physics review - niunicadd.niu.edu/~syphers/uspas/2018w/lect/w1d1l2.pdfmaxwell’s equations...
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![Page 1: Physics Review - NIUnicadd.niu.edu/~syphers/uspas/2018w/Lect/w1d1L2.pdfMaxwell’s Equations Integral Form Differential Form One Consequence: EM Waves speed of waves given by c = (](https://reader034.vdocuments.us/reader034/viewer/2022042310/5ed77877f1b9d0558326569f/html5/thumbnails/1.jpg)
Physics Review
Newtonian MechanicsGravitational vs. Electromagnetic forcesLorentz Force
Maxwell’s EquationsIntegral vs. Differential
Relativity (Special)
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Newtonian Mechanics
v = dx/dtp = mvF = dp/dtdW = F dxFg = G Mm/r2 [ Fe = 1/4!"0 Qq/r2 Fb = qv x B , etc. ]The Simple Harmonic Oscillator + Phase Space
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WinterSession2018MJS USPASFundamentals
Simple Harmonic Motion
3
x = �kx x+ kx = 0
x = a sin(!t) + b cos(!t) = c sin(!t+ �)
x = c ! cos(!t+ �)
x = �c !2 sin(!t+ �) = �!2x
! =pk
x2 + (x/!)2 = c2 circle
x/!
x
phase space diagrams
ellipse
x
xx2 +1
!2x2 = c2
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Maxwell’s EquationsIntegral FormDifferential FormOne Consequence: EM Waves
speed of waves given by c = (�0ϵ0)-1/2
Another Consequence:
If �0 , ϵ0 are fundamental quantities, same in all reference frames, then so should be the speed of light!
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WinterSession2018MJS USPASFundamentals 5
(�E)closed surface =Qencl
✏0(�B)closed surface = 0
I
loop
~B · d~s = µ0
Ienclosed + ✏0
✓d�E
dt
◆
through loop
!
I
loop
~E · d~s = �✓d�B
dt
◆
through loop
�B ⌘I
surface
~B · d ~A
�E ⌘I
surface
~E · d ~A
Flux:
Maxwell’s Equations:
Gauss’ Law
Ampere’s Law
Faraday’s Law
Completed by Maxwell
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WinterSession2018MJS USPASFundamentals
Differential Relationships
6
(�E)closed surface =Qencl
✏0(�B)closed surface = 0
I
loop
~B · d~s = µ0
Ienclosed + ✏0
✓d�E
dt
◆
through loop
!
I
loop
~E · d~s = �✓d�B
dt
◆
through loop
r · ~B = 0
r · ~E = ⇢/✏0
r⇥ ~B = µ0
~J + ✏0
@ ~E
@t
!
r⇥ ~E = �@ ~B
@t
ZZ
Sr⇥ ~A · d~S =
I
@S
~A · d~r
Stoke’s Theorem:
r⇥ ~A = i
✓@Az
@y� @Ay
@z
◆� j
✓@Az
@x� @Ax
@z
◆+ k
✓@Ay
@x� @Ax
@y
◆
r · ~A =@Ax
@x+
@Ay
@y+
@Az
@z
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WinterSession2018MJS USPASFundamentals
Wave Equation and the Speed of Propagation
7
r⇥ ~E = �@ ~B
@tr⇥ ~B = µ0✏0
@ ~E
@t
r⇥r⇥ ~B = r(r · ~B)�r2 ~B = �r2 ~B
r⇥r⇥ ~f = r(r · ~f)�r2 ~f
�r2 ~B = µ0✏0@(r⇥ ~E)
@t= �µ0✏0
@2 ~B
@t2
in general:
Suppose in free space, no current sources…
so,
r2 ~B = µ0✏0@2 ~B
@t2r2 ~E = µ0✏0
@2 ~E
@t2and, likewise,
thus,
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WinterSession2018MJS USPASFundamentals
Wave Equation and the Speed of Propagation
8
r2 ~B = µ0✏0@2 ~B
@t2r2 ~E = µ0✏0
@2 ~E
@t2and wave equation
d2B/dx2 = �k2B
d2B/dt2 = �!2Bd2B/dx2 = �k2B = µ0✏0(�!2B)
µ0✏0 = (k/!)2 = 1/(�f)2 = 1/v2wave
Example: let
speed = 1/pµ0✏0 ⌘ c
c = 1/(4!x10-7 x 8.8x10-12)1/2 m/s = 3.0 x 108 m/s
B = b cos(!t� kx) = b cos(2⇡ft� 2⇡x/�)
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Maxwell’s EquationsIntegral FormDifferential FormOne Consequence: EM Waves
speed of waves given by c = (�0ϵ0)-1/2
Another Consequence:
If �0 , ϵ0 are fundamental quantities, same in all reference frames, then so should be the speed of light!
![Page 10: Physics Review - NIUnicadd.niu.edu/~syphers/uspas/2018w/Lect/w1d1L2.pdfMaxwell’s Equations Integral Form Differential Form One Consequence: EM Waves speed of waves given by c = (](https://reader034.vdocuments.us/reader034/viewer/2022042310/5ed77877f1b9d0558326569f/html5/thumbnails/10.jpg)
Special RelativityThe Principle of Relativity
The Laws of Physics same in all inertial reference frames
The Problem of the Velocity of Light
Simultaneity
Lengths and Clocks
E=mc2
Differential Relationships
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x'x
y'y
P
vt
m1 m2
v
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Simultaneity
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x'x
y'y
v
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!
"
#$ %
!
"
#$ %
!
"
#$ %
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Lengths and Clocks
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x'x
y'y
v
mirror
h
x'
y'
x, x'
y'y
h
s s
∆tv
(a) (b) (c)
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WinterSession2018MJS USPASFundamentals 17
&&&&&&&Rela*vis*c&Momentum&
where&
p = γmu
€
γ =1
1− u2 /c 2
The law of conservation of momentum is valid in all inertial reference frames if the momentum of each particle (with mass m and speed u) is re-defined by:
Principal of relativity: All the laws of physics (not just Newton�s laws) are the same in all inertial reference frames. F%= Δp/Δt&
Ex:%
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WinterSession2018MJS USPASFundamentals 18
&&&&&&&&&&&&&&&&&Rela*vity&Summary&Time dilation:
Length contraction:
Lorentz transformation of velocities:
Relativistic Momentum:
Relativistic Energy:
'/1'
22t
cvtt Δ=
−
Δ=Δ γ
γ/'/1' 22 LcvLL =−=
The%total%energy%is%made%up%of%two%contribu?ons%€
p =mu
1− u2 /c 2= γmu
22 /11cv−
=γ
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E = mc2
The Laws of Physics, and redefining the momentumWhat about Energy?Energy-momentum relationship
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WinterSession2018MJS USPASFundamentals
Work done by a Force Acting on a Mass
20
Work Done by a Force Acting on a Mass
I The work done on a particle is given by
�W =⁄
F · ds =⁄
dp/dt · ds =⁄(ds/dt)dp =
⁄v · dp.
Check: if p = mv then, starting from rest,�W =
s vdp =s v mdv = 1
2mv2.I But, using our new definition of momentum, p = “mv , then
�W =⁄
v d(“mv) =⁄(v/c) m d(“v/c)c2 = mc2
⁄—d(—“)
I For our next step, let’s use the definition of “ to show a new,interesting relationship that we can use to help us. . .
Work, Part III
I Now, from Calculus we know that if y = a + bx2, thendy/dx = 2bx or,
dy = 2bx dxI Thus, di�erentiating each side of our previous equation, we
would have2“d“ = 2(—“)d(—“),
or,d“ = —d(—“)
I So finally, our original integral becomes,
�W = mc2⁄
—d(—“) = mc2⁄
d“ = (“final ≠ “initial)mc2
�2 = 1 + (��)2 �!
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WinterSession2018MJS USPASFundamentals 21
Work, Part IV
I The previous equation tells us that as we do work on a particleits energy will change by an amount �E = �W = �“mc2.Thus, the energy of a particle should be defined as
E = “ mc2.
I If the particle starts from rest, then “initial = 1, and its energyis E = mc2. As it speeds up its kinetic energy will be
KE = �W = (“ ≠ 1)mc2, where here “ © “final .
I So we see that the energy is a combination of a “rest energy”and a “kinetic energy”:
E = “mc2 = mc2 + (“ ≠ 1)mc2.
If no work were done (�W = 0), and the particle were still atrest, the particle would still have energy (rest energy):
E0 = mc2 æ mass is energy!
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WinterSession2018MJS USPASFundamentals 22
&&&&&&&&&&&&&&&&&Rela*vity&Summary&Time dilation:
Length contraction:
Lorentz transformation of velocities:
Relativistic Momentum:
Relativistic Energy:
'/1'
22t
cvtt Δ=
−
Δ=Δ γ
γ/'/1' 22 LcvLL =−=
The%total%energy%is%made%up%of%two%contribu?ons%€
p =mu
1− u2 /c 2= γmu
22 /11cv−
=γ
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Speed, Momentum, vs. Energy
Kinetic Energy Kinetic EnergyM
omen
tum
Spee
d
Electron: 0 0.5 1.0 1.5 MeVProton: 0 1000 2000 3000 MeV
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Relationships Differential, and Otherwise
# = v/c $ = 1/(1-#2)1/2F = dp/dt p = $mv = #$mcE = $mc2 = mc2 + W W = ($-1)mc2$ = 1/(1-#2)1/2 #2 = 1 - 1/$2 $2 = 1 + (#$)2
given E, find p; given p, find W; given W, find vfind: d#/# in terms of dp/p; dE/E in terms of dp/pfind: d#/# in terms of dW/W; dW/W in terms of dE/E
(coefficients only to contain factors of #, $)
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Quantization
Accelerator and Beam Physics are typically “classical” physics; occasionally (esp. wrt synchrotron radiation and light sources), quantum physics comes into playPhoton: E = h% = hc/& = pc (i.e., m = 0)De Broglie Wavelength: & = h/p 'x ∙ 'px = h/4! (Uncertainty Principle)