In my mailbox this morning

Exam questions A uniform magnetic field

B points into the paper inthe indicated region.Outside this region, B = 0.Which of the conductingwire loops, moving withvelocity v, experiences aforce toward the top ofthe page?

– a) A

– b) B

– c) C

– d) D

– e) A and C

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A

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A

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A

B

C D v

v

vv

Two ways of thinking about it. As loop Amoves into magnetic field, more field linesappear in loop A pointing into plane of paper. A counter-clockwise current is induced in A to create field lines coming out of paper. The force on bottom segmentof the loop is pointing up.

Also, the -sign in Lenzs’ law means that anychange is resisted. If the loop is beingpushed into the field, its’ going to push back.

Special relativity

l The laws of physics arethe same in allcoordinate systemseither at rest or movingat constant speed withrespect to one another

l The speed of light in avacuum has the samevalue regardless of thevelocity of the observeror the velocity of thesource emitting the light

Simultaneity

l A basic premise of Newtonianmechanics is that a universaltime scale exists that is thesame for all observers

l “Absolute, true, andmathematical time, of itself,and from its own nature, flowsequably without relation toanything external”

u Isaac Newton

l Newton took the concept ofsimultaneity for granted

l Einstein was forced toabandon this assumption

Was Isaac Newton really Q?

l You decide

Another gedanken

Consider a train moving with velocity v, two observers O and O’,and lightning striking the two ends of the boxcar

Light signals reach O at the same time; she concludes the two lightning strikes were simultaneous.Light from B’ reaches observer O’ before the light from A’; observerO’ concludes that the lightning struck at B’ before it did at A’.Two events which appear to be simultaneous in one frame of reference are not, in another frame

Simultaneity lost

lSimultaneity is not an absolute conceptbut one that depends on the state ofmotion of the observer

Time Dilation: another gedanken

Simultaneity is out the window. Observers in different reference frames may alsomeasure different time intervals between a pair of events. Consider a boxcar moving to the right with a velocity v. Observer O’ shines a laserand observes it reflecting from a mirror on the ceiling. She’s timing this and observes the total time for the laser to reflect and come back to be: Dt=2d/cNow observer O, standing alongside the track, is also observing this event. Becauseof the velocity of the boxcar, observer O sees that the laser light has to travel alonga longer path: d2+(cDt/2)2

Time Dilation: another gedanken

Now the speed of light c is the same in both frames of reference. Thus, observer Omust measure a longer time for the light to reflect from the mirror than observer O’.In other words, time appears to move more slowly in the boxcar according to observer O than according to observer O’.

By how much?

Time Dilation: another gedanken

(cDt/2)2 = (vDt/2)2 + d2

Solve for t:Dt = 2d = 2d c2-v2 c 1-v2/c2

…or Dt = gDtpwhere g = 1/ 1-v2/c2

and Dtp = 2d/c, the time measured by observer O’. We’ll call this the proper time.

Time dilation

l So…the time invervalbetween two eventsmeasured by anobserver moving withrespect to a clock islonger than the timeinterval between twoevents measured by anobserver at rest withrespect to the clock

l …i.e., a clock movingpast an observer at aspeed v runs moreslowly than identicalclock at rest with respectto the observer, by afactor g-1

Dtp, the proper time

Everything depends on g

3.905667329290,000,000 m/s

1.341640787200,000,000 m/s

1.060660172100,000,000 m/s

1.00055601910,000,000 m/s

1.0000055561,000,000 m/s

1.000000056100,000 m/s

1.00000000110,000 m/s

11000 m/s

1100 m/s

gVelocity (m/s)

Cosmic rays

l Some come from the sun(relatively low energy)and some fromcatastrophic eventselsewhere in thegalaxy/universe

Collision of a high energy cosmic ray particle with a photographic emulsion

Cosmic rays interact with the Earth’s upper atmosphere andproduce a shower of particles; eventually only subatomic particlescalled muons are left.Crab nebula

Another example:cosmic ray muons

l Time dilation works forclocks, heartbeats,anything…

l Consider muonsproduced in the upperatmosphere

u they have a lifetime of 2.2ms

u if they travel at the speedof light, you might thinkthey could only travel(3X108 m/s X 2.2 X 10-6 s= 600 m) before decaying

u 2.2 ms is how long theylive in their rest frame

u but if they are travelling at0.99c, then g=7.1 andgcDtp=4800 m (not 600 m)

Let’s back up just a little

Observer O sees observer O’ moving off at velocity v; therefore he sees her clock runs slow.

But, from observer O’s point-of-view, she’s standing still and it’sobserver O who is moving off to the left. And her perspective isperfectly valid. And she sees his clock run slow. Who is right?

Twin paradox

Two twins aged 21. One stayshome on earth. The other heads off in a spaceship travelling close to the speed of light (g=25; 99.9% of the speed of light)

After 50 years have passedfor the twin on earth, only2 years have passed for thetwin in the spaceship. Or is itthe other way around?

Twin paradox

Two twins aged 21. One stayshome on earth. The other heads off in a spaceship travelling close to the speed of light (g=25; 99.9% of the speed of light)

After 50 years have passedfor the twin on earth, only2 years have passed for thetwin in the spaceship. Or is itthe other way around?

Say, how’sthe war inIraq going?

Length contraction

lMeasured distancebetween two pointsdepends on theframe of reference

l Lp is the properlength (in rest frame)

l An observer O seesthe length decreaseto a value of L=gLP

Transformation of velocitiesl Einstein was able to show that his

new ideas of space and time hadconsequences for addition ofvelocities

l Suppose I’m travelling in a convertibleat 60 miles per hour; I stand up andthrow a baseball forward at 30 milesper hour

l How fast is the baseball travelingwith respect to the ground? Easy, 90miles per hour. This is using theGalilean transformation.

l Let u be velocity of car and v thevelocity of the baseball with respect tothe car:

u velocity with respect to the groundthen is u+v

l Using Lorentz transformation, velocityis given by (u+v)/(1+uv/c2) with cbeing the velocity light

l Why haven’t we noticed this before?For the example above, the speed is0.4 millionth of a millionth of apercent less than 90 mph

Relativistic example

l A man on a (veryfast) motorcycletravelling 0.80 cthrows a baseballforward (he has avery good arm) with aspeed of 0.70 c (fromhis persective)l How fast does the

innocent bystandersee the balltravelling?

From Galilean perspective: 0.80 c +0.70 c =1.5 cUsing Lorentz transformation of velocities:[u+v]/[1+uv/c2] = [0.8c+0.7c]/[1+(.8c)(.7c)/c2] =0.96 c

Massl Remember Newtons second law F=mal If I keep a constant force on an object, then I have a constant

acceleration, i.e. the velocity keeps increasing at a constant ratel We’ve seen this doesn’t work at extreme speeds; as the velocity

gets closer to the speed of light, the acceleration decreasesl The object behaves as if its inertia is increasingl But inertia is mass, so the object behaves as if its mass is

increasing; the mass increase becomes very noticeable as theobject approaches the speed of light

l Newtons second law still ok if I reframe it; force is proportional totime rate of change of momentum (mv)

l Changes in momentum and kinetic energy come not only fromchanges in v but also changes in m

l So, mass is a form of energy, just as heat is a form of energy

u E=mc2

Energy and momentum

l Total energy of aparticle

u = kinetic energy + mc2

u = gmc2

u as v->0, g->1; energy=mc2 (just the restmass energy)

l Relativisticexpressions

u energy: E=gmc2

u momentum: p=gmv

u E2=p2c2+(mc2)2

Convenient to quote particle energies in eV1 eV = 1.6 X 10-19 Jme=9.11X10-31 kgmec2=(9.11X10-31 kg)(3.0X108 m/s)2 = 8.2X10-19 J =(8.2X10-14 J)/(1.6X10-19 J/eV)=0.511 MeV

Pair production and annihilation

General Relativity

We said that special relativity applies for inertial frames of reference. What about for non-inertial (accelerating) frames?

That’s the realm of general relativity (also by A. Einstein). His equivalence principle stated that one can not tell the differencebetween gravity and an accelerating frame of reference.

Deflection of starlight

Energy and mass are equivalent. Starlight has energy; starlightshould be affected by gravitational fields.Einstein’s big prediction. Verified by an astronomer (Sir ArthurEddington) in a total eclipse in 1918.

Einstein became a celebrity.