•Hour Exam 2: Wednesday, Oct. 27 •In-class (1300 Sterling...

Mon. Oct 25 Phy107 HE 2 review

• Hour Exam 2: Wednesday, Oct. 27• In-class (1300 Sterling Hall)• Twenty multiple-choice questions• Will cover: 8.1-8.6 (Light and E&M)

9.1-9.5 (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


Today

• Quickly finish material in Chapter 11.

• Review for Hour Exam 2.


From Last Time…• General relativity motivated by Equivalence Principle

– No experiment can distinguish between force of gravityand an accelerated reference frame.

• General relativity does not consider gravity to be aforce 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


Black holes

• Light falls in a gravitational field. A sufficientlystrong field should bind light into orbits.

• When escape velocity equals the speed of light,nothing can emerge.

• Inside the Schwarzschild radius light itselfcannot escape.

• An object condensed within it’s Schwarzschildradius is called a black hole.


How do black holes form?

• Matter in the earth is held apart by electricforces.

• In the Sun, matter is held up by pressureassociated with high temperature by nuclearfusion reactions.

• At higher density when fusion burns out, gravitycan squeeze electrons into protons forming aneutron star.

• At even higher densities, a black hole forms


Black hole formation

• At sufficient density, no knownforce prevents the completecollapse of matter to a point-a black hole.

• Matter falling into black holesprovides the only explanationfor energetic objects seenthroughout the universe.http://antwrp.gsfc.nasa.gov/apod/ap

010921.html

Black hole roaming theMilky Way.


Seeing Holes

• Light cannot escape from a blackhole, so can’t see black hole itself,but can see matter falling into ahole.

• Gravitational forces stretch andrip matter: heats up.

• Very hot objects emit in X-rays(interior of Sun)

• Cygnus X-1.

http://www.owlnet.rice.edu/~spac250/steve/ident.html


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


Chapter 8: Wave properties• Amplitude is the maximum displacement of

string above the equilibrium position

• Wavelength, λ, is the distance between twosuccessive points that behave identically

• Period: time required to complete one cycle

• Frequency = 1/Period = rate at which cyclesare completed

• Velocity = Wavelength/Period,v = λ / T, or v = λf


Different wave motion• Longitudinal wave

– Vibrations are in the direction of motion

• Transverse wave– Vibrations are perpendicular to the direction of

motion.


Superposition & interference• Superposition of waves• Interference of waves on a string• Interference of sound waves

– Constructive interference– Destructive interference

• Superposition of waves with differentfrequencies– Beat frequencies

• Doppler effect– Change in apparent frequency due to motion

of source or observer


Interference of sound waves

• Interference arises when waves change their‘phase relationship’.

• Can vary phase relationship of two waves bychanging physical location of speaker.

Constructive Destructive

‘in-phase’ ‘1/2 λ phase diff’


Superposition & Interference• Consider two harmonic waves A and B meeting at x = 0.

– Same amplitudes, but ω2 = 1.15 x ω1.• The displacement versus time for each is shown below:

A(ω1t)

C(t) = A(t) + B(t) CONSTRUCTIVEINTERFERENCE

DESTRUCTIVEINTERFERENCE

B(ω2t)


Doppler Effect, Source inMotion

• As the source movestoward the observer (A),the wavelength appearsshorter and the frequencyincreases

• As the source moves awayfrom the observer (B),the wavelength appearslonger and the frequencyappears to be lower


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


Coulomb’s Law

• Electrostatic force: FE = k Q1 Q2 /r2

k = 9x109 Nm2/C2

– Force between charges q1 and q2separated by a distance r.

– Direction: Like charges repel unlike attract

Similar to gravitational force: FG=GM1M2/ r2

G=6.7x10-11 Nm2/kg2


Electric Potential• Units Joules/Coulomb≡Volts– Batteries– Power outlets– EKG

• Potential differences• Field lines point down hill• Charge will move

along field lines just asmass falls in gravitationalfield.


Source of magnetic field?

• Current in wire produces magnetic field.• That magnetic field aligns compass needle

Current

Magneticfield


Induction (Faraday)

• Changing magnetic fields produce electric fields

• Here the induced current produces a magneticfield, which repels the bar magnet or the ring


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.


General props of EM waves

xz

y

•A Transverse wave.

•Electric and magnetic fields are perpendicular topropagation direction

•Can travel in empty space

f = v/λv = c = 3 x 108 m/s (186,000 miles/second!)


Chapter 9: Visible light & color• Range of visible light from 400 nm to 700 nm

• Eye interprets different wavelengths as different colorsbut has only three ‘sensors’, cones S, M, L with overlappingsensitivities.

• Lets colors be described as combination of three ‘primaries’

• This makes a ‘color space’, colors can be described as amountof primaries needed.

• Most methods of generating colors have a ‘gamut’, a limitedrange of possible colors that in general to not contain purespectral colors.


• Visible light is onlya narrow range ofthe entire EMspectrum.


Eye spectral sensitivity• Eye’s sensitivity to EM

radiation is through threecones sensitive todifferent spectral ranges.

• Overlapping ranges meansthat different lightsources can be seen as thesame color

• Three cones suggests wecan synthesize colors fromthree primary sources.


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 ofsource emitting light)

• One consequence– Events simultaneous in one frame will not be

simultaneous in another.


Space & time relativistic effects•Events observed to be simultaneous in one

frame may not be simultaneous in another.

•Measured interval between events different fordifferent observers.

— Time dilation. Proper time is that measured inframe where events occur at same spatial location

— All other measured times are longer by factor γ

•Measured distance between events different fordifferent observers.

— Length contraction. Proper length is that measuredin frame where events are simultaneous.

— All other lengths are shorter by factor γ


Time dilationI 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. tjet=tEarth

B. tjet>tEarth

C. tjet<tEarth

Proper time is measured in the jet frame(events occur at same spatial location).

Times measured in other frames are longer(time dilation).


Space-time diagrams• World line : path of particle through space-time.

• World line is made up of sequence of events, plotted aspoints in space-time.

• Different inertial reference frames represented as ‘tilted’coordinate axes on space-time diagram.

• Event has different coordinates (space & time) measuredon these coordinate axes.

• But the combination x2-c2t2 is ‘universal’ in that it ismeasured to be the same for all observers.


Space-time diagrams

x

ct

x1

ct1

General world linex

ct

World line low ofvelocity object

World line of highervelocity object

World lineof light

Const. vel world line


Relativistic Momentum

• Momentum can be increased arbitrarily,but velocity never exceeds c

• We still use

so for constant force we still havemomentum = Force x time,but the velocity never exceeds c

• Momentum has been redefined.

€

prelativistic = γmv =mv

1− (v /c)2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

SPEE

D / S

PEED

OF

LIG

HT

RELATIVISTIC MOMENTUM€

vc

=p / pop / po( )2

+1, po = moc

€

change in momentumchange in time

= Force

Newton’smomentum

Relativisticmomentum

Relativistic momentum fordifferent speeds.


Relativistic mass

• Could say that particlebecomes extremelymassive as speedincreases ( m=γmo )

• But could also say thatrelativistic momentumhas new form ( p= γmov )

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1RELA

TIVI

STIC

MAS

S /

RES

T M

ASS

SPEED / SPEED OF LIGHT


Mass-energy equivalence

• This results in Einstein’s famous relation

• 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, gravitationalenergy, etc, we can talk of mass energy.

€

E = γmoc2, or E = mc 2


General Relativity and Gravity• General relativity motivated by Equivalence Principle

– No experiment can distinguish between force of gravityand an accelerated reference frame.

• General relativity does not consider gravity to be aforce 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

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•Hour Exam 2: Wednesday, Oct. 27 •In-class (1300 Sterling...

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