#### 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.

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**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.

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**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.

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**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.