 9.8 N/kg, p in Pa.
• Floating at rest: upthrust = weight, and the object is less dense than the liquid.
• Sinking: the weight is greater than the upthrust; the object is denser than the liquid.
• Atmospheric pressure decreases as height increases because the weight of air above each unit area is smaller.

Describing motion and motion graphs

• distance = speed × time (s = vt).
• acceleration = change in velocity ÷ time taken (a = Δv/t); unit m/s².
• (final velocity)² − (initial velocity)² = 2 × acceleration × distance (v² − u² = 2as).
• Curved distance–time graph: speed at one time = gradient of the tangent at that time.
• Typical speeds: walking 1.5 m/s, running 3 m/s, cycling 6 m/s, sound in air 330 m/s.
• Free fall near the Earth: acceleration is about 9.8 m/s².
• Terminal velocity: the resistive force equals the weight, the resultant force is zero, and the speed is constant.

Newton's laws and stopping distances

• resultant force = mass × acceleration (F = ma); F in N, m in kg, a in m/s².
• inertial mass = force ÷ acceleration.
• stopping distance = thinking distance + braking distance.
• thinking distance = speed × reaction time.
• Thinking distance increases with speed, tiredness, alcohol, drugs and distractions.
• Braking distance increases with speed, wet or icy roads, worn tyres and worn brakes.
• For the same braking force, doubling the speed gives four times the braking distance.

Momentum

• momentum = mass × velocity (p = mv); unit kg m/s.
• Closed system: total momentum before = total momentum after.
• Objects that stick together: m₁u₁ + m₂u₂ = (m₁ + m₂)v.
• Explosion from rest: total momentum stays zero, so the two momenta are equal in size and opposite in direction.
• force = change in momentum ÷ time taken (F = mΔv/Δt).
• Longer time for the same change in momentum means a smaller force.

Waves

Transverse and longitudinal waves and wave properties

• Amplitude: the maximum displacement of a point on the wave from its undisturbed position, in m.
• Wavelength (λ): the distance from a point on one wave to the equivalent point on the next wave, in m.
• Frequency (f): the number of waves passing a point each second, in Hz.
• period = 1 ÷ frequency (T = 1/f), with T in s and f in Hz.
• wave speed = frequency × wavelength (v = fλ), in m/s.
• Ripple tank: frequency = number of waves ÷ time; wavelength = distance across several waves ÷ number of waves.
• Speed of sound in air: speed = distance ÷ time, using a large distance so the time is long enough to measure.

Reflection, sound and waves for detection and exploration

• Reflection: the wave bounces back. Absorption: its energy is transferred to the material. Transmission: it passes through.
• Human hearing range: 20 Hz to 20 kHz. Ultrasound has a frequency above 20 kHz.
• Echo: distance to the boundary = (speed × time) ÷ 2, because the pulse travels there and back.
• P-waves are longitudinal and travel through solids and liquids.
• S-waves are transverse and cannot travel through a liquid. P-waves travel faster than S-waves.
• The S-wave shadow zone is evidence that the Earth has a liquid outer core.
• Echo sounding (sonar) uses high frequency sound to find the depth of water or objects in it.

Electromagnetic waves: properties, uses and hazards

• Order from long to short wavelength: radio, microwaves, infrared, visible, ultraviolet, X-rays, gamma rays. Frequency increases in the same order.
• All travel at 3.0 × 10⁸ m/s in a vacuum; v = fλ applies.
• Slowing down at a boundary bends the ray towards the normal; speeding up bends it away. Frequency does not change.
• Radio waves are produced by oscillations in electrical circuits; when absorbed by an aerial they create an alternating current of the same frequency.
• Ultraviolet: premature skin ageing and skin cancer. X-rays and gamma rays: ionising, can cause gene mutation and cancer.
• Uses: radio (TV and radio), microwaves (satellites, cooking), infrared (heaters, cooking, infrared cameras), visible (fibre optics).
• Uses: ultraviolet (energy-efficient lamps, sun tanning), X-rays and gamma rays (medical imaging and treatment).

Lenses and visible light

• magnification = image height ÷ object height. It is a ratio, so it has no unit; use the same unit for both heights.
• Convex lens, object beyond the principal focus: real, inverted image. Object inside the focal length: virtual, upright, magnified image.
• Concave lens: the image is always virtual, upright and diminished.
• Ray diagram: a ray parallel to the principal axis passes through the principal focus; a ray through the centre of the lens goes straight on.
• Specular reflection: smooth surface, one direction. Diffuse reflection: rough surface, light scattered.
• An object that reflects all wavelengths equally looks white; one that absorbs all wavelengths looks black.
• Transparent: transmits light clearly. Translucent: transmits but scatters light. Opaque: transmits no light.

Black body radiation

• All bodies emit and absorb infrared radiation, at any temperature.
• Hotter body: more radiation emitted per second.
• Perfect black body: absorbs all incident radiation; best possible emitter.
• Higher temperature: greater intensity, and the peak of the emitted radiation moves to shorter wavelengths.
• Constant temperature: rate of absorbing radiation = rate of emitting radiation.
• Earth's temperature depends on the rates at which radiation is absorbed, emitted and reflected back into space.
• Matt black surfaces are the best absorbers and emitters; shiny surfaces are the worst.

Magnetism and electromagnetism

Magnets, magnetic fields and electromagnets

• Magnetic materials: iron, steel, cobalt and nickel.
• Field lines point from the north pole to the south pole outside the magnet; closer lines mean a stronger field.
• A compass needle is a small bar magnet; it points along the Earth's field, which is evidence that the Earth's core is magnetic.
• Straight wire: field lines are concentric circles around the wire; stronger with more current, weaker further away.
• Solenoid: strong, uniform field inside; outside it has the same shape as a bar magnet's field.
• Stronger electromagnet: more current, more turns, an iron core.
• An electromagnet can be switched on and off; reversing the current reverses its poles.

The motor effect, electric motors and loudspeakers

• force = magnetic flux density × current × length (F = BIl): F in N, B in tesla (T), I in A, l in m.
• F = BIl applies when the conductor is at right angles to the field.
• Left hand: First finger = Field (north to south), seCond finger = Current (positive to negative), thuMb = Motion (force).
• Bigger force: larger current, stronger field, longer length of wire in the field.
• Reversing the current or the field reverses the force; reversing both leaves it unchanged.
• The split-ring commutator reverses the current in the coil every half turn.
• Loudspeaker: an alternating current in a coil in a magnetic field makes the cone vibrate, producing pressure variations (sound).

Induced potential, generators, microphones and transformers

• Larger induced p.d.: faster movement, stronger magnetic field, more turns on the coil.
• Reversing the direction of movement or the poles of the magnet reverses the induced p.d.
• Alternator: slip rings, output is ac. Dynamo: split-ring commutator, output is dc that varies in size but not direction.
• Microphone: sound makes a diaphragm and coil move in a magnetic field, inducing a p.d. that varies like the sound.
• V_p ÷ V_s = n_p ÷ n_s (p.d. ratio = turns ratio).
• 100% efficient transformer: V_p × I_p = V_s × I_s (power in = power out).
• Step-up: more turns on the secondary, p.d. increases. Step-down: fewer turns on the secondary, p.d. decreases.

Space physics

Solar system, life cycle of a star and orbital motion

• Star like the Sun: nebula → protostar → main sequence star → red giant → white dwarf → black dwarf.
• Much more massive star: nebula → protostar → main sequence star → red supergiant → supernova → neutron star or black hole.
• Fusion in stars makes the elements up to iron.
• Elements heavier than iron are made in a supernova, which scatters them through the universe.
• Planets orbit the Sun; moons and artificial satellites orbit planets.
• In a stable orbit, a change of speed must go with a change of radius: a smaller radius needs a higher speed.

Red-shift and the expanding universe

• Red-shift: an observed increase in the wavelength of light from most distant galaxies.
• Longer wavelength means lower frequency; the speed of light does not change.
• Further away = moving away faster = bigger red-shift.
• Red-shift is evidence that the universe is expanding, which supports the Big Bang theory.
• Since 1998, observations of supernovae suggest distant galaxies are receding ever faster.
• Dark mass and dark energy are still not understood.
• A theory is accepted while it explains the evidence; new evidence can lead to it being changed.