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BASIC PHYSICS - SURVEY Stanislav Ďoubal Quantities, units, standards The physical quantities are used for quantification and description of physical processes. Consequently, the methods of measurement must exist for physical quantities. The physical quantity must be carefully defined and referred to common standards. In other words, the quantities are measured in units. Units are the scale with which the dimensions are measured. All terms in equation must have the same dimensions. Units are combination of standards (basic units). Standards are chosen arbitrarily. SI systems of units Standards (basic units) quantity basic unit time second abbreviation s definition 1 s is 9,192,631,770 period of radiation during transition of atom Ce133 between two hypefine levels length meter m 1 m is length of path traveled by light in vacuum during time interval 1/299,792,458 of second mass kilogram kg defined by platinum-iridium standard kept in Paris current ampere A 1 A existing in two long, parallel wires separated 1 m evoked force per unit of length between the wires of exactly 2*10 7 N/m 1 temperature kelvin K 1 K corresponds to 1/273,16 of temperature interval between triple point of water and absolute zero molar mass mol mol The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12 luminous intensity candle cd The candle is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. Derived units All other units (than basic) are called derived units. SI of units is a coherent system. Consequently, the quotient or product of any two SI units in the system yields the SI unit of the resultant quantity. Prefixes for SI symbol name amount E exa 1018 P peta 1015 T tera 1012 G giga 10 9 M mega 10 6 k kilo 10 3 2 c centi 10 2 m milli 10 3 micro 10 6 n nano 10 9 p pico 10 12 f femto 10 15 a atto 10 18 Accepted but not recommended: h hecto 10 2 da deka 101 d deci 10 1 Dimensions and units The dimension of quantity is the powers to which the basic units, expressing that quantity are raised. Example: the dimension of newton (N) is m .kg. s-2 . Extensive and intensive quantities Extensive quantities have character of "amount", such as mass, energy, charge. The extent of extensive quantity of the system is the sum of the amount of this quantity of all subsystems. Intensive quantities describe "state" and cannot be summarized. Examples of intensive quantities: concentration, temperature, pressure. 3 Scalars and vectors Scalar quantities are specified by single number (along with the proper unit). Examples: mass, energy, temperature, charge. Vector quantities are specified by direction and amount or size. Examples: velocity, force, displacement. Symbols for vectors: boldface letters or arrow above the letter (A or A ) Graphical addition of vectors Vector addition does not obey rules of ordinary algebra. The process of addition of vectors is conveniently expressed in graphical terms, according to Fig. 1. X2 X3 X1 Fig. 1. Graphical addition of vectors Vector X3 is the vector sum of vectors X 1 and X 2 . Subtraction of vectors is a reverse process. Addition of vector obey the associative law, consequently, it is possible to add more than two vectors in a subsequent way. 4 Mechanics Kinematics Velocity v v = dr / dt (m/s ; m, s), where r is the vector of displacement, t is the time. If the movement is in direction x than holds v = dx / dt. Speed v v = v . Motion with constant speed If the motion is in direction of x, then for distance of displacement holds: x=vt. Acceleration a (m / s 2 ; m/s, s) a = dv/dt Motion with constant acceleration v = vo + a t, where vo is the initial velocity. 5 If the motion is in direction of x, then for distance of displacement holds: x = xo + vot + 1/2 a t 2 Free fall Free fall is a movement with constant acceleration equal to g = 9,8 m/ s 2 (gravity). If the free fall is combined with another movement (for instance in projectile motion), the resulting movement may be calculated as vector sum of these two movements. Uniform circular motion For the uniform circular motion (see Fig. 2) the following quantities are defined: angular speed : = d/dt (rad/s) frequency f: f = 1/T, (Hz or 1/s) where T is the time for one movement . Consequently: = 2f. For (centripetal) acceleration a0 holds: a0 = 2 t = v 2 /r, where v is the speed of revolution. Also holds: v = s/t = 2r/T = 2rf = r. 6 ao v ado r Fig. 2. Illustration for explanation of circular motion Harmonic motion For simple harmonic motion holds (see Fig. 3): x = A sin (t + 0 ). Quantity A is the amplitude and 0 is the phase at the time t = 0. For velocity holds: v = dx/dt = A cos (t + 0 ). For acceleration holds: a = dv/dt = - 2 A sin (t + 0 ). 7 1,5 1 0,5 0 -0,5 -1 -1,5 Fig. 3. Harmonic motion Dynamics Newton's first law Every body continues its state of rest, or in uniform motion in straight line unless it is compelled to change that state by forces impressed upon it. Newton's second law (N; kg, m/ s 2 ) F = m a, where F is the net force exerted upon the object, m is the mass, a is the acceleration. Newton's third law Fab = - Fba. To every action there is always opposed an equal reaction. A single force cannot exist. 8 Newton's law of universal gravitation: F= m1m2 r2 , where F is the gravitational force between two objects, is the universal constant, m1, m2 are the masses of the objects, r is the distance between objects. Gravitational force F = mg. Kinetic frictional force (Fig. 4) v T -T F N Fig. 4. Kinetic frictional force T = f N, where T is the frictional force, f is the coefficient of kinetic friction, N is the normal force exerted by the surface. 9 Momentum p p = m v, (kg.m/s; kg, m) where m is the mass, v is the velocity. Also holds: p = I = F t , where I is the impulse of the force F. Conservation of momentum pi = p L , where p i is the initial momentum, p L is the momentum any later (provided the net external force is zero). Centripetal force in uniform circular movement F = m a0 = m 2 t = m v 2 /r. Work and energy W = F.s. cos , (J; N, m) W is the work, F is the force, s is the displacement. The angle is according Fig. 5. 10 F s Fig. 5. Illustration to the definition of work Kinetic energy K K = 1/2 m v 2 . Gravitational potential energy U U = m g h, m is the mass, g is the gravity, h is the relative height. Conservation of energy If the net work of external forces exerted on the system is equal to zero, then the energy of the system is constant. Power P P = dA/dt. Average power (W; J, t) P = A/t. 11 Efficiency : = W , W0 where W is the output energy, W0 is the input energy. Also holds: = P , P0 where P is the average output power, P0 is the average input power. Static Rigid body A rigid body is one for which the distance between any pair of points on the object remains fixed. A rigid body retains its shape and size. In the case of a rigid body, for rotational equilibrium holds that the effect of force is the same along the line of action. The line of action is the straight line in the direction of the vector of the force. Torque about axis M (moment of force about axis) M = F r = F.. sin, the meaning of the symbols is according Fig. 6. 12 Static equilibrium Object is in static equilibrium if every point of the object remains in rest. Condition for static equilibrium: net force must be zero (translation equilibrium) and the external net torque must be zero (rotational equilibrium). Translation equilibrium Object is in translation equilibrium if the acceleration of the center of mass is zero. Rotational equilibrium Object is in translation equilibrium if the net external torque is zero: M i Fi ri 0 . i i Center of mass Center of mass is the point of average position of the mass of the system. Center of gravity The point, at which full weight of object can be considered to act. Calculation of net force applied to an rigid body Two forces: a) Line of action with crossing point The forces are translated along lines of action into crossing point and added according rules for adding vectors (Fig. 1). 13 Fig. 6 Addition of non-parallel forces b) Parallel forces Net force is the sum of forces. The position of point of action is calculated with regard to rotational equilibrium (the net torque is zero). If there are more forces, the calculation consists in subsequent addition of pairs of forces. Solids and fluids Solids Tensile (normal) stress and compressive stress (Pa; N, m 2 ) = FN /A, where FN is the force normal to the surface, A is the area. Shear stress = F P /A, where F P is the force parallel to surface, A is area. 14 Pressure p (Pa; N, m 2 ) p = F/A . Strain = L/L, where L is the increase of length, L is the length of unstressed object. Hooke´s law For tensile stress holds: Y= , t where Y is Young´s modulus, t is the tensile strain. For shear stress holds: S= s , where S is shear modulus, s is the shear strain. Density = m/V, where m is the mass, V is the volume. 15 Fluids Pascal's principle Pressure applied to an enclosed incompressible fluid is transmitted undiminished to all parts of the fluid. Static pressure in an incompressible fluid p = p0 + h g , where p0 is the pressure at the top of the surface, h is the depth, is the density, g is the acceleration of gravity. Archimedes´ principle A submerged body is buoyed up by the force equal in magnitude to the weight of the displaced fluid and directed upwards. Equation of continuity for incompressible fluid v1 A1 = v 2 A 2 , v1is the velocity at the point 1, A1is the area at the point 1, v 2 is the velocity at the point 2, A 2 is the area at the point 2. Bernoulli’s equation p + gh + 1/2 v 2 = constant, where p is the pressure, is the density, h is the depth, v is the velocity of the flow of the fluid. 16 Waves Transverse wave A transverse wave is one in which vector of oscillating quantity (position of particles in case of sound) is perpendicular to the direction of propagation. Longitudinal waves A longitudinal wave is one in which vector of oscillating quantity is parallel to the direction of propagation. Wave function for harmonic wave y = A sin (x/v - t) where A is the amplitude, = 2 f is the angular frequency, x is the distance, v is the speed of propagation, f is the frequency. Wavelength =v/f Wave number k k = 2 / 17 Wavefront A wavefront represents a set of points that have the same phase at a given time. Huygen´s principle Propagation of a light wave can be determined by assuming that at every point on a wavefront there arises a spherical wavelet centered on that point. Snell’s law The refracted ray is transmitted into the second media, as shown according to following formula: sin n2 v1 , sin n1 v2 where n1, n2 are the indexes of refraction, v1, v2 are the speeds of propagation. Critical angle C sin C = v1 . v2 Ray that approach the interface with angle less than critical angle are partially reflected and partially refracted. Rays that approach with angle higher than critical one are totally reflected. 18 Sound Sound waves Mechanical waves with frequency between frequency about 20 Hz and 20 kHz are called sound waves. They cause sensation of hearing. The speed of sound waves in the air is about 330m/s. Sound waves consist in compression and rarefaction of fluid. The sound wave in a fluid is longitudinal. Frequency and pitch For harmonic sound wave, the higher the frequency is, the higher is the perception of pitch. Intensity of sound I (W m 2 ; W , m 2 ) I = P/A, where P is the power of sound, A is the area. Sound-intensity level s s = 10 log I , I0 where I 0 is the threshold intensity for human, I 0 = 10 12 W m 2 . 19 (dB) Thermodynamics Temperature and heat transfer Temperature Temperature is proportional to average molecular translation kinetic energy. Temperature scales Kelvin scale: 1 kelvin (K) corresponds to 1/273,16 of temperature interval between triple point of water and absolute zero. Celsius scale: 1 degree of Celsius (°C ) corresponds to 1/100 of temperature interval between normal boiling and melting points of water. Connection between Kelvin and Celsius scales t C T 273,15 , where t C is the temperature in °C, T is the temperature in K. Connection between Fahrenheit (t F ) and Celsius ( t C ) scales t F = 9/5 t C + 32. t C = 5/9( t F - 32) 20 Thermal expansion a) Linear expansion L = L 0 T, where L is the length change, is the coefficient of linear expansion, L 0 is the reference length, T is the temperature interval. b) Volume expansion V = V 0 T, where is the coefficient of volume expansion. The previous formulae hold for moderate temperature changes. Connection between and : =3 Heat conduction a) Heat current H H = Q/t, (J/s; J, s) where Q is the heat, t is the time. b) Steady state heat current along uniform rod T T H=kA 2 1, L 21 where k is the thermal conductivity of the material, A is the cross section area, L is the length, T2 T1 is the temperature difference. Specific heat capacity a) Specific heat capacity at constant pressure c P dQ = m c P dT, where dQ is the infinitesimal change of heat, dT is infinitesimal change of temperature. b) Specific heat capacity at constant volume cV dQ = m cV dT Molar heat capacity a) Molar heat capacity at constant pressure C P dQ = n C P dT, where n is the number of moles. b) Molar heat capacity at constant volume CV dQ = n CV dT 22 Latent heat Latent heat L is the amount of heat per unit of mass, which is added to or removed from a substance undergoing a phase change: L = Q/m (J/kg; J, kg) (Notice that temperature is constant during the phase change). Work done by a system dW = p dV a) Work in isobaric process W = p V Laws of thermodynamics The zeroth law of thermodynamics Two systems in thermal equilibrium with a third system are in thermal equilibrium with each other. The first law of thermodynamics Q - W = U, where U is the change in the internal energy of the system. 23 The second law of thermodynamics There exists no cycle, which extracts heat from reservoir at a single temperature and completely converts it into energy. The third law of thermodynamics The absolute zero is the lowest temperature Ideal gas Properties of ideal gas Ideal gas consists of a large number of molecules. The size of the molecules is negligible compared with the average distance between molecules. The motions of molecules are described by newtonian dynamics. Molecules move freely, forces between molecules are negligible. All collisions (between molecules and between molecules and wall) are elastic. The molecules are in a random motion, and the gas is in equilibrium. Equation of the state of ideal gas p V = n R T, where p is the pressure, V is the volume, n is the number of moles, R is is the universal gas constant, T is the absolute temperature. Equation for isobaric processes V = V0 T T0 , where V0, T0 are the reference volume or temperature. 24 Equation for isothermal processes pV = constant Equation for isochoric processes P = P0 T T0 , where P0 , T0 are the reference pressure or temperature. Volume is constant. Adiabatic processes The system is energetically isolated. No heat is transferred to and from the system. P V = K, where K is the constant given by initial state of the system, = 25 CP . CV Electricity and magnetism Electric field Electric charge Q Q = I t, (C; A, s) where I is the electric current, t is the time. Elementary charge e The elementary charge is the charge of proton. The charge of electron is -e. e = 1,6 10 19 C Coulomb´s law Force between two charges Q A and QB is given by following formula F= 1 4 Q A QB r2 , where is the permitivity, r is the distance between charges. If Q A and QB have the same sign, the force is of a repulsive character. If Q A and QB have an opposite sign, the charges are attracted. = r . 0 , where 0 is the permitivity of vacuum, r is the relative permitivity of space between charges. 26 Electric field intensity E E = F/Q, (V/m; N, C) where F is the force, Q is the charge. For homogenous field also holds E = V/d, where V is the voltage and d is the distance. Electric potential = A/Q, (V; J, C) where A is the electric potential energy. Voltage V Voltage is the difference of electric potentials at points 1 and 2 V = 2 1 Capacitors and capacitance A capacitor consists of two conductors that are insulated from one another. Insulator is referred to as dielectrics. Capacitor provides temporary storage of charge (Q) and energy. Holds: 27 Q = C V, (C; F, V) where C is the capacitance, V is the voltage (voltage is the potential difference). Capacitance of parallel plate capacitor: C r 0. A , d A d Fig. 7. Parallel plate capacitor Electric current The electric current I characterizes flow of charge through material. Holds: I = dQ /dt, . (A; C, t) 28 Current in one direction is called direct current (dc). Current in a sense back and forth is called alternating current (ac). For constant dc holds: I=Q/t. Electric current in metals The electric current in metal is flow of (free) electrons in the crystal lattice of the metal. Resistance and Ohm ´s law I =V / R, (A; V,) (Ohm ´s law) where I is the current, V is the voltage, R is the resistance. The unit of resistance is ohm (). Conductance S : S = 1 / R. The unit of conductance is siemens (S). Production of heat (Joule’s law) W = U I t. Power dissipated in a resistor P = V2 / R = I2 R 29 1. Kirchhoff´s rule (point rule): I1 IN I2 I3 Fig. 8. The point rule: The sum of the currents towards a branch point is equal to the sum of the currents away from the same branch point If the currents “towards” are assigned + and the currents “away” -, then holds: i n I i 1 i 0, 2. Kirchhoff´s rule (loop rule): V1 V2 V3 VN Fig. 9. The loop rule: The sum of the voltage differences in the round-trip around any closed loop is zero 30 in V i 1 i 0. Resistors in series R1 R2 RN RC Fig. 10. Resistors in series í n RC Ri , i 1 31 Resistors in parallel R1 R2 RN RC Fig. 11. Resistors in parallel í n 1 / RC 1/ Ri , i 1 Electric current in vacuum The vacuum is a ideal insulator. The electric current in vacuum is possible only if charged particles are injected into vacuum. The practically possible ways of injection of charge carriers are as follows: Thermal electron emission The carriers are electrons discharged from metallic cathode by thermal movement of electrons (heating the cathode) 32 Photoelectric emission The carriers are electrons discharged from metallic cathode by photons (elimination of the cathode) Electric current in gases The gases are also insulators. The electric current in vacuum is possible only if charged particles are injected into gas. The practically possible ways of injection of charge carriers are as follows: a) Thermal electron emission b) Photoelectric emission c) Ionization of the gas (by ionizing radiation or heating the gas) I zone of saturation zone of self-maintained discharge zone of recombination U Fig. 12. Volt-ampere characteristic of the discharge in gas Electric current in liquids The electric current is possible only if carriers are presented in the liquid. The ions are the carriers in the case of liquid. The dc current causes the electrolysis: The ions are loosing electrons on anode or accepting electrons on cathode. The carriers are gradually changed into non-charged 33 particles. Cations (metal, hydrogen) are eliminating on cathode, anions on cathode. The liquid is gradually losing the ability to conduct electric current. For the amount of material (M) accumulating on electrode holds: M=kQ, (1. Faraday’s law) where k, is the electrochemical equivalent, Q is the charge. For electrochemical equivalent holds: k A , F .z (2. Faraday’s law) where A is the ion mass, F is the Faraday’s constant, z is the number of elementary charges of the ion. Electric current in semiconductors Intrinsic semiconductors: Materials with small number of carriers and low conductivity. Extrinsic semiconductors: Crystal material (as Si, Ge -their atoms have 4 electrons in outer sphere) with small amount of impurities in the lattice. Impurities: atoms with 3 electrons in outer sphere (P semiconductors) or atoms with 5 electrons in the outer sphere (N semiconductors). 34 Magnetic field Magnetic induction (B) F B I Fig. 13. Force on conductor in magnetic fields: direction of the vectors B F , I .d (Tesla - T; N, A, m) where F is the force, I is the current, d the length of conductor. Force on moving charge in magnetic field (Fig. 14) F Q .v B , Magnetic field intensity (H) H B , r . 0 (A / m) where r relative permeability of material, 0. is permeability of vacuum. 35 F B v Fig. 14. Force on moving charge in magnetic fields: direction of the vectors Magnetic flux ( ) B. A , (Wb – weber) where A is the area (perpendicular to vector B) Magnetic line of force Magnetic field lines have the direction of vector B, each point. I B N B B S I Fig. 14. Magnetic field lines and right hand rule 36 Paramagnetic materials: r 1 (slight attenuation of magnetic field and force) Diamagnetic materials: r 1 (slight amplification of magnetic field and force) Ferromagnetic materials (Fe, Ni, Co) r 1, (strong amplification of magnetic field and force), materials for magnets. The relationship between H and B is non-linear, according hysteresis curve. BF I B HI Fig. 15. Hysteresis curve Induced electric field and voltage u d . dt where u is the voltage Inductance (L) L / I . 37 Alternating current Generators – principle: The coil is revolving magnetic field. The voltage u is induced in the coil u N. d , dt as for holds: B S 0 cos , and =t the voltage has sinusoidal character: u U 0 sin ( t 0 ) , = .t + O Fig. 16. Origin of alternating current 38 Effective value of sinusoidal voltage (Uef) and current (Ief) U ef Um , 2 I ef Im , 2 and where Um and Im are the maximum values. For energy production P in ac circuits holds: P = Uef. Ief . t . Resistor in ac circuits UR IR IR UR =0° Fig. 17. Resistors: voltage-current relationships UR / IR = R 39 Condenser in ac circuits iC uC 90° / 2 Fig. 18. Condensers: voltage-current relationships UC / IC = X C 1 C Condensers in serial: 1 / C = 1/C1 + 1/C2 + .... Condensers in parallel: C = C1 + C2 + .... Coil in ac circuits uL 90° / 2 iL Fig. 19. Coils: voltage-current relationships 40 UL / IL = X L L Coils in parallel: 1 / L = 1/L1 + 1/L2 + .... Coils in serial: L = L1 + L2 + .... Power (P) in ac circuits P U I cos , where is the phase shift between I and U Transformer Transformer is a device for changing magnitudes of current and voltage. For use in ac circuits only. N2 N1 I1 U1 U2 I2 Fig. 20. Transformer 41 N1 U I 1 2 , N2 U2 I1 where N1 and N2 are the numbers of loops in the primary and the secondary coils Optics The subject of optics is a visible light. The visible light are electromagnetic waves of wavelength from 400 to 760 nm. Electromagnetic waves Transversal waves of vectors B and E. Speed of light in vacuum c = 3 . 108 m/s. E c (direction of propagation) B Fig. 21. Electromagnetic waves 42 Photons The light behaves partially like particles: photons. Energy of photon: E h , where h is Planck’s constant, is the frequency of waves. The color of light The color of monochromatic light (of one frequency or wavelength) is determined by wavelength. The red color corresponds to the longest wavelength, the blue color to the shortest wavelength. Origin of light and emission spectrum Visible light is emitted during shift of electrons between energy levels in outer sphere of electron shell of atoms. Line emission spectrum: emitted from ionized atoms (see Fig.26) Continuous spectrum: emitted from solids and liquids 43 Survey of electromagnetic waves wavelength (m) 10 - 16 10 -10 10 -4 gamma rays X rays 10 +9 TV and radio UV 10 +6 10 +3 10 +2 VIS microwaves IR 10 0 10 -3 energy (eV) min. ionizing energy Fig. 22. Survey of electromagnetic waves Geometrical optics Fig. 23. Basic rules of geometrical optics: rays in converging lens 44 10 +8 Thin lenses Power of lens (D): D 1 , f where f is the focal distance. Lens-maker equation: 1 1 1 , / a a f where a is the distance of object (from lens), a' is the distance of image. Simple magnifier The simple magnifier is a converging lens. Angular magnification (M): M 1 d , f where d is the near point of the eye, usually d = 0,25 m. 45 Compound light microscope Fig. 24. Compound light microscope Magnification: M t.d f OB . f OK 46 optical interval object observer fOB fOK image eye piece objective Fig. 25. Compound light microscope: scheme Absorption of light - . Lambert’s law I I 0 . e .d , where I, I0 are the intensities of light (before and after absorption), coefficient of absorption Atoms Size of atom: 10-9 to 10-10 m, zero charge Structure of atom: nucleus (99,95% of mass, + charge, size: 10-14 to 10-15 m) electron shell (- charge) Energy levels of electrons- quantum approach 47 The electron in electron shell may be only in certain energy levels (W). The shift between levels leads to absorption or emission of photon. Energy of photons: Wn Ws h ionization limit of series energy levels zero state absorption of photon emission of photon hrana série příklad čárového spektra (He) Fig. 26. Line spectrum of atom Radioactive radiation (origin in nucleus) 4 Alfa particles: nuclei of helium ( 2 He ). Beta particles: electrons or positrons (plus charged electrons). Gamma particles: short wave electromagnetic radiation. 48 Radioactive decay Spontaneous process of change of nucleus. No No/2 half-life T t Fig. 27. Kinetics of radioactive decay Absorption of radioactive radiation The kinetics of absorption is described by Lambert’s law. I I 0 . e .d , 49 Io Io/2 d half-thickness Fig. 28. Kinetics of radioactive absorption 50