1. Introduction


1.1 Purpose of this syllabus

This syllabus lists topics which may be used for the IPhO. Guidance about the level of each topic within the syl­labus is to be found from past IPhO questions.


1.2 Character of the problems

Problems should focus on testing creativity and under­standing of physics rather than testing mathematical vir­tuosity or speed of working. The proportion of marks al­located for mathematical manipulations should be kept small. In the case of mathematically challenging tasks, alternative approximate solutions should receive partial credit. Problem texts should be concise; the theoreti­cal and the experimental examination texts should each contain fewer than 12000 characters (including white spaces ,but excluding cover sheets and answer sheets).


1.3 Exceptions 

Questions may contain concepts and phenomena not mentioned in the Syllabus providing that  sufficient in­formation is given in the problem text so that students without previous knowledge of these topics would not be at a noticeable disadvantage. Such new concepts must be closely related to the topics included in the syllabus. Such new concepts should be explained in terms of top­ics in the Syllabus.


1.4 Units

Numerical values are to be given using SI units, or units officially accepted for use with the SI. 
It is assumed that the contestants are familiar with the phenomena, concepts, and methods listed below, and are able to apply their knowledge creatively.


2. Theoretical skills


2.1 General 

The ability to make appropriate approximations, while modelling real life problems. Recognition of and ability to exploit symmetry in problems.


2.2 Mechanics

2.2.1 Kinematics 

Velocity and acceleration of a point particle as the deriva­tives of its displacement vector. Linear speed; centripetal and tangential acceleration. Motion of a point particle with a constant acceleration. Addition of velocities and angular velocities; addition of accelerations without the Coriolis term; recognition of the cases when the Coriolis acceleration is zero. Motion of a rigid body as a rota­tion around an instantaneous center of rotation; veloci­ties and accelerations of the material points of rigid ro­tating bodies.


2.2.2 Statics 

Finding the center of mass of a system via summation or via integration. Equilibrium conditions: force balance (vectorially or in terms of projections), and torque bal­ance (only for one-and two-dimensional geometry). Nor­mal force, tension force, static and kinetic friction force; Hooke’s law, stress, strain, and Young modulus. Stable and unstable equilibria.

2.2.3 Dynamics 

Newton’s second law (in vector form and via projections (components)); kinetic energy for translational and rota­tional motions. Potential energy for simple force fields (also as a line integral of the force field). Momentum, angular momentum, energy and their conservation laws. Mechanical work and power; dissipation due to friction. Inertial and non-inertial frames of reference: inertial force, centrifugal force, potential energy in a rotating frame. Moment of inertia for simple bodies (ring, disk, sphere, hollow sphere, rod), parallel axis theorem; find­ing a moment of inertia via integration.

2.2.4 Celestial mechanics 

Law of gravity, gravitational potential, Kepler’s laws (no derivation needed for first and third law). Energy of a point mass on an elliptical orbit.

2.2.5 Hydrodynamics

Pressure, buoyancy, continuity law. the Bernoulli equa­tion. Surface tension and the associ­ated energy, capillary pressure.


2.3 Electromagnetic fields

2.3.1 Basic concepts 

Concepts of charge and current; charge conservation and Kirchhoff’s current law. Coulomb force; electrostatic field as a potential field; Kirchhoff’s voltage law. Mag­netic B-field; Lorentz force; Ampère’s force; Biot-Savart law and B-field on the axis of a circular current loop and for simple symmetric systems like straight wire, circular loop and long solenoid.


2.3.2 Integral forms of Maxwell’s equations 

Gauss’law (for E-and B-fields); Ampère’s law; Faraday’s law; using these laws for the calculation of fields when the integrand is almost piece-wise constant. Boundary conditions for the electric field (or electrostatic potential) at the surface of conductors and at infinity; concept of grounded conductors. Superposition principle for electric and magnetic fields.


2.3.3 Interaction of matter with electric and magnetic fields 

Resistivity and conductivity; differential form of Ohm’s law. Dielectric and magnetic permeability; relative per­mittivity and permeability of electric and magnetic ma­terials; energy density of electric and magnetic fields; fer­romagnetic materials; hysteresis and dissipation; eddy currents; Lenz’s law. Charges in magnetic field: helicoidal motion, cyclotron frequency, drift in crossed E-and B-fields. Energy of a magnetic dipole in a magnetic field; dipole moment of a current loop.


2.3.4 Circuits 

Linear resistors and Ohm’s law; Joule’s law; work done by an electromotive force; ideal and non-ideal batter­ies, constant current sources, ammeters, voltmeters and ohmmeters. Nonlinear elements of given V -I characteristic. Capacitors and capacitance(also for a single electrode with respect to infinity); self-induction and inductance; energy of capacitors and inductors; mutual inductance; time con­stants for RL and RC circuits. AC circuits: complex amplitude; impedance of resistors, inductors, capacitors, and combination circuits; phasor diagrams; current and voltage resonance; active power.


2.4 Oscillations and waves

2.4.1 Single oscillator 

Harmonic oscillations: equation of motion, frequency, angular frequency and period. Physical pendulum and its reduced length. Behavior near unstable equilib­ria. Exponential decay of damped oscillations; resonance of sinusoidally forced oscillators: amplitude and phase shift of steady state oscillations. Free oscillations of LC-circuits; mechanic-electrical analogy; positive feedback as a source of instability; generation of sine waves by feed back in a LC-resonator.


2.4.3 Waves 

Propagation of harmonic waves: phase as a linear func­tion of space and time; wave length, wave vector, phase and group velocities; exponential decay for waves propa­gating in dissipative media; transverse and longitudinal waves; the classical Doppler effect. Waves in inhomo­geneous media: Fermat’s principle, Snell’s law. Sound waves: speed as a function of pressure (Young’s or bulk modulus) and density, Mach cone. Energy carried by waves: proportionality to the square of the amplitude, continuity of the energy flux.


2.4.4 Interference and diffraction

Superposition of waves: coherence, beats, standing waves, Huygens’ principle, interfer­ence due to thin films (conditions for intensity minima and maxima only). Diffraction from one and two slits, diffraction grating, Bragg reflection.


2.4.5 Interaction of electromagnetic waves with mat­ter 

Dependence of electric permittivity on frequency (qual­itatively); refractive index; dispersion and dissipation of electromagnetic waves in transparent and opaque ma­terials. Linear polarization; Brewster angle; polarizers; Malus’ law.


2.4.6 Geometrical optics and photometry 

Approximation of geometrical optics: rays and optical images; a partial shadow and full shadow. Thin lens ap­proximation; construction of images created by ideal thin lenses; thin lens equation Lu­minous flux and its continuity; illuminance; luminous intensity.


2.4.7 Optical devices 

Telescopes and microscopes: magnification and resolv­ing power; diffraction grating and its resolving power; interferometers.


2.5 Relativity 

Principle of relativity and Lorentz transformations for the time and spatial coordinate, and for the energy and momentum; mass-energy equivalence; invariance of the space time interval and of the rest mass. Addition of par­allel velocities; time dilation; length contraction; relativ­ity of simultaneity; energy and momentum of photons and relativistic Doppler effect; relativistic equation of motion; conservation of energy and momentum for elas­tic and non-elastic interaction of particles.


2.6 Quantum Physics

2.6.1 Probability waves

Particles as waves: relationship between the frequency and energy, and between the wave vector and momen­tum. Energy levels of hydrogen-like atoms (circular orbits only) and of parabolic potentials; quantization of angular momentum. Uncertainty principle for the con­jugate pairs of time and energy, and of coordinate and momentum(as a theorem, and as a tool for estimates).


2.6.2 Structure of matter

Emission and absorption spectra for hydrogen-like atoms (for other atoms —qualitatively), and for molecules due to molecular oscillations; spectral width and lifetime of excited states. Pauli exclusion principle for Fermi parti­cles. Particles (knowledge of charge and spin): electrons, electron neutrinos, protons, neutrons, photons; Comp­ton scattering. Protons and neutrons as compound par­ticles. Atomic nuclei, energy levels of nuclei (qualita­tively); alpha-, beta-and gamma-decays; fission, fusion and neutron capture; mass defect; half-life and exponen­tial decay. Photoelectric effect.


2.7 Thermodynamics and statistical physics

2.7.1 Classical thermodynamics 

Concepts of thermal equilibrium and reversible pro­cesses; internal energy, work and heat; Kelvin’s tem­perature scale; entropy; open, closed, isolated systems; first and second laws of thermodynamics. Kinetic the­ory of ideal gases: Avogadro number, Boltzmann factor and gas constant; translational motion of molecules and pressure; ideal gas law; translational, rotational and os­cillatory degrees of freedom; equipartition theorem; in­ternal energy of ideal gases; root-mean-square speed of molecules. Isother­mal, isobaric, isochoric, and adiabatic processes; specific heat for isobaric and isochoric processes; forward and reverse Carnot cycle on ideal gas and its efficiency; ef­ficiency of non-ideal heat engines.


2.7.2 Heat transfer and phase transitions 

Phase transitions (boiling, evaporation, melting, subli­mation) and latent heat; saturated vapor pressure, rel­ative humidity; boiling; Dalton’s law; concept of heat conductiv­ity; continuity of heat flux.


2.7.3 Statistical physics 

Planck’s law (explained qualitatively, does not need to be remembered), Wien’s displacement law;the Stefan-Boltzmann law.


3. Experimental skills

3.1 Introduction 

The theoretical knowledge required for carrying out the experiments must be covered by Section 2 of this Syl­labus. 
The experimental problems should contain at least some tasks for which the experimental procedure (setup, the list of all the quantities subject to direct measure­ments, and formulae to be used for calculations) is not described in full detail. 
The experimental problems may contain implicit the­oretical tasks (deriving formulae necessary for calcula­tions); there should be no explicit theoretical tasks unless these tasks test the understanding of the operation prin­ciples of the given experimental setup or of the physics of the phenomena to be studied, and do not involve long mathematical calculations. 
The expected number of direct measurements and the volume of numerical calculations should not be so large as to consume a major part of the allotted time: the exam should test experimental creativity, rather than the speed with which the students can perform technical tasks. 
The students should have the following skills.


3.2 Safety 

Knowing standard safety rules in laboratory work. Nev­ertheless, if the experimental set-up contains any safety hazards, the appropriate warnings should be included in the text of the problem. Experiments with major safety hazards should be avoided.


3.3 Measurement techniques and apparatus 

Being familiar with the most common experimental tech­niques for measuring physical quantities mentioned in the theoretical part. 
Knowing commonly used simple laboratory in­struments and digital and analog versions of sim­ple devices, such as calipers, the Vernier scale, stop­watches, thermometers, multimeters (including ohmme­ters and AC/DC voltmeters and ammeters), potentiome­ters, diodes, transistors, lenses, prisms, optical stands, calorimeters, and so on. 
Sophisticated practical equipment likely to be unfa­miliar to the students should not dominate a problem. In the case of moderately sophisticated equipment (such as oscilloscopes, counters, rate meters, signal and function generators, photo gates, etc), instructions must be given to the students.


3.4 Accuracy 

Being aware that instruments may affect the outcome of experiments. 
Being familiar with basic techniques for increasing experimental accuracy (e.g. measuring many periods in­stead of a single one, minimizing the influence of noise, etc). 
Knowing that if a functional dependence of a physi­cal quantity is to be determined, the density of taken data points should correspond to the local characteristic scale of that functional dependence. 
Expressing the final results and experimental uncer­tainties with a reasonable number of significant digits, and rounding off correctly.


3.5 Experimental uncertainty analysis 

Identification of dominant error sources, and reasonable estimation of the magnitudes of the experimental uncer­tainties of direct measurements (using rules from docu­mentation, if provided).
Distinguishing between random and systematic er­rors; being able to estimate and reduce the former via repeated measurements. 
Finding absolute and relative uncertainties of a quan­tity determined as a function of measured quantities us­ing any reasonable method (such as linear approxima­tion, addition by modulus or Pythagorean addition).


3.6 Data analysis

Transformation of a dependence to a linear form by ap­propriate choice of variables and fitting a straight line to experimental points. Finding the linear regression pa­rameters (gradient, intercept and uncertainty estimate) either graphically, or using the statistical functions of a calculator (either method acceptable). 
Selecting optimal scales for graphs and plotting data points with error bars.


4. Mathematics

4.1 Algebra 

Simplification of formulae by factorization and expan­sion. Solving linear systems of equations. Solving equa­tions and systems of equations leading to quadratic and biquadratic equations; selection of physically meaning­ful solutions. Summation of arithmetic and geometric series.


4.2 Functions 

Basic properties of trigonometric, inverse-trigonometric, exponential and logarithmic functions and polynomials. 
This includes formulae regarding trigonometric func­tions of a sum of angles. Solv­ing simple equations involving trigonometric, inverse-trigonometric, logarithmic and exponential functions.


4.3 Geometry and stereometry 

Degrees and radians as alternative measures of angles. Equality of alternate interior and exterior angles, equal­ity of corresponding angles. Recognition of similar trian­gles. Areas of triangles, trapezoids, circles and ellipses; surface areas of spheres, cylinders and cones; volumes of spheres, cones, cylinders and prisms. Sine and cosine rules, property of inscribed and central angles, Thales’ theorem. Medians and centroid of a triangle. Students are expected to be familiar with the properties of conic sections including circles, ellipses, parabolae and hyper­bolae.


4.4 Vectors 

Basic properties of vectorial sums, dot and cross products. Double cross product and scalar triple product. Ge­ometrical interpretation of a time derivative of a vector quantity.


4.5 Complex numbers 

Summation, multiplication and division of complex numbers; separation of real and imaginary parts. Con­version between algebraic, trigonometric, and exponen­tial representations of a complex number. Complex roots of quadratic equations and their physical interpretation.


4.6 Statistics 

Calculation of probabilitiesas the ratio of the number of objects or event occurrence frequencies. Calculation of mean values, standard deviations, and standard devia­tion of group means.


4.7 Calculus 

Finding derivatives of elementary functions, their sums, products, quotients, and nested functions. Integration as the inverse procedure to differentiation. Finding defi­nite and indefinite integrals in simple cases: elementary functions, sums of functions, and using the substitution rule for a linearly dependent argument. Making definite integrals dimensionless by substitution. Geometric in­terpretation of derivatives and integrals. Finding constants of integration using initial con­ditions. Concept of gradient vectors (partial derivative formalism is not needed).


4.8 Approximate and numerical methods

Using linear and polynomial approximations based on Taylor series. Linearization of equations and expressions. Perturbation method: calculation of corrections based on unperturbed solutions. Numerical integration using the trapezoidal rule or adding rectangles.