Physics 2D Lecture Slides
Lecture 8
Jan. 28, 2008
Velocity Transformation Perpendicular to S-S’ motion
v
dy ' = dy, dt ' = ! (dt " 2 dx)
c
dy '
dy
'
uy =
=
v
dt '
! (dt " 2 dx)
c
divide by dt on RHS
uy
'
uy =
v
! (1 " 2 ux )
c
There is a change in velocity in the
direction # to S-S' motion !
Similarly
Z component of
Ant' s velocity
transforms as
uz
'
uz =
v
! (1 " 2 ux )
c
Inverse Lorentz Velocity Transformation
Inverse Velocity Transform:
ux ' + v
ux =
vux '
1+ 2
c
As usual,
u 'y
replace
uy =
v '
v⇒-v
! (1 + 2 ux )
c
'
uz
uz =
v '
! (1 + 2 ux )
c
Relativistic Kinetic Energy & Newtonian Physics
Relativistic KE K = ! mc 2 " mc 2
# Remember Binomial Theorem
&
%
(
nx
n(n-1)
% for x << 1; (1+x) n = (1+
+
x 2 +smaller terms) (
%$
('
1!
2!
"
1
2
# u2 &
1 u2
) When u << c, %1- 2 (
* 1+
+ ...smaller terms
2
2c
$ c '
1 u2
1 2
2
2
so K * mc [1 +
]
"
mc
=
mu (classical form recovered)
2
2c
2
Total Energy of a Particle E = ! mc 2 = KE + mc 2
For a particle at rest, u = 0 " Total Energy E= mc 2
E=mc2 ---> Sunshine Won’t Be Forever !
Q:
Solar Energy reaches earth at rate of 1.4kW per square
meter of surface perpendicular to the direction of the sun.
by how much does the mass of sun decrease per second
owing to energy loss? The mean radius of the Earth’s
orbit is 1.5 x 1011m.
r
• Surface area of a sphere of radius r is A = 4πr2
• Total Power radiated by Sun = power received by a sphere whose
radius is equal to earth’s orbit radius
E= mc2 ---> Sunshine Won’t Be Forever !
Total Power radiated by Sun
= power received by a
sphere with radius equal to
earth-sun orbit radius( r in figure)
r
sun
Plost
=
sun
Plost
P Earth
incident
Aearth! sun =
A
= 4.0 # 1026 W
P Earth
incident
A
2
3
2
11 2
4" rearth!
=
(1.4
#
10
W
/
m
)(4
"
)(1.5
#
10
)
sun
So Sun loses E = 4.0 # 1026 J of rest energy per second
E 4.0 # 1026 J
9
Its mass decreases by m = 2 =
=
4.4
#
10
kg per sec!!
8 2
c
(3.0 # 10 )
If the Sun's Mass = 2.0 # 1030 kg So how long with the Sun last ?
One day the sun will be gone and the solar system will not be a
hospitable place for life
E = ! mc 2 " E 2 = ! 2 m2 c 4
Relationship between P and E
p = ! mu " p 2 c 2 = ! 2 m2u 2 c 2
" E 2 # p 2 c 2 = ! 2 m2 c 4 # ! 2 m2u 2 c 2 = ! 2 m2 c 2 (c 2 # u 2 )
2 4
m2 c 2 2
m
c
2
2
2
2 4
=
(c
#
u
)
=
(c
#
u
)
=
m
c
2
2
2
u
c #u
1# 2
c
E 2 = p 2 c 2 + (mc 2 )2 ........important relation
For particles with zero rest mass like photon (EM waves)
E
E= pc or p =
(light has momentum!)
c
Relativistic Invariance : E 2 # p 2 c 2 = m2 c 4 : In all Ref Frames
Rest Mass is a "finger print" of the particle
Mass Can “Morph” into Energy & Vice Versa
• Unlike in Newtonian mechanics
• In relativistic physics : Mass and Energy are the same
thing
• New word/concept : MassEnergy , just like SpaceTime
• It is the mass-energy that is always conserved in every
reaction : Before & After a reaction has happened
• Like squeezing a balloon : Squeeze here, it grows
elsewhere
– If you “squeeze” mass, it becomes (kinetic) energy &
vice verca !
• CONVERSION FACTOR = C2
• This exchange rate never changes !
Relativistic Force & Acceleration
r
p=
r
mu
1 ! (u / c)2
r
= " mu
Relativistic
Force
And
Acceleration
Reason why you cant
quite get up to the speed
of light no matter how
hard you try!
r
%
r dpr d "
mu
F=
= $
'
dt dt $# 1 ! (u / c)2 '&
(
m
F=*
*
2
*) 1 ! (u / c)
+
- du : Relativistic Force
3/ 2 dt
-,
r
r du
Since Acceleration a =
,[rate of change of velocity]
dt
(
)
r
3/ 2
F
r
2
(1 ! (u / c) +
. a=
,
m)
r
Note: As u / c / 1, a / 0 !!!!
Its harder to accelerate when you get
closer to speed of light
Linear Particle Accelerator : 50 GigaVolts Accelating Potential
r
3/ 2
r eE
2
a = "#1 ! (u / c) $%
m
PEP-II accelerator schematic and tunnel view
Magnetic Confinement & Circular Particle Accelerator
r Br
V
Classically
v2
F=m
r
v2
qvB = m
r
B
Fr
r
dp d(! mu)
du
=
=!m
= quB
dt
dt
dt
du u 2
=
(Centripetal accelaration)
dt
r
u2
! m = quB " ! mu = qBr " p = qBr
r
F=
Charged Form of Matter & Anti-Matter in a B Field
Antimatter form of electron = Positron (e+)
Same Mass but opposite Charge
Positron curls the other way from electron in a B Field
Accelerating Electrons Thru RF Cavities
A Circular Accelerator : Using B Field to Confine the electron and RF cavity to power it
Circular Particle Accelerator: LEP @ CERN, Geneve
Accelerated electron through an effective voltage of 100 Billion Volts !
To be upgraded to 7 trillion Volts by 2007
circular track for accelerating electron
French
Border
Swiss
Border
Geneva Airport
In Tunnel 150m underground, 27km ring of Magnets Keep electron in Circular Orbit
Inside A Circular Particle Accelerator Tunnel : Monorail !
Test of Relativistic Momentum In Circular Accelerator
r
mu
r
r
p=
= " mu
1 ! (u / c)2
" mu = qBr = p
qBr
" =
mu
Proof That
p ≠ mu
Relativistic Work Done & Change in Energy
x2
W =
!
r r
F .dx =
x1
X2 , u=u
x1 , u=0
x2
!
x1
r
dp r
.dx
dt
du
r
m
mu
dp
dt
p=
#
=
, substitute in W,
3/ 2
2
2
dt
u
$
u '
1" 2
&1 " 2 )
c
c (
%
du
u
m
udt
dt
#W = !
(change in var x * u)
3/ 2
0 $
u2 '
&1 " 2 )
c (
%
u
W =
!$
mudu
2
3/ 2
=
mc 2
2
1/ 2
" mc 2 = + mc 2 " mc 2
$
u '
u '
1
"
1
"
&
&
2 )
2 )
c
c
%
(
%
(
Work done is change in Kinetic energy K
0
K = + mc 2 " mc 2 or
Total Energy E= + mc 2 = K + mc 2
Relativistic Kinetic Energy & Newtonian Physics
Relativistic KE K = ! mc 2 " mc 2
# Remember Binomial Theorem
&
%
(
nx
n(n-1)
% for x << 1; (1+x) n = (1+
+
x 2 +smaller terms) (
%$
('
1!
2!
"
1
2
# u2 &
1 u2
) When u << c, %1- 2 (
* 1+
+ ...smaller terms
2
2c
$ c '
1 u2
1 2
2
2
so K * mc [1 +
]
"
mc
=
mu (classical form recovered)
2
2c
2
Total Energy of a Particle E = ! mc 2 = KE + mc 2
For a particle at rest, u = 0 " Total Energy E= mc 2
Mass is Energy, Energy is Mass : Mass-Energy Conservation
Examine Kinetic energy Before and After Inelastic Collision: Conserved?
K=0
S
K = mu2
Before
v
1
v
V=0
2
1
2
After
Mass-Energy Conservation: sum of mass-energy of a system of particles
before interaction must equal sum of mass-energy after interaction
E before
mc 2
2
+
=
mc 2
2
=
Eafter
Mc 2 " M =
2m
2
> 2m
u
u
u
1
!
1
!
c2
c2
c2
Kinetic energy has been transformed into mass increase
1!
$
'
)
2K 2 && mc 2
2)
#M = M - 2m = 2 = 2
! mc
2
&
)
c
c
u
& 1! 2
)
%
(
c
Kinetic energy is not lost,
its transformed into
more mass in final state
Creation and Annihilation of Particles
µ-
γ
µ+
Sequence of events in a matter-antimatter collision:
+
e + e "# " µ + µ
!
+
!
Relativistic Kinematics of Subatomic Particles
Reconstructing Decay of a π Meson
The decay of a stationary ! + " µ +# happens quickly,
# is invisible, has m $ 0; µ + leaves a trace in a B field
Pν , Εν
µ + mass=106 MeV/c 2 , KE = 4.6 MeV
What was mass of the fleeting ! + ?
Energy Conservation:
ν
π+
At+rest
µ
E! = Eµ + E% & m ! c 2 = (mµ c 2 )2 + pµ2 c 2 + p# c
Momentum Conservation : pµ = p%
& m! c 2 = (mµ c 2 )2 + pµ2 c 2 + pµ c
Pµ , Εµ
Conservation of Mass-Energy: Nuclear Fission
M1
M
Mc 2 =
M1c 2
1!
u12
c
2
<1
+
+
M2
M 2c2
1!
<1
u22
c
2
+
+
M 3c 2
1!
u32
M3
Nuclear Fission
" M > M1 + M 2 + M 3
c2
<1
Loss of mass shows up as kinetic energy of final state particles
Disintegration energy per fission Q=(M – (M1+M2+M3))c2 =ΔMc2
236
92
90
1
U ! 143
Cs+
Rb+3
n
55
92
0
(1 AMU= 1.6605402 " 10-27 kg = 931.49 MeV)
#m=0.177537u=2.9471 " 10-28 kg = 165.4 MeV= energy release/fission =peanuts
What makes it explosive is 1 mole of Uranium = 6.023 x 1023 Nuclei !!
Energy Released by 1 Kg of Fissionable Uranium
1 Mole of Uranium = 236 gm, Avagadro''s # = 6.023 ! 1023 Nuclei
6.023 ! 1023
So in 1 kg N =
! 1000g = 2.55 ! 1024 nuclei
236g / mole
1 Nuclear fission = 165.4 MeV " 103 g = 2.55 ! 1024 ! 165.4 MeV
Note 1 MeV = 4.45 ! 10-20 kWh
If the power plant has conversion efficiency = 40%
Energy Transformed = 748 ! 106 kWh
# 1 100W lamp can be lit for 8500 years !
Nuclear Fission Schematic : Tickling a Nucleus
Excited U
Absorption of Neutron
Unstable
Nucleus
Oscillation
Deforms Nucleus
Sustaining Chain Reaction: 1st three Fissions
Average # of Neutrons/Fission = 2.5
Neutron emitted in fission of one U
Needs to be captured by another
To control reaction => define factor K
Supercritical K >> 1 in a Nuclear Bomb
Critical
K = 1 in a Nuclear Reactor
Schematic of a Pressurized-Water Reactor
Water in contact with reactor core serves as a moderator and heat transfer
Medium. Heat produced in fission drives turbine
Lowering Fuel Core in a Nuclear Reactor
First Nuke Reactor :Pennsylvania
1957
Pressure Vessel contains :
14 Tons of Natural Uranium
+ 165 lb of enriched Uranium
Power plant rated at 90MW,
Retired (82)
Pressure vessel packed with
Concrete now sits in Nuclear Waste
Facility in Hanford, Washington
Nuclear Fusion : What Powers the Sun
Opposite of Fission
Mass of a Nucleus < mass of its component protons+Neutrons
Nuclei are stable, bound by an attractive "Strong Force"
Think of Nuclei as molecules and proton/neutron as atoms making it
Binding Energy: Work/Energy required to pull a bound system (M) apart leaving its
components (m) free of the attractive force and at rest:
n
Mc 2 +BE=! m ic 2
i=1
2
1
H
Deuterium
+
2
1
H
=
4
2
He + 23.9 MeV =
Deuterium = Helium + Released Energy
Think of energy released in Fusion as Dissociation energy of Chemistry
Sun's Power Output = 4 " 1026 Watts # 1038 Fusion/Second !!!!
Nuclear Fusion: Wishing For The Star
• Fusion is eminently desirable because
– More Energy/Nucleon
• (3.52 MeV in fusion Vs 1 MeV in fission)
• 2H + 3H  4He + n + 17.6 MeV
– Relatively abundant fuel supply, No danger like nuclear reactor going
supercritical
• Unfortunately technology not commercially available
– What’s inside nuclei => protons and Neutrons
– Need Large KE to overcome Coulomb repulsion between nuclei
• About 1 MeV needed to bring nuclei close enough together for Strong
Nuclear Attraction  fusion
• Need to
– heat particle to high temp such that thermal energy E= kT ≈ 10keV 
tunneling thru coulomb barrier
– Implies heating to T ≈ 108 K ( like in stars)
– Confine Plasma (± ions) long enough for fusion
» In stars, enormous gravitational field confines plasma
Inertial Fusion Reactor : Schematic
Pellet of frozen-solid Deuterium & tritium bombarded
from all sides with intense pulsed laser beam with
energy ≈106 Joules lasting 10-8 S
Momentum imparted by laser beam compresses
pellet by 1/10000 of normal density and heats it to
temp T ≈ 108 K for 10-10 S
Burst of fusion energy transported away by liquid Li
World’s Most Powerful Laser : NOVA @ LLNL
Size of football field, 3 stories tall
Generates 1.0 x 1014 watts (100 terawatts)
10 laser beams converge onto H pellet (0.5mm diam)
Fusion reaction is visible as a starlight lasting 10-10 S
Releasing 1013 neutrons
ITER: The Next Big Step in Nuclear Fusion
Visit www.iter.org for Details of this mega Science & Engineering Project
This may be future of cheap, clean Nuclear Energy for Earthlings