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Physics equation sheets

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42 equations

Plumi’s copy of the AQA GCSE Physics equations (specification Appendix A). Check the board’s own sheet for the exam you sit. AQA GCSE Physics (8463), Version 1.1, September 2019. The board’s own document

Appendix A: Physics equations

In solving quantitative problems, students should be able to recall and apply the following equations, using standard SI units. Equations required for Higher Tier papers only are indicated by HT in the left hand column.

Equation numberWord equationSymbol equation
1weight = mass × gravitational field strength (g)
2work done = force × distance (along the line of action of the force)
3force applied to a spring = spring constant × extension
4moment of a force = force × distance (normal to direction of force)
5
6distance travelled = speed × time
7
8resultant force = mass × acceleration
9 HTmomentum = mass × velocity
10kinetic energy = 0.5 × mass × (speed)²
11gravitational potential energy = mass × gravitational field strength (g) × height
12
13
14
15
16wave speed = frequency × wavelength
17charge flow = current × time
18potential difference = current × resistance
19power = potential difference × current
20power = (current)² × resistance
21energy transferred = power × time
22energy transferred = charge flow × potential difference
23

Students should be able to select and apply the following equations from the Physics equation sheet. Equations required for Higher Tier papers only are indicated by HT in the left hand column.

Equation numberWord equationSymbol equation
1 HTpressure due to a column of liquid = height of column × density of liquid × gravitational field strength (g)
2(final velocity)² − (initial velocity)² = 2 × acceleration × distance
3 HT
4elastic potential energy = 0.5 × spring constant × (extension)²
5change in thermal energy = mass × specific heat capacity × temperature change
6
7
8 HTforce on a conductor (at right angles to a magnetic field) carrying a current = magnetic flux density × current × length
9thermal energy for a change of state = mass × specific latent heat
10 HT
11 HTpotential difference across primary coil × current in primary coil = potential difference across secondary coil × current in secondary coil
12For gases: pressure × volume = constant
About this sheet

Typed and laid out by Plumi from AQA’s own document: GCSE Physics specification (8463), Appendix A: Physics equations (Version 1.1, September 2019). It is not the board’s sheet and may differ from the copy you are given in the exam. Plumi is independent of the exam boards.

This is the equation list in the board’s specification. Which equations an exam prints on its equation sheet depends on the board’s arrangements for your exam year: check the board’s own notice.

AQA: equation sheets for 2027 and from 2028

All boards and levels

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271 equations in this selection

GCSE & A-level · All boards · All topics · Both GCSE tiers · All equation statuses

Plumi · tryplumi.app/physics-equations · sources checked 8 November 2026

Motion and forces

Average speed

v=stv=\frac{s}{t}

v: average speed (m/s); s: distance (m); t: elapsed time (s).

Use the total distance and total time for an average speed. This is also written s = vt.

Board notes

Acceleration

a=v−ut=Δvta=\frac{v-u}{t}=\frac{\Delta v}{t}

a: acceleration (m/s²); u and v: initial and final velocities (m/s); t: time (s).

Board notes

Momentum

p=mvp=mv

p: momentum (kg m/s); m: mass (kg); v: velocity (m/s).

Board notes

Spring force

F=kxF=kx

F: force (N); k: spring constant (N/m); x: extension from the natural length (m).

Applies in the linear part of a force–extension graph, before the limit of proportionality.

Board notes

Moment of a force

M=FdM=Fd

M: moment (N m); F: force (N); d: perpendicular distance from the pivot to the force's line of action (m).

Board notes

Energy

Change in gravitational potential energy

ΔEp=mgΔh\Delta E_p=mg\Delta h

ΔEₚ: energy change (J); m: mass (kg); g: field strength (N/kg); Δh: vertical height change (m).

Use this near a planet's surface where g can be treated as constant.

Board notes

Power

P=Et=WtP=\frac{E}{t}=\frac{W}{t}

P: power (W); E: energy transferred (J); W: work done (J); t: time (s).

Board notes

Energy efficiency

η=EusefulEinput\eta=\frac{E_{\mathrm{useful}}}{E_{\mathrm{input}}}

η: efficiency (no unit); Euseful: useful energy output (J); Einput: total energy input (J).

Multiply the ratio by 100 to express efficiency as a percentage.

Board notes

Efficiency using power

η=PusefulPinput\eta=\frac{P_{\mathrm{useful}}}{P_{\mathrm{input}}}

η: efficiency (no unit); Puseful: useful output power (W); Pinput: total input power (W).

Multiply the ratio by 100 for a percentage.

Board notes

Elastic energy stored

Ee=12kx2E_e=\frac12kx^2

Eₑ: stored energy (J); k: spring constant (N/m); x: extension (m).

For a spring with F = kx, this is also ½Fx: the triangular area under its force–extension graph.

Board notes

Matter and thermal physics

Heating without a change of state

ΔE=mcΔθ\Delta E=mc\Delta\theta

ΔE: thermal energy change (J); m: mass (kg); c: specific heat capacity (J/(kg °C)); Δθ: temperature change (°C).

A temperature difference has the same numerical value in kelvins and degrees Celsius. Boards also write Q or ΔQ for the energy transferred.

Board notes

Gas pressure and volume

pV=constantp1V1=p2V2pV=\text{constant}\qquad p_1V_1=p_2V_2

p: pressure (Pa); V: volume (m³). Subscripts identify the two states of the same gas sample.

For a fixed mass of gas at a constant temperature.

Board notes

Gas pressure, volume and temperature

pVT=constant\frac{pV}{T}=\text{constant}

p: pressure (Pa); V: gas volume (m³); T: absolute temperature (K).

For a fixed amount of gas. Use kelvins, including when one variable is held constant.

Board notes

Waves and light

Image magnification

m=himagehobjectm=\frac{h_{\mathrm{image}}}{h_{\mathrm{object}}}

m: magnification (no unit); himage and hobject: image and object heights, measured in the same units.

Board notes

Electricity

Potential difference and resistance

V=IRV=IR

V: potential difference (V); I: current (A); R: resistance (Ω).

Board notes

Electrical energy and charge

E=QVE=QV

E: energy transferred or work done (J); Q: charge (C); V: potential difference (V).

The definition of potential difference is the equivalent V = W/Q.

Board notes

Series resistance

R=R1+R2R=R_1+R_2

R: total resistance (Ω); R₁ and R₂: the two component resistances (Ω).

Board notes

Parallel resistance

1R=1R1+1R2\frac1R=\frac1{R_1}+\frac1{R_2}

R: combined resistance (Ω); R₁ and R₂: resistances of the two parallel branches (Ω).

Board notes

Electrical energy in kilowatt-hours

EkWh=PkW thE_{\mathrm{kWh}}=P_{\mathrm{kW}}\,t_{\mathrm{h}}

E: electrical energy (kWh); P: power (kW); t: time (h). One electricity 'unit' is one kWh.

Board notes

Electricity cost

cost=energy in kWh×price per kWh\text{cost}=\text{energy in kWh}\times\text{price per kWh}

Use one consistent currency: a price in pence/kWh gives a cost in pence.

Board notes

Electric current

I=ΔQΔtI=\frac{\Delta Q}{\Delta t}

I: current (A); Q: charge (C); t: time (s).

Board notes

Potential difference

V=WQV=\frac{W}{Q}

V: potential difference (V); W: energy transferred (J); Q: charge (C).

Board notes

Resistance

R=VIR=\frac{V}{I}

R: resistance (Ω); V: potential difference (V); I: current (A).

Board notes

Electrical power

P=IV=I2R=V2RP=IV=I^2R=\frac{V^2}{R}

P: power (W); I: current (A); V: potential difference (V); R: resistance (Ω).

The resistance forms apply to a resistor.

Board notes

Electrical energy transfer

W=VQ=VItW=VQ=VIt

W: energy (J); V: potential difference (V); Q: charge (C); I: current (A); t: time (s).

Constant potential difference and current for VIt.

Board notes

Resistivity

R=ρLAR=\frac{\rho L}{A}

R: resistance (Ω); ρ: resistivity (Ω m); L: length (m); A: cross-sectional area (m²).

Board notes

Resistors in series

R=R1+R2+⋯R=R_1+R_2+\cdots

R: total resistance (Ω); R₁, R₂: individual resistances (Ω).

Board notes

Resistors in parallel

1R=1R1+1R2+⋯\frac1R=\frac1{R_1}+\frac1{R_2}+\cdots

R: total resistance (Ω); R₁, R₂: individual resistances (Ω).

Board notes

Internal resistance

E=I(R+r)=V+Ir\mathcal E=I(R+r)=V+Ir

ℰ: emf (V); I: current (A); R: load resistance (Ω); r: internal resistance (Ω); V: terminal pd (V).

Board notes

Potential divider

Vout=R2R1+R2VinV_{\mathrm{out}}=\frac{R_2}{R_1+R_2}V_{\mathrm{in}}

Vᵢₙ, Vₒᵤₜ: input and output pd (V); R₁, R₂: series resistances (Ω).

Output is measured across R₂, with negligible load current.

Board notes

Potential-divider ratio

V1V2=R1R2\frac{V_1}{V_2}=\frac{R_1}{R_2}

V₁, V₂: potential differences (V); R₁, R₂: series resistances (Ω).

The same current flows through both resistors.

Board notes

Magnetism

Force on a current-carrying wire

F=BIlF=BIl

F: force (N); B: magnetic flux density (T); I: current (A); l: length of wire in the field (m).

The wire and magnetic field must be perpendicular.

Board notes

Transformer turns ratio

VpVs=NpNs\frac{V_p}{V_s}=\frac{N_p}{N_s}

Vₚ and Vₛ: primary and secondary potential differences (V); Nₚ and Nₛ: numbers of turns on those coils.

Board notes

Power in an ideal transformer

VpIp=VsIsV_pI_p=V_sI_s

Vₚ and Vₛ: primary and secondary potential differences (V); Iₚ and Iₛ: primary and secondary currents (A).

Assumes 100% efficiency: output power equals input power.

Board notes

Mechanics

Velocity

v=ΔsΔtv=\frac{\Delta s}{\Delta t}

v: velocity (m s⁻¹); s: displacement (m); t: time (s).

The instantaneous velocity is the gradient of a displacement–time graph.

Board notes

Acceleration

a=ΔvΔta=\frac{\Delta v}{\Delta t}

a: acceleration (m s⁻²); v: velocity (m s⁻¹); t: time (s).

Board notes

Velocity after constant acceleration

v=u+atv=u+at

u, v: initial and final velocity (m s⁻¹); a: acceleration (m s⁻²); t: time (s).

Uniform acceleration in one direction.

Board notes

Displacement from mean velocity

s=12(u+v)ts=\tfrac12(u+v)t

s: displacement (m); u, v: initial and final velocity (m s⁻¹); t: time (s).

Uniform acceleration.

Board notes

Displacement after constant acceleration

s=ut+12at2s=ut+\tfrac12at^2

s: displacement (m); u: initial velocity (m s⁻¹); a: acceleration (m s⁻²); t: time (s).

Uniform acceleration.

Board notes

Velocity and displacement

v2=u2+2asv^2=u^2+2as

u, v: initial and final velocity (m s⁻¹); a: acceleration (m s⁻²); s: displacement (m).

Uniform acceleration.

Board notes

Perpendicular vector components

Fx=Fcos⁡θ,Fy=Fsin⁡θF_x=F\cos\theta,\quad F_y=F\sin\theta

F: vector magnitude; θ: angle measured from the x direction. Components have the same units as F.

Board notes

Newton’s second law

∑F=ma\sum F=ma

F: force (N); m: mass (kg); a: acceleration (m s⁻²).

Constant mass; use the resultant force.

Board notes

Weight

W=mgW=mg

W: weight (N); m: mass (kg); g: gravitational field strength (N kg⁻¹).

Board notes

Momentum

p=mvp=mv

p: momentum (kg m s⁻¹); m: mass (kg); v: velocity (m s⁻¹).

Non-relativistic motion.

Board notes

Force as a rate of momentum change

F=ΔpΔtF=\frac{\Delta p}{\Delta t}

F: resultant force (N); p: momentum (kg m s⁻¹); t: time (s).

For a varying force use dp/dt; the finite-change form gives average force.

Board notes

Impulse

FΔt=ΔpF\Delta t=\Delta p

F: average force (N); Δt: interaction time (s); Δp: momentum change (N s).

Board notes

Kinetic energy

Ek=12mv2E_k=\tfrac12mv^2

Eₖ: kinetic energy (J); m: mass (kg); v: speed (m s⁻¹).

Non-relativistic speeds.

Board notes

Gravitational potential energy change

ΔEp=mgΔh\Delta E_p=mg\Delta h

ΔEₚ: energy change (J); m: mass (kg); g: field strength (N kg⁻¹); Δh: height change (m).

A uniform gravitational field.

Board notes

Power

P=ΔEΔt=WtP=\frac{\Delta E}{\Delta t}=\frac{W}{t}

P: power (W); E, W: energy or work (J); t: time (s).

Board notes

Efficiency

η=EusefulEinput=PusefulPinput\eta=\frac{E_{\mathrm{useful}}}{E_{\mathrm{input}}}=\frac{P_{\mathrm{useful}}}{P_{\mathrm{input}}}

η: efficiency (no unit); E: energy (J); P: power (W).

Multiply by 100 for a percentage.

Board notes

Density

ρ=mV\rho=\frac{m}{V}

ρ: density (kg m⁻³); m: mass (kg); V: volume (m³).

Board notes

Pressure difference in a fluid

Δp=ρgΔh\Delta p=\rho g\Delta h

Δp: pressure difference (Pa); ρ: fluid density (kg m⁻³); g: field strength (N kg⁻¹); Δh: depth difference (m).

Board notes

Materials

Hooke’s law

F=kxF=kx

F: force (N); k: spring stiffness (N m⁻¹); x: extension (m).

Within the limit of proportionality.

Board notes

Elastic strain energy

E=12Fx=12kx2E=\tfrac12Fx=\tfrac12kx^2

E: stored energy (J); F: final force (N); x: extension (m); k: stiffness (N m⁻¹).

Linear elastic loading.

Board notes

Tensile stress

σ=FA\sigma=\frac{F}{A}

σ: stress (Pa); F: tensile force (N); A: original cross-sectional area (m²).

Board notes

Tensile strain

ε=ΔLL\varepsilon=\frac{\Delta L}{L}

ε: strain (no unit); ΔL: extension (m); L: original length (m).

Board notes

Young modulus

E=σε=FLAΔLE=\frac{\sigma}{\varepsilon}=\frac{FL}{A\Delta L}

E: Young modulus (Pa); σ: stress (Pa); ε: strain; F: force (N); L, ΔL: lengths (m); A: area (m²).

Use the linear elastic region.

Board notes

Waves and optics

Wave speed

v=fλv=f\lambda

v: wave speed (m s⁻¹); f: frequency (Hz); λ: wavelength (m).

Board notes

Frequency and period

f=1Tf=\frac1T

f: frequency (Hz); T: period (s).

Board notes

Phase difference and path difference

Δϕ=2πΔxλ\Delta\phi=\frac{2\pi\Delta x}{\lambda}

Δφ: phase difference (rad); Δx: path difference (m); λ: wavelength (m).

Board notes

Intensity

I=PAI=\frac{P}{A}

I: intensity (W m⁻²); P: power (W); A: normal area (m²).

Board notes

Intensity and amplitude

I∝A2I\propto A^2

I: intensity (W m⁻²); A: wave amplitude (appropriate displacement unit).

For the same medium and frequency.

Board notes

Inverse-square spreading

I=P4πr2I=\frac{P}{4\pi r^2}

I: intensity (W m⁻²); P: source power (W); r: distance from source (m).

An isotropic point source without absorption.

Board notes

Absolute refractive index

n=cvn=\frac{c}{v}

n: refractive index (no unit); c: light speed in vacuum (m s⁻¹); v: speed in material (m s⁻¹).

Board notes

Refraction

n1sin⁡θ1=n2sin⁡θ2n_1\sin\theta_1=n_2\sin\theta_2

n₁, n₂: refractive indices; θ₁, θ₂: angles to the normal.

Board notes

Critical angle

sin⁡θc=n2n1\sin\theta_c=\frac{n_2}{n_1}

θc: critical angle; n₁: higher refractive index; n₂: lower refractive index.

For a material–air boundary this becomes sin θc = 1/n₁.

Board notes

Double-slit fringe spacing

w=λDsw=\frac{\lambda D}{s}

w: fringe spacing (m); λ: wavelength (m); D: screen distance (m); s: slit separation (m).

Small-angle approximation; D is much larger than s.

Board notes

Diffraction grating

dsin⁡θ=nλd\sin\theta=n\lambda

d: slit spacing (m); θ: diffraction angle; n: integer order; λ: wavelength (m).

Normal incidence.

Board notes

Thin-lens equation (Cartesian convention)

1v=1u+1f\frac1v=\frac1u+\frac1f

f: focal length (m); u, v: signed object and image positions (m).

OCR B uses a Cartesian convention; the signed object distance is negative for a real object to the left.

Board notes

Linear magnification

m=hiho=vum=\frac{h_i}{h_o}=\frac{v}{u}

m: magnification (no unit); hᵢ, hₒ: image and object heights; v, u: image and object distances (same length unit).

Use the board’s distance sign convention when a signed magnification is needed.

Board notes

Quantum and particle physics

Photon energy

E=hf=hcλE=hf=\frac{hc}{\lambda}

E: photon energy (J); h: Planck constant (J s); f: frequency (Hz); c: light speed (m s⁻¹); λ: wavelength (m).

Board notes

Energy-level transition

hf=E2−E1hf=E_2-E_1

h: Planck constant (J s); f: photon frequency (Hz); E₂, E₁: higher and lower energy levels (J).

Board notes

Photoelectric equation

hf=ϕ+Ek,max⁡hf=\phi+E_{k,\max}

hf: incident photon energy (J); φ: work function (J); Eₖ,ₘₐₓ: maximum electron kinetic energy (J).

Board notes

Threshold frequency

f0=ϕhf_0=\frac{\phi}{h}

f₀: threshold frequency (Hz); φ: work function (J); h: Planck constant (J s).

Board notes

De Broglie wavelength

λ=hp\lambda=\frac hp

λ: wavelength (m); h: Planck constant (J s); p: momentum (kg m s⁻¹).

Board notes

Acceleration through a potential difference

ΔEk=qV\Delta E_k=qV

ΔEₖ: kinetic energy gain (J); q: charge magnitude (C); V: accelerating potential difference (V).

Neglect other energy transfers; for an electron q = e.

Board notes

Electronvolt conversion

1 eV=1.60×10−19 J1\,\mathrm{eV}=1.60\times10^{-19}\,\mathrm J

eV: electronvolt; J: joule. Use the board’s supplied value of e for greater precision.

Board notes

Circular motion and oscillations

Angular velocity

ω=ΔθΔt=2πf=2πT\omega=\frac{\Delta\theta}{\Delta t}=2\pi f=\frac{2\pi}{T}

ω: angular velocity (rad s⁻¹); θ: angle (rad); t, T: time and period (s); f: frequency (Hz).

The frequency forms describe uniform circular motion.

Board notes

Tangential speed

v=ωrv=\omega r

v: speed (m s⁻¹); ω: angular velocity (rad s⁻¹); r: radius (m).

Board notes

Centripetal acceleration

a=v2r=ω2ra=\frac{v^2}{r}=\omega^2r

a: inward acceleration (m s⁻²); v: speed (m s⁻¹); r: radius (m); ω: angular velocity (rad s⁻¹).

Board notes

Centripetal force

F=mv2r=mω2rF=\frac{mv^2}{r}=m\omega^2r

F: resultant inward force (N); m: mass (kg); v: speed (m s⁻¹); r: radius (m); ω: angular velocity (rad s⁻¹).

Board notes

Simple harmonic acceleration

a=−ω2xa=-\omega^2x

a: acceleration (m s⁻²); ω: angular frequency (rad s⁻¹); x: displacement from equilibrium (m).

Board notes

Simple harmonic displacement

x=Acos⁡(ωt+ϕ)x=A\cos(\omega t+\phi)

x: displacement (m); A: amplitude (m); ω: angular frequency (rad s⁻¹); t: time (s); φ: initial phase (rad).

Sine is an equivalent solution with a different phase origin.

Board notes

Simple harmonic velocity

v=−Aωsin⁡(ωt+ϕ)v=-A\omega\sin(\omega t+\phi)

v: velocity (m s⁻¹); A: amplitude (m); ω: angular frequency (rad s⁻¹); t: time (s); φ: initial phase (rad).

Board notes

Simple harmonic speed at a displacement

v=±ωA2−x2v=\pm\omega\sqrt{A^2-x^2}

v: velocity (m s⁻¹); ω: angular frequency (rad s⁻¹); A: amplitude (m); x: displacement (m).

Choose the sign from the direction of motion.

Board notes

Maximum speed and acceleration

vmax⁡=ωA,amax⁡=ω2Av_{\max}=\omega A,\quad a_{\max}=\omega^2A

vₘₐₓ: maximum speed (m s⁻¹); aₘₐₓ: maximum acceleration magnitude (m s⁻²); A: amplitude (m).

Board notes

Simple-pendulum period

T=2πLgT=2\pi\sqrt{\frac Lg}

T: period (s); L: pendulum length (m); g: gravitational field strength (N kg⁻¹).

Small-amplitude oscillations.

Board notes

Energy in a spring oscillator

E=12kA2=12mv2+12kx2E=\tfrac12kA^2=\tfrac12mv^2+\tfrac12kx^2

E: total energy (J); k: stiffness (N m⁻¹); A, x: amplitude and displacement (m); m: mass (kg); v: speed (m s⁻¹).

Undamped ideal oscillator.

Board notes

Thermal physics

Heating without a change of state

Q=mcΔTQ=mc\Delta T

Q: energy transferred (J); m: mass (kg); c: specific heat capacity (J kg⁻¹ K⁻¹); ΔT: temperature change (K).

Board notes

Celsius to kelvin

T/K=θ/∘C+273.15T/\mathrm K=\theta/{}^\circ\mathrm C+273.15

T: thermodynamic temperature (K); θ: Celsius temperature (°C).

Board notes

Ideal gas: amount of substance

pV=nRTpV=nRT

p: pressure (Pa); V: volume (m³); n: amount (mol); R: molar gas constant (J mol⁻¹ K⁻¹); T: temperature (K).

Board notes

Ideal gas: number of particles

pV=NkTpV=NkT

p: pressure (Pa); V: volume (m³); N: particle count; k: Boltzmann constant (J K⁻¹); T: temperature (K).

Board notes

Particle number and amount

N=nNA,n=mMN=nN_A,\quad n=\frac mM

N: particle count; n: amount (mol); Nₐ: Avogadro constant (mol⁻¹); m: sample mass (kg); M: molar mass (kg mol⁻¹).

Board notes

Gas constants

k=RNAk=\frac R{N_A}

k: Boltzmann constant (J K⁻¹); R: molar gas constant (J mol⁻¹ K⁻¹); Nₐ: Avogadro constant (mol⁻¹).

Board notes

Kinetic theory of an ideal gas

pV=13Nm⟨c2⟩pV=\tfrac13Nm\langle c^2\rangle

p: pressure (Pa); V: volume (m³); N: molecule count; m: one molecule’s mass (kg); ⟨c²⟩: mean-square speed (m² s⁻²).

Board notes

Mean translational kinetic energy

12m⟨c2⟩=32kT\tfrac12m\langle c^2\rangle=\tfrac32kT

m: molecular mass (kg); ⟨c²⟩: mean-square speed (m² s⁻²); k: Boltzmann constant (J K⁻¹); T: temperature (K).

Board notes

Fields and capacitance

Capacitor-discharge halving time

T1/2=RCln⁡2≈0.69RCT_{1/2}=RC\ln2\approx0.69RC

T½: time to halve charge, pd or current (s); R: resistance (Ω); C: capacitance (F).

Board notes

Gravitational attraction

F=GMmr2F=\frac{GMm}{r^2}

F: force magnitude (N); G: gravitational constant (N m² kg⁻²); M, m: masses (kg); r: centre separation (m).

Point masses or spherical bodies outside their surfaces; force is attractive.

Board notes

Gravitational field strength

g=Fm=GMr2g=\frac Fm=\frac{GM}{r^2}

g: inward field magnitude (N kg⁻¹); F: force (N); m: test mass (kg); M: source mass (kg); r: radial distance (m).

Board notes

Gravitational potential

Vg=−GMrV_g=-\frac{GM}{r}

Vg: potential (J kg⁻¹); G: gravitational constant; M: source mass (kg); r: distance (m).

Zero potential at infinity.

Board notes

Gravitational potential energy

Ep=−GMmrE_p=-\frac{GMm}{r}

Eₚ: potential energy (J); M, m: masses (kg); r: separation (m).

Zero energy at infinite separation.

Board notes

Energy change in a gravitational field

ΔEp=mΔVg\Delta E_p=m\Delta V_g

ΔEₚ: energy change (J); m: mass (kg); ΔVg: potential change (J kg⁻¹).

Board notes

Field as potential gradient

gr=−dVgdrg_r=-\frac{\mathrm dV_g}{\mathrm dr}

gᵣ: signed radial field component (N kg⁻¹); Vg: potential (J kg⁻¹); r: radial distance (m).

Board notes

Circular-orbit period

T2=4π2r3GMT^2=\frac{4\pi^2r^3}{GM}

T: period (s); r: orbital radius (m); M: central mass (kg); G: gravitational constant.

Orbiting mass much smaller than central mass.

Board notes

Coulomb’s law

F=Qq4πε0r2F=\frac{Qq}{4\pi\varepsilon_0r^2}

F: signed force (N); Q, q: charges (C); ε₀: vacuum permittivity (F m⁻¹); r: separation (m).

In vacuum; opposite charges attract.

Board notes

Electric field strength

E=FqE=\frac Fq

E: electric field strength (N C⁻¹); F: force (N); q: test charge (C).

Board notes

Radial electric field

E=Q4πε0r2E=\frac{Q}{4\pi\varepsilon_0r^2}

E: signed radial field (N C⁻¹); Q: source charge (C); ε₀: vacuum permittivity; r: distance (m).

Board notes

Uniform electric field

E=VdE=\frac Vd

E: field magnitude (V m⁻¹); V: pd (V); d: plate separation (m).

Parallel plates, away from edges.

Board notes

Electric potential

V=Q4πε0rV=\frac{Q}{4\pi\varepsilon_0r}

V: potential (V); Q: source charge (C); r: distance (m); ε₀: vacuum permittivity.

Zero potential at infinity.

Board notes

Electric potential energy

Ep=Qq4πε0rE_p=\frac{Qq}{4\pi\varepsilon_0r}

Eₚ: energy (J); Q, q: charges (C); r: separation (m); ε₀: vacuum permittivity.

Board notes

Energy change in an electric field

ΔEp=qΔV\Delta E_p=q\Delta V

ΔEₚ: energy change (J); q: charge (C); ΔV: potential change (V).

Board notes

Electric potential gradient

Er=−dVdrE_r=-\frac{\mathrm dV}{\mathrm dr}

Eᵣ: signed radial electric field (V m⁻¹); V: potential (V); r: distance (m).

Board notes

Parallel-plate capacitor

C=ε0εrAdC=\frac{\varepsilon_0\varepsilon_r A}{d}

C: capacitance (F); εr: relative permittivity; A: plate overlap area (m²); d: separation (m).

For vacuum or air take εr ≈ 1; neglect fringing.

Board notes

Energy stored in a capacitor

W=12QV=12CV2=Q22CW=\tfrac12QV=\tfrac12CV^2=\frac{Q^2}{2C}

W: energy (J); Q: charge (C); V: pd (V); C: capacitance (F).

Board notes

Capacitor discharge

Q=Q0e−t/(RC)Q=Q_0e^{-t/(RC)}

Q, Q₀: remaining and initial charge (C); t: time (s); R: resistance (Ω); C: capacitance (F).

V and I decay with the same time constant.

Board notes

Capacitor charging

Q=Qmax⁡(1−e−t/(RC))Q=Q_{\max}\left(1-e^{-t/(RC)}\right)

Q: charge (C); Qₘₐₓ: final charge (C); R: resistance (Ω); C: capacitance (F); t: time (s).

Initially uncharged capacitor connected to a constant-voltage source.

Board notes

RC time constant

τ=RC\tau=RC

τ: time constant (s); R: resistance (Ω); C: capacitance (F).

Board notes

Magnetism and induction

Flux through a perpendicular surface

Φ=BA,Λ=NΦ\Phi=BA,\quad\Lambda=N\Phi

Φ: flux (Wb); B: flux density (T); A: area (m²); N: turn count; Λ: flux linkage (Wb turns).

Surface normal parallel to the field.

Board notes

Charged-particle orbit in a magnetic field

r=pBqr=\frac p{Bq}

r: radius (m); p: momentum (kg m s⁻¹); B: flux density (T); q: charge magnitude (C).

Motion perpendicular to a uniform field.

Board notes

Magnetic flux and linkage

Φ=BAcos⁡θ,Λ=NΦ\Phi=BA\cos\theta,\quad \Lambda=N\Phi

Φ: flux (Wb); B: flux density (T); A: area (m²); θ: angle between field and area normal; N: turn count; Λ: flux linkage (Wb turns).

Board notes

Faraday’s and Lenz’s laws

E=−d(NΦ)dt\mathcal E=-\frac{\mathrm d(N\Phi)}{\mathrm dt}

ℰ: induced emf (V); NΦ: flux linkage (Wb turns); t: time (s).

The minus sign describes opposition to the flux change.

Board notes

Rotating-coil emf

E=NBAωsin⁡(ωt)\mathcal E=NBA\omega\sin(\omega t)

ℰ: emf (V); N: turns; B: flux density (T); A: coil area (m²); ω: angular speed (rad s⁻¹); t: time (s).

Phase depends on the chosen coil orientation at t = 0.

Board notes

Sinusoidal rms values

Vrms=V02,Irms=I02V_{\mathrm{rms}}=\frac{V_0}{\sqrt2},\quad I_{\mathrm{rms}}=\frac{I_0}{\sqrt2}

Vᵣₘₛ, Iᵣₘₛ: rms voltage (V) and current (A); V₀, I₀: peak values.

Board notes

Nuclear physics

Activity

A=λNA=\lambda N

A: activity (Bq); λ: decay constant (s⁻¹); N: number of undecayed nuclei.

Board notes

Radioactive decay rate

dNdt=−λN\frac{\mathrm dN}{\mathrm dt}=-\lambda N

N: undecayed nuclei; t: time (s); λ: decay constant (s⁻¹).

Board notes

Radioactive decay

N=N0e−λt,A=A0e−λtN=N_0e^{-\lambda t},\quad A=A_0e^{-\lambda t}

N, N₀: remaining and initial nuclei; A, A₀: activities (Bq); λ: decay constant (s⁻¹); t: time (s).

Board notes

Half-life

T1/2=ln⁡2λT_{1/2}=\frac{\ln2}{\lambda}

T½: half-life (s); λ: decay constant (s⁻¹).

Board notes

Mass–energy equivalence

ΔE=Δm c2\Delta E=\Delta m\,c^2

ΔE: energy equivalent (J); Δm: mass difference (kg); c: light speed (m s⁻¹).

Board notes

Astrophysics

Stefan–Boltzmann luminosity

L=4πR2σT4L=4\pi R^2\sigma T^4

L: luminosity (W); R: stellar radius (m); σ: Stefan constant (W m⁻² K⁻⁴); T: surface temperature (K).

Black-body approximation.

Board notes

Wien’s displacement law

λmax⁡T=b\lambda_{\max}T=b

λₘₐₓ: peak wavelength (m); T: temperature (K); b: Wien constant (m K).

Board notes

Redshift at low speed

z=λobs−λemitλemit≈vcz=\frac{\lambda_{\mathrm{obs}}-\lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit}}}\approx\frac vc

z: redshift; λ: wavelengths (same unit); v: recession speed (m s⁻¹); c: light speed (m s⁻¹).

The speed approximation requires v much smaller than c.

Board notes

Hubble’s law

v=H0dv=H_0d

v: recession speed (m s⁻¹); H₀: Hubble constant (s⁻¹); d: distance (m).

Keep units consistent if H₀ is given in km s⁻¹ Mpc⁻¹.

Board notes

Expansion-age estimate

t≈H0−1t\approx H_0^{-1}

t: age estimate (s); H₀: Hubble constant (s⁻¹).

Assumes a constant expansion rate.

Board notes

Digital signals

Resolution limited by noise

b=log⁡2 ⁣(VtotalVnoise)b=\log_2\!\left(\frac{V_{\mathrm{total}}}{V_{\mathrm{noise}}}\right)

b: maximum usable bits per sample; Vₜₒₜₐₗ: total signal range (V); Vₙₒᵢₛₑ: noise voltage (V).

Board notes

Image information

B=Npixels bB=N_{\mathrm{pixels}}\,b

B: image information (bits); Nₚᵢₓₑₗₛ: pixel count; b: bits per pixel.

Board notes

Minimum sampling rate

fs>2fmax⁡f_s>2f_{\max}

fₛ: sampling frequency (Hz); fₘₐₓ: highest signal frequency (Hz).

Board notes

Digital transmission rate

Rb=fsbR_b=f_s b

Rᵦ: bit rate (bit s⁻¹); fₛ: samples per second (s⁻¹); b: bits per sample.

Board notes

Relativity

Radiation risk model

risk=probability×consequence\mathrm{risk}=\mathrm{probability}\times\mathrm{consequence}

Use a consistent measure of consequence; this is a model for comparing risks.

Board notes

Option: astrophysics

Telescope angular magnification

M=θimageθobject=fofeM=\frac{\theta_{\mathrm{image}}}{\theta_{\mathrm{object}}}=\frac{f_o}{f_e}

M: angular magnification; θ: angular sizes (rad); fₒ, fₑ: objective and eyepiece focal lengths (m).

Normal adjustment; magnitude shown.

Board notes

Medical physics

Exponential attenuation

I=I0e−μxI=I_0e^{-\mu x}

I, I₀: transmitted and initial intensity (W m⁻²); μ: linear attenuation coefficient (m⁻¹); x: thickness (m).

Board notes

Acoustic impedance

Z=ρcZ=\rho c

Z: acoustic impedance (kg m⁻² s⁻¹); ρ: density (kg m⁻³); c: sound speed (m s⁻¹).

Board notes

Ultrasound intensity reflection

IrIi=(Z2−Z1Z2+Z1)2\frac{I_r}{I_i}=\left(\frac{Z_2-Z_1}{Z_2+Z_1}\right)^2

Iᵣ, Iᵢ: reflected and incident intensities; Z₁, Z₂: acoustic impedances (same unit).

Normal incidence at a boundary.

Board notes

Doppler ultrasound

Δff0=2vcos⁡θc\frac{\Delta f}{f_0}=\frac{2v\cos\theta}{c}

Δf: frequency shift (Hz); f₀: transmitted frequency (Hz); v: blood speed (m s⁻¹); c: sound speed (m s⁻¹); θ: beam–flow angle.

Board notes

Option: rotational mechanics

Rotational kinetic energy

Ek=12Iω2E_k=\tfrac12I\omega^2

Eₖ: energy (J); I: moment of inertia (kg m²); ω: angular speed (rad s⁻¹).

Board notes

Torque and angular acceleration

τ=Iα\tau=I\alpha

τ: resultant torque (N m); I: moment of inertia (kg m²); α: angular acceleration (rad s⁻²).

Board notes

Angular momentum

L=IωL=I\omega

L: angular momentum (kg m² s⁻¹); I: moment of inertia (kg m²); ω: angular velocity (rad s⁻¹).

Board notes

Option: engineering thermodynamics

Option: turning points

Millikan force balance

QVd=mg\frac{QV}{d}=mg

Q: droplet charge magnitude (C); V: pd (V); d: plate spacing (m); m: droplet mass (kg); g: field strength (N kg⁻¹).

Stationary charged droplet, with buoyancy neglected in this form.

Board notes

Accelerated-electron wavelength

λ=h2meV\lambda=\frac h{\sqrt{2meV}}

λ: wavelength (m); h: Planck constant (J s); m: electron rest mass (kg); e: charge magnitude (C); V: accelerating pd (V).

Non-relativistic electrons.

Board notes

Option: electronics

LC resonant frequency

f0=12πLCf_0=\frac1{2\pi\sqrt{LC}}

f₀: resonance frequency (Hz); L: inductance (H); C: capacitance (F).

Board notes

Inverting amplifier

VoutVin=−RfRin\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}=-\frac{R_f}{R_{\mathrm{in}}}

Vₒᵤₜ, Vᵢₙ: voltages (V); R𝒇: feedback resistance (Ω); Rᵢₙ: input resistance (Ω).

Board notes

Summing amplifier

Vout=−Rf(V1R1+V2R2+⋯ )V_{\mathrm{out}}=-R_f\left(\frac{V_1}{R_1}+\frac{V_2}{R_2}+\cdots\right)

V: voltages (V); R₁, R₂: input resistances (Ω); R𝒇: feedback resistance (Ω).

Board notes

Option: alternating currents

Series-resonance Q-factor

Q=ω0LR=VLVRQ=\frac{\omega_0L}{R}=\frac{V_L}{V_R}

Q: quality factor; ω₀: resonant angular frequency (rad s⁻¹); L: inductance (H); R: resistance (Ω); Vᴸ, Vᴿ: rms voltages (V).

At resonance; Vᴸ = Vᶜ in magnitude.

Board notes

Option: physics of sport

Coefficient of restitution

e=relative separation speedrelative approach speede=\frac{\text{relative separation speed}}{\text{relative approach speed}}

e: coefficient of restitution (no unit); use positive relative speed magnitudes.

Board notes

Option: energy and the environment

Thermal conduction

ΔQΔt=−kAΔTΔx\frac{\Delta Q}{\Delta t}=-kA\frac{\Delta T}{\Delta x}

ΔQ/Δt: heat-transfer rate (W); k: thermal conductivity (W m⁻¹ K⁻¹); A: area (m²); ΔT/Δx: temperature gradient (K m⁻¹).

Heat flows down the temperature gradient.

Board notes
Sources and exam notes

Sources checked 8 November 2026

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