Your student ID 1 page notes, written double sided Calculator Pencil for marking answer sheet

Transcription

1 Hour Exam 2: Wednesday, Oct. 27 In-class (1300 Sterling Hall) Twenty multiple-choice questions Will cover: (Light and E&M) (E&M waves and color) 10, 11 (Relativity) You should bring Your student ID 1 page notes, written double sided Calculator Pencil for marking answer sheet

2 Today Quickly finish material in Chapter 11. Review for Hour Exam 2.

3 From Last Time General relativity motivated by Equivalence Principle No experiment can distinguish between force of gravity and an accelerated reference frame. General relativity does not consider gravity to be a force between massive objects Mass tells space-time how to curve Curvature of space-time dictates how objects move. Objects move along straight line in space-time Motion independent of mass of object (even light bends) Leads to effects such as Gravitational lensing Black holes

4 Black holes Light falls in a gravitational field. A sufficiently strong field should bind light into orbits. When escape velocity equals the speed of light, nothing can emerge. Inside the Schwarzschild radius light itself cannot escape. An object condensed within it s Schwarzschild radius is called a black hole.

5 How do black holes form? Matter in the earth is held apart by electric forces. In the Sun, matter is held up by pressure associated with high temperature by nuclear fusion reactions. At higher density when fusion burns out, gravity can squeeze electrons into protons forming a neutron star. At even higher densities, a black hole forms

6 Black hole formation html At sufficient density, no known force prevents the complete collapse of matter to a pointa black hole. Matter falling into black holes provides the only explanation for energetic objects seen throughout the universe. Black hole roaming the Milky Way.

7 Seeing Holes Light cannot escape from a black hole, so can t see black hole itself, but can see matter falling into a hole. Gravitational forces stretch and rip matter: heats up. Very hot objects emit in X-rays (interior of Sun) Cygnus X-1.

8 Exam Review: Topics by chapter Chap 8: Waves, interference, & electromagnetism Chap 9: EM radiation, light, color Chap 10: Special Relativity time & space Chap 11: Special Rel. energy, & General Relativity

9 Chapter 8: Wave properties Amplitude is the maximum displacement of string above the equilibrium position Wavelength, λ, is the distance between two successive points that behave identically Period: time required to complete one cycle Frequency = 1/Period = rate at which cycles are completed Velocity = Wavelength/Period, v = λ / T, or v = λf

10 Different wave motion Longitudinal wave Vibrations are in the direction of motion Transverse wave Vibrations are perpendicular to the direction of motion.

11 Superposition & interference Superposition of waves Interference of waves on a string Interference of sound waves Constructive interference Destructive interference Superposition of waves with different frequencies Beat frequencies Doppler effect Change in apparent frequency due to motion of source or observer

12 Interference of sound waves Interference arises when waves change their phase relationship. Can vary phase relationship of two waves by changing physical location of speaker. in-phase 1/2 λ phase diff Constructive Destructive

13 Superposition & Interference Consider two harmonic waves A and B meeting at x = 0. Same amplitudes, but ω 2 = 1.15 x ω 1. The displacement versus time for each is shown below: A(ω 1 t) B(ω 2 t) C(t) = A(t) + B(t) CONSTRUCTIVE INTERFERENCE DESTRUCTIVE INTERFERENCE

14 Doppler Effect, Source in Motion As the source moves toward the observer (A), the wavelength appears shorter and the frequency increases As the source moves away from the observer (B), the wavelength appears longer and the frequency appears to be lower

15 Chap. 8: Electromagnetism Coulomb force between charged particles Same form as gravitational force Electric field lines: path followed by charged particle Electric current: flow of charged particles Electrostatic potential: Measured in volts. Analogous to gravitational potential. Magnetic field: arises from electric currents (moving charges) also results in force on an electric current Faraday effect: changing magnetic field induces electric current Magnetic field from induced currents opposes change in applied field

16 Coulomb s Law Electrostatic force: F E = k Q 1 Q 2 /r 2 Force between charges q 1 and q 2 separated by a distance r. Direction: Like charges repel unlike attract k = 9x10 9 Nm 2 /C 2 Similar to gravitational force: F G =GM 1 M 2 / r 2 G=6.7x10-11 Nm 2 /kg 2

17 Electric Potential Units Joules/Coulomb Volts Batteries Power outlets EKG Potential differences Field lines point down hill Charge will move along field lines just as mass falls in gravitational field.

18 Source of magnetic field? Current in wire produces magnetic field. That magnetic field aligns compass needle Current Magnetic field

19 Induction (Faraday) Changing magnetic fields produce electric fields Here the induced current produces a magnetic field, which repels the bar magnet or the ring

20 Chapter 9: Electromagnetic waves EM radiation is a transverse wave Amplitudes are perpendicular to propagation dir. Contains both electric field and magnetic field EM radiation propagates at c, speed of light EM radiation generated by accelerating charges Visible light is a narrow band of entire EM spectrum. Radio waves are EM radiation. Can be generated by spark jumping a gap.

21 General props of EM waves y z x A Transverse wave. Electric and magnetic fields are perpendicular to propagation direction Can travel in empty space f = v/λ v = c = 3 x 10 8 m/s (186,000 miles/second!)

22 Chapter 9: Visible light & color Range of visible light from 400 nm to 700 nm Eye interprets different wavelengths as different colors but has only three sensors, cones S, M, L with overlapping sensitivities. Lets colors be described as combination of three primaries This makes a color space, colors can be described as amount of primaries needed. Most methods of generating colors have a gamut, a limited range of possible colors that in general to not contain pure spectral colors.

23 Visible light is only a narrow range of the entire EM spectrum.

24 Eye spectral sensitivity Eye s sensitivity to EM radiation is through three cones sensitive to different spectral ranges. Overlapping ranges means that different light sources can be seen as the same color Three cones suggests we can synthesize colors from three primary sources.

25 Chapter 10: Basic relativity Galilean Relativity Laws of mechanics identical in all inertial ref. frames Einstein s Relativity All laws of physics identical in inertial ref. frames Speed of light=c in all inertial ref. Frames (e.g. independent of velocity of observer, velocity of source emitting light) One consequence Events simultaneous in one frame will not be simultaneous in another.

26 Space & time relativistic effects Events observed to be simultaneous in one frame may not be simultaneous in another. Measured interval between events different for different observers. Time dilation. Proper time is that measured in frame where events occur at same spatial location All other measured times are longer by factor γ Measured distance between events different for different observers. Length contraction. Proper length is that measured in frame where events are simultaneous. All other lengths are shorter by factor γ

27 Time dilation I am on jet traveling at 500 mph and throw a ball up and catch it in my hand. You are on the ground and watch me. How do the time intervals compare for you and I? A. t jet =t Earth B. t jet >t Earth C. t jet <t Earth Proper time is measured in the jet frame (events occur at same spatial location). Times measured in other frames are longer (time dilation).

28 Space-time diagrams World line : path of particle through space-time. World line is made up of sequence of events, plotted as points in space-time. Different inertial reference frames represented as tilted coordinate axes on space-time diagram. Event has different coordinates (space & time) measured on these coordinate axes. But the combination x 2 -c 2 t 2 is universal in that it is measured to be the same for all observers.

29 Space-time diagrams ct World line low of velocity object ct World line of higher velocity object ct 1 World line of light x 1 General world line x x Const. vel world line

30 Relativistic Momentum Momentum can be increased arbitrarily, but velocity never exceeds c We still use change in momentum = Force change in time so for constant force we still have momentum = Force x time, but the velocity never exceeds c Momentum has been redefined. p relativistic = γmv = mv 1 (v /c) 2 SPEED / SPEED OF LIGHT Newton s momentum v c = Relativistic momentum p / p o ( p / p o ) 2 +1, p o = m o c RELATIVISTIC MOMENTUM Relativistic momentum for different speeds.

31 Relativistic mass Could say that particle becomes extremely massive as speed increases ( m=γm o ) But could also say that relativistic momentum has new form ( p= γm o v ) RELATIVISTIC MASS / REST MASS SPEED / SPEED OF LIGHT

32 Mass-energy equivalence This results in Einstein s famous relation E = γm o c 2, or E = mc 2 This says that the total energy of a particle is related to its mass. Even when the particle is not moving it has energy. We could also say that mass is another form of energy Just as the text talks of chemical energy, gravitational energy, etc, we can talk of mass energy.

33 General Relativity and Gravity General relativity motivated by Equivalence Principle No experiment can distinguish between force of gravity and an accelerated reference frame. General relativity does not consider gravity to be a force between massive objects Mass tells space-time how to curve Curvature of space-time dictates how objects move. Objects move along straight line in space-time Motion independent of mass of object (even light bends) Leads to effects such as Gravitational lensing Black holes