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1
Special RelativityOne of the biggest surprises in our
understanding of Physics
2
Extend Inertial Coordinate SystemSymmetry of Galileo and Newton
• THE EINSTEIN PRINCIPLE OFRELATIVITYo "ALL OF THE LAWS OF PHYSICS are
the same for every inertial observer."• In particular,
o "The speed of light is the SAME for allinertial observers, regardless of themotion of the source.”
• But this required some big changesin our basic understanding of time.
3
Speed of Light• The speed of light (c) was the central question
that gave rise to the theory of relativity.• The speed of light is very large compared to the
speeds we experience.o We have no physical intuition about speeds
approaching c.• We can however measure the speed of light with
a rotating mirror.o Light bounces off the mirror twice with a long distance
in betweeno A small deflection results !x =
8"DD ' f
c#25 10m( )
2
600 / s
3$108m / s
= 0.005m
4
In what frame is the speed of light c?• The equations of EM were consistent if the speed
of light is constant in one fixed frame .o Physicists thought EM waves must propagate in some
medium.o Postulated the “ether” (aether).
→ They thought, space is filled with “the ether” in which EM wavespropagate at a fixed speed.
o Ether gave one fixed frame for EM.• But experiments disagreed.• And we would loose the symmetry found in
Newton’s laws; “any inertial frame”.
5
The Sun and Earth are Moving
• We should see some velocity of the ether.• We should see a seasonal variation.
o MM set up to be sensitive even to the motion of the earth.
6
The Michelson-Morley ExperimentAlbert Abraham Michelson (1852-1931) was a German-born U.S. physicist whoestablished the speed of light as a fundamental constant. He received the 1907 NobelPrize for Physics. In 1878 Michelson began work on the passion of his life, themeasurement of the speed of light. His attempt to measure the effect of the earth'svelocity through the supposed ether laid the basis for the theory of relativity. FirstAmerican scientist to win Nobel Prize.
Edward Williams Morley (1838-1923) was an American chemist whose reputation asa skilled experimenter attracted the attention of Michelson. In 1887 the pair performedwhat has come to be known as the Michelson-Morley experiment to measure themotion of the earth through the ether.
7
Michelson-Morley Experiment• Michelson interferometer on
a block of granite.o Light waves from two mirrors
will interfere.• Granite block floats in
mercury to greatly reducevibration.
• Slowly rotate apparatus andmeasure interference.o MM found no change as they
rotated.o Speed of light is the same even
though the earth is moving.
Wavelength of light is about 400nm so interferometermeasurement is very accurate.
8
MM calculation with ether• Ether with just the earth’s velocity would show
interference.• As apparatus is rotated, interference changes.
t1 =L '
c+L '
c in direction perpendicular to ether motion
L ' = L2+ vt1( )
2! L 1+
v2
c2
t2 =L
c + v+
L
c " v in direction parallel to ether motion
t2 " t1 =L c " v( )
c2 " v2
+L c + v( )
c2 " v2
-2L '
c=
2Lc
c2 " v2
-2L '
c=
2L
c 1"v
2
c2
#$%
&'(
-
2L 1+v
2
c2
c!
2L
c1+
v2
c2"1"
v2
2c2
#$%
&'(=L
c
v2
c2
c(t2 " t1) = 11m( )3)104
3)108
#$%
&'(
2
= 11)10"8m = 110nm a large fraction of the wavelength of light
9
Heaviside, Lorentz, FitzgeraldOliver Heaviside (1850-1925) was a telegrapher, but deafness forced him to retire and devote himself toinvestigations of electricity. He became an eccentric recluse, befriended by FitzGerald and (by correspondence)by Hertz. In 1892 he introduced the operational calculus (Laplace transforms) to study transient currents innetworks and theoretical aspects of problems in electrical transmission. In 1902, after wireless telegraphyproved effective over long distances, Heaviside theorized that a conducting layer of the atmosphere existed thatallows radio waves to follow the Earth's curvature. He invented vector analysis and wrote Maxwell’s equationsas we know them today. He showed how EM fields transformed to new inertial frames.
Hendrik Antoon Lorentz (1853-1928), a professor of physics at the University of Leiden, sought to explain theorigin of light by the oscillations of charged particles inside atoms. Under this assumption, a strong magneticfield would effect the wavelength. The observation of this effect by his pupil, Zeeman, won a Nobel prize for1902 for the pair. However, the Lorentz theory could not explain the results of the Michelson-Morley experiment.Influenced by the proposal of Fitzgerald, Lorentz arrived at the formulas known as the Lorentz transformationsto describe the relation of mass, length and time for a moving body. These equations form the basis forEinstein's special theory of relativity.
George Francis FitzGerald (1851-1901), a professor at Trinity College, Dublin, was the first to suggest that anoscillating electric current would produce radio waves, laying the basis for wireless telegraphy. In 1892FitzGerald suggested that the results of the Michelson-Morley experiment could be explained by the contractionof a body along its its direction of motion.Einstein
Working on Electrodynamics of moving bodies
Einstein read Lorentz’s book
10
“On the Electrodynamics ofMoving Bodies”
• Einstein had readLorentz’s book andworked for a few yearson the problem.• He did not believe
there should be onefixed frame.• He had a breakthrough,
“the step” in 1905.
While Einstein didn’t read much about the MM experiment, Lorentz did.
11
Einstein
Albert Einstein (1879-1955), one of the great geniuses of physics, grew up inMunich where his father and his uncle had a small electrical plant andengineering works. Einstein's special theory of relativity, first printed in 1905with the title "On the Electrodynamics of Moving Bodies" had its beginnings inan essay Einstein wrote at age sixteen. The special theory is often regardedas the capstone of classical electrodynamic theory.
Assistant patentagent, third classEinstein
Einstein did not get a Nobelprize for SR. He got one forcontributions to theoreticalphysics including thephotoelectric effect. Thecommittee did not think SR hadbeen proved correct until the1940s.
12
Velocity Addition• Einstein wanted the speed of light to be the
same in every frame.o This would work for EM equations.o It would agree with experiment.
→ Einstein did consider experiment but maybe not MichelsonMorley.
• But velocity addition didn’t make sense toanyone.o How could an observer in an inertial frame moving at
0.9c measure light to move at the same speed as wedo in our frame at rest? x ' = x + 0.9ct
v ' =dx '
dt=dx
dt+ 0.9c = v + 0.9c
13
• Einstein realized that bydiscarding the concept of auniversal time, the speed of lightcould be the same in every frame.o In going from one inertial frame to
another, both x and t transform.o The time is different in different
inertial frames of reference.o He derived the previously stated
Lorentz transformation fromrequirement that the speed of light isthe same in every inertial frame.
“The Step”
x ' = ! x " vt( )
t ' = ! t "v
c2x
#$%
&'(
! =1
1"v2
c2
14
Minkowski Space (1907)
• Hermann Minkowski proposed that specialrelativity could be best expressed in 4dimensionso With a new and unusual dot producto (Important that c=1 or else t→ct )
• We do not notice this geometry becausewe move at very small angles from the taxis.
aµ !bµ = axbx + ayby + azbz " atbt
angle =#x
#t=v
c!1 for humans
xµ = x, y, z,t( ) 4-vector has a greek subscribt
15
Future, past and light-cone• Humans move almost
directly along the time axis.o Photons move on the light-
coneo We can shoot other particles
close to the light-cone• Spacelike vectors have
positive normo (Dot product with itself)
• Timelike vectors havenegative norm.
• Lightlike vectors are on thelight-cone and have zeronorm.
Our path throughspace-time is
nearly on time axis
16
Example: Norm of a 4-vector
• Dot the 4-vector with itself
• Norm>0, spacelike• Norm<0, timelike• Norm=0, lightlike
x = x, y, z,ct( ) = 1,2,3,4( )
norm = x ! x = 1+ 4 + 9 "16 = "2
this is a "future pointing" timelike vector
17
4-vectors
• 4-vectors transform under rotations• The dot product of 4-vectors is a scalar
and is invariant under rotations• The norm of a vector can be positive,
negative or zeroo Spacelike, timelike or lightlike respectively
• Transforming to an inertial frame movingin the x direction is a rotation in the x-tplane.
18
Rotations in 4DRotation in the xy plane.
Rotation in the xt plane
! =v
c
" =1
1# ! 2
cosh$ = "
tanh$ = !
sinh$ = !"
A rotation in the xt plane is a “boost” along the x direction to another inertial frame
All 4-vectors transform the same way.
19
Rotation Symmetry in 4D
• Includes 3D rotationso Rotations in xy, yz, xz planes
• Plus symmetry that all the laws of physicsare the same in any inertial frame.o Rotation in xt plane is boost in the x directiono Rotation in yt plane is boost in the y directiono Rotation in zt plane is boost in the z direction
• All 4-vectors transform with the same matrix
20
Question: Michelson Morley
• Which of the following statements is false?
E) None of the above.
D) MM tends to show that the speed of light is aconstant independent of velocity of the observer.
C) MM used the interference of light waves.
B) MM should have detected the ether had it existed.
A) MM compares the speed of light in perpendiculardirections.
C
21
Effects of Lorentz Transformation
• Time dilationo A fast moving particle lives
much longer in the lab than inits rest frame.
• Lorentz contractiono Objects are shorter in the lab
frame in which they are movingthan they are in their restframe.
x ' = ! x " vt( )
t ' = ! t "v
c2x
#$%
&'(
! =1
1"v2
c2
22
Energy Momentum 4-vector
xµ = x, y, z,ct( ) keep c in case c ! 1
pµ = px, p
y, p
z,E
c
"#$
%&'
xµ ( xµ scalar giving distance to event
pµ ( pµ =p2 -E
2
c2= -m2
c2 scalar mass
E = mc2 when particle is at rest
pµ ( xµ =!p (!x ) Et phase in Quantum Mechanics
23
Special Relativity (1905-1908)
• Laws of physics are independent of whichinertial frame of reference we choose(Newtonian mechanics).• Extend this to speed of light.
o Speed of light the same in any inertial frame.o Michelson Morley experiment.
• It turns out we really live in 4 Dimensionso 3 space, 1 timeo Symmetry under rotations in 4D
24
Spark Chamber
• It was felt (at least by the Nobel committee)that there was no good experimentalevidence in favor of Special Relativity.• Some of the first clear evidence was from
the muons produced by cosmic rays.o While muons have a lifetime measured in their
rest frame of about 2 microsecons,o They can travel 100 km without decaying
25
A Boost in the x Direction(x, y, z,t) initial coordinates in 4D
transform to inertial system moving with velovity v along x axis
(x ', y ', z ',t ') transformed coordinates in 4D
x ' = ! x " vt( ) Lorentz Transformation
y ' = y
z ' = z
t ' = ! t "v
c2x
#$%
&'(
Lorentz Transformation
! =1
1"v
2
c2
depends on v
26
Example: Time Dilationx ' = ! x " vt( )
t ' = ! t "v
c2x
#$%
&'(
! =1
1"v2
c2
Assume the muon lifetime is τ. If it has avelocity of 0.999c, how far does it travelbefore it decays?start in a frame at rest with the muon
the muon is at the origin
assume the muon decays at time !
now boost to the frame in which the earth is at rest
x ' = " x # vt( ) = " v!( )
t ' = " !( )
" =1
1#v
2
c2
=1
1# .9992$
1
1# .998$
1
0.002$
1
0.002$ 22
E = mc2 in rest frame
E = "mc2 in earth frame
! =2"r
v=
2"r3/ 2
GM
G = 6.673#10-11 m3
kg s2
! GM
2"= r
3/ 2
r3=!
2GM
4" 2
r =!
2GM
4" 23 =
[(24)(3600)]2(6.67 #10-11)(6.0 #1024 )
4" 23 = 4.2 #107 m
r = 42000 km
27
Problem: Muon Lifetime
• The mean muonlifetime is about 2microseconds.Without the timedilation of SR, howfar would a muonmoving at avelocity of 0.999ctypically travelbefore it decays?
E) Not enoughinfo to answer.
D) 6000 m
C) 600 m
B) 60 m
A) 6 mx ' = ! x " vt( )
t ' = ! t "v
c2x
#$%
&'(
! =1
1"v2
c2
25
28
Problem: Muon Lifetime
• The mean muonlifetime is about 2microseconds.Without the timedilation of SR, howfar would a muonmoving at avelocity of 0.99ctypically travelbefore it decays?
E) Not enoughinfo to answer.
D) 6000 m
C) 600 m
B) 60 m
A) 6 mx = vt + x
0
x ' = ! x " vt( )
t ' = ! t "v
c2x
#$%
&'(
! =1
1"v2
c2
25
29
Tough Problem: Muon Lifetime
• The mean muonlifetime is about2 microseconds.According to SR,how far would amuon moving ata velocity of0.999c typicallytravel before itdecays? E) 1.3 m
D) 13 m
C) 130 m
B) 1300 m
A) 13000 mx = vt + x
0
x ' = ! x " vt( )
t ' = ! t "v
c2x
#$%
&'(
! =1
1"v2
c2
26
30
Tough Problem: Muon Lifetime
• The mean muonlifetime is about 2microseconds.According to SR,how far would amuon moving at avelocity of 0.9999ctypically travelbefore it decays?
E) 130000 m
D) 42000 m
C) 32000 m
B) 22000 m
A) 13000 mx = vt + x
0
x ' = ! x " vt( )
t ' = ! t "v
c2x
#$%
&'(
! =1
1"v2
c2
26
31
End Quiz 3
32
F =GMm
r12
2
a =v
2
r
! =2"r
v=
2"r3/ 2
GM
G = 6.673#10-11 m3
kg s2
PE = $GMm
r
KE =1
2mv
2
E = KE + PE
rearth$orbit
= 1.5X1011 m
mearth
= 6 #1024 kg
x ' = ! x " vt( )
t ' = ! t "v
c2x
#$%
&'(
! =1
1"v
2
c2
c = 3)108 m
s
aµ !bµ = axbx + ayby + azbz " atbtxµ = x, y, z,ct( )
pµ = px , py , pz ,E
c
#$%
&'(
xµ ! xµ = s
pµ ! pµ =p2
-E
2
c2= -m
2c
2
E= pc( )2
+ mc2( )
2
Erest" frame = mc2
pµ ! xµ =!p !!x " Et
33
Question: Units
• To make the laws ofphysics simple and tomake rotations in 4D mosteasily understandable,what should the speed oflight be?
E)1
D)1 cm/s
C)1 m/s
B)3X1010 cm/s
A)3X108 m/s
C
34
Example: Distance in 4D• Assume an observer is at rest at the origin. If the coordinates
of an event in that frame are (xe,ye,ze,cte) meters, at whattime did/will he first see the light from that event?light has to travel a distance of d = xe
2+ ye
2+ ze
2
the event happened at time te
light should then arrive at t = te +d
c
we could use the spacetime interval t ! te =xe
2+ ye
2+ ze
2
c
ct ! cte = xe2+ ye
2+ ze
2
ct ! cte( )2= xe
2+ ye
2+ ze
2
light interval is zero s = xe2+ ye
2+ ze
2! ct ! cte( )
2= 0
35
Problem: Distance in 4D
• Assume an observer is at rest atthe origin. If the coordinates of anevent in that frame are(x,y,z,ct)=(1500,0,0,1500) m, atwhat time did/will he first see thelight from that event?
E) t=5 µsec
D) t=10 µsec
C) t=10 msec
B) t=-5 µsec
A) t=-5 msec
x ' = ! x " vt( )
t ' = ! t "v
c2x
#$%
&'(
! =1
1"v2
c2
s = xe
2+ y
e
2+ z
e
2 " ct " cte( )2
27
36
Problem: Distance in 4D
• Assume an observer is at rest atthe origin. If the coordinates of anevent in that frame are(x,y,z,ct)=(0,6000,0,-1500) m, atwhat time did/will he first see thelight from that event?
E) t=25 µsec
D) t=15 µsec
C) t=10 µsec
B) t=5 µsec
A) t=-15 µsec
x ' = ! x " vt( )
t ' = ! t "v
c2x
#$%
&'(
! =1
1"v2
c2
s = xe
2+ y
e
2+ z
e
2 " ct " cte( )2
27
37
Question: Minkowski Space
• Assume an observer is at rest atthe origin and t=0. If thecoordinates of an event in thatframe are(x,y,z,ct)=(500,0,0,-1500) m,where does the event appear inthe observer’s space-timediagram?
E) Causallydisconnected
D) On Past lightcone
C) On Futurelight cone
B) past
A) future
Our path throughspace-time isnearly on time
axis
C
38
Energy and Momentum
pµc = pxc, p
yc, p
zc,E( )
pµc( ) ! pµc( )= pc( )2
- E2 = -(mc2 )2 scalar mass
E2 = pc( )
2+ mc
2( )2
E = mc2 in rest frame
pµc = 0,0,0,mc2( ) in rest frame
pµc = "#mc2 ,0,0,#mc2( ) boosting in x direction
pc = "#mc2
E = #mc2
39
Problem: Mass into Energy
• If we could convert 10-3 kgof matter into energy, howfast could we make a1000 kg car go?
E) 0.8 m/s
D) 8 m/s
C) 130 m/s
B) 13000 m/s
A) 420000 m/s
P.E. = mgh
K .E. =1
2mv
2
E = mc2
27
40
Space-Time• We see the universe
expanding• We can look back in time
to see the early universe• The universe is about 13.6
billion years old• Space-Time itself was
created in the big bang
41
Special Relativity• Minkowski space (3 space, 1 time)• Laws of physics invariant under translations and
rotations in Minkowski space• Boost to a new inertial frame of reference is just
a rotation in Minkowski space!o To see this, its important to get the units right (c=1)
• Some vectors in Minkowski spaceo (x,y,z;t)o (px,py,pz;E)
• This is all there is to Special Relativity.
42
Subsequent Developments
• By now, SR, MM areextremely well tested.o Don’t believe
crackpots on the web.• Einstein went on to
use curved 4Dspacetime to developGR, a new theory ofgravity that is testedfairly well too.
43
General Relativity• General Relativity puts us in curved space-time.• Gravity is equivalent to acceleration.• It is a well tested theory of gravity with new
phenomenao Black holeso Bending of lighto Cosmology
• It is not easy to make a theory of QuantumGravity• Curvature of space-time is related to mass and
energy density
Gµ!
= 8"Tµ!Einstein Equation
44
Black Holes
• Light cannot escape a blackhole (nothing can).• From the outside, we see a
collapse to R=2M (in someunits).• It takes light an infinite amount
of time to get out from R=2M.• If you fall inside, the black hole
collapses to a point!o But you get torn apart by tidal
forces before you see much.
45
The Electromagnetic Field
Magic,Invisible force fields…
Ancient Greek’sobserved rubbedamber attractingstraw…
Electrum: Latinfor amber
46
New Physics: E&M
• Mechanics was relatively simple.o Explained the motion of bodieso Didn’t explain existence of matter, light, elements,
chemistry…o Didn’t explain rather minor effects of static electricity and
magnetism.• Studies of these effects changed the world and our
understanding of Physics.• They led to our understanding of all the areas listed
above.
47
Static Electricity with Rods
• We can see the existence of charge rathersimply.• By rubbing some objects we can transfer
charge between them leaving one objectwith positive charge and another withnegative.• Charge somehow creates Electric Field• There is a force on a charge in an Electric
Field.
48
Electrons
• An electron is a relatively light, fundamentalparticle with a charge of -1.60X10-19
Coulombs.• It is the particle that is transferred to charge
up rods…• Bound electrons determine the properties of
atoms and molecules.• It is the particle that moves in wires,
transistors, ICs, radios, computers… tomake those devices work.
49
Van de Graaff
• With a Van de Graaff generator we canreally charge up an object.• Electrons are moved from one object to
another building up a large charge and alarge electric field.
50
Van de Graaf
• Can produce a largevoltage but not muchcurrent.o Current = charge per
second.• Need very dry
conditions to do this.
51
Electric Field Lines
• We wish to see the nature of theElectric field.• We can see the field lines.
o Field lines start on positive chargesand end on negative charges.
• It is a vector field• The field lines are conserved,
only beginning and ending oncharges
52
Field Lines
• Field Lines start on+ charges and endon - charges.• It is a vector field.
!E!x,t( ) field is a function of position and time
vector field!E !d!S = qenclosed"" field strength (number of field lines) is proportional to charge
field lines are continuous
53
What is a Field
• A field is a function of position and time.• A vector field is a vector function of
position and time.o
o
• These fields are natural phenomena thatwe must understand
!E x, y, z,t( )!B x, y, z,t( )
54
We Live in an Electromagnetic World
• E field binds atoms togethero Binds atoms into moleculeso Metals, semiconductors…o Biologyo (nuclear force determines what elements we have)
• EM waveso Light is our primary energy sourceo Radio waveso EM waves have both E and B fields.
55
Question: Fields
• Whatpropertiesdo boththe E andB fieldshave?
E)All of the above.
D)A and B.
C)They are present in EM radiation(light…).
B)They are both vectors havingmagnitude and direction.
A)They can depend on position and time.
C
56
Magnetic Field Lines in 2D and 3D• There are also magnetic field lines.• Its also a vector field.• Magnetic field lines circulate around current
o Current is a flow of electric chargeo Moving charges generate loops of field
Field lines due to multipleloops of current
!B!x,t( ) field is a function of position and time
vector field!B !d!S = 0"" as many field lines enter any volume as exit it
field lines are continuous
57
Maxwell’s EquationsFields obey equations.
These equations fully describe E&M.
Charge density
Current density
Force on an electron
VectorDerivative
!! =
""x,""y,""z
#$%
&'(
Equations are fairly complex with 3 constants.
58
Maxwell’s Equations (1864)
• Were the result of many studies byCoulomb (1784), Ampere (1820), Faraday(1831), Gauss (1837) and others.• Unified electricity and magnetism• Predicted EM waves• His 20 equations were later rewritten by
Heaviside in vector notation as 4equations (1887).
59
Static Charge Gives Static E-field
Charge density
DivergenceOf E
!! "!E =
#Ex
#x+#E
y
#y+#E
z
#z
E-field lines start on positive charges,And end on negative charges.Field lines are continuous.E-field is proportional to the number of field lines per unit area.
60
E Field of Point Chargeor Outside a Charged Sphere
!E =
1
4!"0
q
r2r field of a point charge
!F = q2
!E force on a charge in a E-field
#e = #1.60 $10#19 Coulombs charge on an electron
"0 = 8.85 $10#12 C2
N m2=
C2s
2
kg m3
The SI unit of charge is the Coulomb.All I can say is that it is defined in a stupid way.
61
Direction of the E-field
• The field of a point charge or a chargedsphere points radially outward from thecharge.• We denote this by the unit vector in the
radial direction• We may use the unit vectors to
give the direction of vectors in Cartesiancoordinates.
r
x, y. z
62
Problem: E-field
• If a small chargedsphere withq=3.0X10-12
Coulombs isplaced at theorigin, what is theElectric field at(x,y,z)=(0,0.2,0)?
E)
D)
C)
B)
A)
!E =
1
4!"0
q
r2r
!F = q2
!E
#e = #1.60 $10#19 C
"0 = 8.85 $10#12 C2s
2
kg m3
0.67kg m
s2 Cy
!0.67kg m
s2 Cy
!0.67kg m
s2 Cx
!0.67kg m
s2 Cz
0.67kg m
s2 Cx
28
63
Problem: E-field
• If a small chargedsphere withq=-3.0X10-12
Coulombs isplaced at theorigin, what is theElectric field at(x,y,z)=(0,0,-0.5)?
E)
D)
C)
B)
A)
!E =
1
4!"0
q
r2r
!F = q2
!E
#e = #1.60 $10#19 C
"0 = 8.85 $10#12 C2s
2
kg m3
0.67kg m
s2 Cy
0.67kg m
s2 Cz
!0.11kg m
s2 C
!z
!0.67kg m
s2 Cz
28
0.11kg m
s2 C
!z
64
Lorentz Force Equation
Force on an electron
with charge -e
Force is parallel to
E-field
Force is perpendicular
to B-field and v
65
Example: Force between Point Charges
• Assumeq1=q2=1µCoulombat adistance of0.2 m.
!E =
1
4!"0
q1
r2r field of a point charge
!F = q2
!E force on a charge in a E-field
#e = #1.60 $10#19 Coulombs
"0 = 8.85 $10#12 C2s
2
kg m3
!F =
1
4!"0
q1q2
r2r opposites attract
!F =
1
4! 8.85 $10#12 C2s
2
kg m3
%&'
()*
10#6C( )
2
0.2m( )2r =
1
4! 8.85s
2
kg m
%&'
()*
0.2( )2
r = 0.22kg m
s2
66
Problem: Force on point charge
• A charge q1=3X10-6 C islocated at the origin. Whatis the force on a chargeq2=5X10-6 C which islocated at (x,y,z)=(0,0.05,0)m?
E)
D)
C)
B)
A)
!E =
1
4!"0
q
r2r
!F = q2
!E
#e = #1.60 $10#19 C
"0 = 8.85 $10#12 C2s
2
kg m3
2.7kg m
s2x
54kg m
s2x
54kg m
s2y
!54kg m
s2x
!54kg m
s2y
29
67
Problem: Force on point charge
• A charge q1=2X10-6 C islocated at the origin. What isthe force on a chargeq2=5X10-6 C which is locatedat (x,y,z)=(0,0,-0.05) m?
E)
D)
C)
B)
A)
!E =
1
4!"0
q
r2r
!F = q2
!E
#e = #1.60 $10#19 C
"0 = 8.85 $10#12 C2s
2
kg m3
2.7kg m
s2y
36kg m
s2y
!36kg m
s2y
!36kg m
s2z
36kg m
s2z
29
68
Electric Potential
• Since electric force fallswith r2 just like gravity,the potential energycan just be writtendown by analogy.• PE divided by q2 is the
electric potential whichhas units known asVolts.
!F =
1
4!"0
q1q2
r2r force between two charges
PE =1
4!"0
q1q2
r potential energy
# r( ) =1
4!"0
q1
r electric potential due to q1
69
Some Electrical Terms
• Charge: a new fundamental quantity (C)o Charge Density: Charge per unit volume
• Current: Charge per second flowingthrough some plane (A=C/s).• Voltage: Electric Potential Energy per unit
charge.• Power=(Voltage)(Current)
70
Capacitor
• A capacitor can storecharge• Attraction between
positive and negativecharges helps keep thecharge• Energy is stored in the
field
71
Parallel Plate Capacitor
• Electric field is nearlyconstant betweenplates.• Q=VC defines
Capacitance C
C =A!
0
d
72
No Magnetic ChargesThere are no
magnetic charges.
DivergenceOf B
!! "!B =
#Bx
#x+#B
y
#y+#B
z
#z
B-field lines do not start or end.Field lines are continuous.B-field is proportional to the number of field lines per unit area.
73
Moving Charges give B-field Loops
Current density
Curl of B
!! "!B =
#Bz
#y$#B
y
#z,#B
x
#z$#B
z
#x,#B
y
#x$#B
x
#y%&'
()*
B-field lines loop around electric currents.Right thumb along current, B-field along direction fingers curl.Field lines are continuous.
74
B Field of a Current in a Long Wire
!B =
µ0I
2!r" field of a long wire with current I
!F = I
!l #!B( ) force on wire of length l with current I in B-field
µ0 = 4! #10$7 Tesla m/A
Current (I) is the flow of electric charge.The Ampere is one Coulomb per second.The Ampere is defined to give a force of2X10-7 Newton per meter between parallelwires at a distance of one meter, eachhaving one Ampere of current.
A current produces loops of magnetic field.Use the right hand to get the field direction.
75
Question: B-field Lines
• Magneticfieldlines:
E)Start and end on acceleratingcharges.
D)Form circles around movingcharges.
C)Form circles around charges.
B)Start and end on moving charges.
A)Start and end on charges.
C
76
Example: B-field of a long wire
• A very long wirehas +1000 A ofcurrent passingthrough it alongthe x axis. Whatis the magneticfield at a point at(x,y,z)=(0,0,0.1)meters?
!B =
µ0I
2!r"
!F = I
!l #!B( )
µ0 = 4! #10$7 Tesla m/A
!B =
µ0I
2!r" =
4! #10$7 Tesla m/A( ) 1000A( )
2! 0.1 m( )"
!B = 2 #10$3" Tesla!B = $2 #10$3
y Tesla using right hand
This is larger than the field of the earth.For large fields, electromagnets have manyturns of wire in a coil.Iron can multiply fields by a large number.
77
Problem: Field of a long wire
• A very long wire has I=+5000A of current passing through italong the y axis. What is themagnetic field at a point at(x,y,z)=(0,0,0.02) meters?
E)
D)
C)
B)
A)
!B =
µ0I
2!r"
!F = I
!l #!B( )
µ0= 4! #10
$7 Tesla m/A
!0.004 x Tesla
0.05 x Tesla
1.0 y Tesla
0.05z Tesla
!0.05z Tesla
x
y
z
I
30
78
Problem: Field of a long wire
• A very long wire has I=+5000A of current passing through italong the x axis. What is themagnetic field at a point at(x,y,z)=(0,0,0.02) meters?
E)
D)
C)
B)
A)
!B =
µ0I
2!r"
!F = I
!l #!B( )
µ0= 4! #10
$7 Tesla m/A
!0.05 x Tesla
0.05 x Tesla
0.05 y Tesla
0.05z Tesla
!0.05
!y Tesla
x
y
z
I
30
79
Simple Electromagmet
• Solenoid• Nearly constant B-field
inside.• B=µ0nI• Field can be greatly
enhanced by wrappingcoil around iron.
80
EXB
• Force on a moving charge in a magneticfield
81
Magnet in Tube
• Permanent magnet induces eddy currentsthat try to keep the magnet from moving• Lenz’s law• Permanent magnets are caused by
interesting Quantum Mechanical phenomena.• Attractive and repulsive forces between
magnets.• Forces on currents.• Can be used in motors and generators.
82
Swinging Plates in B Field
• Another illustration of Lenz’s lawo EM always tries to oppose change
83
Generating Electric PowerTime varying
B-fieldCurl of E
!! "!E =
#Ez
#y$#E
y
#z,#E
x
#z$#E
z
#x,#E
y
#x$#E
x
#y%&'
()*
E-field lines loop around changing B-fields.These lines do not start or end on charges.Field lines are continuous.We use this to generate electricity.
84
Motor and Generator
• A time varying magnetic field generates anelectric field.o A time varying flow of B-field through a loop of
wire can be used to generate electric power.o Use some source of energy to turn a generator
and produce electric power.• Similarly, we can use electric power to
generate a magnetic force to make a motor.
85
Question: Generator
• We cangenerate electricpower by:
E)All of the above
D)Both A and B
C)Passing a current through a coilof wire.
B)Turning a coil of wire in a E field.
A)Turning a coil of wire in a B field.
C
86
End Quiz 4
87
B-field loops from varying E-field
Curl of B
!! "!B =
#Bz
#y$#B
y
#z,#B
x
#z$#B
z
#x,#B
y
#x$#B
x
#y%&'
()*
Time varying E-field
B-field lines loop around changing E-fields.Field lines are continuous.Effectively keeps flow of (displacement) current through capacitor.
88
Electromagnetic Waves
• Changing E field generates Bfield perpendicular.• Changing B field generates E
field perpendicular.• Transverse vector wave
o E, B, and direction ofpropagation are allperpendicular.
• Two possible polarizations.
89
Question: Light Wave
• If a light waveis moving inthe y direction,in whichdirection canthe Electricfield point?
E)Any of the above
D)A or C
C)z direction
B)y direction
A)x direction
C
90
Polarized Light
• Select light polarized in the x direction• Rotate second polarizer to get maximum
transmission or zero transmission• Corn Syrup rotates polarization
91
But Light Wave is Composed of Quanta
• Diffraction is a property of waves.• If we turn down the intensity of the
light we detect single photonshitting the detector with E=hf.• They slowly fill up the diffraction
pattern when the distribution issummed up over time.o Indicates that single photon interferes
with itself.o Probability (amplitude) waves
interfere→ We will study this later.
92
Question: Diffraction
• The diffraction oflight is evidencethat light:
E)comes from a laser
D)is a transverse wave
C)has two polarizations
B)is a wave
A)is composed of particles
C
93
Radio Waves
• Static Charges cause static E-fields.• Constant currents (of charge) cause
static B-fields.• Accelerating charges cause EM
radiation.o Put oscillating signal onto antennao Radiates waves with the frequency of
the signalo E-field polarized in vertical direction.
• Use another antenna to pick upE-field in waves.
94
Maxwell’s EquationsFields obey equations
Charge density
Current density
Force on an electron
VectorDerivative
!! =
""x,""y,""z
#$%
&'(
Equations are fairly complex with 3 constants.But they are very practical for real problems (for slow humans).
95
Heaviside, Lorentz, FitzgeraldOliver Heaviside (1850-1925) was a telegrapher, but deafness forced him to retire and devote himself toinvestigations of electricity. He became an eccentric recluse, befriended by FitzGerald and (by correspondence)by Hertz. In 1892 he introduced the operational calculus (Laplace transforms) to study transient currents innetworks and theoretical aspects of problems in electrical transmission. In 1902, after wireless telegraphyproved effective over long distances, Heaviside theorized that a conducting layer of the atmosphere existed thatallows radio waves to follow the Earth's curvature. He invented vector analysis and wrote Maxwell’s equationsas we know them today. He showed how EM fields transformed to new inertial frames.
Hendrik Antoon Lorentz (1853-1928), a professor of physics at the University of Leiden, sought to explain theorigin of light by the oscillations of charged particles inside atoms. Under this assumption, a strong magneticfield would effect the wavelength. The observation of this effect by his pupil, Zeeman, won a Nobel prize for1902 for the pair. However, the Lorentz theory could not explain the results of the Michelson-Morley experiment.Influenced by the proposal of Fitzgerald, Lorentz arrived at the formulas known as the Lorentz transformationsto describe the relation of mass, length and time for a moving body. These equations form the basis forEinstein's special theory of relativity.
George Francis FitzGerald (1851-1901), a professor at Trinity College, Dublin, was the first to suggest that anoscillating electric current would produce radio waves, laying the basis for wireless telegraphy. In 1892FitzGerald suggested that the results of the Michelson-Morley experiment could be explained by the contractionof a body along its its direction of motion.Einstein
Working on Electrodynamics of moving bodies
Einstein read Lorentz’s book
96
Field Equations in 4D
• Aν is a 4-vector field• xν is the coordinate 4-
vector• jν is the charge-current
4-vectoro Souce of EM field
• c=1• Heaviside Lorentz units• No constants• Wave equation with source
term.
One equation in 4-vector fieldwith a 4-vector source
(charge-current density)replaces Maxwell’s equations
Einstein notation: sum over repeated index
Field tensor
4-vector equation
97
Question: EM in 4D
• The 4Dequationbelow:
E)All of the above.
D)Both A and B
C) Replaces all 4, 3D Maxwell equations.
B)Is a 4-vector equation.
A)Has a source term jµ
C
98
And there was Light• Visible light, IR, UV, microwaves, radio waves, x-rays,
γ-rays are all Electromagnetic waves.• Light and Electromagnetic Field are the same thing.• Electric Field comes from charges.• Magnetic Field comes from moving charges.
o Really just Electric field transformed to moving coordinatesystem.
o Symmetry of space-time.• Vector Field• EM field is also composed of photons
Aµ
!
!x"
Fµ"
= jµ
Fµ!
="A
!
"xµ
#"A
µ
"x!
99
EM Field Does Amazing Things• Changing magnetic field has an effect
on charged particles which are outsidethe field.• While the B field doesn’t touch the wire,
the potential A-field does.
!B
!t
Wire
current
Generator with noB field touching coil
100
Field Equations in 4D
• EM equation is quite simple in4D with the right units.
• Wave equation with sourceterm.
• Probably the simplest vectorfield theory possible
• A strange new symmetry called“gauge symmetry”.
• Einstein wanted to unify E&Mwith General Relativity.o He failedo But Kaluza did it and it may have
some truth to it.
One equation in 4-vector fieldwith a 4-vector source
(charge-current density)replaces Maxwell’s equations
Einstein notation: sum over repeated index
Field tensor
101
Question: 3D or 4D
• Which is false?
E)None of the above.
D)The simplicity of the 4D equations is further evidence thatwe live in 4D.
C)The SI units chosen for Coulombs, Amperes and fields werenot chosen well.
B)Its easier to use the 3D Maxwell equations to solve realproblems like capacitors and electromagnets.
A)Maxwell’s equations in 4D are relatively simple.
C
102
Kaluza-Klein Theory• Kaluza (1919) postulated an invisible fifth
dimension.• Klein said it was invisible because it was
curled up into a small circle.• There is a circle at every point in 4D.
o The circle is very small.• There is a symmetry that we can be
anywhere on the circle U(1).• And there is a symmetry under rotations in
5D.• E&M may just be GR in this invisible
dimension.• In modern theories there are more extra
dimensions to accommodate the otherfields.
o There is no proof of extra dimensions yet.
103
Quantum MechanicsProbability Amplitude Waves
for all particles
Probability(
!r ,t) = ! (
!r ,t)
2 ! is a "probability amplitude"
electron probabilityin H states
104
The Other Big Surprise of 1905
• Physics described mechanics, gravity,E&M fairly well.• But there were many things that were not
described.o Why all the elements and their properties?o Atomic emission energies.o Photoelectric effect (Einstein 1905).
• There was still a lot of physics to learn.
105
Photoelectric Effect• Photons are particles of light with E=hf.
o We need high frequency light to knock electrons out ofmetal plate.
• EM waves would not have this property.o Waves of sufficient intensity would knock electrons out
independent of frequency.
106
Problems with Classical Physics
• Too many states in the EM field.o All energy would be (rather instantly) radiated into EM
field leaving everything at zero temperature.• Atomic electrons should radiate and fall into
nucleus.o Atoms would not exist.
• Atoms observed to emit light at quantizedwavelengths.• Light hitting metals could eject electrons from
the metals as if E=hf.o h is Plank’s constant (small number).
107
Question: Classical Physics
• Which effect wasunderstood inclassical physics?
(before quantummechanics)
E) None of the above.
D) Size of atoms
C) Photoelectic effect
B) Emission of light atquantized frequency by atoms
A) Interference of light waves
C
108
Electromagnetic Waves?• EM waves travel at speed of light
o Maximum possible speed• Waves can interfere as evidenced by
diffraction.• However light is always detected in quanta
(particles) with E=hf.
o Einstein also published photoelectric effect in 1905o Particles are called photons
• We can get diffraction patterns using onephoton at a time!
• Each photon is described by a wave.• Quantum Mechanics!
! =h
2!= 1.05 "10#20
kg m2
s
109
Two Slit Diffraction
• Diffraction is a wave phenomenon.• With 2 slit diffraction, we can get several
maxima and minima.
110
Diffraction•Turn down beam intensityso there is only one photonin the apparatus at a time.•Always detect one photon.•Probability distributionshows same diffractionpattern!
111
A-field Satisfies Wave Equation
!2Aµ
!x"
2= # jµ
Prob. ! A2
• Solutions are waves• E and B are just derivatives of A• Energy in field proportional to squares
of fields.• Probability to find a photon goes like
square of fields.• Probability Amplitude Waves
o Amplitudes can interfereo Probability gives diffraction patterno Yet we detect quantized photonso With E=hf
Sum for ν=1,2,3,4
112
Other Particles alsoProbability Amplitude Waves• Electrons, quarks, neutrinos…• Waves of probability amplitude ψ
o P=|ψ|2
ψ complex wave function→ Complex number: a+ib→
• Satisfy Schrödinger equation (wave)i = !1
!!2
2m"2#
"x,t( ) +V
"x( )#
"x,t( ) = i!
$#
$t
H# = i!$#
$t
113
Electron Diffraction
• Use crystal model too• Electron diffraction with 2 slits is a very
difficult experiment• We can more easily see electron
diffraction from a crystal with many atomsarranged in a regular array.• Strong evidence that electrons are also
waves.
114
Probability Amplitudes can Interfere
• Amplitudes for particles to arrive at onplace from two different sources caninterfere.
• Describes diffraction and various otherexperiments well.• If one observes which slit the particle went
through, the diffraction pattern is spoiled.
! =!1+!
2
prob. = !1+!
2
2
115
Question: Particle Wave Duality
• How do weexplain the wavebehavior ofelectrons andphotons at thesame time as thefact that they arealways detectedas quantizedparticles. E) Waves are always made of
particles.
D) Particles are always waves.
C) Probability to find a particleproportional to wave.
B) Probability to find a particle isproportional to square of a wave.
A) Sometimes particles arewaves and sometimes not.
C
116
Photons
• Photons have zero mass• Energy is related to
frequencyo E=hf according to Plank
and Einsteino Frequency is related to
wavelength: λf=c
• E=pc since mass is zero• λ=h/p relation between
momentum andwavelength.
E = hf
! =h
p
h = 2"!
117
Wavefunctions for Electrons…
• Wavefunctions for matter particles can be purewaves with one frequency and one wavelength.
• Momentum related to wavelength• Energy related to frequency• Particle’s probability is completely spread out
over all space.
! !p= e
i!p"!x#Et( )/"
ei$= cos$ + i sin$
i = #1
P = !2
= 1
E = hf
! =h
p
h = 2"!
118
Localize Electrons with Wavepackets
• Wavepackets can be localized by combining waveswith different wavelengths.o Principle of superposition.
• To localize well, we need a wide spectrum ofwavelengths.• This gives rise to an uncertainty principle which is a
basic property of waves.o Math too complicated to show here.o But result is simple.
!p!x "!
2
119
Fast Fourier Transform
• We can describe a pulse either as g(t) oras h(f).
120
Wave Nature ImpliesUncertainty Principle
• Heisenberg uncertainty principle
• We cannot know a particle’s momentumand position at the same time.• This is a basic property of waves.
!p!x "!
2
!k!x "1
2
p = !k
121
Question: Uncertainty
• If we try to localizean electron’sposition very well,what other propertymust become veryuncertain.
E) None of the above
D) All of th above
C) Its momentum
B) Its mass
A) Its charge
C
122
Electron Volts (Unit of Energy)• One electron Volt is the energy a particle with
charge e gets by going through one Volt ofElectric potential difference.o eV is a convenient atomic physics unit.o It is quite convenient for accelerators.
• MeV (million electron Volts) is a good unit fornuclei and accelerators.• eV, keV, MeV, GeV, TeV
o 1, 1000, 1000000, 1000000000, 1000000000000
123
Estimate Hydrogen Energyusing Uncertainty Principle
E =1
2mv
2!
e2
4"#0r Energy=KE+PE
E =p
2
2m!
e2
4"#0r use momentum mv
$p$r % ! uncertainty
p % $p min. p is $p
r % $r min. r is $r
pr % !
E =p
2
2m!
e2p
4"#0! replace r
dE
dp=p
m!
e2
4"#0!= 0 min E
p =me
2
4"#0!
solve
E =p
2
2m!
e2p
4"#0!
E =me
4
2 4"#0!( )
2!
me4
4"#0!( )
2
$ =e
2
4"#0!c
=1
137 EM constant
E = !me
4
2 4"#0!( )
2= !
$2mc
2
2
E=-13.6 eV, the right answer
124
Atoms
• Atomic energies and sizes are set byuncertainty principle.o E=-13.6 eV Hydrogen ground state energyo a0=0.53X10-10 m Bohr radius
• Comes from wavefunction for electron.o Can’t make both p and r small
125
Another Uncertainty Principle
• Can violate energy conservation for shortperiods of timeo Quantum tunnelingo Virtual particles
• Stable states have definite energieso ΔE=0o Δt=infinityo Eigenvalue Solution to wave equationo If E is measured, must get one of the eigenvalues
!E!t "!
2
!!2
2m"2#
"x,t( ) +V
"x( )#
"x,t( ) = E#
"x,t( )
H# = E#
126
Schrödinger Equation
• The Schrödinger Equation is a wave equation.• Equation for stationary states replaces time
derivative by energy.
• Solutions exist only for some energies.o Quantized energies.o Only these energies can be measured.
!!2
2m"2#
"x,t( ) +V
"x( )#
"x,t( ) = E#
"x,t( )
H# = E#
127
Hydrogen Energies• A solution to the Schrödinger equation
gives the Hydrogen energies in terms ofquantum numbers.
!n!m
Definite energy wavefunction
n = 1,2,3,... Principle quantum number for Hydrogen
! = 0,1,...,n "1 integer: total angular momentum quantum number
"! # m # ! integer: z component of angular momentum is m"
En!m
= "$ 2mc
2
2n2 energies only depend on n (+ small corrections)
En!m
= "13.6
n2
eV
$ =1
137 dimensionless
128
Energy “Eigenstates”
• Atoms have definiteenergy states.• Ground state lives
forever.• Excited states decay
o Have some small Ewidth
!E!t "!
2
129
Question: Quantized Energies
• We cancalculatethequantizedenergiesallowed foratoms by:
E) Using the science of Chemistry.
D) Solving Maxwell’s equations.
C) They can only be determined byexperiment.
B) Solving the Schrödinger equation.
A) Using the uncertainty principle.
C
130
Diffraction Grating• A diffraction grating is a large array of evenly spaced
scratches,• Light of one wavelength from each scratch will interfere
constructively at the correct angle.• We can see the energies of light from atomic decays
using a diffraction grating.
Emission spectrum
Absorption spectrum
131
Question: Energy Spectra
• We canusediffractiongratingsto
E) All of the above
D) A and B
C) Separate light of different wavelengths
B) Measure absorption energies of atoms
A) Measure emission energies of atoms
C
132
Electrons are Spin 1/2• Internal angular momentum• Two spin states (up or down) for any axis
o Spin is a new internal coordinate for electrons• Identical particles
o Fermions for spin 1/2o No two particles in the same stateo Atomic levels fill upo Matter particles
133
Other Phenomena of QuantumMechanics
• Atomic energy levels, nuclei…• Atomic decay rates• Permanent magnets• Metallic conductors• Energy bands in crystals• Semiconductors• Tunneling• Superconductivity• lasers
134
Photons are Spin 1 (vector)
• Internal angular momentumo The two polarizations
• Identical particleso Bosons for integer spino Can be emitted or absorbed by charged
particleso Many particles in the same stateo Energy particleso EM Force carriers
135
Question: Identical Particles
• Which is true about the states of identical particles?
E) None of the above
D) No two particles in the same state
C) No two identical Bosons in the same state
B) No two identical Fermions in the same state
A) No two identical particles in the same state
C
136
Gauge Symmetry
• A strange symmetry of wave functions and EM field• Extremely important in our theories of EM, and other
fields• We can change the phase of the wavefunction by a
different amount at every point in space-time.o We must make a related change in the EM potential Aµ at
every point.• Gauge symmetry keeps photon massless.• Is this really some space-time symmetry in extra
dimensions?o A curled up extra dimension can give EM gauge theory
137
Relativity and Quantum Mechanics
• It was hard to merge relativity and Quantum Mechanics.• Physicists were puzzled by the energy relation in relativity.• When they solved the wave equations they got solutions
with both positive and negative energy.o And they could show they had to keep the negative energy
solutions.
• Dirac made a big effort to find an equation that was linearin the energy (operator).o He found a matrix equation that represents electrons wello But it still had negative energy solutions.
E = ± pc( )2
+ mc2( )2
138
Relativistic Quantum Mechanics
• The wave equations of E&Malready represent therelativistic equation forphotons well.o They just need to be quantized
• Dirac’s equation is good forelectrons.• But it has negative energy
solutions.
!
!x"
Fµ"
= jµ
Fµ!
="A
!
"xµ
#"A
µ
"x!
(! µ "
"xµ
+m)# =0
139
Dirac Equation
• γ is a 4X4 matrix• ψ is a 4 component spinor
o 2 spin stateso Times 2 energy solutions, + and -
• Automatically includes spin.• Negative energy states reinterpreted.
(! µ "
"xµ
+m)# =0
140
Antiparticles• Negative energy could be reinterpreted as
positive energy moving backward in time.
o Time reversal symmetry in solutions• Particles that move backward in time, appear to
us as having the opposite charge…o Gives us pair production of e+e-
o Pair annihilation• Dirac predicted Antimatter.
! !
p= e
i!p"!x#Et( )/"
141
Quantum ElectroDynamics
• QED is quantum field theory of electrons and photons.• Written in terms of electron field ψ and photon field Aµ.• Fields are quantized.
o Able to create or annihilate photons with E=hf.o Able to create or annihilate electron positron pairs.
• Gauge (phase) symmetry transformationo Only gauge theory we could have with 1 photon!
• Extremely good calculational accuracyo 15 digits
!
!x"
Fµ"
= jµ
Fµ!
="A
!
"xµ
#"A
µ
"x!
(! µ "
"xµ
+m)# =0
142
Relativistic Quantum Field Theory• Dirac Equation: Relativistic QM for electrons
o Matrix (γ) eq. Includes Spin, Antiparticleso Negative E solutions understood
• Quantum Electrodynamicso Field theory for electrons and photonso Rules of QFT developed and tested
→ Lamb Shift→ Vacuum Polarization
o Example of a “Gauge Theory”
• Weak and Strong Interactions understood as gauge theorieslike QED around 1970
0mcx
µ
µ!
"# $% & =' (") *
Feynman Diagrams
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