Download - 60 years ago…
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60 years ago…
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The explosion in high-tech medical imaging
& nuclear medicine
(including particle beam cancer treatments)
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The constraints of limited/vanishing fossils fuels in the face of an exploding population
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…together with undeveloped or under-developed new technologies
The constraints of limited/vanishing fossils fuels
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Nuclear
will renew interest in nuclear power
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Fission power generators
will be part of the political
landscape again
as well as the Holy Grail of FUSION.
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…exciting developments in theoretical astrophysics
The evolution of stars is well-understood in terms of stellar models
incorporating known nuclear processes.
The observed expansion of the universe (Hubble’s Law) lead Gamow to postulate a Big Bang which predicted the
Cosmic Microwave Background Radiation
as well as made very specific predictions of the relative abundance of the elements
(on a galactic or universal scale).
Applying well established nuclear physics to the epoch of nuclear formation - ~3 -15 minutes after the big bang - allows the abundances of deuterium, helium, lithium and other light elements to be predicted.
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1896
1899
1912
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Henri Becquerel (1852-1908) 1903 Nobel Prize
discovery of natural radioactivity
Wrapped photographic plate showed distinct silhouettes of uranium salt samples stored atop it.
1896 While studying fluorescent & phosphorescent materials, Becquerel finds potassium-uranyl sulfate spontaneously emits radiation that can penetrate thick opaque black paper aluminum plates copper plates
Exhibited by all known compounds of uranium (phosphorescent or not) and metallic uranium itself.
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1898 Marie Curie discovers thorium (90Th) Together Pierre and Marie Curie discover polonium (84Po) and radium (88Ra)
1899 Ernest Rutherford identifies 2 distinct kinds of rays emitted by uranium - highly ionizing, but completely absorbed by 0.006 cm aluminum foil or a few cm of air
- less ionizing, but penetrate many meters of air or up to a cm of aluminum.
1900 P. Villard finds in addition to rays, radium emits - the least ionizing, but capable of penetrating many cm of lead, several ft of concrete
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B-fieldpoints
into page
1900-01 Studying the deflection of these rays in magnetic fields, Becquerel and the Curies establish rays to be charged particles
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F
R
mvF
2
190sin o
R
mvqvB
2
mvqBR
BR
v
m
q
R
mvqvB
2
sin
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1900-01 Using the procedure developed by J.J. Thomson in 1887 Becquerel determined the ratio of charge q to mass m for
: q/m = 1.76×1011 coulombs/kilogram identical to the electron!
: q/m = 4.8×107 coulombs/kilogram 4000 times smaller!
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RCteQtQ /
0)( RCteVtV /
0)(
/
0)( xeNxN
/0
)( teAtA
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teNtN 0)(
RCteQtQ /
0)( RCteVtV /
0)( /
0)( xeNxN
/0
)( teAtA N
um
be
r su
rviv
ing
Ra
dio
act
ive
ato
ms
What does stand for?
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teNtN 0)(N
um
ber
su
rviv
ing
Rad
ioac
tive
ato
ms
time
tNN 0logloglogN
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!4!3!2
1432 xxx
xex
!7!5!3
sin753 xxx
xx
for x measured in radians (not degrees!)
!6!4!2
1cos642 xxx
x
32
!3
)2)(1(
!2
)1(1)1ln( x
pppx
pppxx p
What if
x was a measurementthat carried
units?
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)2sin()( ftAty
!7
)2(
!5
)2(
!3
)2(22sin
753 ftftftftft
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Let’s complete the table below (using a calculator) to check the “small angle approximation” (for angles not much bigger than ~1520o)
xx sinwhich ignores more than the 1st term of the series
Note: the x or (in radians) = (/180o) (in degrees)
Angle (degrees) Angle (radians) sin
25o
0 0 0.0000000001 0.017453293 0.0174524062 0.034906585 3 0.052359878 4 0.069813170 6810152025
0.1047197550.1396263400.1745329520.2617993880.3490658500.436332313
0.0348994970.0523359560.0697564730.1045284630.1391731010.1736482040.2588190450.3420201430.42261826297% accurate!
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y = sin x
y = xy = x3/6
y = x - x3/6
y = x5/120
y = x - x3/6 + x5/120
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...718281828.2eAny power of e can be expanded as an infinite series
!4!3!2
1432 xxx
xex
Let’s compute some powers of e using just the above 5 terms of the series
e0 = 1 + 0 + + + =
e1 = 1 + 1 +
e2 = 1 + 2 +
0 0 0 1
0.500000 + 0.166667 + 0.041667
2.708334
2.000000 + 1.333333 + 0.666667
7.000000
e2 = 7.3890560989…
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Piano, Concert C
Clarinet, Concert C
Miles Davis’ trumpet
violin
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A Fourier series can be defined for any function over the interval 0 x 2L
1
0 sincos2
)(n
nn L
xnb
L
xna
axf
where dxL
xnxf
La
L
n
2
0cos)(
1
dxL
xnxf
Lb
L
n
2
0sin)(
1
Ofteneasiestto treat
n=0 casesseparately
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Compute the Fourier series of the SQUARE WAVE function f given by
)(xf
2,1
0,1
x
x
2
Note: f(x) is an odd function ( i.e. f(-x) = -f(x) )
so f(x) cos nx will be as well, while f(x) sin nx will be even.
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dxL
xnxf
La
L
n
2
0cos)(
1)(xf
2,1
0,1
x
x
dxxfa 0cos)(1 2
00
dxdx 0cos)1(0cos11 2
0
0
dxnxdxnxan
2
0cos)1(cos1
1
dxnnxdxnx ( )coscos1
00
dxnxdxnx
00coscos
1
change of variables: x x' = x-
periodicity: cos(X+n) = (-1)ncosX
for n = 1, 3, 5,…
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dxL
xnxf
La
L
n
2
0cos)(
1)(xf
2,1
0,1
x
x
00 a
dxnxan
0cos
2for n = 1, 3, 5,…
0na for n = 2, 4, 6,…
change of variables: x x' = nx
dxxn
an
n
0cos
2 0
IF f(x) is odd, all an vanish!
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dxL
xnxf
Lb
L
n
2
0sin)(
1)(xf
2,1
0,1
x
x
00sin)(1 2
00 dxxfb
dxnxdxnxbn
2
0sinsin
1
dxnnxdxnx ( )sinsin1
00
periodicity: cos(X±n) = (-1)ncosX
dxnxdxnx
00sinsin
1
for n = 1, 3, 5,…
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)(xf
2,1
0,1
x
x
00 b
dxnxbn
0sin
2for n = 1, 3, 5,…
0nb for n = 2, 4, 6,…
change of variables: x x' = nx
dxxn
n
0sin
2
dxL
xnxf
Lb
L
n
2
0sin)(
1
dxxn
0sin
1
for odd n
nxn
40cos
2 for n = 1, 3, 5,…
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)5
5sin
3
3sin
1
sin(
4)( xxx
xf
1
2x
y
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http://www.jhu.edu/~signals/fourier2/
http://www.phy.ntnu.edu.tw/java/sound/sound.html
http://mathforum.org/key/nucalc/fourier.html
http://www.falstad.com/fourier/
Leads you through a qualitative argument in building a square wave
Add terms one by one (or as many as you want) to build fourier series approximation to a selection of periodic functions
Build Fourier series approximation to assorted periodic functionsand listen to an audio playing the wave forms
Customize your own sound synthesizer