HLIBSciences

Physics

A concept-based Higher Level syllabus extending motion, matter, waves, fields, and nuclear and quantum physics through rotational mechanics, relativity, thermodynamics, induction, and quantum theory.

Course units

Estimated study time

180 hours

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Course Overview

IB Physics HL is organized into 5 units and 24 focused topics with 299 Key Concepts sections, 0 questions, and 480 flashcards.

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Begin with Topic A.1: Kinematics.

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Course units

Each unit groups focused topics. Every topic follows the same clear rhythm: review Key Concepts, use the Question Bank, then strengthen recall.

Unit 1
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Space, Time and Motion
Theme A begins with classical mechanics and then extends the idea of motion into rotation and relativistic spacetime. The HL-only A.4 and A.5 subtopics are major additions and should be learned as conceptual extensions of familiar linear motion, momentum and energy.
42 hours estimated ·5 topics
Unit 1 progress0%
Topics in this unit
Topic A.1Not started
Kinematics
Kinematics describes motion through position, displacement, velocity and acceleration. Students connect equations with vector representations and motion graphs, then apply the same ideas to projectile motion.
540 min estimated
Topic A.1 progress0%
Key Concepts0/10 sections complete Question Bank0/0 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Distinguish scalar distance and speed from vector displacement and velocity.
Calculate average and instantaneous velocity, speed and acceleration.
Use gradients and areas on position–time, velocity–time and acceleration–time graphs.
Apply constant-acceleration equations with a consistent sign convention.
Resolve projectile motion into independent horizontal and vertical components.
Describe qualitatively how resistance in a fluid changes a trajectory and produces terminal speed.
Key formulas
Displacement from position
$\Delta x=x_{\mathrm{f}}-x_{\mathrm{i}}$
Average acceleration
$\bar{a}=\frac{\Delta v}{\Delta t}$
Average speed
$\text{average speed}=\frac{\text{total distance}}{\Delta t}$
Average velocity
$\bar{v}=\frac{\Delta x}{\Delta t}$
Instantaneous velocity
$v=\frac{\mathrm{d}x}{\mathrm{d}t}$
Instantaneous acceleration
$a=\frac{\mathrm{d}v}{\mathrm{d}t}$
Speed from one-dimensional velocity
$\text{speed}=|v|$
Velocity from a position–time graph
$v=\text{gradient of the }x\text{-}t\text{ graph}$
Displacement from a velocity–time graph
$\Delta x=\int_{t_1}^{t_2}v\,\mathrm{d}t$
Distance from a velocity–time graph
$\text{distance}=\int_{t_1}^{t_2}|v|\,\mathrm{d}t$
Velocity change from an acceleration–time graph
$\Delta v=\int_{t_1}^{t_2}a\,\mathrm{d}t$
Displacement from mean velocity
$s=\frac{1}{2}(u+v)t$
Velocity after time
$v=u+at$
Displacement after time
$s=ut+\frac{1}{2}at^2$
Time-independent equation
$v^2=u^2+2as$
Vertical acceleration with upward positive
$a_y=-g\approx-9.81\,\mathrm{m\,s^{-2}}$
Initial velocity components
$u_x=u\cos\theta,\qquad u_y=u\sin\theta$
Ideal horizontal motion
$x=x_0+u_xt,\qquad v_x=u_x$
Ideal vertical motion
$y=y_0+u_yt-\frac{1}{2}gt^2,\qquad v_y=u_y-gt$
Speed from perpendicular components
$v=\sqrt{v_x^2+v_y^2}$
Flight time, range and maximum rise: check the conditions
$T=\frac{2u\sin\theta}{g},\quad R=\frac{u^2\sin(2\theta)}{g},\quad H=\frac{u^2\sin^2\theta}{2g}$
Terminal-speed condition
$F_{\mathrm{resultant}}=0\quad\Rightarrow\quad a=0$
Simplified vertical balance
$F_{\mathrm{drag}}=mg\quad\text{at terminal speed (if buoyancy is negligible)}$
Topic A.2Not started
Forces and momentum
This subtopic uses Newton’s laws and momentum to explain changes in motion. It also introduces circular motion as a continuous change of velocity direction caused by a resultant inward force.
600 min estimated
Topic A.2 progress0%
Key Concepts0/11 sections complete Question Bank0/0 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Apply Newton’s three laws and identify valid action–reaction pairs.
Draw free-body diagrams and resolve forces into useful components.
Model normal force, tension, friction, spring force, viscous drag and buoyancy.
Use momentum and impulse in collisions, explosions and force–time situations.
Apply conservation of momentum to isolated systems.
Analyse uniform circular motion using centripetal acceleration and resultant force.
Key formulas
Resultant external force
$\vec F_{\mathrm{net}}=\vec F_1+\vec F_2+\cdots+\vec F_n$
Newton's second law in momentum form
$\vec F_{\mathrm{net}}=\frac{\mathrm{d}\vec p}{\mathrm{d}t}$
Newton's second law for constant mass
$\vec F_{\mathrm{net}}=m\vec a$
Translational equilibrium
$\vec F_{\mathrm{net}}=0\quad\Rightarrow\quad \vec a=0$
Weight near Earth's surface
$W=mg$
Newton's second law by components
$F_{\mathrm{net},x}=ma_x,\qquad F_{\mathrm{net},y}=ma_y$
Newton's third law
$\vec F_{A\to B}=-\vec F_{B\to A}$
Weight components on an incline
$W_{\parallel}=mg\sin\theta,\qquad W_{\perp}=mg\cos\theta$
Normal force on a simple incline
$N=mg\cos\theta\quad\text{if }a_\perp=0\text{ and no other perpendicular force acts}$
Static-friction range
$0\le f_s\le \mu_sN$
Limiting and kinetic friction
$f_{s,\max}=\mu_sN,\qquad f_k=\mu_kN$
Ideal-string tension
$T_1=T_2=T$
Hooke's law
$\vec F_s=-k\vec x\qquad\text{and}\qquad F_s=k|\vec x|\ \text{in magnitude}$
Archimedes' principle
$F_b=\rho_{\mathrm{fluid}}V_{\mathrm{displaced}}g$
Stokes' law
$F_d=6\pi\eta rv$
Terminal-motion balance
$W=F_b+F_d\quad\text{for steady downward motion in still fluid with no other forces}$
Momentum
$\vec p=m\vec v$
Net external impulse
$\vec J_{\mathrm{net}}=\Delta\vec p=\int_{t_1}^{t_2}\vec F_{\mathrm{net}}\,\mathrm{d}t$
Average resultant external force
$\vec F_{\mathrm{net,avg}}=\frac{\Delta\vec p}{\Delta t}$
Momentum conservation
$\vec P_{\mathrm{before}}=\vec P_{\mathrm{after}}\quad\text{when external impulse is negligible}$
Translational kinetic energy
$K=\frac{1}{2}mv^2$
Speed, period and angular speed (uniform circular motion)
$v=\frac{2\pi r}{T}=\omega r$
Angular speed and period (uniform circular motion)
$\omega=\frac{2\pi}{T}$
Centripetal acceleration
$a_c=\frac{v^2}{r}=\omega^2r,\qquad a_c=\frac{4\pi^2r}{T^2}\ \text{for uniform motion}$
Required radial resultant
$F_{\mathrm{radial}}=ma_c=\frac{mv^2}{r}$
Topic A.3Not started
Work, energy and power
Energy methods provide an alternative to force-by-force analysis. Students use work, kinetic and gravitational potential energy, efficiency and power to understand how systems transfer and transform energy.
480 min estimated
Topic A.3 progress0%
Key Concepts0/11 sections complete Question Bank0/0 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Calculate work done by a constant force, including forces at an angle to displacement.
Use the work–energy principle and conservation of energy.
Calculate kinetic and near-Earth gravitational potential energy.
Interpret force–displacement graphs as work transferred.
Calculate power from work or energy transfer rate and from force and velocity.
Analyse efficiency and Sankey-type energy-transfer representations.
Key formulas
Conservation of total energy
$E_{\mathrm{initial}}=E_{\mathrm{final}}$
Average power
$P_{\mathrm{avg}}=\frac{\Delta E}{\Delta t}=\frac{W}{\Delta t}$
Efficiency
$\eta=\frac{E_{\mathrm{useful}}}{E_{\mathrm{input}}}=\frac{P_{\mathrm{useful}}}{P_{\mathrm{input}}}$
Work by a constant force
$W=Fs\cos\theta$
Work by the parallel component
$W=F_{\parallel}s$
Work from a variable force
$W=\text{signed area under the }F_{\parallel}\text{ against }s\text{ graph}$
Translational kinetic energy
$E_k=\frac12mv^2=\frac{p^2}{2m}$
Work–energy theorem
$W_{\mathrm{net}}=\Delta E_k=\frac12m(v^2-u^2)$
Change in gravitational potential energy
$\Delta E_g=mg\Delta h$
Gravitational potential energy from a chosen zero
$E_g=mgh$
Work done by gravity
$W_g=-\Delta E_g$
Hooke's law
$F_s=-kx$
Elastic potential energy
$E_s=\frac12kx^2$
Change in elastic potential energy
$\Delta E_s=\frac12k(x_2^2-x_1^2)$
Mechanical energy
$E_{\mathrm{mech}}=E_k+E_g+E_s$
Ideal mechanical-energy conservation
$E_{k,i}+E_{g,i}+E_{s,i}=E_{k,f}+E_{g,f}+E_{s,f}$
System energy change when work is the only boundary energy transfer
$W_{\mathrm{ext}}=\Delta E_{\mathrm{system}}$
Thermal transfer by constant kinetic friction
$\Delta E_{\mathrm{th}}=f_k s$
Non-conservative work
$W_{\mathrm{nc}}=\Delta E_{\mathrm{mech}}$
Idealized braking distance
$d=\frac{mv^2}{2F_b}$
Average power
$P_{\mathrm{avg}}=\frac{W}{\Delta t}=\frac{\Delta E}{\Delta t}$
Instantaneous mechanical power
$P=\vec F\cdot\vec v=Fv\cos\theta$
Parallel force form
$P=F_{\parallel}v$
Energy efficiency
$\eta=\frac{E_{\mathrm{useful}}}{E_{\mathrm{input}}}$
Power efficiency
$\eta=\frac{P_{\mathrm{useful}}}{P_{\mathrm{input}}}$
Percentage efficiency
$\eta_{\%}=100\eta$
Overall efficiency of a simple useful-output cascade
$\eta_{\mathrm{overall}}=\eta_1\eta_2\eta_3\cdots$
Gravitational power of a mass flow
$P_g=\dot m gh$
Mechanical-to-thermal balance with no external energy transfer
$E_{\mathrm{mech},i}=E_{\mathrm{mech},f}+\Delta E_{\mathrm{th}}$
Spring constant from a graph
$k=\frac{\Delta F}{\Delta x}$
Elastic work from graph area
$W=\frac12Fx=\frac12kx^2$
Useful climbing power
$P=\frac{mg\Delta h}{\Delta t}$
Topic A.4Not started
Rigid body mechanics
Rigid body mechanics is an HL-only extension from linear motion to rotation. Torque plays a role analogous to force, moment of inertia plays a role analogous to mass, and angular momentum provides a rotational conservation law.
420 min estimated
Topic A.4 progress0%
Key Concepts0/10 sections complete Question Bank0/0 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Calculate torque from force magnitude, lever arm and angle.
Apply rotational equilibrium by setting the net torque to zero.
Use angular displacement, angular velocity and angular acceleration.
Apply constant-angular-acceleration equations in rotational motion.
Understand moment of inertia as a measure that depends on both total mass and how that mass is distributed relative to the axis.
Use τ = Iα to connect resultant torque and angular acceleration.
Calculate angular momentum and apply conservation of angular momentum when external torque is negligible.
Relate torque acting over time to angular impulse.
Calculate rotational kinetic energy.
Analyse rolling without slipping as simultaneous translation and rotation.
Key formulas
Radian definition
$\theta=\frac{s}{r}$
Average angular velocity
$\omega_{\mathrm{av}}=\frac{\Delta\theta}{\Delta t}$
Average angular acceleration
$\alpha_{\mathrm{av}}=\frac{\Delta\omega}{\Delta t}$
Angular-to-linear relationships
$v=r|\omega|,\qquad a_{\mathrm{t}}=r\alpha,\qquad a_{\mathrm{c}}=r\omega^2$
Angular velocity–time
$\omega=\omega_0+\alpha t$
Angular displacement–time
$\Delta\theta=\omega_0t+\frac12\alpha t^2$
No-time equation
$\omega^2=\omega_0^2+2\alpha\Delta\theta$
Mean-angular-velocity equation
$\Delta\theta=\frac{\omega_0+\omega}{2}t$
Torque about a chosen origin (vector extension)
$\boldsymbol{\tau}=\mathbf{r}\times\mathbf{F}$
Planar torque magnitude about the perpendicular axis
$\tau=rF\sin\phi=Fd_\perp$
Torque of a couple
$\tau_{\mathrm{couple}}=Fd$
Translational equilibrium
$\mathbf F_{\mathrm{net}}=\mathbf{0}$
Rotational equilibrium
$\tau_{\mathrm{net}}=0$
Centre of mass for discrete masses
$\mathbf r_{\mathrm{CM}}=\frac{\sum\limits_{i=1}^{n} m_i\mathbf r_i}{\sum\limits_{i=1}^{n} m_i}$
Discrete masses
$I=\sum\limits_{i=1}^{n} m_ir_i^2$
Continuous mass distribution
$I=\int r^2\,dm$
Parallel-axis theorem
$I=I_{\mathrm{CM}}+Md^2$
Thin uniform ring or hoop about its central symmetry axis
$I=MR^2$
Uniform solid disk or cylinder about its central symmetry axis
$I=\frac12MR^2$
Uniform thin rod about a perpendicular axis through its centre
$I=\frac1{12}ML^2$
Uniform thin rod about a perpendicular axis through one end
$I=\frac13ML^2$
Uniform solid sphere about an axis through its centre
$I=\frac25MR^2$
Rotational Newton's second law
$\tau_{\mathrm{net}}=I\alpha$
Net external angular impulse about a fixed axis
$J_{\mathrm{angular,net}}=\int\tau_{\mathrm{ext,net}}\,dt=\Delta L$
Constant net-torque angular impulse
$\tau_{\mathrm{ext,net}}\Delta t=\Delta L$
Particle angular momentum
$\mathbf L=\mathbf r\times\mathbf p$
Angular-momentum component about a fixed axis
$L=I\omega$
Torque and angular momentum about a fixed inertial origin
$\boldsymbol\tau_{\mathrm{ext,net}}=\frac{d\mathbf L}{dt}$
Angular-momentum conservation
$L_{\mathrm{before}}=L_{\mathrm{after}}$
Work by constant torque
$W=\tau\Delta\theta$
Work by variable torque
$W=\int_{\theta_i}^{\theta_f}\tau\,d\theta$
Fixed-axis rotational kinetic energy
$K_{\mathrm{rot}}=\frac12I\omega^2=\frac{L^2}{2I}$
Rotational power
$P=\tau\omega$
No-slip speed constraint
$v_{\mathrm{CM}}=R|\omega|$
No-slip acceleration constraint
$a_{\mathrm{CM}}=\alpha R$
Total rolling kinetic energy
$K=\frac12Mv_{\mathrm{CM}}^2+\frac12I_{\mathrm{CM}}\omega^2$
Free gravity-driven rolling down a fixed incline (no slip or losses)
$a=\frac{g\sin\beta}{1+\frac{I_{\mathrm{CM}}}{(MR^2)}}$
Coupled linear dynamics
$F_{\mathrm{net}}=ma$
Coupled rotational dynamics
$\tau_{\mathrm{net}}=I\alpha$
No-slip coupling
$a=\alpha R$
Topic A.5Not started
Galilean and special relativity
This HL-only subtopic begins with classical relative motion and then shows why the invariance of the speed of light requires a new relationship between space and time. Students use Lorentz transformations and spacetime ideas quantitatively.
480 min estimated
Topic A.5 progress0%
Key Concepts0/12 sections complete Question Bank0/0 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Describe inertial reference frames and use Galilean transformations at low speeds.
Use classical relative velocity and recognize its limitations near the speed of light.
State and apply the two postulates of special relativity.
Calculate the Lorentz factor γ.
Use Lorentz transformations for position and time in one-dimensional motion.
Apply relativistic velocity addition.
Use the spacetime interval and distinguish proper time from coordinate time.
Calculate time dilation and length contraction.
Explain the relativity of simultaneity.
Interpret basic spacetime diagrams and relate their axes to relative speed.
Use high-speed particle observations such as atmospheric muons as evidence for relativistic effects.
Key formulas
Classical relative velocity
$v_{\text{B/A}}=v_{\text{B/S}}-v_{\text{A/S}}$
Uniform frame motion
$x_{\mathrm{origin\ of\ }S'}=vt$
Galilean position transformation
$x'=x-vt,\qquad t'=t$
Inverse Galilean transformation
$x=x'+vt,\qquad t=t'$
Galilean velocity transformation
$u'_x=u_x-v$
Acceleration invariance classically
$a'_x=a_x$
Vacuum light speed
$c\approx2.998\times10^8\ \mathrm{m\,s^{-1}}$
Stationary light-clock path
$L=\frac{c\Delta\tau}{2}$
Moving light-clock geometry
$\left(\frac{c\Delta t}{2}\right)^2=L^2+\left(\frac{v\Delta t}{2}\right)^2$
Dimensionless speed
$\beta=\frac{v}{c}$
Lorentz factor
$\gamma=\frac{1}{\sqrt{1-\beta^2}}=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
Speed from Lorentz factor
$\frac{v}{c}=\sqrt{1-\frac{1}{\gamma^2}}$
Lorentz position transformation
$x'=\gamma(x-vt)$
Lorentz time transformation
$t'=\gamma\left(t-\frac{vx}{c^2}\right)$
Transverse coordinates
$y'=y,\qquad z'=z$
Inverse position transformation
$x=\gamma(x'+vt')$
Inverse time transformation
$t=\gamma\left(t'+\frac{vx'}{c^2}\right)$
Forward collinear velocity addition
$u=\frac{u'+v}{1+\frac{u'v}{c^2}}$
Inverse velocity transformation
$u'=\frac{u-v}{1-\frac{uv}{c^2}}$
Dimensionless velocity form
$\frac{u}{c}=\frac{\frac{u'}{c}+\frac{v}{c}}{1+(\frac{u'}{c})(\frac{v}{c})}$
One-dimensional spacetime interval
$\Delta s^2=c^2\Delta t^2-\Delta x^2$
Three-dimensional spacetime interval
$\Delta s^2=c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2$
Inertial-clock proper time from a one-dimensional interval
$\Delta\tau=\sqrt{\Delta t^2-\frac{\Delta x^2}{c^2}}$
Interval classification
$\Delta s^2>0\ \mathrm{timelike},\quad \Delta s^2=0\ \mathrm{lightlike},\quad \Delta s^2<0\ \mathrm{spacelike}$
Time dilation for uniform relative motion
$\Delta t=\gamma\Delta\tau$
Uniform-motion proper time
$\Delta\tau=\frac{\Delta t}{\gamma}$
Travel distance in a frame
$d=v\Delta t$
Length contraction
$L=\frac{L_0}{\gamma}=L_0\sqrt{1-\frac{v^2}{c^2}}$
Proper length from contracted length
$L_0=\gamma L$
Simultaneous endpoint condition
$\Delta t_{\mathrm{measuring\ frame}}=0$
Time-separation transformation
$\Delta t'=\gamma\left(\Delta t-\frac{v\Delta x}{c^2}\right)$
Simultaneous in S
$\Delta t=0\quad\Longrightarrow\quad\Delta t'=-\gamma\frac{v\Delta x}{c^2}$
Einstein synchronization condition
$t_B=\frac{t_{\mathrm{send}}+t_{\mathrm{return}}}{2}$
Light-ray worldlines through the origin
$x=\pm ct$
Constant-velocity worldline and angle to the time axis
$x=x_0+vt=x_0+\frac{v}{c}(ct),\qquad \tan\theta=\frac{|v|}{c}$
Moving time axis
$x=vt$
Moving space axis
$ct=\frac{v}{c}x$
Causal speed condition
$\left|\frac{\Delta x}{\Delta t}\right|\le c$
Core relativity chain
$\beta=\frac vc\ \Longrightarrow\ \gamma=\frac1{\sqrt{1-\beta^2}}$
One-dimensional inertial-clock proper-time check
$v=\frac{\Delta x}{\Delta t},\qquad \Delta\tau=\frac{\Delta t}{\gamma}=\sqrt{\Delta t^2-\frac{\Delta x^2}{c^2}}$
Low-speed consistency
$v\ll c\quad\Longrightarrow\quad\gamma\approx1$
Unit 2
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The Particulate Nature of Matter
Theme B connects microscopic particle models with macroscopic thermal behaviour, energy balance, gases, thermodynamic processes and electric circuits. At HL, thermodynamics formalizes heat, work, internal energy and entropy.
32 hours estimated ·5 topics
Unit 2 progress0%
Topics in this unit
Topic B.1Not started
Thermal energy transfers
Thermal physics links temperature and internal energy to microscopic particle behaviour, then examines heating, phase changes, conduction and thermal radiation.
360 min estimated
Topic B.1 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Distinguish temperature from internal energy using a particle model.
Use the Kelvin temperature scale and convert between kelvin and degrees Celsius when appropriate.
Calculate energy transfers using specific heat capacity and specific latent heat.
Describe conduction through particle interactions.
Use black-body radiation concepts and the Stefan–Boltzmann relationship.
Connect temperature and peak wavelength using Wien’s displacement relationship.
Relate luminosity and distance to apparent brightness.
Key formulas
Energy conservation for an isolated system
$E_{\mathrm{before}}=E_{\mathrm{after}}$
Average rate of energy transfer
$P=\frac{\Delta E}{\Delta t}$
Density
$\rho=\frac{m}{V}$
Mass from density
$m=\rho V$
Celsius to Kelvin
$T=\theta+273.15$
Equal temperature intervals
$\Delta T\,(\mathrm{K})=\Delta\theta\,({}^\circ\mathrm{C})$
Mean translational kinetic energy of an ideal-gas molecule
$\overline{E_k}=\frac32k_BT$
Internal-energy components
$U=E_{\mathrm{microscopic\,kinetic}}+E_{\mathrm{intermolecular\,potential}}$
Simple stationary-sample energy balance
$\Delta U=E_{\mathrm{in}}-E_{\mathrm{out}}$
Energy for a temperature change
$Q=mc\Delta T$
Heat capacity
$C=mc$
Ideal calorimetry balance
$Q_{\mathrm{lost}}=Q_{\mathrm{gained}}$
Magnitude of latent-heat transfer
$Q=mL$
Multi-stage heating
$Q_{\mathrm{total}}=mc_1\Delta T_1+mL+mc_2\Delta T_2$
Latent heat from constant power
$L=\frac{P\Delta t}{m}$
Steady thermal conduction through a slab
$P=\frac{kA\Delta T}{L}$
Thermal resistance of one uniform layer
$R_{\mathrm{th}}=\frac{L}{kA}$
Power from thermal resistance
$P=\frac{\Delta T}{R_{\mathrm{th}}}$
Emissivity
$\varepsilon=\frac{P_{\mathrm{emitted}}}{P_{\mathrm{black\ body}}}$
Absorptivity
$a=\frac{P_{\mathrm{absorbed}}}{P_{\mathrm{incident}}}$
Physical range of emissivity
$0\leq\varepsilon\leq1$
Stefan–Boltzmann law
$P=\varepsilon\sigma AT^4$
Net radiative power to an isothermal black-body enclosure
$P_{\mathrm{net}}=\varepsilon\sigma A(T^4-T_s^4)$
Black-body stellar luminosity
$L=4\pi R^2\sigma T^4$
Wien's displacement law
$\lambda_{\max}T=b_W$
Temperature from peak wavelength
$T=\frac{b_W}{\lambda_{\max}}$
Apparent brightness
$b=\frac{L}{4\pi d^2}$
Luminosity from observations
$L=4\pi d^2b$
Stellar radius from luminosity and temperature
$R=\sqrt{\frac{L}{4\pi\sigma T^4}}$
Electrical energy input
$E=Pt=VIt$
Experimental specific heat capacity
$c=\frac{VIt}{m\Delta T}$
Experimental specific latent heat
$L=\frac{P\Delta t}{m}$
Topic B.2Not started
Greenhouse effect
The greenhouse-effect subtopic applies radiation and energy-balance ideas to planetary atmospheres. It emphasizes modelling, absorption and emission rather than memorizing a single climate narrative.
360 min estimated
Topic B.2 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Distinguish incoming short-wavelength solar radiation from longer-wavelength thermal radiation emitted by Earth.
Explain how atmospheric gases absorb and re-emit infrared radiation.
Use energy-balance reasoning to discuss equilibrium temperature.
Interpret absorption spectra and identify how molecular properties affect greenhouse behaviour.
Evaluate simplified greenhouse models and recognize their assumptions and limitations.
Key formulas
Top-of-atmosphere energy imbalance
$F_{\mathrm{net}}=F_{\mathrm{absorbed}}-F_{\mathrm{outgoing}}$
Radiative equilibrium
$F_{\mathrm{absorbed}}=F_{\mathrm{outgoing}}$
Wien's displacement law
$\lambda_{\max}T=b_W$
Peak wavelength
$\lambda_{\max}=\frac{b_W}{T}$
Projected area of a spherical planet
$A_{\mathrm{projected}}=\pi R^2$
Total incident solar power
$P_{\mathrm{solar}}=S\pi R^2$
Global-mean incident solar flux
$\overline F_{\mathrm{solar}}=\frac{S}{4}$
Albedo
$\alpha=\frac{P_{\mathrm{reflected}}}{P_{\mathrm{incident}}}$
Absorbed shortwave power
$P_{\mathrm{absorbed}}=(1-\alpha)P_{\mathrm{incident}}$
Global-mean absorbed solar flux
$F_{\mathrm{absorbed}}=\frac{(1-\alpha)S}{4}$
Stefan–Boltzmann emission
$P=\varepsilon\sigma AT^4$
Emitted longwave flux
$F_{\mathrm{emitted}}=\varepsilon\sigma T^4$
Emissivity range
$0\leq\varepsilon\leq1$
Uniform-emitter planetary equilibrium balance
$(1-\alpha)S\pi R^2=4\pi R^2\varepsilon\sigma T^4$
Uniform grey-emitter equilibrium temperature
$T=\left[\frac{(1-\alpha)S}{4\varepsilon\sigma}\right]^{\frac{1}{4}}$
Photon energy
$E_\gamma=hf=\frac{hc}{\lambda}$
Atmospheric spectral energy accounting (negligible scattering/reflection)
$P_{\mathrm{incident}}=P_{\mathrm{transmitted}}+P_{\mathrm{absorbed}}$
Surface–effective temperature difference
$\Delta T_{\mathrm{greenhouse}}\approx T_{\mathrm{surface}}-T_e$
Initial enhanced-effect imbalance
$F_{\mathrm{net}}=F_{\mathrm{absorbed}}-F_{\mathrm{outgoing}}>0$
Albedo-change forcing estimate
$\Delta F_{\alpha}=-\frac{S\Delta\alpha}{4}$
Simplified climate response
$\Delta T=\lambda_c\Delta F$
Stored energy per unit area
$\frac{\Delta E}{A}=C_A\Delta T$
Constant-imbalance response time
$\Delta t=\frac{C_A\Delta T}{F_{\mathrm{net}}}$
Absorbed fraction from measured albedo
$f_{\mathrm{absorbed}}=1-\alpha$
Planetary absorbed power
$P_{\mathrm{absorbed}}=(1-\alpha)S\pi R^2$
Equilibrium test
$P_{\mathrm{absorbed}}\stackrel{?}{=}P_{\mathrm{outgoing}}$
Topic B.3Not started
Gas laws
This subtopic connects pressure, volume and temperature to a microscopic model of gas particles. Students move between empirical gas laws, the ideal-gas equation and kinetic-theory interpretations.
360 min estimated
Topic B.3 progress0%
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Learning objectives
Use the ideal-gas equation for a fixed amount of gas.
Interpret pressure as the result of particle collisions with container walls.
Connect absolute temperature to mean random kinetic energy.
Interpret p–V and related graphs for gas processes.
Solve problems involving changes of state for an ideal gas.
Key formulas
Definition of pressure
$p=\frac{F_{\perp}}{A}$
Absolute and gauge pressure
$p_{\mathrm{absolute}}=p_{\mathrm{gauge}}+p_{\mathrm{atmospheric}}$
Normal momentum change at an elastic rebound from a stationary smooth wall
$|\Delta p_x|=2m|v_x|$
Average force from momentum transfer
$F_{\mathrm{average}}=\frac{\Delta p_{\mathrm{total}}}{\Delta t}$
Microscopic route to pressure
$p=\frac{1}{A}\frac{\Delta p_{\mathrm{total}}}{\Delta t}$
Mean translational kinetic energy
$\langle E_k\rangle=\frac{1}{2}m\langle v^2\rangle=\frac{3}{2}k_{\mathrm B}T$
Root-mean-square molecular speed
$v_{\mathrm{rms}}=\sqrt{\frac{3k_{\mathrm B}T}{m}}=\sqrt{\frac{3RT}{M}}$
Internal energy of a monatomic ideal gas
$U=\frac{3}{2}Nk_{\mathrm B}T=\frac{3}{2}nRT$
Particle number and amount
$N=nN_{\mathrm A}$
Amount from mass
$n=\frac{m_{\mathrm{sample}}}{M}$
Boyle's law proportionality
$p\propto\frac{1}{V}\quad(T,n\text{ constant})$
Boyle's law between two states
$p_1V_1=p_2V_2$
Constant-pressure relationship
$\frac{V}{T}=\text{constant}\quad\Rightarrow\quad\frac{V_1}{T_1}=\frac{V_2}{T_2}$
Constant-volume relationship
$\frac{p}{T}=\text{constant}\quad\Rightarrow\quad\frac{p_1}{T_1}=\frac{p_2}{T_2}$
Molar form of the ideal-gas equation
$pV=nRT$
Particle form and constant bridge
$pV=Nk_{\mathrm B}T,\qquad R=N_{\mathrm A}k_{\mathrm B}$
Combined gas law for fixed amount
$\frac{p_1V_1}{T_1}=\frac{p_2V_2}{T_2}$
Quasistatic boundary work done by a gas
$W_{\mathrm{by}}=\int_{V_1}^{V_2}p\,\mathrm dV$
Work sign convention and constant-pressure quasistatic work
$W_{\mathrm{on}}=-W_{\mathrm{by}},\qquad W_{\mathrm{by}}=p\Delta V\;\text{at constant }p$
Kinetic-theory pressure equation
$pV=\frac{1}{3}Nm\langle v^2\rangle$
Density form of the kinetic equation
$p=\frac{1}{3}\rho v_{\mathrm{rms}}^2$
Mechanical meaning of temperature
$\frac{1}{2}m\langle v^2\rangle=\frac{3}{2}k_{\mathrm B}T$
Compressibility factor
$Z=\frac{pV}{nRT}$
Percentage deviation from ideal behaviour
$\text{percentage deviation}=|Z-1|\times100\%$
First-order maximum-uncertainty estimate for a product
$\frac{\Delta(pV)}{pV}\approx\frac{\Delta p}{p}+\frac{\Delta V}{V}$
Boyle-law linear form
$p=(pV)\left(\frac{1}{V}\right)$
Topic B.4Not started
Thermodynamics
Thermodynamics is an HL-only treatment of energy transfer in macroscopic systems. It combines the first law, p–V work, ideal-gas internal energy, entropy, the second law and heat-engine limits.
480 min estimated
Topic B.4 progress0%
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Learning objectives
Use the first law with a consistent sign convention for heat transferred to a system and work done by the system.
Calculate work in constant-pressure expansion or compression and interpret work as area under a p–V curve.
Calculate the internal-energy change of a monatomic ideal gas.
Analyse isovolumetric, isobaric, isothermal and adiabatic processes.
Use the monatomic ideal-gas adiabatic relationship.
Apply entropy change ΔS = ΔQ/T for reversible heat transfer at constant temperature.
Interpret entropy statistically using S = kB lnΩ.
Use the second law to explain the direction of spontaneous processes and the impossibility of complete conversion of heat into work in a cyclic engine.
Analyse cyclic heat engines and calculate efficiency.
Calculate the maximum theoretical efficiency of a Carnot engine operating between two temperatures.
Key formulas
Ideal-gas equation
$pV=nRT$
Particle form of the ideal-gas equation
$pV=Nk_{\mathrm B}T$
Change in a state function
$\Delta U=U_{\mathrm f}-U_{\mathrm i}$
First law
$\Delta U=Q-W$
Heat from the first law
$Q=\Delta U+W$
Complete-cycle condition
$\Delta U_{\mathrm{cycle}}=0$
Quasi-static p–V work
$W=\int_{V_{\mathrm i}}^{V_{\mathrm f}}p\,\mathrm dV$
Constant-pressure work
$W=p(V_{\mathrm f}-V_{\mathrm i})$
Straight-path graph area
$W=\frac{p_{\mathrm i}+p_{\mathrm f}}{2}(V_{\mathrm f}-V_{\mathrm i})$
Monatomic internal energy
$U=\frac32nRT$
Particle form
$U=\frac32Nk_{\mathrm B}T$
Monatomic internal-energy change
$\Delta U=\frac32nR\Delta T$
State-point form
$U=\frac32pV$
Isochoric work
$W_{V}=0$
Isochoric first law
$Q_V=\Delta U$
Isobaric work for an ideal gas
$W_p=p\Delta V=nR\Delta T$
Isobaric heat for a monatomic ideal gas
$Q_p=\frac52nR\Delta T$
Isothermal state relation
$p_{\mathrm i}V_{\mathrm i}=p_{\mathrm f}V_{\mathrm f}$
Isothermal internal-energy change
$\Delta U=0$
Reversible isothermal work
$W=nRT\ln\!\left(\frac{V_{\mathrm f}}{V_{\mathrm i}}\right)$
Isothermal energy balance
$Q=W$
Adiabatic first law
$Q=0\quad\Rightarrow\quad\Delta U=-W$
Reversible adiabatic pressure relation
$pV^{\gamma}=\mathrm{constant}$
Reversible adiabatic temperature relation
$TV^{\gamma-1}=\mathrm{constant}$
Monatomic heat-capacity ratio
$\gamma=\frac53$
Reversible adiabatic work
$W=\frac{p_{\mathrm i}V_{\mathrm i}-p_{\mathrm f}V_{\mathrm f}}{\gamma-1}$
Cycle internal-energy change
$\Delta U_{\mathrm{cycle}}=0$
Cycle energy balance
$Q_{\mathrm{net}}=W_{\mathrm{net}}$
Rectangular-cycle work
$\lvert W_{\mathrm{net}}\rvert=(p_{\mathrm{high}}-p_{\mathrm{low}})(V_{\mathrm{high}}-V_{\mathrm{low}})$
Reversible constant-temperature entropy change
$\Delta S=\frac{Q_{\mathrm{rev}}}{T}$
General reversible entropy change
$\Delta S=\int_{\mathrm i}^{\mathrm f}\frac{\mathrm dQ_{\mathrm{rev}}}{T}$
Total entropy change
$\Delta S_{\mathrm{total}}=\Delta S_{\mathrm{system}}+\Delta S_{\mathrm{surroundings}}$
Boltzmann entropy
$S=k_{\mathrm B}\ln\Omega$
Entropy change from multiplicity
$\Delta S=k_{\mathrm B}\ln\!\left(\frac{\Omega_{\mathrm f}}{\Omega_{\mathrm i}}\right)$
Boltzmann constant
$k_{\mathrm B}=1.380649\times10^{-23}\ \mathrm{J\,K^{-1}}$
Second-law inequality
$\Delta S_{\mathrm{total}}\ge 0$
Reversible limit
$\Delta S_{\mathrm{total}}=0$
Irreversible spontaneous process
$\Delta S_{\mathrm{total}}>0$
Heat-engine energy balance
$W=Q_H-Q_C$
Thermal efficiency
$\eta=\frac{W}{Q_H}=1-\frac{Q_C}{Q_H}$
Carnot efficiency
$\eta_C=1-\frac{T_C}{T_H}$
Real-engine limit
$0<\eta_{\mathrm{real}}<\eta_C<1$
Topic B.5Not started
Current and circuits
Circuit physics models the movement of charge and the transfer of electrical energy. Students use potential difference, current, resistance, power and Kirchhoff-type reasoning in direct-current networks.
360 min estimated
Topic B.5 progress0%
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Learning objectives
Define current as rate of flow of charge and potential difference as energy transferred per unit charge.
Use resistance and Ohm’s law for ohmic components.
Interpret current–voltage characteristics for common components.
Combine resistors in series and parallel.
Use conservation of charge at junctions and energy around closed loops.
Calculate electrical power and energy transfer.
Include electromotive force and internal resistance in simple source models.
Key formulas
Average current (or steady current)
$I=\frac{\Delta Q}{\Delta t}$
Quantization of charge
$Q=Ne$
Current in a conductor
$I=nqAv_{\mathrm d}$
Potential difference with specified polarity
$V_{AB}=\frac{W_{\mathrm{transferred}}}{Q}$
Electromotive force
$\varepsilon=\frac{E_{\mathrm{supplied}}}{Q}$
Resistance at an operating point
$R=\frac{V}{I}$
Ohm's law
$V=IR\quad(\text{constant physical conditions})$
Resistance of a uniform conductor
$R=\rho\frac{L}{A}$
Area of a circular wire
$A=\frac{\pi d^2}{4}$
Linear temperature approximation
$\rho=\rho_0[1+\alpha(T-T_0)]$
Electrical power
$P=VI$
Power in a resistor
$P=I^2R=\frac{V^2}{R}$
Energy transferred at constant voltage and current
$E=P\Delta t=VI\Delta t$
Kirchhoff's junction rule
$\sum\limits_{j=1}^{m}I_{j,\mathrm{in}}=\sum\limits_{k=1}^{n}I_{k,\mathrm{out}}$
Kirchhoff's loop rule
$\sum\limits_{k=1}^{n}\Delta V_k=0$
Equivalent series resistance
$R_{\mathrm{eq}}=R_1+R_2+\cdots$
Common voltage across a parallel group
$V_{\mathrm{group}}=V_1=V_2=\cdots$
Total current in parallel
$I_{\mathrm{total}}=I_1+I_2+\cdots$
Equivalent parallel resistance
$\frac{1}{R_{\mathrm{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots$
Terminal voltage while delivering current
$V_{\mathrm{terminal}}=\varepsilon-Ir$
Current from a real cell
$I=\frac{\varepsilon}{R+r}$
Internal power transfer
$P_{\mathrm{internal}}=I^2r$
Unloaded potential divider
$V_{\mathrm{out}}=V_{\mathrm{in}}\frac{R_2}{R_1+R_2}$
Uniform potentiometer ratio
$\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}=\frac{x}{L}$
Unloaded divider output measured across a sensor
$V_{\mathrm{out}}=V_{\mathrm{in}}\frac{R_{\mathrm{sensor}}}{R_{\mathrm{fixed}}+R_{\mathrm{sensor}}}$
Qualitative sensor trends
$T\uparrow\Rightarrow R_{\mathrm{NTC}}\downarrow,\qquad L\uparrow\Rightarrow R_{\mathrm{LDR}}\downarrow$
Parallel loading by a voltmeter
$R_{\mathrm{loaded}}=\frac{R R_{\mathrm V}}{R+R_{\mathrm V}}$
First-order maximum fractional uncertainty in resistance
$\frac{\Delta R}{R}\approx\frac{\Delta V}{V}+\frac{\Delta I}{I}$
Two resistors in parallel
$R_p=\frac{R_1R_2}{R_1+R_2}$
Steady dc operating current from electrical input power
$I=\frac{P}{V}$
Unit 3
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Wave Behaviour
Theme C moves from oscillation to wave propagation, interference, diffraction, standing waves, resonance and Doppler shifts. Higher Level adds a fully quantitative treatment of SHM, diffraction patterns and moving-source/observer Doppler effects.
29 hours estimated ·5 topics
Unit 3 progress0%
Topics in this unit
Topic C.1Not started
Simple harmonic motion
Simple harmonic motion combines the shared SL model with HL phase constants, trigonometric state equations, signed velocity-displacement analysis and quantitative energy through the cycle.
420 min estimated
Topic C.1 progress0%
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Learning objectives
Identify the conditions for SHM and use a = − $ω^{2}$ x.
Relate period, frequency and angular frequency.
Describe qualitatively the exchange between kinetic and potential energy in an oscillator.
Use a phase constant in sinusoidal SHM equations.
Calculate instantaneous displacement and velocity using trigonometric models.
Use the velocity–displacement relationship.
Calculate total, kinetic and potential energy quantitatively through an SHM cycle.
Use radians naturally in phase and angular-frequency calculations.
Key formulas
Displacement from equilibrium
$x=s-s_{\mathrm{eq}}$
Direction test for a restoring force
$F_{\mathrm{restoring}}x<0\quad(x\ne0)$
Defining SHM relation
$a=-\omega^2x$
Gradient of an acceleration-displacement graph
$\frac{\Delta a}{\Delta x}=-\omega^2$
Maximum acceleration magnitude
$a_{\max}=\omega^2A$
Period-frequency relationship
$f=\frac{1}{T}$
Angular frequency
$\omega=2\pi f=\frac{2\pi}{T}$
Phase-offset magnitude from time-offset magnitude
$|\Delta\phi|=\omega|\Delta t|=2\pi\frac{|\Delta t|}{T}$
SHM displacement
$x=A\cos(\omega t+\phi)$
SHM velocity
$v=-\omega A\sin(\omega t+\phi)$
SHM acceleration
$a=-\omega^2A\cos(\omega t+\phi)=-\omega^2x$
Speed-displacement relation
$v^2=\omega^2(A^2-x^2)$
Cosine-form initial conditions
$x_0=A\cos\phi,\qquad v_0=-\omega A\sin\phi$
Amplitude from an instantaneous state
$A=\sqrt{x_0^2+\left(\frac{v_0}{\omega}\right)^2}$
Phase constant with quadrant control
$\phi=\operatorname{atan2}\!\left(-\frac{v_0}{\omega},x_0\right)$
Normalized SHM phase ellipse
$\frac{x^2}{A^2}+\frac{v^2}{\omega^2A^2}=1$
Phase-plane intercepts
$(x,v)=(\pm A,0)\quad\text{or}\quad(0,\pm\omega A)$
SHM phase separations
$|\Delta\phi_{xv}|=\frac{\pi}{2},\qquad|\Delta\phi_{xa}|=\pi$
Net restoring force about spring equilibrium
$F_{\mathrm{rest}}=-kx$
Spring angular frequency
$\omega=\sqrt{\frac{k}{m}}$
Mass-spring period
$T=2\pi\sqrt{\frac{m}{k}}$
Small-angle approximation
$\sin\theta\approx\theta\quad(|\theta|\ll1\text{ rad})$
Pendulum angular frequency
$\omega=\sqrt{\frac{g}{L}}$
Simple-pendulum period
$T=2\pi\sqrt{\frac{L}{g}}$
Total energy of ideal SHM
$E_{\mathrm{total}}=\frac{1}{2}m\omega^2A^2$
Potential energy relative to equilibrium
$E_{\mathrm p}=\frac{1}{2}m\omega^2x^2$
Kinetic energy in ideal SHM
$E_{\mathrm k}=\frac{1}{2}m\omega^2(A^2-x^2)$
Spring-model equivalence
$k=m\omega^2$
Potential-energy fraction
$\frac{E_{\mathrm p}}{E_{\mathrm{total}}}=\frac{x^2}{A^2}$
Kinetic-energy fraction
$\frac{E_{\mathrm k}}{E_{\mathrm{total}}}=1-\frac{x^2}{A^2}$
Energy against phase for cosine-form SHM
$E_{\mathrm p}=E_{\mathrm{total}}\cos^2\Theta,\qquad E_{\mathrm k}=E_{\mathrm{total}}\sin^2\Theta$
Equal-energy displacement
$|x|=\frac{A}{\sqrt2}$
Experimental SHM proportionality test
$\frac{a}{x}=-\omega^2=\text{constant}$
Percentage deviation from a model
$\text{percentage deviation}=\frac{|y_{\mathrm{measured}}-y_{\mathrm{model}}|}{|y_{\mathrm{model}}|}\times100\%$
Period from multiple-cycle timing
$T=\frac{t_N}{N}$
Linearized mass-spring relationship
$T^2=\frac{4\pi^2}{k}m$
Linearized pendulum relationship
$T^2=\frac{4\pi^2}{g}L$
Topic C.2Not started
Wave model
The wave model describes how oscillations transfer energy without net transport of matter. Students work with transverse and longitudinal waves, wave speed, intensity and graphical representations.
180 min estimated
Topic C.2 progress0%
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Learning objectives
Distinguish transverse and longitudinal waves.
Relate wavelength, frequency, period and wave speed.
Interpret displacement–position and displacement–time graphs.
Use amplitude and intensity relationships.
Describe wavefronts and rays and connect their geometry to propagation direction.
Key formulas
Average wave propagation speed
$v=\frac{\Delta x}{\Delta t}$
Energy transferred by a wave
$E=P\Delta t$
Transverse-direction condition
$\vec{s}\cdot\vec{v}=0$
Longitudinal-direction condition
$\vec{s}\parallel\vec{v}$
Frequency-period relationship
$f=\frac{1}{T}$
Frequency from counting cycles
$f=\frac{N}{\Delta t}$
Optional phase model: angular frequency
$\omega=2\pi f=\frac{2\pi}{T}$
Wave equation
$v=f\lambda=\frac{\lambda}{T}$
Optional model: Wave number
$k=\frac{2\pi}{\lambda}$
Optional model: Positive-direction sinusoidal wave
$y=A\sin(kx-\omega t+\phi)$
Optional model: Phase speed
$v=\frac{\omega}{k}$
Optional: spatial phase-offset magnitude
$\lvert\Delta\phi\rvert=\frac{2\pi\lvert\Delta x\rvert}{\lambda}$
Optional: temporal phase-offset magnitude
$\lvert\Delta\phi\rvert=\frac{2\pi\lvert\Delta t\rvert}{T}$
Optional: particle velocity for a fixed-shape right-moving transverse wave
$u_y=-v\frac{\partial y}{\partial x}$
Optional: wave speed on an ideal uniform string
$v=\sqrt{\frac{F_T}{\mu}}$
Linear mass density
$\mu=\frac{m}{L}$
Echo-ranging distance
$d=\frac{v\Delta t}{2}$
Sound intensity
$I=\frac{P}{A}$
Optional: sinusoidal travelling pressure variation
$\Delta p=\Delta p_{\max}\cos(kx-\omega t)$
Electromagnetic wave relation in vacuum
$c=f\lambda$
Quantum-physics link: photon energy
$E=hf$
Optional: plane-wave field-amplitude ratio in vacuum
$\frac{E_0}{B_0}=c$
Intensity definition
$I=\frac{P}{A}$
Uniform spherical spreading
$I=\frac{P}{4\pi r^2}$
Intensity ratio: same medium, frequency and amplitude definition
$\frac{I_2}{I_1}=\left(\frac{A_2}{A_1}\right)^2$
Mean wavelength over several intervals
$\lambda=\frac{L}{n}$
Percentage uncertainty
$\%\,u=\frac{\Delta q}{\lvert q\rvert}\times100\%$
Speed from repeated distance-time measurements
$v=\frac{\Delta d}{\Delta t}$
Topic C.3Not started
Wave phenomena
Wave phenomena combines the shared reflection, refraction, diffraction and interference core with HL quantitative single-slit analysis, diffraction envelopes, grating orders and intensity-pattern control.
660 min estimated
Topic C.3 progress0%
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Learning objectives
Use the laws of reflection and qualitative/quantitative refraction.
Apply Snell’s law and discuss refractive index.
Explain diffraction and how it depends on wavelength relative to aperture size.
Use the principle of superposition.
Analyse two-source interference and Young double-slit fringes.
Analyse the angular structure of a single-slit diffraction pattern and use the first-minimum approximation.
Explain how the single-slit diffraction envelope modifies a two-slit interference pattern.
Use the diffraction-grating equation for multiple-slit interference.
Interpret how changing slit width, spacing, wavelength or number of slits changes observed intensity patterns.
Key formulas
Ray-wavefront geometry
$\vec{v}\perp\text{wavefront}$
Wave speed from wavefront spacing
$v=f\lambda$
Law of reflection
$\theta_i=\theta_r$
Frequency continuity at a stationary boundary
$f_i=f_r=f_t$
Power balance at a passive boundary in steady state
$P_i=P_r+P_t+P_{\mathrm{abs}}$
Refractive index
$n=\frac{c}{v}$
Snell’s law
$n_1\sin\theta_1=n_2\sin\theta_2$
Speed–angle–wavelength ratio: nonzero ray angles; use undivided Snell at normal incidence
$\frac{\sin\theta_1}{\sin\theta_2}=\frac{v_1}{v_2}=\frac{\lambda_1}{\lambda_2}=\frac{n_2}{n_1}$
Wavelength in a refractive medium
$\lambda=\frac{\lambda_0}{n}$
Critical-angle form of Snell’s law
$n_1\sin c=n_2\sin90^\circ$
Critical-angle equation
$\sin c=\frac{n_2}{n_1}$
Total-internal-reflection conditions
$n_1>n_2\quad\text{and}\quad\theta_i>c$
Diffraction-control ratio
$\mathcal{D}=\frac{\lambda}{a}$
Strong-spreading condition
$a\lesssim\lambda$
Same-medium wave relation
$v=f\lambda=\text{constant}$
Principle of superposition
$y_{\mathrm{resultant}}=\sum\limits_{i=1}^{N}y_i$
In-phase amplitude addition
$A_{\max}=A_1+A_2$
Opposite-phase resultant amplitude
$A_{\min}=\left|A_1-A_2\right|$
Path difference
$\Delta r=\left|r_2-r_1\right|$
Constructive and destructive conditions
$\Delta r=m\lambda\quad\text{or}\quad\Delta r=\left(m+\frac12\right)\lambda$
Path-induced unwrapped phase-lag magnitude
$\Delta\phi=\frac{2\pi\Delta r}{\lambda}$
Double-slit path difference
$\Delta r\approx s\lvert\sin\theta\rvert\approx\frac{s\lvert y\rvert}{D}$
Bright-fringe position: far field, small angles
$y_m=\frac{m\lambda D}{s}$
Fringe spacing: far field, small angles
$w=\frac{\lambda D}{s}$
Wavelength from a near-axis fringe pattern
$\lambda=\frac{ws}{D}$
Far-field rectangular-slit first-minimum condition at normal incidence
$b\sin\theta_1=\lambda$
Small-angle first-minimum approximation
$\theta_1\approx\frac{\lambda}{b}$
First-minimum screen position
$y_1\approx\frac{\lambda D}{b}$
Central-maximum screen width
$W\approx\frac{2\lambda D}{b}$
Interference-factor maxima: normal incidence, far field
$d\sin\theta=m\lambda$
Single-slit envelope minima
$b\sin\theta=p\lambda$
Missing-order condition
$m=p\frac{d}{b}$
Grating phase-matching equation at normal incidence
$m\lambda=d\sin\theta$
Spacing from line density
$d=\frac{1}{q}$
Wavelength from a nonzero grating order at normal incidence
$\lambda=\frac{d\sin\theta}{m}$
Highest possible grating-order bound
$m_{\max}\le\frac{d}{\lambda}$
Envelope angular scale
$\theta_{\mathrm{env}}\sim\frac{\lambda}{b}$
Interference angular spacing near the centre
$\Delta\theta_{\mathrm{fringe}}\approx\frac{\lambda}{d}$
Angular-to-linear conversion
$y\approx D\theta$
Principal intensity for equal slit amplitudes
$I_{\max}=N^2 I_1$
Mean fringe spacing
$w=\frac{L}{N}$
Double-slit graph gradient
$\frac{\Delta w}{\Delta D}=\frac{\lambda}{s}$
Percentage uncertainty
$\%\,u=\frac{\Delta q}{\lvert q\rvert}\times100\%$
Topic C.4Not started
Standing waves and resonance
Standing waves arise from superposition of waves travelling in opposite directions. Resonance occurs when a system is driven near one of its natural frequencies.
240 min estimated
Topic C.4 progress0%
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Learning objectives
Identify nodes, antinodes and wavelength in standing-wave patterns.
Relate boundary conditions to permitted harmonics in strings and pipes.
Calculate fundamental and overtone frequencies in simple systems.
Describe resonance and the effect of damping on amplitude and resonance width.
Explain energy transfer in driven oscillations.
Key formulas
Right-moving component
$y_1=A\sin(kx-\omega t)$
Left-moving component
$y_2=A\sin(kx+\omega t)$
Resultant standing wave
$y=2A\sin(kx)\cos(\omega t)$
Adjacent-node or adjacent-antinode spacing
$\Delta x=\frac{\lambda}{2}$
Node-to-nearest-antinode spacing
$\Delta x=\frac{\lambda}{4}$
Neighbouring-loop phase difference
$\Delta\phi=\pi\,\mathrm{rad}$
Fixed–fixed string: allowed wavelengths
$\lambda_n=\frac{2L}{n}$
Fixed–fixed string: harmonic frequencies
$f_n=\frac{nv}{2L}=nf_1$
Optional supporting model: wave speed on a uniform ideal string
$v=\sqrt{\frac{F_T}{\mu}}$
Optional uniform fixed–fixed string: first-harmonic frequency
$f_1=\frac{1}{2L}\sqrt{\frac{F_T}{\mu}}$
Open–open pipe allowed wavelengths
$\lambda_n=\frac{2L}{n}$
Open–open pipe frequencies
$f_n=\frac{nv}{2L}$
Open–open pipe first-harmonic length
$L=\frac{\lambda_1}{2}$
Closed-open allowed wavelengths
$\lambda_m=\frac{4L}{2m-1}$
Closed-open resonance frequencies
$f_m=\frac{(2m-1)v}{4L}$
Closed-open first harmonic
$f_1=\frac{v}{4L}$
Light-damping resonance: approximate frequency condition
$f_d\approx f_0$
Natural period
$T_0=\frac{1}{f_0}$
Energy supplied by the driver over an interval, not net stored-energy gain
$\Delta E=P_{\mathrm{in}}\Delta t$
Optional supporting model: linear viscous damping
$F_d=-bv$
Optional supporting model: free underdamped amplitude envelope
$A(t)=A_0e^{-\gamma t}$
Optional narrow-resonance bandwidth estimate
$Q\approx\frac{f_0}{\Delta f}$
Travelling-wave intensity at fixed medium and frequency
$I\propto A^2$
Energy transferred by a driver
$E=P\Delta t$
Wavelength from string loops
$\lambda=\frac{2L}{n}$
Closed–open pipe first-harmonic sound speed
$v=4Lf_1$
Optional combined model: uniform fixed–fixed string
$f_n=\frac{n}{2L}\sqrt{\frac{F_T}{\mu}}$
Equal ideal length and sound speed: pipe first-harmonic ratio
$\frac{f_{\mathrm{open-open}}}{f_{\mathrm{closed-open}}}=2$
Percentage difference from a model
$\%\,\mathrm{difference}=\frac{\left|q_{\mathrm{measured}}-q_{\mathrm{model}}\right|}{\left|q_{\mathrm{model}}\right|}\times100\%$
Topic C.5Not started
Doppler effect
The Doppler effect is the change in observed frequency caused by relative motion between source, observer and wave medium or by relative motion in electromagnetic-wave contexts.
240 min estimated
Topic C.5 progress0%
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Learning objectives
Explain qualitatively why approach increases observed frequency and recession decreases it.
Use the low-speed fractional-shift approximation for waves when appropriate.
Interpret redshift and blueshift in electromagnetic contexts.
Calculate the observed frequency for a moving source in a stationary medium.
Calculate the observed frequency for a moving observer in a stationary medium.
Choose signs by physical reasoning: approach should raise frequency and recession should lower it.
Key formulas
Wave equation using quantities in the same frame
$v_w=f\lambda$
Frequency and period
$f=\frac{1}{T}$
Fractional frequency shift
$\text{fractional shift}=\frac{\Delta f}{f_s}=\frac{f_o-f_s}{f_s}$
Wavelength ahead of a moving source
$\lambda_{ahead}=(v_w-v_s)T_s=\frac{v_w-v_s}{f_s}$
Wavelength behind a moving source
$\lambda_{behind}=(v_w+v_s)T_s=\frac{v_w+v_s}{f_s}$
Frequency received by a stationary observer
$f_o=\frac{v_w}{\lambda}$
Observer approaching a stationary source
$f_o=f_s\left(\frac{v_w+v_o}{v_w}\right)$
Observer receding from a stationary source
$f_o=f_s\left(\frac{v_w-v_o}{v_w}\right)$
Radial component
$v_r=v\cos\theta$
Optional extension: Direct approach in a stationary medium
$f_o=f_s\left(\frac{v_w+v_o}{v_w-v_s}\right)$
Optional extension: Direct recession in a stationary medium
$f_o=f_s\left(\frac{v_w-v_o}{v_w+v_s}\right)$
Optional extension: Mach number
$M=\frac{v_s}{v_w}$
Approaching moving observer
$f'=f\frac{v+v_o}{v}$
Receding moving observer
$f'=f\frac{v-v_o}{v}$
Signed moving-observer form
$f'=f\frac{v+v_o}{v}$
Approaching moving source
$f'=f\frac{v}{v-v_s}$
Receding moving source
$f'=f\frac{v}{v+v_s}$
Moving-source wavelength
$\lambda'=\frac{v\mp v_s}{f}$
Optional beyond assessed scope: combined signed sound relation
$f'=f\frac{v+v_o}{v-v_s}$
Low-speed fractional-shift magnitude
$\left|\frac{\Delta f}{f}\right|\approx\frac{|v_{relative}|}{v}$
Frequency-ratio check
$\frac{f'}{f}>1\ \text{for approach},\qquad \frac{f'}{f}<1\ \text{for recession}$
Light wave relationship
$c=f\lambda$
Small-speed wavelength shift
$\frac{\Delta\lambda}{\lambda_0}\approx\frac{v_r}{c}$
Small-speed frequency shift
$\frac{\Delta f}{f_0}\approx-\frac{v_r}{c}$
Opposite fractional changes
$\frac{\Delta f}{f_0}\approx-\frac{\Delta\lambda}{\lambda_0}$
Redshift parameter
$z=\frac{\lambda_{obs}-\lambda_0}{\lambda_0}=\frac{\Delta\lambda}{\lambda_0}$
Low-speed Doppler radial-velocity estimate
$v_r\approx zc$
Observed wavelength from redshift
$\lambda_{obs}=(1+z)\lambda_0$
Optional quantitative application: Low-speed reflected electromagnetic shift
$\frac{|\Delta f|}{f_0}\approx\frac{2|u_r|}{c}$
Optional quantitative application: Doppler ultrasound echo shift
$|\Delta f|\approx\frac{2f_0u|\cos\theta|}{v_w}$
Optional quantitative application: Blood speed inferred from shift
$u\approx\frac{|\Delta f|v_w}{2f_0|\cos\theta|},\quad |\cos\theta|>0$
Wavelength change from radial velocity
$\Delta\lambda\approx\frac{\lambda_0}{c}v_r$
Signed radial velocity component
$u_r=u\cos\theta$
Percentage wavelength shift
$\text{percentage shift}=100\frac{\lambda_{obs}-\lambda_0}{\lambda_0}$
Approaching-source frequency
$f_o=f_s\frac{v_w}{v_w-v_s}$
Source speed from observed frequency
$v_s=v_w\left(1-\frac{f_s}{f_o}\right)$
First-order light shift: total relative speed much smaller than c
$\left|\frac{\Delta f}{f_0}\right|\approx\left|\frac{v_r}{c}\right|\ll1$
Unit 4
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Fields
Theme D develops gravitational, electric and magnetic field models and then studies charged-particle motion and electromagnetic induction. Higher Level introduces potential-energy and potential-field descriptions as well as induction.
38 hours estimated ·4 topics
Unit 4 progress0%
Topics in this unit
Topic D.1Not started
Gravitational fields
Gravitational fields describe how masses interact at a distance. HL extends force and field strength into gravitational potential, potential energy, orbital energy and escape conditions.
720 min estimated
Topic D.1 progress0%
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Learning objectives
Apply Newton’s law of gravitation.
Calculate gravitational field strength around a point or spherical mass.
Interpret gravitational field lines.
Use Kepler’s laws qualitatively and quantitatively in orbital contexts.
Calculate gravitational potential energy and gravitational potential.
Connect field strength to the spatial gradient of potential.
Use equipotential surfaces to reason about work and field direction.
Calculate work done in moving a mass through a potential difference.
Derive or use orbital speed for a circular orbit.
Calculate escape speed from a spherical body.
Analyse energy changes in satellites and other gravitational systems.
Key formulas
Definition of gravitational field strength
$g=\frac{F}{m}$
Weight
$W=mg$
Equivalent field-strength units
$1\ \mathrm{N\,kg^{-1}}=1\ \mathrm{m\,s^{-2}}$
Newton's law of gravitation
$F=G\frac{m_1m_2}{r^2}$
Gravitational constant
$G\approx6.67\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}$
Inverse-square force ratio
$\frac{F_2}{F_1}=\frac{m_{1,2}m_{2,2}}{m_{1,1}m_{2,1}}\left(\frac{r_1}{r_2}\right)^2$
Radial gravitational field strength
$g=G\frac{M}{r^2}$
Force on a mass in a field
$\vec F=m\vec g$
Source mass from field data
$M=\frac{gr^2}{G}$
Inverse-square field ratio
$\frac{g_2}{g_1}=\left(\frac{r_1}{r_2}\right)^2$
Radial vector direction
$\vec g=-\frac{GM}{r^2}\hat r$
Area of a sphere
$A=4\pi r^2$
Field strength at altitude
$g(h)=\frac{GM}{(R+h)^2}$
Altitude-to-surface field ratio
$\frac{g(h)}{g_0}=\left(\frac{R}{R+h}\right)^2$
Optional first-order estimate of small-height fractional change
$\frac{\Delta g}{g_0}\approx-\frac{2h}{R}\quad(h\ll R)$
Superposition of gravitational fields
$\vec g_{net}=\sum\limits_{i=1}^{N}\vec g_i$
Two-mass cancellation condition
$\frac{GM_1}{x^2}=\frac{GM_2}{(d-x)^2}$
Zero-field position between two masses
$x=\frac{d\sqrt{M_1}}{\sqrt{M_1}+\sqrt{M_2}}$
Gravitational potential energy
$E_p=-\frac{GMm}{r}$
Gravitational potential
$V_g=-\frac{GM}{r}$
Energy-potential relation
$E_p=mV_g$
Radial field-potential relation
$g_r=-\frac{dV_g}{dr}$
Finite-difference estimate
$g_r\approx-\frac{\Delta V_g}{\Delta r}$
Spherical-mass radial field
$g_r=-\frac{GM}{r^2}$
Potential-energy change
$\Delta E_p=m\Delta V_g=m(V_B-V_A)$
Work done by gravity
$W_{gravity}=-\Delta E_p=-m\Delta V_g$
External transfer with no kinetic-energy change or other losses
$W_{external}=\Delta E_p=m\Delta V_g$
Optional ellipse geometry: periapsis and apoapsis distances
$r_p=a(1-e),\qquad r_a=a(1+e)$
Kepler's equal-area statement
$\frac{\Delta A}{\Delta t}=\text{constant}$
Optional small-sector area approximation
$\Delta A\approx\frac{1}{2}r^2\Delta\theta$
Kepler's third-law ratio
$\frac{T_1^2}{a_1^3}=\frac{T_2^2}{a_2^3}$
Newton–Kepler orbital relationship
$T^2=\frac{4\pi^2a^3}{GM}$
Central mass from an orbit
$M=\frac{4\pi^2a^3}{GT^2}$
Circular-orbit speed
$v_{orbit}=\sqrt{\frac{GM}{r}}$
Circular-orbit force balance
$\frac{GMm}{r^2}=\frac{mv^2}{r}$
Circular-orbit period
$T=2\pi\sqrt{\frac{r^3}{GM}}$
Circular-orbit kinetic energy
$E_k=\frac{GMm}{2r}$
Circular-orbit potential energy
$E_p=-\frac{GMm}{r}=-2E_k$
Circular-orbit total energy
$E=-\frac{GMm}{2r}=-E_k$
Escape-energy condition
$\frac12mv_{escape}^2-\frac{GMm}{r}=0$
Escape speed
$v_{escape}=\sqrt{\frac{2GM}{r}}$
Escape-orbit relation
$v_{escape}=\sqrt2\,v_{orbit}$
Circular-orbit energy change
$\Delta E=\frac{GMm}{2}\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$
Mechanical energy: gravity alone, fixed source and no drag
$E_k+E_p=\text{constant}$
Bound-or-escape test
$E<0\ \text{bound},\qquad E\ge0\ \text{escape possible}$
Orbital period from central mass
$T=2\pi\sqrt{\frac{a^3}{GM}}$
Gradient of T squared against a cubed
$S=\frac{4\pi^2}{GM}$
Small-angle simple-pendulum estimate of g (SHM link)
$g=\frac{4\pi^2L}{T_p^2}$
Inverse-square scaling check
$g\propto\frac{M}{r^2}$
Orbital-period scaling
$\frac{T_2}{T_1}=\left(\frac{a_2}{a_1}\right)^{\frac{3}{2}}$
Percentage difference from surface g
$\text{percentage change}=100\frac{g(h)-g_0}{g_0}$
Topic D.2Not started
Electric and magnetic fields
This subtopic treats electrostatic and magnetic interactions through force and field models. HL adds electric potential and potential energy, allowing direct comparison with gravitational fields.
840 min estimated
Topic D.2 progress0%
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Learning objectives
Apply Coulomb’s law for point charges.
Define electric field strength as force per unit positive test charge.
Use field-line representations and superposition ideas.
Use $E = \frac{V}{d}$ for a uniform field between parallel plates.
Describe magnetic fields around magnets, wires and solenoids.
Interpret evidence for quantized charge such as the Millikan experiment.
Calculate electric potential energy for pairs of point charges.
Calculate electric potential due to point or spherical charge distributions where the point-charge model applies.
Connect electric field to the negative gradient of electric potential.
Use equipotential surfaces and potential differences to analyse work on charges.
Compare gravitational and electric potential models, including the role of sign.
Key formulas
Net charge from particle counts
$Q=(N_p-N_e)e$
Elementary charge
$e\approx1.60\times10^{-19}\ \mathrm{C}$
Third-law force pair
$\vec F_{1\to2}=-\vec F_{2\to1}$
Charge conservation
$Q_{\text{total,before}}=Q_{\text{total,after}}$
Equal charge sharing under equivalent electrical conditions
$Q_{\text{each}}=\frac{Q_1+Q_2+\cdots}{N}$
Transferred-electron count
$N_e=\frac{|\Delta Q|}{e}$
Neutral polarized conductor
$Q_{\text{near}}+Q_{\text{far}}=0$
Charge gained through grounding
$Q_{\text{final}}=Q_{\text{initial}}+\Delta Q_{\text{Earth}}$
Optional two-point polarization model: equal opposite induced charges
$\frac{F_{\text{near}}}{F_{\text{far}}}=\left(\frac{r_{\text{far}}}{r_{\text{near}}}\right)^2$
Coulomb's law
$F=k\frac{|q_1q_2|}{r^2}$
Coulomb constant
$k=\frac1{4\pi\epsilon_0}\approx8.99\times10^9\ \mathrm{N\,m^2\,C^{-2}}$
Force superposition
$\vec F_{\text{net}}=\sum\limits_{i=1}^{N}\vec F_i$
Charge quantization
$Q=ne,\qquad n=0,\pm1,\pm2,\ldots$
Stationary oil drop: upward electric force, negligible buoyancy
$|q|E=mg$
Elementary-charge count
$N=\frac{|Q|}{e}$
Electric field definition
$\vec E=\frac{\vec F}{q_{\text{test}}}\quad(q_{\text{test}}>0)$
Force in an electric field
$\vec F=q\vec E$
Electric-field unit equivalence
$1\ \mathrm{N\,C^{-1}}=1\ \mathrm{V\,m^{-1}}$
Point-charge field strength
$E=k\frac{|Q|}{r^2}$
Electric-field superposition
$\vec E_{\text{net}}=\sum\limits_{i=1}^{N}\vec E_i$
Inverse-square field ratio
$\frac{E_2}{E_1}=\left(\frac{r_1}{r_2}\right)^2$
Uniform parallel-plate field
$E=\frac{V}{d}$
Electric force between plates
$\vec F=q\vec E$
D.3 link: non-relativistic acceleration when electric force dominates
$\vec a=\frac{q\vec E}{m}$
Point-charge potential energy
$E_p=\frac{kq_1q_2}{r}$
Potential-energy change
$\Delta E_p=E_{p,f}-E_{p,i}$
Electrostatic energy conservation
$\Delta E_k=-\Delta E_p$
Point-charge potential
$V_e=\frac{kQ}{r}$
Potential superposition
$V_{net}=\sum\limits_{i=1}^{N}\frac{kQ_i}{r_i}$
Energy from potential
$E_p=qV_e$
One-dimensional field-potential relation
$E_x=-\frac{dV_e}{dx}$
Radial field-potential relation
$E_r=-\frac{dV_e}{dr}$
Finite-difference estimate
$E_x\approx-\frac{\Delta V_e}{\Delta x}$
Electric potential-energy change
$\Delta E_p=q\Delta V_e=q(V_B-V_A)$
Work by electric field
$W_{field}=-q\Delta V_e=q(V_A-V_B)$
Uniform field with sign
$E_x=-\frac{\Delta V_e}{\Delta x}$
Electric and gravitational potentials
$V_e=\frac{kQ}{r},\qquad V_g=-\frac{GM}{r}$
Electric and gravitational fields
$E_r=-\frac{dV_e}{dr},\qquad g_r=-\frac{dV_g}{dr}$
Pair-potential energies
$E_{p,e}=\frac{kQq}{r},\qquad E_{p,g}=-\frac{GMm}{r}$
Magnetic-field superposition
$\vec B_{\text{net}}=\sum\limits_{i=1}^{N}\vec B_i$
External bar-magnet direction
$N\longrightarrow S\quad(\text{outside})$
Internal return direction
$S\longrightarrow N\quad(\text{inside})$
Optional long-wire exterior model: Current scaling
$B\propto I\quad(r\text{ fixed})$
Optional long-wire exterior model: Distance scaling
$B\propto\frac{1}{r}\quad(I\text{ fixed})$
Optional long-wire exterior model: Two-condition field ratio
$\frac{B_2}{B_1}=\frac{I_2r_1}{I_1r_2}$
Turn density
$n=\frac{N}{L}$
Optional long air-core model: Solenoid field scaling
$B\propto nI$
Optional long air-core model: Solenoid comparison
$\frac{B_2}{B_1}\approx\frac{n_2I_2}{n_1I_1}$
Electrostatic-force scaling
$F\propto\frac{|q_1q_2|}{r^2}$
Field-force connection
$\vec F=q\vec E$
Combined-source rule
$\vec X_{\text{net}}=\sum\limits_{i=1}^{N}\vec X_i\quad(\vec X=\vec F,\vec E,\vec B)$
Topic D.3Not started
Motion in electromagnetic fields
This subtopic applies electric and magnetic forces to moving charges and current-carrying conductors. It includes circular particle motion, velocity selection and forces between currents.
360 min estimated
Topic D.3 progress0%
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Learning objectives
Calculate the force on a charge in a uniform electric field.
Calculate magnetic force on a moving charge and determine its direction.
Analyse circular motion of charged particles in a uniform magnetic field.
Use crossed electric and magnetic fields as a velocity selector.
Calculate force on a current-carrying conductor in a magnetic field.
Analyse the force per unit length between parallel current-carrying wires.
Key formulas
Electric force
$\vec F_E=q\vec E$
Magnetic force
$\vec F_B=q\vec v\times\vec B$
Combined Lorentz force
$\vec F=q(\vec E+\vec v\times\vec B)$
Electric acceleration
$\vec a=\frac{q\vec E}{m}$
Velocity under constant electric acceleration
$\vec v=\vec u+\frac{q\vec E}{m}t$
Displacement under constant electric acceleration
$\vec s=\vec u t+\frac{1}{2}\frac{q\vec E}{m}t^2$
Transit time through plates
$t=\frac{L}{v_x}$
Transverse deflection
$y=\frac{1}{2}\frac{qE}{m}t^2$
Electric-field trajectory
$y=\frac{qE}{2mv_x^2}x^2$
Magnetic-force magnitude
$F_B=|q|vB\sin\theta$
Perpendicular speed
$v_\perp=v\sin\theta$
Magnetic power
$P_B=\vec F_B\cdot\vec v=0$
Magnetic centripetal balance
$|q|vB=\frac{mv^2}{r}$
Magnetic orbit radius
$r=\frac{mv}{|q|B}$
Magnetic orbit period
$T=\frac{2\pi m}{|q|B}$
Velocity components
$v_\parallel=v\cos\theta,\qquad v_\perp=v\sin\theta$
Helix radius
$r=\frac{mv_\perp}{|q|B}$
Helix pitch
$p=|v_{\parallel}|T=\frac{2\pi m|v_{\parallel}|}{|q|B}$
Selector force balance
$|q|E=|q|vB$
Selected speed
$v=\frac{E}{B}$
Initial transverse force magnitude in the standard selector geometry
$F_{\text{net}}=|q|\,|E-vB|$
Charge-to-mass ratio from a magnetic orbit
$\frac{|q|}{m}=\frac{v}{Br}$
Selector plus analyser
$\frac{|q|}{m}=\frac{E}{B_sB_ar}$
Mass-to-radius comparison
$\frac{r_2}{r_1}=\frac{m_2|q_1|}{m_1|q_2|}$
Conductor-force magnitude
$F=BIL\sin\theta$
Vector conductor force
$\vec F=I\vec L\times\vec B$
Force per unit active length
$\frac{F}{L}=BI\sin\theta$
Field of wire 1 at wire 2
$B_1=\frac{\mu_0I_1}{2\pi d}$
Parallel-wire force per length
$\frac{F}{L}=\frac{\mu_0I_1I_2}{2\pi d}$
Permeability of free space
$\mu_0\approx4\pi\times10^{-7}\ \mathrm{T\,m\,A^{-1}}$
Field from current balance: nonzero current, length and sine of angle
$B=\frac{F}{IL\sin\theta}$
Current-balance gradient
$\frac{\Delta F}{\Delta I}=BL\quad(\theta=90^\circ)$
Force reversal checks
$I\to-I\Rightarrow\vec F\to-\vec F,\qquad \vec B\to-\vec B\Rightarrow\vec F\to-\vec F$
Electromagnetic decision equation
$\vec F=q(\vec E+\vec v\times\vec B)$
Magnetic-radius scaling
$r\propto\frac{mv_\perp}{|q|B}$
Parallel-wire scaling
$\frac{F}{L}\propto\frac{I_1I_2}{d}$
Topic D.4Not started
Induction
Electromagnetic induction is an HL-only subtopic describing how changing magnetic flux produces an emf. It links generators, moving conductors and energy conservation through Faraday’s and Lenz’s laws.
360 min estimated
Topic D.4 progress0%
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Learning objectives
Define magnetic flux and calculate it for a uniform field through a planar loop.
Apply Faraday’s law to changing flux and multi-turn coils.
Use Lenz’s law to determine the direction of induced current or emf.
Calculate motional emf for a conductor moving through a magnetic field.
Analyse the sinusoidal emf produced by a rotating coil in a uniform magnetic field.
Explain how changing rotation frequency affects the induced emf.
Describe self-induction qualitatively and relate it to opposition to changes in current.
Key formulas
Emf as energy per charge
$\mathcal E=\frac{W}{q}$
Current magnitude: passive resistive loop, negligible self-inductance
$I=\frac{\lvert\mathcal E\rvert}{R}$
Nonzero net emf around a specified circuit contour
$\frac{\mathrm d(N\Phi)}{\mathrm dt}\ne0$
Magnetic flux
$\Phi=BA\cos\theta$
Maximum flux magnitude
$\lvert\Phi\rvert_{\max}=BA$
Weber equivalence
$1\ \mathrm{Wb}=1\ \mathrm{T\,m^2}$
Flux linkage
$\lambda=N\Phi=NBA\cos\theta$
Change in linkage for a fixed number of equal-flux turns
$\Delta\lambda=N(\Phi_{\mathrm f}-\Phi_{\mathrm i})$
Average graph gradient over a finite interval
$\text{gradient}=\frac{\Delta(N\Phi)}{\Delta t}$
Faraday’s law
$\mathcal E=-\frac{\mathrm d(N\Phi)}{\mathrm dt}$
Average induced emf
$\mathcal E_{\mathrm{avg}}=-N\frac{\Delta\Phi}{\Delta t}$
Magnitude form
$\lvert\mathcal E\rvert=\left\lvert\frac{\mathrm d(N\Phi)}{\mathrm dt}\right\rvert$
Faraday–Lenz law
$\mathcal E=-N\frac{\mathrm d\Phi}{\mathrm dt}$
Magnetic field at a loop centre direction cue
$I_{\mathrm{CCW}}\Rightarrow\vec B_{\mathrm{ind}}\ \text{out of page}$
Clockwise direction cue
$I_{\mathrm{CW}}\Rightarrow\vec B_{\mathrm{ind}}\ \text{into page}$
Magnetic force on charge
$\vec F=q\vec v\times\vec B$
Force magnitude
$F=|q|vB\sin\alpha$
Field-direction rule for a loop
$\text{thumb}=\vec B_{\mathrm{ind}},\quad\text{fingers}=I$
Motional-emf magnitude: rod, velocity and uniform field mutually perpendicular
$|\mathcal E|=B\ell v$
Magnetic force on the rod
$F=BI\ell$
Input-power balance: constant-speed, frictionless resistive rail circuit
$P=Fv=|\mathcal E|I=I^2R$
Optional extension: motional emf along a specified directed path
$\mathcal E=\int(\vec v\times\vec B)\boldsymbol\cdot\mathrm d\vec\ell$
Rotating-coil flux linkage
$N\Phi=NBA\cos(\omega t)$
Sinusoidal generator emf
$\mathcal E=NBA\omega\sin(\omega t)$
Peak generator emf
$\mathcal E_0=NBA\omega$
Frequency and angular speed
$\omega=2\pi f$
Period–frequency relation
$T=\frac1f$
Adjacent extrema time separation: sinusoidal linkage and emf
$\Delta t=\frac{T}{4}$
Peak-emf proportionality
$\mathcal E_0\propto f$
Magnitude of sinusoidal linkage–emf phase separation
$\Delta\varphi=\frac{\pi}{2}$
Optional: fixed linear inductor flux–current relation
$N\Phi=LI$
Optional: self-induced emf for constant inductance
$\mathcal E_L=-L\frac{\mathrm dI}{\mathrm dt}$
Optional: magnetic energy of a linear lossless inductor
$U_B=\frac12LI^2$
Optional: henry unit equivalence
$1\ \mathrm H=1\ \mathrm{V\,s\,A^{-1}}$
Core induction chain
$\Phi=BA\cos\theta\quad\longrightarrow\quad N\Phi\quad\longrightarrow\quad\mathcal E=-\frac{\mathrm d(N\Phi)}{\mathrm dt}$
Current magnitude: passive resistive circuit, negligible self-inductance
$I=\frac{\lvert\mathcal E\rvert}{R}$
Rotating-coil maximum
$\mathcal E_0=2\pi fNBA$
Unit 5
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Nuclear and Quantum Physics
Theme E follows the development of atomic and nuclear models, then extends into wave–particle duality, radioactive decay, fission, fusion and stellar physics. The HL-only quantum subtopic is one of the most conceptually distinctive parts of the course.
39 hours estimated ·5 topics
Unit 5 progress0%
Topics in this unit
Topic E.1Not started
Structure of the atom
This subtopic traces evidence for atomic structure from scattering and spectra to photons and quantized energy levels. HL adds quantitative nuclear-size and Bohr-model relationships.
720 min estimated
Topic E.1 progress0%
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Learning objectives
Describe how Rutherford scattering supports a small, dense, positively charged nucleus.
Use nuclear notation and distinguish protons, neutrons, nucleons and electrons.
Interpret emission and absorption spectra as evidence for discrete atomic energy levels.
Use photon energy E = hf.
Use the nuclear-radius relationship R = $R_{0}$ A^( $\frac{1}{3}$ ) and discuss approximately constant nuclear density.
Use closest-approach energy reasoning in charged-particle scattering.
Calculate hydrogen energy levels with the Bohr model.
Use quantization of angular momentum in the Bohr model.
Connect transitions between levels to photon frequencies and wavelengths.
Key formulas
Neutral-atom charge balance
$Q_{\text{atom}}=Ze-Ze=0$
Alpha-particle charge
$q_\alpha=+2e$
Nuclear-to-atomic radius scale
$\frac{r_{\text{nucleus}}}{r_{\text{atom}}}\sim10^{-5}$
Detected fraction
$f=\frac{N_{\text{detected}}}{N_{\text{incident}}}$
Percentage of events
$p=100\frac{N_{\text{selected}}}{N_{\text{total}}}\%$
Alpha-particle composition
${}^{4}_{2}\mathrm{He}^{2+}=2p+2n$
Alpha–nucleus Coulomb force magnitude outside the nucleus
$F=k\frac{(2e)(Ze)}{r^2}$
Inverse-square force comparison
$\frac{F_2}{F_1}=\left(\frac{r_1}{r_2}\right)^2$
Average count rate
$R=\frac{N}{\Delta t}$
Atomic-to-nuclear radius ratio
$\frac{r_{\text{atom}}}{r_{\text{nucleus}}}\sim\frac{10^{-10}}{10^{-15}}=10^5$
Atomic mass unit
$1\,\mathrm u\approx1.66\times10^{-27}\,\mathrm{kg}$
Approximate nuclear mass
$m_{\text{nucleus}}\approx A\mathrm{u}$
Nuclear notation
${}^{A}_{Z}X$
Neutron number
$N=A-Z$
Nucleon count
$A=Z+N$
Ion charge
$Q_{\text{ion}}=(Z-N_e)e$
Electron count from ion charge
$N_e=Z-\frac{Q_{\text{ion}}}{e}$
Isotope condition
$Z_1=Z_2,\qquad A_1\ne A_2$
Nuclear-radius relationship
$R=R_0A^{\frac{1}{3}}$
Nuclear-radius ratio
$\frac{R_2}{R_1}=\left(\frac{A_2}{A_1}\right)^{\frac{1}{3}}$
Mean nuclear density
$\rho\approx\frac{Am_{\mathrm n}}{\frac43\pi R^3}=\frac{3m_{\mathrm n}}{4\pi R_0^3}$
Energy at closest approach
$E_{\mathrm{k}}=\frac{k(2e)(Ze)}{r_{\min}}=\frac{2kZe^2}{r_{\min}}$
Closest-approach distance
$r_{\min}=\frac{2kZe^2}{E_{\mathrm{k}}}$
Electronvolt conversion
$1\,\mathrm{MeV}\approx1.60\times10^{-13}\,\mathrm J$
Excitation energy
$\Delta E=E_{\text{higher}}-E_{\text{lower}}>0$
Upper–lower level-pair count: an upper bound on transition lines
$N_{\text{transitions}}=\frac{n(n-1)}{2}$
Ionization energy from a bound level
$E_{\text{ion}}=E_{\text{limit}}-E_{\text{bound}}$
Photon energy
$E_\gamma=hf$
Emission condition
$hf=E_i-E_f\quad(E_i>E_f)$
Absorption condition
$E_f=E_i+hf$
Photon energy from wavelength
$E_\gamma=hf=\frac{hc}{\lambda}$
Energy-gap wavelength
$\lambda=\frac{hc}{|\Delta E|}$
Electronvolt conversion
$1\,\mathrm{eV}\approx1.60\times10^{-19}\,\mathrm J$
Hydrogen energy levels
$E_n=-\frac{13.6\ \mathrm{eV}}{n^2}$
Ionization energy from level n
$E_{\mathrm{ion},n}=0-E_n=|E_n|$
Joule-electronvolt conversion
$1\,\mathrm{eV}\approx1.60\times10^{-19}\,\mathrm J$
Bohr angular-momentum condition
$m_{\mathrm{e}}vr=\frac{nh}{2\pi}=n\hbar$
Reduced Planck constant
$\hbar=\frac{h}{2\pi}$
Coulomb force as centripetal force
$\frac{ke^2}{r^2}=\frac{m_{\mathrm{e}}v^2}{r}$
Photon energy from a transition
$E_\gamma=|E_f-E_i|=hf=\frac{hc}{\lambda}$
Hydrogen transition energy
$E_\gamma=13.6\ \mathrm{eV}\left|\frac{1}{n_f^2}-\frac{1}{n_i^2}\right|$
Electronvolt-nanometre shortcut
$E_\gamma\lambda\approx1240\,\mathrm{eV\,nm}$
Spectral frequency
$f=\frac{c}{\lambda}$
Line energy difference
$|\Delta E|=\frac{hc}{\lambda}$
Wavelength match condition
$\lambda_{\text{abs}}=\lambda_{\text{em}}\quad(\text{same level pair})$
C.3 link: normal-incidence diffraction-grating equation
$d\sin\theta=m\lambda$
Grating spacing from line density
$d=\frac{1}{n_g}$
Percentage deviation from a positive reference wavelength
$\text{percentage difference}=100\frac{|\lambda_{\text{meas}}-\lambda_{\text{ref}}|}{\lambda_{\text{ref}}}\%$
Particle-count checklist
$p=Z,\qquad n=A-Z,\qquad Q=(Z-N_e)e$
One-photon bound-bound energy checklist: consistent units, vacuum wavelength
$E_\gamma=|E_i-E_f|=hf=\frac{hc}{\lambda}$
Energy-wavelength scaling
$E_\gamma\propto f\propto\frac{1}{\lambda}$
Topic E.2Not started
Quantum physics
Quantum physics is HL-only. It uses experimental evidence to show that electromagnetic radiation and matter each display both wave-like and particle-like behaviour, depending on the measurement context.
480 min estimated
Topic E.2 progress0%
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Learning objectives
Explain the photoelectric effect and why it conflicts with a purely classical wave model of light.
Use work function, threshold frequency and maximum photoelectron kinetic energy.
Apply the de Broglie wavelength to matter particles.
Interpret electron or particle diffraction as evidence for matter waves.
Discuss wave–particle duality without assuming a classical particle or classical wave picture is universally adequate.
Analyse Compton scattering as a photon–electron collision and use the Compton wavelength-shift equation.
Key formulas
Planck relation
$E=hf$
de Broglie relation
$\lambda=\frac{h}{p}$
Photon momentum
$p_{\gamma}=\frac{h}{\lambda}$
Photon energy
$E=hf=\frac{hc}{\lambda}$
Photon momentum
$p_{\gamma}=\frac{E}{c}=\frac{h}{\lambda}$
Photon rate
$\dot N=\frac{P}{hf}$
Electronvolt conversion
$1\ \mathrm{eV}=1.602176634\times10^{-19}\ \mathrm J$
Threshold condition
$hf_0=\phi$
Emission condition
$hf\ge\phi$
Photocurrent
$I=\frac{\Delta Q}{\Delta t}$
Einstein photoelectric equation
$hf=\phi+K_{\max}$
Maximum kinetic energy
$K_{\max}=hf-\phi$
Threshold frequency
$f_0=\frac{\phi}{h}$
Threshold wavelength
$\lambda_0=\frac{hc}{\phi}$
Stopping-potential relation
$K_{\max}=eV_s$
Kinetic-energy graph
$K_{\max}=hf-\phi$
Stopping-potential graph
$V_s=\frac{h}{e}f-\frac{\phi}{e}$
de Broglie wavelength
$\lambda=\frac{h}{p}$
Non-relativistic matter wavelength
$\lambda=\frac{h}{mv}$
Kinetic energy from accelerating potential
$qV=\frac{p^2}{2m}$
Accelerated-particle wavelength
$\lambda=\frac{h}{\sqrt{2mqV}}$
Supporting crystal method: Bragg maxima
$n\lambda=2d\sin\theta$
Voltage–wavelength proportionality
$\lambda\propto\frac{1}{\sqrt V}$
Momentum from measured wavelength
$p=\frac{h}{\lambda}$
de Broglie consistency
$p\lambda=h$
Photon energy consistency
$\frac{E}{f}=h$
Optional explanation: coherent indistinguishable two-path amplitudes
$\mathcal A_{\mathrm{total}}=\mathcal A_1+\mathcal A_2$
Photon energy
$E_{\gamma}=\frac{hc}{\lambda}$
Collision energy balance
$E_{\gamma,\mathrm i}=E_{\gamma,\mathrm f}+K_e$
Vector momentum balance
$\vec p_{\gamma,\mathrm i}=\vec p_{\gamma,\mathrm f}+\vec p_e$
Compton shift
$\Delta\lambda=\lambda'-\lambda=\frac{h}{m_ec}(1-\cos\theta)$
Electron Compton wavelength
$\lambda_C=\frac{h}{m_ec}\approx2.426\times10^{-12}\,\mathrm m$
Maximum Compton shift
$\Delta\lambda_{\max}=2\lambda_C$
Recoil kinetic energy
$K_e=hc\left(\frac{1}{\lambda}-\frac{1}{\lambda'}\right)$
Photoelectric chain
$\frac{hc}{\lambda}=\phi+K_{\max}=\phi+eV_s$
Matter-wave chain
$qV=\frac{p^2}{2m}\quad\longrightarrow\quad\lambda=\frac{h}{p}$
Compton chain
$\lambda'=\lambda+\lambda_C(1-\cos\theta)$
Topic E.3Not started
Radioactive decay
Radioactive decay links nuclear stability, random decay processes and radiation to energy and conservation laws. HL adds the exponential-decay model and a deeper treatment of nuclear stability and spectra.
900 min estimated
Topic E.3 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Distinguish isotopes and use nuclear notation.
Calculate mass defect, binding energy and energy released using E = $mc^{2}$ .
Describe the strong nuclear interaction qualitatively.
Write and balance alpha, beta-minus and beta-plus decay equations where applicable.
Compare penetration and ionization for alpha, beta and gamma radiation.
Use activity, count rate and half-life in simple decay problems.
Explain the role of neutrinos/antineutrinos in beta decay and conservation laws.
Use neutron-to-proton ratio and binding-energy ideas to discuss nuclear stability.
Explain why alpha and gamma emissions have discrete energy spectra while beta decay produces a continuous spectrum.
Use the exponential law for the number of undecayed nuclei.
Relate activity to decay constant and number of nuclei.
Relate decay constant to half-life.
Key formulas
Neutron number
$N=A-Z$
Nucleon count
$A=Z+N$
Ion charge
$Q=(Z-N_e)e$
Mass–energy equivalence
$\Delta E=\Delta m c^2$
Atomic mass unit
$1\,u\approx1.66054\times10^{-27}\,\mathrm{kg}$
Energy equivalent of u
$1\,u\,c^2\approx931.5\,\mathrm{MeV}$
Nuclear-mass defect
$\Delta m=Zm_p+(A-Z)m_n-m_{\text{nucleus}}$
Atomic-mass defect
$\Delta m\approx Zm_{\mathrm H}+(A-Z)m_n-m_{\text{atom}}$
Binding energy
$E_b=\Delta m c^2$
Binding energy per nucleon
$\overline{E}_b=\frac{E_b}{A}$
Total binding from average
$E_b=A\overline{E}_b$
Binding-energy change: unchanged total proton and neutron counts
$Q=E_{b,\text{final}}-E_{b,\text{initial}}$
Nuclear length scale
$1\,\mathrm{fm}=10^{-15}\,\mathrm{m}$
Neutron-to-proton ratio
$R_{np}=\frac{N}{Z}$
Proton-pair count
$N_{pp}=\frac{Z(Z-1)}{2}$
Neutron–proton ratio
$\text{neutron–proton ratio}=\frac{N}{Z}=\frac{A-Z}{Z}$
Binding energy per nucleon
$\frac{E_{\mathrm{b}}}{A}$
Nucleon-number conservation
$A_{\text{parent}}=A_{\text{daughters}}+A_{\text{emitted}}$
Charge-number conservation
$Z_{\text{parent}}=Z_{\text{daughters}}+Z_{\text{emitted}}$
Nuclear reaction energy
$Q=(m_{\text{initial}}-m_{\text{final}})c^2$
General alpha decay
${}^{A}_{Z}X\rightarrow{}^{A-4}_{Z-2}Y+{}^{4}_{2}\mathrm{He}$
Momentum conservation from rest
$\vec p_{\alpha}+\vec p_{\text d}=0$
Released kinetic energy
$Q=K_{\alpha}+K_{\text d}$
Beta-minus decay
${}^{A}_{Z}X\rightarrow{}^{A}_{Z+1}Y+{}^{0}_{-1}e+\bar\nu_e$
Beta-plus decay
${}^{A}_{Z}X\rightarrow{}^{A}_{Z-1}Y+{}^{0}_{+1}e+\nu_e$
Nucleon changes in energetically allowed beta decays
$n\rightarrow p+e^-+\bar\nu_e,\qquad p\rightarrow n+e^++\nu_e$
Gamma transition
${}^{A}_{Z}X^*\rightarrow{}^{A}_{Z}X+\gamma$
Gamma photon energy
$E_\gamma=hf=\frac{hc}{\lambda}$
Photon momentum
$p_\gamma=\frac{E_\gamma}{c}$
Gamma-transition energy: negligible nuclear recoil
$E_\gamma=hf\approx E_i-E_f$
Alpha/beta energy balance: Q referenced to ground-state products
$Q=K_{\mathrm{products}}+E_{\mathrm{excitation}}$
Beta energy sharing: selected daughter state and negligible neutrino rest energy
$Q=K_\beta+K_\nu+K_{\mathrm{recoil}}$
Electric force
$\vec F=q\vec E$
Magnetic-path radius
$r=\frac{p}{|q|B}$
Steady isotropic point source: negligible attenuation and background
$I\propto\frac{1}{r^2}$
Average activity
$A=\frac{\Delta N_{\text{decays}}}{\Delta t}$
Measured count rate
$C=\frac{N_{\text{counts}}}{\Delta t}$
Idealized net detection: one relevant emission per decay
$C_{\text{source}}=\varepsilon g A$
Parents after integer half-lives
$N=N_0\left(\frac{1}{2}\right)^n$
Activity after integer half-lives
$A=A_0\left(\frac{1}{2}\right)^n$
Elapsed complete half-lives
$n=\frac{t}{T_{\frac{1}{2}}}$
Rate equation for undecayed nuclei
$\frac{dN}{dt}=-\lambda N$
Exponential decay law
$N=N_0e^{-\lambda t}$
Time from population ratio
$t=\frac{1}{\lambda}\ln\left(\frac{N_0}{N}\right)$
Activity and undecayed nuclei
$A=-\frac{dN}{dt}=\lambda N$
Exponential activity law
$A=A_0e^{-\lambda t}$
Common remaining fraction
$\frac{A}{A_0}=\frac{N}{N_0}=e^{-\lambda t}$
Half-life and decay constant
$T_{\frac{1}{2}}=\frac{\ln 2}{\lambda}$
Decay constant from half-life
$\lambda=\frac{\ln 2}{T_{\frac{1}{2}}}$
Repeated-halving exponential form
$N=N_0\left(\frac{1}{2}\right)^{\frac{t}{T_{\frac{1}{2}}}}$
Background correction
$C_{\text{net}}=C_{\text{measured}}-C_{\text{background}}$
Supporting counting statistics: independent raw Poisson counts
$\sigma_N\approx\sqrt{N}$
Supporting fractional uncertainty: sufficiently many raw counts
$\frac{\sigma_N}{N}\approx\frac{1}{\sqrt{N}}$
Steady isotropic point source, negligible attenuation: distance trend
$\frac{I_2}{I_1}=\left(\frac{r_1}{r_2}\right)^2$
Time after complete half-lives
$t=nT_{\frac{1}{2}}$
Single parent, fixed response: integer-half-life net count model
$C_{\text{net}}=(C_{\text{measured}}-C_{\text{background}})=C_0\left(\frac{1}{2}\right)^n$
Topic E.4Not started
Fission
Nuclear fission releases energy when a heavy nucleus splits into more tightly bound products. The subtopic connects binding energy, neutron multiplication, chain reactions and reactor control.
240 min estimated
Topic E.4 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Explain energy release through changes in binding energy per nucleon.
Write and interpret representative fission reaction equations.
Explain how emitted neutrons can sustain a chain reaction.
Distinguish subcritical, critical and supercritical conditions qualitatively.
Describe the functions of moderator, control rods, coolant and shielding in a thermal-fission reactor.
Discuss practical benefits and challenges including waste, safety and resource use.
Key formulas
Fission bookkeeping: spontaneous parent or already formed compound nucleus
${}^{A}_{Z}X\rightarrow{}^{A_1}_{Z_1}Y+{}^{A_2}_{Z_2}W+x{}^{1}_{0}n+\text{energy}$
Nucleon-number conservation
$A=A_1+A_2+x$
Charge-number conservation
$Z=Z_1+Z_2$
Compound-nucleus formation
${}^{235}_{92}\mathrm U+{}^{1}_{0}n\rightarrow{}^{236}_{92}\mathrm U^*$
Representative fission channel
${}^{236}_{92}\mathrm U^*\rightarrow{}^{141}_{56}\mathrm{Ba}+{}^{92}_{36}\mathrm{Kr}+3{}^{1}_{0}n+Q$
Missing-neutron count
$x=A_{\text{initial}}-A_1-A_2$
Fission mass difference
$\Delta m=m_{\text{initial}}-m_{\text{final}}$
Energy from SI mass
$Q=\Delta mc^2$
Energy from mass in u
$\frac{Q}{\mathrm{MeV}}\approx931.5\frac{\Delta m}{\mathrm u}$
Binding energy per nucleon
$\overline{E}_b=\frac{E_b}{A}$
Binding increase: unchanged separate total proton and neutron counts
$Q=E_{b,\text{products}}-E_{b,\text{initial}}$
Converted mass fraction
$f_m=\frac{\Delta m}{m_{\text{initial}}}$
Energy from many fissions
$E_{\text{total}}=N_fE_f$
Steady thermal-power model: mean recoverable energy per fission
$R_f=\frac{P_{\text{th}}}{E_f}$
Mass represented by consumed isotope nuclei, not loaded fuel
$m=\frac{N_fM}{N_A}$
Supporting model: expected next-generation population
$N_{g+1}=kN_g$
Supporting model: constant factor, no independent source
$N_g=N_0k^g$
Supporting illustration: exactly two next-generation fissions
$N_g=N_0\,2^g$
Supporting model: expected-population multiplication factor
$k=\frac{N_{g+1}}{N_g}$
Supporting model: expected generation change
$N_{g+1}-N_g=(k-1)N_g$
Supporting model: percentage generation change
$\%\Delta N=100(k-1)$
Nonrelativistic neutron kinetic energy
$K=\frac{1}{2}m_nv^2$
Nonrelativistic speed from kinetic energy
$v=\sqrt{\frac{2K}{m_n}}$
Same nonrelativistic neutron: positive initial energy
$\frac{K_f}{K_i}=\left(\frac{v_f}{v_i}\right)^2$
Supporting toy model: extra fractional loss, all other effects fixed
$k_{\text{after}}=k_0(1-f_{\text{abs}})$
Control change
$\Delta k=k_{\text{after}}-k_{\text{before}}$
Mean absorption rate over a positive interval
$R_{\text{abs}}=\frac{N_{\text{absorbed}}}{\Delta t}$
Mean thermal-energy transfer over a positive interval
$P=\frac{Q}{\Delta t}$
Steady single-phase coolant: constant heat capacity, negligible other changes
$P=\dot m c\Delta T$
Power-station efficiency
$\eta=\frac{P_{\text{electrical}}}{P_{\text{thermal}}}$
Supporting model: steady isotropic point source, negligible attenuation/background
$\frac{I_2}{I_1}=\left(\frac{r_1}{r_2}\right)^2$
Supporting concept: mean absorbed dose in the stated mass
$D=\frac{E_{\text{absorbed}}}{m}$
Supporting concept: equivalent dose to one tissue, one radiation type
$H=w_RD$
Single isolated parent, no replenishment: integer-half-life activity
$A=A_0\left(\frac{1}{2}\right)^n$
Elapsed whole half-lives, not a universal safe-storage duration
$t=nT_{\frac{1}{2}}$
Electrical-energy audit: Q is mean recoverable heat per fission here
$E_{\text{electrical}}=\eta N_fQ$
Topic E.5Not started
Fusion and stars
This subtopic applies nuclear physics and radiation to stars. Students connect fusion, stellar equilibrium, luminosity, temperature, distance and stellar evolution.
360 min estimated
Topic E.5 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Explain why fusion requires high temperature and density and why it can release energy for light nuclei.
Describe the competition between inward gravitational effects and outward pressure/energy transport in a stable star.
Use luminosity, surface area and temperature to model stellar radiation.
Use parallax to estimate stellar distance.
Interpret the Hertzsprung–Russell diagram.
Describe how initial stellar mass influences broad evolutionary pathways and final states.
Use astronomical distance units including AU, light-year and parsec.
Key formulas
Mean density
$\rho = \frac{M}{V}$
Volume of a spherical star
$V = \frac{4}{3}\pi R^3$
Newtonian spherical-star surface field: supporting relation
$g = \frac{GM}{R^2}$
Supporting spherical hydrostatic model: total pressure
$\frac{\mathrm{d}P}{\mathrm{d}r} = -\rho g$
Supporting thin-layer estimate: nearly constant density and field
$|\Delta P| \approx \rho g\Delta r$
Radial force on a local layer element: outward positive
$F_{\text{net}} = F_{\text{pressure}} - F_{\text{gravity}}$
Supporting nonrelativistic classical-particle mean translational energy
$\overline{E_k} = \frac{3}{2}k_{\text{B}}T$
Supporting point-charge Coulomb energy: zero at infinite separation
$E_{\text{electric}} = \frac{kQ_1Q_2}{r}$
Supporting identical-species rate scaling: same temperature and reaction
$R \propto n^2$
Supporting net proton–proton nuclear equation, before positron annihilation
$4\,{}^{1}_{1}\mathrm{H} \rightarrow {}^{4}_{2}\mathrm{He} + 2e^{+} + 2\nu_e + \mathrm{energy}$
Completed-chain count
$N_{\text{chain}} = \frac{N_p}{4}$
Average completed-chain rate over the stated interval
$r = \frac{N_{\text{chain}}}{\Delta t}$
Reaction mass defect
$\Delta m = m_{\text{initial}} - m_{\text{final}}$
Mass–energy equivalence
$E = \Delta m c^2$
Energy conversion
$\frac{E}{\mathrm J}\approx1.602\times10^{-13}\frac{E}{\mathrm{MeV}}$
Average emitted luminosity over the stated interval
$L = \frac{E}{\Delta t}$
Isotropic inverse-square flux without intervening extinction
$F = \frac{L}{4\pi d^2}$
Flux ratio: same luminosity and no extinction
$\frac{F_2}{F_1} = \left(\frac{d_1}{d_2}\right)^2$
Wien law: black-body spectrum per unit wavelength
$\lambda_{\max}T = b$
Total black-body emission, or stellar effective-temperature definition
$\frac{P}{A} = \sigma T^4$
Black-body temperature estimate from wavelength-spectrum peak
$T = \frac{b}{\lambda_{\max}}$
Luminosity in solar units
$L = \left(\frac{L}{L_{\odot}}\right)L_{\odot}$
Logarithmic coordinate when this luminosity axis is specified
$y = \log_{10}\left(\frac{L}{L_{\odot}}\right)$
Inverse of the specified logarithmic luminosity coordinate
$\frac{L}{L_{\odot}} = 10^y$
Approximate mass–luminosity relation
$\frac{L}{L_{\odot}} \approx \left(\frac{M}{M_{\odot}}\right)^{3.5}$
Supporting lifetime scaling: comparable usable-fuel fraction and energy per mass
$t \propto \frac{M}{L}$
Approximate lifetime–mass relation
$\frac{t}{t_{\odot}} \approx \left(\frac{M}{M_{\odot}}\right)^{-2.5}$
Supporting Newtonian gravitational binding-energy scale, not an exact value
$E_{\text{g}} \sim -\frac{GM^2}{R}$
Surface area
$A = 4\pi R^2$
Black-body luminosity
$L = A\sigma T^4$
Supporting Schwarzschild horizon: non-rotating, uncharged black hole
$r_{\text{s}} = \frac{2GM}{c^2}$
Supporting Euclidean spherical mean density, not a black-hole interior model
$\rho = \frac{3M}{4\pi R^3}$
Supporting Newtonian escape speed: weak field and no further propulsion
$v_{\text{esc}} = \sqrt{\frac{2GM}{R}}$
Parallax distance
$\frac{d}{\mathrm{pc}}=\frac{1}{\frac{p}{\mathrm{arcsec}}}$
Parallax from maximum annual full diameter, other motions removed
$p = \frac{\theta_{\text{full}}}{2}$
Small-angle geometry
$\frac{p}{\mathrm{rad}}\approx\frac{1\,\mathrm{AU}}{d}$
Astronomical unit
$1\,\mathrm{AU}\approx1.496\times10^{11}\,\mathrm m$
Light-year
$1\,\mathrm{ly} \approx 9.46\times10^{15}\,\mathrm{m}$
Parsec conversions
$1\,\mathrm{pc}\approx3.086\times10^{16}\,\mathrm m\approx3.26\,\mathrm{ly}\approx206265\,\mathrm{AU}$
Stellar luminosity
$L = 4\pi R^2\sigma T^4$
Solar ratio form
$\frac{L}{L_{\odot}} = \left(\frac{R}{R_{\odot}}\right)^2\left(\frac{T}{T_{\odot}}\right)^4$
Radius ratio
$\frac{R}{R_{\odot}} = \sqrt{\frac{L}{L_{\odot}}}\left(\frac{T_{\odot}}{T}\right)^2$

Physics

Course Overview

Continue where you left off

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Space, Time and Motion

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Kinematics

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Forces and momentum

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Work, energy and power

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Rigid body mechanics

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Galilean and special relativity

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The Particulate Nature of Matter

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Thermal energy transfers

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Greenhouse effect

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Gas laws

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Thermodynamics

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Current and circuits

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Wave Behaviour

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Simple harmonic motion

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Wave model

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Wave phenomena

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Standing waves and resonance

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Doppler effect

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Fields

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Gravitational fields

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Electric and magnetic fields

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Motion in electromagnetic fields

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Induction

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Nuclear and Quantum Physics

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Structure of the atom

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Quantum physics

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Radioactive decay

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