Vågrörelselära och optik

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1 Vågrörelselära och optik Harmonic oscillation: Experiment Experiment to find a mathematical description of harmonic oscillation Kapitel 14 Harmonisk oscillator 1 2 Harmonic oscillation: Experiment Harmonic oscillation: Function x x = A sin(bt + C) or x = A cos(bt + C /2) x : Vertical displacement. Unit: meters t : Time. Unit: seconds Conclusion: Harmonic oscillation can be described by the function: x = A sin(bt + C) where t is time and A, B and C are constants describing the motion. A : Amplitude (maximum movement). Unit: meters B = : Angular frequency (number of oscillations per second times 2 ). Unit: Radians per second C = : Phase angle that determines position at time = 0. Unit: radians 3 4

2 Harmonic oscillation: f and T Harmonic oscillation: Phase angle x x = A sin( t + ) or x = A cos( t + ) The phase angle ( ) determines the position at time = 0 since then x = Asin( ) or x = Acos( ) T: Period = The time it takes for the weight to go up and down. Unit: seconds f: Frequency = The number of periods per second. Unit: 1/Seconds f = 1 / T = 2 f x t x X= A sin( t) X = A cos( t - ) x x t t t X= A cos( t) X = A sin( t + ) X= A cos( t + ) X = A sin( t - ) 5 6 Harmonic oscillation: velocity & acceleration Harmonic oscillation: velocity & acceleration We now have a mathematical description of the displacement. What is the velocity and acceleration? 7 8

3 Harmonic oscillation: Summary Harmonic oscillation: The spring Properties of a spring Hookes law & Forces 9 10 Harmonic oscillation: The spring Harmonic oscillation: The spring Gravity will stretch the spring to a new eqilibrium position. This is not the case when the spring is horizonthal. However, the oscillations will be the same

4 Harmonic oscillation: Forces Harmonic oscillation: Forces x = 0 F total = 0 a x = 0 x > 0 F total < 0 a x < 0 x < 0 F total > 0 a x > Harmonic oscillation: Forces Harmonic oscillation: Forces Old formulas: a x = - 2 x New formula: Combine: - 2 = -k/m The frequency depends on the spring constant and the mass 15 16

5 Harmonic oscillation: Forces Harmonic oscillation: Forces An alternative way to look at it: This is a differential equation with the following solution: Gravity will stretch the spring to a new eqilibrium position. This is not the case when the spring is horizonthal Harmonic oscillation: Forces Harmonic oscillation: Forces The mass hangs in the spring without oscillations: The mass hangs in the spring and oscillates: 19 20

6 Harmonic oscillation: Forces Harmonic oscillation: Frequency Spring at rest: Newton s second law: Note: f and T depends only on k and m but not on the amplitude! This is a differential equation with the following solution: m k A Harmonic oscillation: Summary Forces Harmonic oscillation: Energy The differential equation describing the motion: Energy in harmonic oscillation 23 24

7 Harmonic oscillation: Energy Harmonic oscillation: Energy The total mechanical energy is constant E t E p E t E p E k E k Vågrörelselära och optik Mechanical waves: Transverse waves Transverse waves Kapitel 15 Mekaniska vågor 27 28

8 Mechanical waves: Transverse waves Mechanical waves: Transverse waves A sinusoidal transverse wave is when the waves have a periodic sinus shape. Transverse wave: The medium moves transverse to the wave direction Mechanical waves: Transverse waves Mechanical waves: Transverse waves Transversal sinusoidal wave: Definitions: y x Every point on the wave moves up and down like an harmonic oscillator with the period T. y x 31 32

9 Mechanical waves: Longitudinal waves Mechanical waves: Longitudinal waves Longitudinal wave: The medium moves in the wave direction. Longitudinal waves Mechanical waves Mechanical waves: Longitudinal waves Longitudinal sinusoidal wave What is the wavelength ( ) for a sinusoidal wave? What is the wave speed ( )? Amplitude y Every point on the wave moves sideways like an harmonic oscillator with the period T. x = / T 35 36

10 Mechanical waves: The wavefunction The height of the wave as a function of distance x Mechanical waves: The wavefunction The height of the wave as a function of time t The wavefunction Wavefunction y(x,t): Function that describs the height of the wave as a function of time and distance Mechanical waves: The wavefunction Mechanical waves: The wavefunction Amplitude: A Wavenumber: = / T f = 1 / T Angular frequency: + if moving in the x direction = / T = (2 k 2 = / k 39 40

11 Mechanical waves: Summary Mechanical waves: Wave speed The wavefunction: Velocity and acceleration up and down: The wave equation: Wave speed and the string characteristics = / T = (2 k 2 = / k Mechanical waves: Wave speed Mechanical waves: Reflections The wave speed in a string depends on two things: Force (or string trension) String mass per unit length Reflections More generally: 43 44

12 Mechanical waves: Reflections Mechanical waves: Reflections Reflections of a wave Boundary conditions The wavefunction of two waves is typically the sum of the individual wavefunctions. The support provides an opposite force which produces and inverted wave. This is called the principle of superposition. This is true if the wave equations for the waves are linear (they contain the function y(x,t) only to the first power). For example can sinusoidal waves be superimposed like this because their wave equation is linear Mechanical waves: Standing waves Mechanical waves: Standing waves Standing waves 47 48

13 Mechanical waves: Standing waves Mechanical waves: Standing waves Wavefunction from superposition of two waves: Trigonometrical relationship: Different times Wavefunction: Nodes are given by sin(kx) = Mechanical waves: Standing waves Mechanical waves: Stringed instrument Wavefunction: Velocity: Stringed instrument Acceleration: 51 52

14 Mechanical waves: Stringed instrument Mechanical waves: Stringed instrument Instrument with strings of length L has nodes at both ends. = f Long string: Low frequency Thick string: Low frequency Large tension: High frequency f 1, f 2, f 3. Harmonic frequencies f 1 : Fundamental frequency f 2, f 3, f 4. Overtones A stringed instrument does not produce only harmonic frequencies but a superposition of many normal modes Vågrörelselära och optik Sound & Pressure Sound as pressure waves Kapitel 16 - Ljud 55 56

15 Sound & Pressure Sound & Pressure Longitudinal sinusoidal wave Piston moving in and out: Amplitude y x Air molecule movement: y x Pressure: p x Sound & Pressure Sound & Pressure Bulk modulus p = -B V/V The change in pressure after a change of volume: p = -B V/V y p x x Pressure increase: p > 0 and V <

16 Sound - velocity Sound - velocity General: The velocity of sound waves String: Liquid: Solid: Gas: F: String tension : Mass per unit length B: Bulk modulus : Density Y: Young s module : Density B: Bulk modulus : Density Mechanical waves: Power & Intensity Mechanical waves: Power The power in general: Power of mechanical waves on strings Wave power (P): y is the only direction where the velocity is not zero The instantaneous rate at which energy is transfered along the wave. Unit: W or J/s 63 64

17 Mechanical waves: Power Mechanical waves: Power y y = x 2 The power in general: Wave power (P): y = 4x - 4 x The ratio of the force in the y-direction to the force in the x-direction is the slope of the string: The instantaneous rate at which energy is transfered along the wave. Unit: W or J/s The derivative gives the slope of the tangent Mechanical waves: Power Mechanical waves: Power The wave power: The wave power: 67 68

18 Sound power & intensity Sound power The wave power: The pressure function: The wave function: The power of sound Pressure is equal to force per unit area The wave power per unit area: Sound power Sound power The wave power: Power in general: Wave power - string: Wave power - sound: 71 72

19 Sound - Intensity Sound - Intensity The power in general: Intensity of sound Wave power (P): The instantaneous rate at which energy is transfered along the wave. Unit: W or J/s Wave intensity (I): Average power per unit area through a surface perpendicular to the wave direction. Unit: W/m Sound - Intensity Mechanical waves: Power & Intensity Wave intensity (I): The rate at which energy is transported by a wave through a surface perpendicular to the wave direction per unit surface area (average power per unit area). Unit: W/m 2 The pressure function: The intensity through a sphere with radius r 1 The pressure amplitude: If there is no loss of power: The intensity is proportional to the square of the pressure amplitude 75 76

20 Sound - Decibel Sound - Decibel Intensity in the unit of decibel (db) The decibel scale of the intensity I 0 = W/m 2 is a reference intensity It is roughfly the threshold of human hearing = 0 db for I = I 0 = 120 db for I = 1 W/m Sound Standing waves Sound Standing waves Sound and standing waves Antinode Antinode 79 80

21 Sound Standing waves Sound Standing waves Organpipe: Airflow from below. time = 0 time = T/2 Here the pressure is atmospheric giving displacement antinode (pressure node) Standing wave: If the airspeed and pipelengths are choosen correctly. Mouth: Pipe is open at the bottom and gives a pressure node (displacement antinode). Airflow: Depending on time the air flow will either go into the pipe or out through the mouth Sound Standing waves Sound Doppler effect An organ pipe can be open-open or open-closed. Remember: The distance between two nodes is /2 The Doppler effect

22 Sound Doppler effect Sound Doppler effect Doppler effect The time for a sound wave to reach a listener (L) gets longer if the source (S) is moving away. L f s s L The time for a sound wave to reach a listener (L) gets shorter if the source is moving closer. behind longer in front shorter Sound Doppler effect Sound Doppler effect What if the listener is also moving? always works if the positive direction is defined as going from the listener to the source. positive direction positive direction L S S L L S S L The wave speed relative to L is + L change in frequency L L S S S S L L 87 88

23 Sound shockwave Sound shockwave Shock waves Shockwave : Speed of sound s : Speed of the plane s > Shockwave is created (not only when s = s > No sound in front of the plane Sound Vågrörelselära och optik A conical shock wave is produced if a plane flies faster than the speed of sound. A series of circular wave crests from the plane interfere constructively along a line that is given by an angle. : Speed of sound s : Speed of the plane Speed of the plane in Mach number: s / Kapitel 32 Elektromagnetiska vågor 91 92

24 Electromagnetic waves Maxwell s equations The implications of Maxwell s Equations for magnetic and electric fields: 1. A static electric field can exist in the absence of a magnetic field e.g. a capacitor with a static charge has an electric field without a magnetic field. Electromagnetic waves Maxwell s equations The speed of light from Maxwell s equations 2. A constant magnetic field can exist without an electric field e.g. a conductor with constant current has a magnetic field without an electric field. 3. Where electric fields are time-variable, a non-zero magnetic field must exist. = 8.85 x F/m = 1.26 x 10-6 N/A 2 4. Where magnetic fields are time-variable, a non-zero electric field must exist 5. Magnetic fields can be generated by permanent magnets, by an electric current or by a changing electric field. 6. Magnetic monopoles cannot exist. All lines of magnetic flux are closed loops. Permittivity: A mediums ability to form an electric field in itself. Permeability: A mediums ability to form a magnetic field in itself Electromagnetic waves Maxwell s equations Electromagnetic waves Maxwell s equations The electromagnetic wave Electromagnetic waves are produced by the vibration of charged particles. An electromagnetic wave is a wave that is capable of transmitting its energy through a vacuum. B The propagation of an electromagnetic wave, which has been generated by a discharging capacitor or an oscillating molecular dipole. The field is strongest at 90 degrees to the moving charge and zero in the direction of the moving charge. As the current oscillates up and down in the spark gap a magnetic field is created that oscillates in a horizontal plane. The changing magnetic field, in turn, induces an electric field so that a series of electrical and magnetic oscillations combine to produce a formation that propagates as an electromagnetic wave

25 Electromagnetic waves Electromagnetic waves The electromagnetic spectrum = c / f Electromagnetic waves Electromagnetic waves Electromagnetic waves Wavefronts: surfaces with constant phase Wavefronts depends on the distance to the source

26 Electromagnetic waves A plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector. Electromagnetic waves The wave function At a particular point and time all E and B vectors in the plane have the same magnitude. No true plane waves exist since only a plane wave of infinite extent will propagate as a plane wave. However, many waves are approximately plane waves in a localized region of space. In a plane electromagnetic wave the E and B fields are perpendicular to the direction of propagation so it is a transverse wave. The wavefunction B Electromagnetic waves The wave function Electromagnetic waves The wave function The electromagnetic wavefunction Amplitude: E max = c B max Wavenumber: not the same k c= / T f = 1 / T Angular frequency: c= / T = (2 k 2 = / k 104

27 Compare wavefunctions Mechanical waves Electromagnetic waves Electromagnetic waves The wave function In a dielectric medium the speed of light is smaller than c! Amplitude: A Wavenumber: Amplitude: E max = c B max Wavenumber: Electromagnetic waves in matter: Dielectric constant K = / 0 Angular frequency: = / T = / k Angular frequency: c= / T = / k Relative permeability K m = / Electromagnetic waves The wave function Electromagnetic waves Power & Intensity Electromagnetic wave in vacuum Electromagnetic wave in matter Power & Intensity Permettivity Permability Refractive index Dielectric constant Relative permeability K = / 0 K m = /

28 Mechanical waves: Power & Intensity Electromagnetic waves Power & Intensity The power in general: Wave power (P): The instantaneous rate at which energy is transfered along the wave. Unit: W or J/s Wave intensity (I): Average power per unit area through a surface perpendicular to the wave direction. Unit: W/m 2 Total energy density (u): Energy per unit volume due to an electric and magnetic field. Unit: J/m 3 Power (P): The instantaneous rate at which energy is transfered along a wave. Unit: W or J/s The Poynting vector (S): Energy transferred per unit time per unit area = Power per unit area. Unit: W/m 2 Intensity (I): Average power per unit area through a surface perpendicular to the wave direction = the average value of S. Unit: W/m Electromagnetic waves Power & Intensity Electromagnetic waves Power & Intensity The total energy density (energy per unit volume) due to an electric and magnetic field is B Energy transfer = energy transferred per unit time per unit area. S = Power per unit area = Energy transfer = Energy flow + B 2 = 0 0 E 2 where Energy E-field Energy B-field Amplitude = maximum energy transfer Conclusions: The electric and magnetic fields carry the same amount of energy. The energy density varies with position and time

29 Electromagnetic waves Power & Intensity Vågrörelselära och optik Intensity = the average value of S The average of cos 2 (x) = 1/2 Electromagnetic waves in matter: Kapitel 33 - Ljus The nature of light The nature of light Source of electromagnetic radiation is electric charges in accelerated motion Thermal radiation: Thermal motions of molecules create electromagnetic radiation. Wave front: surface with constant phase. Plane wave: is a wave whose wave fronts are infinite parallel planes. Ray: an imaginary line along the direction of the wave s propagation. Lamp: A current heats the filament which then sends out thermal radiation with many wavelengths. Laser: Atoms emits light coherently giving (almost) monocromatic radiation

30 The nature of light The nature of light Reflection & Refraction Reflection and refraction The nature of light The nature of light Conclusions: At the surface between air and glass the angle is always 90 degrees and then the reflected and refracted light is also at 90 degrees. n a n b The plane of incident: The plane of the incident ray and the normal to the surface. The reflected and refracted rays are in the plane of incident. At the surface between glass and air some of the light is reflected and some is refracted. The angle of reflection is the same as the incident angle. The angle of refraction is larger than the incident angle. n = 1 in vacuum n > 1 in a material Snell s law:

31 The nature of light The nature of light Snell s law: n a < n b Light intensity n a > n b Rule: Large n Small angle The nature of light The nature of light Intensity The intensity of the reflected light increases from almost 0% at = 0 o to 100% at = 90 o. The intensity of the reflected light also depends on n and on polarization of the incoming light. Total internal reflection The sum of the intensity of the reflected and refracted light is equal to the intensity of the incoming light

32 The nature of light Total Internal Reflection when light goes to a medium with smaller n The nature of light Total Internal Reflection optical fiber Porro prism 90 o The nature of light The nature of light Optical fibers Principle n 2 < n 1 Structure Dependency on frequency and wavelength

33 The nature of light The nature of light n a n b Frequency and wavelength : The speed is larger in a material with a small n. f: The frequency does not depend on n. : The wavelength is longer in a material with a small n. Dispersion n = 1 in vacuum n > 1 in a material / f n > 1 c / f n = 1 / n The nature of light The nature of light How is this possible? Dispersion Answer: n must depend on! n = c / so the speed in a material must then depend on

34 The nature of light Vågrörelselära och optik Rainbow Kapitel 34 - Optik Virtual Images: outgoing rays diverge Mirrors Real Images: outgoing rays converge to an image that can be shown on a screen

35 Point object positive Sign rules: Object distance (s) positive if same side as incoming light. Image distance (s ) positive if same side as outgoing light. Flat mirror negative Virtual image Extended object Spherical mirror An infinite number of rays can be drawn from an object to its image. But only two rays are needed to determine the location of the image

36 How to find the image in a concave mirror The bottom of the object is on the optical axis and so the bottom of the image will also be on the optical axis. The top of the image can be found with any two rays. Use for example two rays that goes through the focal point. y s y f s Summary spherical mirrors Sign rules: Object distance (s) positive if same side as incoming light. Image distance (s ) positive if same side as outgoing light. Radius of curvature (R) positive if center is on same side as outgoing light. y negative y, s, s, f positive y s y R f s y negative y, s, s, f positive s y negative y, s, s, f positive s y negative y, s, s, f positive s negative y, y, s, f positive Magnification (m) positive if direction of object and image is the same

37 Convex mirrors Virtual Focal Point s, f negative y, y, s positive Spherical surface

38 Sign rules: Object distance (s) positive if same side as incoming light. Image distance (s ) positive if same side as outgoing light. Radius of curvature (R) positive if center is on same side as outgoing light. Spherical surface -Summary s positive s positive R positive Special case: flat surface n a /s = -n b /s -s /s = n b /n a Different type of lenses Lenses

39 Useful rays An object placed at the focal point appear to be at infinity

40 Sign rules: Object distance (s) positive if same side as incoming light. Image distance (s ) positive if same side as outgoing light. Convex lenses -Summary s is positive f is positive m is negative s Gauss formula Newton s formula Focal length (f) positive for converging lenses (convex lenses) s s is negative f is positive m is positive Two lenses s 1 s 1 s 2 s 2 EXAMPLE Known: s 1, f 1, f 2 and L Calculate s 2 and m s 1 s 1 s 2 s 2 L

41 Lenses Lens formula for concave lenses s s f is negative for diverging lenses s is negative for diverging lenses m is positive

42 Lenses The lensmaker s equation Rule: A lens that is thicker at the center than the edges is converging (positive f) A lens that is thinner at the center than the edges is diverging (negative f) = Sign rule: Radius of curvature positive if center is on same side as outgoing light. f = positive R 1 = positive R 2 = positive s = positive or negative f = positive R 1 = positive R 2 = negative s = positive or negative f = negative R 1 = negative R 2 = positive s = negative The eye

43 Near point: Closest distance to the eye at which people can see clear (7cm at age 10 to 40cm at age 50 for normal eye). Normal reading distance: Assumed to be 25 cm when designing correction lenses. When the person puts an object at s = 25 cm from the correcting lens we want the image to end up at s = 100 cm because this is the nearest point the eye can see sharply. Lenses for corrections are given in diopter. Lens power = 1/f (unit diopter = m -1 ) The lens should move the actual far point from 50 cm to infinity. The correcting lens should therefore have s = infinity for s = 50-2 = 48 cm. The magnifying glass

44 A magnifying glass is a convex lens. The magnifying glass If you hold a magnifying glass far away from the eye (arms lengths distance) you can see a magnified and up-side down image. s Near point ( ): Closest distance an eye can focus (approximatively 25 cm) 25 cm The normal use of a magnifying glass is to put the object between the focal point and the lens to get a magnified up-right image. s When the object is at the focal point one uses angular magnification (M) instead of lateral magification (m) Magnifying glass (f is a couple of cm) The microscope Creates magnified image close to the focal point of the eye piece (f < 1 cm)

45 s 1 s 1 L Eyepiece Angular magnification of magnifying glass The telescope Objective Eyepiece Objective Microscope Magnification is the nearpoint which is typically 25 cm Comparing microscopes with telescopes Object at a close distance s 1 s 1 L The first image will be in the focal point of the first lens. The eye piece works as a magnifying glass with I in its focal point. Object at infinity Small f 1 & Small f 2 is the nearpoint which is typically 25 cm The angular magnification of a telescope is defined as the ratio of the angle of the image to that of the incoming light. Large f 1 & Small f

46 Vågrörelselära och optik Interference Interference: Wave overlap in space Coherent sources: Same frequency (or wavelength) and constant phase relationship (not necessarily in phase). Kapitel 35 - Interferens Interference Interference Contructive interference Antinodal curves = Contructive interference Contructive interference A path difference of one wavelength corresponds to a phase difference of 2 Destructive interference Destructive interference

47 Interference Interference Geometry: m=-1 m=0 m=1 m=2 m=3 R y y Contructive Contructive interference: Destructive Interference Interference A path difference of one wavelength corresponds to a phase difference of 2 Introduce y in the formula Path difference y small

48 Interference Interference m=-1 m=0 m=1 Intensity: Summary Contructive interference: m=2 m=3 y Intensity: Interference Interference The Michelson Interferometer The Michelson Interferometer y The compensator plate compensates for this The observer will see an interference pattern with rings. The fringes in the pattern will move when the mirror is moved. The number of fringes (m) can be used to calculate y or

49 Interference Vågrörelselära och optik Kapitel 36 - Diffraktion Diffraction Diffraction Interference: Double slit experiment Diffraction: single slit experiment

50 Diffraction Diffraction For every point in the top half of the slit there is a corresponding point in the bottom half Diffraction Diffraction Bright bands: Destructive Interference: Geometry: Dark bands: Small angles:

51 Diffraction Diffraction r 2 r 1 A path difference of one wavelength corresponds to a phase difference of 2 Path difference: r 2 r 1 = a sin( ) r 2 -r 1 is the path difference between a ray at the top and bottom of the slit Diffraction Summary Intensity Intensity: = 6 = 4 = 2 where = 0 = -2 = -4 =

52 Diffraction Diffraction In the analysis of interference from two slits it was assumed that they were very narrow. What if they are broad? Two narrow slits: One broad slit: Two broad slits Two broad slits: where Diffraction Diffraction Two narrow slits: One broad slit: Multiple slits Two broad slits:

53 Diffraction Diffraction The path difference between adjacent slits that gives maximum intensity with many slits is always: N = 2 slits N = 2 N = 8 N = 16 N = 8 slits N-1 minima N-2 small peaks Principal maxima: Diffraction Diffraction Diffraction In diffraction grating one uses devices with thousands of slits or reflecting surfaces. This gives very narrow principal maximum that can be used to determine the wavelength of light. Transmission grating Reflection grating Spectrometers

54 Diffraction Spectrometer for astronomy Diffraction Chromatic resolving power: The minimum wavelength difference ( ) that can be distinguished by a spectrograph. Number of slits in the grating Order of maximum in diffraction pattern Light incident on a grating is dispursed into a spectrum. The angles of deviations of the maxima are measured to calculate the wave length. R is higher for many slits and higher orders! Diffraction Diffraction Pinhole diffraction

55 Diffraction Diffraction limits the angular resolution of optical intruments. Diffraction Rayleigh s criterion: Two point objects can be resolved by an optical system if their angular separation is larger than 1 where The limit for two objects to be resolved is when the center of one diffraction pattern is in the first minimum of the other. D