Physics 212 Lecture 23
Physics 212 Lecture 23, Slide 1
Music Who is the Artist? A) B) C) D) E)
Bob Wills and the Texas Playboys The Viper and His Famous Orchestra Big Sandy and the Fly-Rite Boys Pokey LaFarge and the South City Three Blind Boy Paxton
Why? They were the headliners at the CU Folk and Roots Festival this past weekend. They were wonderful !!
See who you missed at: http://www.folkandroots.org/site/2011-artists
Your Comments “This lecture was fun!” “super hard” “I don't understand what the equations mean with the (kz+-wt).” “Can you explain the E and B electromagnetic waves graph in more detail? I'm still confused. ”
It certainly can be confusing.. We will try to make it clear !! “Sometimes, you can't deny that physics is just awesome. Doppler effect seen from moving stars?? So cool!” “So, according to the Doppler Effect, the only thing I have to do to get XRay Vision is run away from whatever I want to see through? Also: would it work if said thing was running away from me? (Always looking for ways to minimize the work I have to do)” We’ll do a Doppler shift example anyway (& discuss approximations)
05
“Seriously? You mispell "pointing" and then expect us to run at our iclickers so as to see the magical waves they emit? SMH” Physics 212 Lecture 23, Slide 3
Your Comments “PLEASE READ MEEEEEEEEEEEEEEE. <----- Don't post this annoying part =D I set the equations of (energy density )*(Volume) = planck's constant *(c/lambda) . Solving for lambda I see that the wavelength decreases as Volume increases. Does that mean that EM waves loose their wavelength as the Universe expands?” YES! Photons left over from the Big Bang (starting out with the energy that binds electrons to protons = hydrogen atom) have lost a lot of energy as the universe expands (from t = 400,000 years to now) λinitial ≅ 100 nm = 10-7 m, λtoday ≅ 1 mm: the cosmic microwave background
05
Physics 212 Lecture 23, Slide 4
Plane Waves from Last Time
E and B are perpendicular and in phase Oscillate in time and space Direction of propagation given by E X B E0 = cB0 Argument of sin/cos gives direction of propagation
Physics 212 Lecture 23, Slide 5
Understanding the speed and direction of the wave Ex = Eosin(kz-ωt) Ex
t=0
z
Ex
sin( kz − π ) = − cos( kz)
z
2
t = π/2ω
What has happened to the waveform in this time interval? It has MOVED TO THE RIGHT by λ/4
λ/4 ω speed = c = =λ = λf π / 2ω 2π Physics 212 Lecture 23, Slide 6
Checkpoint 1a
“The value at t=0 is 0 for this function and it is positive in the x direction right after” “Since the sine wave is going up, its - the omega t” “The terms within the sin function show that the wave is moving in the +z direction and B is in the y plane so it's By.”
Physics 212 Lecture 23, Slide 7
Checkpoint 1a
No – moving in the minus z direction No – has Ey rather than Ex
Physics 212 Lecture 23, Slide 8
Checkpoint 2a
c=3.0 x 108 m/s
Wavelength is equal to the speed of light divided by the frequency.
c 300, 000, 000 1 λ= = = f 900, 000, 000 3 Check: Look at size of antenna on base unit
Physics 212 Lecture 23, Slide 9
Doppler Shift
The Big Idea As source approaches: Wavelength decreases Frequency Increases
Physics 212 Lecture 23, Slide 10
Doppler Shift for e-m Waves What’s Different from Sound or Water Waves ? Sound /Water Waves : You can calculate (no relativity needed) BUT Result is somewhat complicated: is source or observer moving wrt medium?
Electromagnetic Waves : You need relativity (time dilation) to calculate BUT Result is simple: only depends on relative motion of source & observer
1+ β f′= f 1 − β
1 2
β = v/c β > 0 if source & observer are approaching β < 0 if source & observer are separating Physics 212 Lecture 23, Slide 11
Doppler Shift for e-m Waves f
f’
v or
f
f’
v The Doppler Shift is the SAME for both cases ! f’/f ONLY DEPENDS ON THE RELATIVE VELOCITY
1+ β f′= f 1− β
1 2
Physics 212 Lecture 23, Slide 12
Doppler Shift for e-m Waves A Note on Approximations
1+ β f′= f − β 1
1 2
f ′ ≈ f (1 + β )
β << 1
Remember β > 0 for approach β < 0 for separation
WHY ?? 1/ 2
Taylor Series: Expand
F ( β ) = F (0) + Evaluate:
F ( 0) = 1 F ′(0) = 1
1 + β F (β ) = 1− β
around β = 0
F ′(0) F ′′(0) 2 β+ β + ... 1! 2!
F (β ) ≈ 1 + β
NOTE: F ( β ) = (1 ± β ) n F (β ) ≈ 1± nβ
Physics 212 Lecture 23, Slide 13
Red Shift
Wavelengths shifted higher
wavelength
Frequencies shifted lower
Star separating from us (Expanding Universe)
Our Sun
Star in a distant galaxy Physics 212 Lecture 23, Slide 14
Red Shift “the dopper effect in relation to magnetic fields. Is this used in any practical applications? Is anything moving fast enough on earth that relative to the speed of light that there would be any significant shift? I understand that this makes sense in looking at the universe and motion of planets and stars and such but there is no way you can "run" fast enough to ever see an iclicker signal.”
Star Clusters and Distant, Red Galaxies Near Edge of Galaxy NGC 3370 (Hubble Space Telescope)
Physics 212 Lecture 23, Slide 15
Example
Police radars get twice the effect since the EM waves make a round trip:
f ′ ≈ f (1 + 2β ) If f = 24,000,000,000 Hz (k-band radar gun) c = 300,000,000 m/s v
β
f’
f’-f
30 m/s (67 mph)
1.000 x 10-7
24,000,004,800
4800 Hz
31 m/s (69 mph)
1.033 x 10-7
24,000,004,959
4959 Hz Physics 212 Lecture 23, Slide 16
Checkpoint 2b A) B) C)
ficlicker = 900 MHz
“Running away decreases the frequency that your eyes would see.”
“Running toward the iclicker will cause the frequency we observe to be greater than the source frequency.”
“Moving relative to the iclicker would only change its velocity in relation to you, but not its frequency.”
Physics 212 Lecture 23, Slide 17
Checkpoint 2b ficlicker = 900 MHz
A) B) C)
Need to approach i>clicker
Need to shift frequency UP
(β > 0)
How fast would you need to run to see the i>clicker radiation? 1/ 2
f ′ 1014 1 + β = 9 = 105 = f 10 1 − β
1 + β 10 = − 1 β 10
1010 − 1 1 − 10 −10 β = 10 = 10 + 1 1 + 10 −10
Approximation Exercise:
β ≈ 1 − (2 × 10−10 ) Physics 212 Lecture 23, Slide 18
Waves Carry Energy
Physics 212 Lecture 23, Slide 19
Intensity Intensity = Average energy delivered per unit time, per unit area
1 dU I≡ A dt
Length = c dt
Area = A
dU = u ⋅ volume = u Acdt
I =c u
Sunlight on Earth:
I ~ 1000J/s/m2 ~ 1 kW/m2 Physics 212 Lecture 23, Slide 20
Waves Carry Energy
Physics 212 Lecture 23, Slide 21
Comment on Poynting Vector Just another way to keep track of all this - Its magnitude is equal to I – Its direction is the direction of propagation of the wave
Physics 212 Lecture 23, Slide 22
Light has Momentum! Momentum of light is p = U/c → light can exert pressure!! Recall force from mechanics:
Approximate ∆p by p:
But intensity of light is:
∆p F F= ; P= ∆t A
p U P= = A∆t cA∆t
U I= A∆t
Therefore:
I P= c Physics 212 Lecture 23, Slide 23
Light has Momentum! Momentum of light is p = U/c → light can exert pressure!!
Optical Tweezers
Physics 212 Lecture 23, Slide 24
Checkpoint 1b Which of the following actions will increase the energy carried by an electromagnetic wave? A. Increase E keeping ω constant C. Both of the above will increase the energy
B. Increase ω keeping E constant D. Neither of the above will increase the energy
“Only the electric field changing can increase energy. ” “Energy doesn't depend on the amplitude, just the frequency.” “Increasing both will add energy since it has a greater base energy if you increase E and more waves per a period of time if you incease w.” “only velocity of the wave influences energy”
Physics 212 Lecture 23, Slide 25
Checkpoint 1b Which of the following actions will increase the energy carried by an electromagnetic wave? A. Increase E keeping ω constant C. Both of the above will increase the energy
B. Increase ω keeping E constant D. Neither of the above will increase the energy
But then again, what are we keeping constant here? WHAT ABOUT PHOTONS? The energy of one photon is Ephoton = hf = hω/2π Uwave = Nphotons Ephoton = 1/2 ε0E02 Physics 212 Lecture 23, Slide 26
PHOTONS We believe the energy in an e-m wave is carried by photons Question: What are Photons? Answer: Photons are Photons. Photons possess both wave and particle properties Particle: Energy and Momentum localized Wave: They have definite frequency & wavelength (fλ = c) Connections seen in equations: E = hf p = h/λ
Planck’s constant h = 6.63e-34 J-s
Question: How can something be both a particle and a wave? Answer: ~It can’t (when we observe it) What we see depends on how we choose to measure it ! The mystery of quantum mechanics: More on this in PHYS 214 Physics 212 Lecture 23, Slide 27
Calculation 1
y x
An electromagnetic wave is described by: ˆjE0 cos(kz − ωt ) E = where ˆj is the unit vector in the +y direction.
z
Which of the following graphs represents the z-dependence of Bx at t = 0?
X (A)
X (B)
(C)
(D)
E and B are “in phase” (or 180o out of phase)
E = ˆjE0 cos(kz − ωt )
Wave moves in +z direction y E E × B points in direction of propagation x
ˆ 0 cos(kz − ωt ) B = −iB
B
zPhysics 212
Lecture 23, Slide 28
Calculation 2
y
iˆ + ˆj E= E0 cos(kz + ωt ) 2
An electromagnetic wave is described by:
x z
What is the form of B for this wave? −iˆ + ˆj B= ( E0 / c) cos(kz + ωt ) 2
(A)
iˆ + ˆj B= ( E0 / c)cos(kz + ωt ) 2
(C)
(B)
iˆ − ˆj B= ( E0 / c) cos(kz + ωt ) 2
(D) B = −i − j ( E0 / c) cos(kz + ωt )
iˆ + ˆj E= E0 cos(kz + ωt ) 2
ˆ
ˆ
2
Wave moves in –z direction
y E x
+z points out of screen -z points into screen
B E × B points in negative z-direction Physics 212 Lecture 23, Slide 29
Calculation 3 An electromagnetic wave is described by:
E = ˆjE0 sin(kz + ωt )
Which of the following plots represents Bx(z) at time t = π/2ω ω?
(A)
(B)
Wave moves in negative z-direction y +z points out of screen
E
-z points into screen
B
x
E × B points in negative z-direction
(C)
(D)
B = iˆ( E0 / c)sin(kz + ωt ) at ωt = π/2:
Bx = ( E0 / c)sin(kz + π / 2) Bx = ( E0 / c){sin kz cos(π / 2) + cos kz sin(π / 2)}
Bx = ( E0 / c) cos(kz ) Physics 212 Lecture 23, Slide 30
Calculation 4 A certain unnamed physics professor was arrested for running a stoplight. He said the light was green. A pedestian said it was red. The professor then said: “We are both being truthful; you just need to account for the Doppler effect !” Is it possible that the professor’s argument is correct? (λ λgreen = 500 nm, λred = 600 nm)
(A) YES
(B) NO
• As professor approaches stoplight, the frequency of its emitted light will be shifted UP • The speed of light does not change • Therefore, the wavelength (c/f) would be shifted to a smaller value • If he goes fast enough, he could observe a green light !
Physics 212 Lecture 23, Slide 31
Follow-Up A certain unnamed physics professor was arrested for running a stoplight. He said the light was green. A pedestian said it was red. The professor then said: “We are both being truthful; you just need to account for the Doppler effect !” How fast would the professor have to go to see the light as green? (λ λgreen = 500 nm, λred = 600 nm)
(A) 540 m/s
(B) 5.4 X104 m/s
(C) 5.4 X 107 m/s
Relativistic Doppler effect: f ′ = f
f ′ 600 1+ β = = f 500 1− β
1+ β 1− β
36(1 − β ) = 25(1 + β )
Note approximation for small β is not bad: c = 3 X 108 m/s fl v = 5.4 X 107 m/s
(D) 5.4 X 108 m/s
β=
f ′ = f (1 + β )
11 = 0.18 61 1 β = = 0.2 5
Change the charge to SPEEDING! Physics 212 Lecture 23, Slide 32
Physics 212 Lecture 23
Physics 212 Lecture 23, Slide 1
Music Who is the Artist? A) B) C) D) E)
Bob Wills and the Texas Playboys The Viper and His Famous Orchestra Big Sandy and the Fly-Rite Boys Pokey LaFarge and the South City Three Blind Boy Paxton
Why? They were the headliners at the CU Folk and Roots Festival this past weekend. They were wonderful !!
See who you missed at: http://www.folkandroots.org/site/2011-artists
Your Comments “This lecture was fun!” “super hard” “I don't understand what the equations mean with the (kz+-wt).” “Can you explain the E and B electromagnetic waves graph in more detail? I'm still confused. ”
It certainly can be confusing.. We will try to make it clear !! “Sometimes, you can't deny that physics is just awesome. Doppler effect seen from moving stars?? So cool!” “So, according to the Doppler Effect, the only thing I have to do to get XRay Vision is run away from whatever I want to see through? Also: would it work if said thing was running away from me? (Always looking for ways to minimize the work I have to do)” We’ll do a Doppler shift example anyway (& discuss approximations)
05
“Seriously? You mispell "pointing" and then expect us to run at our iclickers so as to see the magical waves they emit? SMH” Physics 212 Lecture 23, Slide 3
Your Comments “PLEASE READ MEEEEEEEEEEEEEEE. <----- Don't post this annoying part =D I set the equations of (energy density )*(Volume) = planck's constant *(c/lambda) . Solving for lambda I see that the wavelength decreases as Volume increases. Does that mean that EM waves loose their wavelength as the Universe expands?” YES! Photons left over from the Big Bang (starting out with the energy that binds electrons to protons = hydrogen atom) have lost a lot of energy as the universe expands (from t = 400,000 years to now) λinitial ≅ 100 nm = 10-7 m, λtoday ≅ 1 mm: the cosmic microwave background
05
Physics 212 Lecture 23, Slide 4
Plane Waves from Last Time
E and B are perpendicular and in phase Oscillate in time and space Direction of propagation given by E X B E0 = cB0 Argument of sin/cos gives direction of propagation
Physics 212 Lecture 23, Slide 5
Understanding the speed and direction of the wave Ex = Eosin(kz-ωt) Ex
t=0
z
Ex
sin( kz − π ) = − cos( kz)
z
2
t = π/2ω
What has happened to the waveform in this time interval? It has MOVED TO THE RIGHT by λ/4
λ/4 ω speed = c = =λ = λf π / 2ω 2π Physics 212 Lecture 23, Slide 6
Checkpoint 1a
“The value at t=0 is 0 for this function and it is positive in the x direction right after” “Since the sine wave is going up, its - the omega t” “The terms within the sin function show that the wave is moving in the +z direction and B is in the y plane so it's By.”
Physics 212 Lecture 23, Slide 7
Checkpoint 1a
No – moving in the minus z direction No – has Ey rather than Ex
Physics 212 Lecture 23, Slide 8
Checkpoint 2a
c=3.0 x 108 m/s
Wavelength is equal to the speed of light divided by the frequency.
c 300, 000, 000 1 λ= = = f 900, 000, 000 3 Check: Look at size of antenna on base unit
Physics 212 Lecture 23, Slide 9
Doppler Shift
The Big Idea As source approaches: Wavelength decreases Frequency Increases
Physics 212 Lecture 23, Slide 10
Doppler Shift for e-m Waves What’s Different from Sound or Water Waves ? Sound /Water Waves : You can calculate (no relativity needed) BUT Result is somewhat complicated: is source or observer moving wrt medium?
Electromagnetic Waves : You need relativity (time dilation) to calculate BUT Result is simple: only depends on relative motion of source & observer
1+ β f′= f 1 − β
1 2
β = v/c β > 0 if source & observer are approaching β < 0 if source & observer are separating Physics 212 Lecture 23, Slide 11
Doppler Shift for e-m Waves f
f’
v or
f
f’
v The Doppler Shift is the SAME for both cases ! f’/f ONLY DEPENDS ON THE RELATIVE VELOCITY
1+ β f′= f 1− β
1 2
Physics 212 Lecture 23, Slide 12
Doppler Shift for e-m Waves A Note on Approximations
1+ β f′= f − β 1
1 2
f ′ ≈ f (1 + β )
β << 1
Remember β > 0 for approach β < 0 for separation
WHY ?? 1/ 2
Taylor Series: Expand
F ( β ) = F (0) + Evaluate:
F ( 0) = 1 F ′(0) = 1
1 + β F (β ) = 1− β
around β = 0
F ′(0) F ′′(0) 2 β+ β + ... 1! 2!
F (β ) ≈ 1 + β
NOTE: F ( β ) = (1 ± β ) n F (β ) ≈ 1± nβ
Physics 212 Lecture 23, Slide 13
Red Shift
Wavelengths shifted higher
wavelength
Frequencies shifted lower
Star separating from us (Expanding Universe)
Our Sun
Star in a distant galaxy Physics 212 Lecture 23, Slide 14
Red Shift “the dopper effect in relation to magnetic fields. Is this used in any practical applications? Is anything moving fast enough on earth that relative to the speed of light that there would be any significant shift? I understand that this makes sense in looking at the universe and motion of planets and stars and such but there is no way you can "run" fast enough to ever see an iclicker signal.”
Star Clusters and Distant, Red Galaxies Near Edge of Galaxy NGC 3370 (Hubble Space Telescope)
Physics 212 Lecture 23, Slide 15
Example
Police radars get twice the effect since the EM waves make a round trip:
f ′ ≈ f (1 + 2β ) If f = 24,000,000,000 Hz (k-band radar gun) c = 300,000,000 m/s v
β
f’
f’-f
30 m/s (67 mph)
1.000 x 10-7
24,000,004,800
4800 Hz
31 m/s (69 mph)
1.033 x 10-7
24,000,004,959
4959 Hz Physics 212 Lecture 23, Slide 16
Checkpoint 2b A) B) C)
ficlicker = 900 MHz
“Running away decreases the frequency that your eyes would see.”
“Running toward the iclicker will cause the frequency we observe to be greater than the source frequency.”
“Moving relative to the iclicker would only change its velocity in relation to you, but not its frequency.”
Physics 212 Lecture 23, Slide 17
Checkpoint 2b ficlicker = 900 MHz
A) B) C)
Need to approach i>clicker
Need to shift frequency UP
(β > 0)
How fast would you need to run to see the i>clicker radiation? 1/ 2
f ′ 1014 1 + β = 9 = 105 = f 10 1 − β
1 + β 10 = 1 − β 10
1010 − 1 1 − 10 −10 β = 10 = 10 + 1 1 + 10 −10
Approximation Exercise:
β ≈ 1 − (2 × 10−10 ) Physics 212 Lecture 23, Slide 18
Waves Carry Energy
Physics 212 Lecture 23, Slide 19
Intensity Intensity = Average energy delivered per unit time, per unit area
1 dU I≡ A dt
Length = c dt
Area = A
dU = u ⋅ volume = u Acdt
I =c u
Sunlight on Earth:
I ~ 1000J/s/m2 ~ 1 kW/m2 Physics 212 Lecture 23, Slide 20
Waves Carry Energy
Physics 212 Lecture 23, Slide 21
Comment on Poynting Vector Just another way to keep track of all this - Its magnitude is equal to I – Its direction is the direction of propagation of the wave
Physics 212 Lecture 23, Slide 22
Light has Momentum! Momentum of light is p = U/c → light can exert pressure!! Recall force from mechanics:
Approximate ∆p by p:
But intensity of light is:
∆p F F= ; P= ∆t A
p U P= = A∆t cA∆t
U I= A∆t
Therefore:
I P= c Physics 212 Lecture 23, Slide 23
Light has Momentum! Momentum of light is p = U/c → light can exert pressure!!
Optical Tweezers
Physics 212 Lecture 23, Slide 24
Checkpoint 1b Which of the following actions will increase the energy carried by an electromagnetic wave? A. Increase E keeping ω constant C. Both of the above will increase the energy
B. Increase ω keeping E constant D. Neither of the above will increase the energy
“Only the electric field changing can increase energy. ” “Energy doesn't depend on the amplitude, just the frequency.” “Increasing both will add energy since it has a greater base energy if you increase E and more waves per a period of time if you incease w.” “only velocity of the wave influences energy”
Physics 212 Lecture 23, Slide 25
Checkpoint 1b Which of the following actions will increase the energy carried by an electromagnetic wave? A. Increase E keeping ω constant C. Both of the above will increase the energy
B. Increase ω keeping E constant D. Neither of the above will increase the energy
But then again, what are we keeping constant here? WHAT ABOUT PHOTONS? The energy of one photon is Ephoton = hf = hω/2π Uwave = Nphotons Ephoton = 1/2 ε0E02 Physics 212 Lecture 23, Slide 26
PHOTONS We believe the energy in an e-m wave is carried by photons Question: What are Photons? Answer: Photons are Photons. Photons possess both wave and particle properties Particle: Energy and Momentum localized Wave: They have definite frequency & wavelength (fλ = c) Connections seen in equations: E = hf p = h/λ
Planck’s constant h = 6.63e-34 J-s
Question: How can something be both a particle and a wave? Answer: ~It can’t (when we observe it) What we see depends on how we choose to measure it ! The mystery of quantum mechanics: More on this in PHYS 214 Physics 212 Lecture 23, Slide 27
Calculation 1
y x
An electromagnetic wave is described by: ˆjE0 cos(kz − ωt ) E = where ˆj is the unit vector in the +y direction.
z
Which of the following graphs represents the z-dependence of Bx at t = 0?
X (A)
X (B)
(C)
(D)
E and B are “in phase” (or 180o out of phase)
E = ˆjE0 cos(kz − ωt )
Wave moves in +z direction y E E × B points in direction of propagation x
ˆ 0 cos(kz − ωt ) B = −iB
B
zPhysics 212
Lecture 23, Slide 28
Calculation 2
y
iˆ + ˆj E= E0 cos(kz + ωt ) 2
An electromagnetic wave is described by:
x z
What is the form of B for this wave? −iˆ + ˆj B= ( E0 / c) cos(kz + ωt ) 2
(A)
iˆ + ˆj B= ( E0 / c)cos(kz + ωt ) 2
(C)
(B)
iˆ − ˆj B= ( E0 / c) cos(kz + ωt ) 2
(D) B = −i − j ( E0 / c) cos(kz + ωt )
iˆ + ˆj E= E0 cos(kz + ωt ) 2
ˆ
ˆ
2
Wave moves in –z direction
y E x
+z points out of screen -z points into screen
B E × B points in negative z-direction Physics 212 Lecture 23, Slide 29
Calculation 3 An electromagnetic wave is described by:
E = ˆjE0 sin(kz + ωt )
Which of the following plots represents Bx(z) at time t = π/2ω ω?
(A)
(B)
Wave moves in negative z-direction y +z points out of screen
E
-z points into screen
B
x
E × B points in negative z-direction
(C)
(D)
B = iˆ( E0 / c)sin(kz + ωt ) at ωt = π/2:
Bx = ( E0 / c)sin(kz + π / 2) Bx = ( E0 / c){sin kz cos(π / 2) + cos kz sin(π / 2)}
Bx = ( E0 / c) cos(kz ) Physics 212 Lecture 23, Slide 30
Calculation 4 A certain unnamed physics professor was arrested for running a stoplight. He said the light was green. A pedestian said it was red. The professor then said: “We are both being truthful; you just need to account for the Doppler effect !” Is it possible that the professor’s argument is correct? (λ λgreen = 500 nm, λred = 600 nm)
(A) YES
(B) NO
• As professor approaches stoplight, the frequency of its emitted light will be shifted UP • The speed of light does not change • Therefore, the wavelength (c/f) would be shifted to a smaller value • If he goes fast enough, he could observe a green light !
Physics 212 Lecture 23, Slide 31
Follow-Up A certain unnamed physics professor was arrested for running a stoplight. He said the light was green. A pedestian said it was red. The professor then said: “We are both being truthful; you just need to account for the Doppler effect !” How fast would the professor have to go to see the light as green? (λ λgreen = 500 nm, λred = 600 nm)
(A) 540 m/s
(B) 5.4 X104 m/s
(C) 5.4 X 107 m/s
Relativistic Doppler effect: f ′ = f
f ′ 600 1+ β = = f 500 1− β
1+ β 1− β
36(1 − β ) = 25(1 + β )
Note approximation for small β is not bad: c = 3 X 108 m/s fl v = 5.4 X 107 m/s
(D) 5.4 X 108 m/s
β=
f ′ = f (1 + β )
11 = 0.18 61 1 β = = 0.2 5
Change the charge to SPEEDING! Physics 212 Lecture 23, Slide 32
Physics 212 Lecture 23
Physics 212 Lecture 23, Slide 1
Music Who is the Artist? A) B) C) D) E)
Bob Wills and the Texas Playboys The Viper and His Famous Orchestra Big Sandy and the Fly-Rite Boys Pokey LaFarge and the South City Three Blind Boy Paxton
Why? They were the headliners at the CU Folk and Roots Festival this past weekend. They were wonderful !!
See who you missed at: http://www.folkandroots.org/site/2011-artists
Your Comments “This lecture was fun!” “super hard” “I don't understand what the equations mean with the (kz+-wt).” “Can you explain the E and B electromagnetic waves graph in more detail? I'm still confused. ”
It certainly can be confusing.. We will try to make it clear !! “Sometimes, you can't deny that physics is just awesome. Doppler effect seen from moving stars?? So cool!” “So, according to the Doppler Effect, the only thing I have to do to get XRay Vision is run away from whatever I want to see through? Also: would it work if said thing was running away from me? (Always looking for ways to minimize the work I have to do)” We’ll do a Doppler shift example anyway (& discuss approximations)
05
“Seriously? You mispell "pointing" and then expect us to run at our iclickers so as to see the magical waves they emit? SMH” Physics 212 Lecture 23, Slide 3
Your Comments “PLEASE READ MEEEEEEEEEEEEEEE. <----- Don't post this annoying part =D I set the equations of (energy density )*(Volume) = planck's constant *(c/lambda) . Solving for lambda I see that the wavelength decreases as Volume increases. Does that mean that EM waves loose their wavelength as the Universe expands?” YES! Photons left over from the Big Bang (starting out with the energy that binds electrons to protons = hydrogen atom) have lost a lot of energy as the universe expands (from t = 400,000 years to now) λinitial ≅ 100 nm = 10-7 m, λtoday ≅ 1 mm: the cosmic microwave background
05
Physics 212 Lecture 23, Slide 4
Plane Waves from Last Time
E and B are perpendicular and in phase Oscillate in time and space Direction of propagation given by E X B E0 = cB0 Argument of sin/cos gives direction of propagation
Physics 212 Lecture 23, Slide 5
Understanding the speed and direction of the wave Ex = Eosin(kz-ωt) Ex
t=0
z
Ex
sin( kz − π ) = − cos( kz)
z
2
t = π/2ω
What has happened to the waveform in this time interval? It has MOVED TO THE RIGHT by λ/4
λ/4 ω speed = c = =λ = λf π / 2ω 2π Physics 212 Lecture 23, Slide 6
Checkpoint 1a
“The value at t=0 is 0 for this function and it is positive in the x direction right after” “Since the sine wave is going up, its - the omega t” “The terms within the sin function show that the wave is moving in the +z direction and B is in the y plane so it's By.”
Physics 212 Lecture 23, Slide 7
Checkpoint 1a
No – moving in the minus z direction No – has Ey rather than Ex
Physics 212 Lecture 23, Slide 8
Checkpoint 2a
c=3.0 x 108 m/s
Wavelength is equal to the speed of light divided by the frequency.
c 300, 000, 000 1 λ= = = f 900, 000, 000 3 Check: Look at size of antenna on base unit
Physics 212 Lecture 23, Slide 9
Doppler Shift
The Big Idea As source approaches: Wavelength decreases Frequency Increases
Physics 212 Lecture 23, Slide 10
Doppler Shift for e-m Waves What’s Different from Sound or Water Waves ? Sound /Water Waves : You can calculate (no relativity needed) BUT Result is somewhat complicated: is source or observer moving wrt medium?
Electromagnetic Waves : You need relativity (time dilation) to calculate BUT Result is simple: only depends on relative motion of source & observer
1+ β f′= f 1 − β
1 2
β = v/c β > 0 if source & observer are approaching β < 0 if source & observer are separating Physics 212 Lecture 23, Slide 11
Doppler Shift for e-m Waves f
f’
v or
f
f’
v The Doppler Shift is the SAME for both cases ! f’/f ONLY DEPENDS ON THE RELATIVE VELOCITY
1+ β f′= f 1− β
1 2
Physics 212 Lecture 23, Slide 12
Doppler Shift for e-m Waves A Note on Approximations
1+ β f′= f − β 1
1 2
f ′ ≈ f (1 + β )
β << 1
Remember β > 0 for approach β < 0 for separation
WHY ?? 1/ 2
Taylor Series: Expand
F ( β ) = F (0) + Evaluate:
F ( 0) = 1 F ′(0) = 1
1 + β F (β ) = 1− β
around β = 0
F ′(0) F ′′(0) 2 β+ β + ... 1! 2!
F (β ) ≈ 1 + β
NOTE: F ( β ) = (1 ± β ) n F (β ) ≈ 1± nβ
Physics 212 Lecture 23, Slide 13
Red Shift
Wavelengths shifted higher
wavelength
Frequencies shifted lower
Star separating from us (Expanding Universe)
Our Sun
Star in a distant galaxy Physics 212 Lecture 23, Slide 14
Red Shift “the dopper effect in relation to magnetic fields. Is this used in any practical applications? Is anything moving fast enough on earth that relative to the speed of light that there would be any significant shift? I understand that this makes sense in looking at the universe and motion of planets and stars and such but there is no way you can "run" fast enough to ever see an iclicker signal.”
Star Clusters and Distant, Red Galaxies Near Edge of Galaxy NGC 3370 (Hubble Space Telescope)
Physics 212 Lecture 23, Slide 15
Example
Police radars get twice the effect since the EM waves make a round trip:
f ′ ≈ f (1 + 2β ) If f = 24,000,000,000 Hz (k-band radar gun) c = 300,000,000 m/s v
β
f’
f’-f
30 m/s (67 mph)
1.000 x 10-7
24,000,004,800
4800 Hz
31 m/s (69 mph)
1.033 x 10-7
24,000,004,959
4959 Hz Physics 212 Lecture 23, Slide 16
Checkpoint 2b A) B) C)
ficlicker = 900 MHz
“Running away decreases the frequency that your eyes would see.”
“Running toward the iclicker will cause the frequency we observe to be greater than the source frequency.”
“Moving relative to the iclicker would only change its velocity in relation to you, but not its frequency.”
Physics 212 Lecture 23, Slide 17
Checkpoint 2b ficlicker = 900 MHz
A) B) C)
Need to approach i>clicker
Need to shift frequency UP
(β > 0)
How fast would you need to run to see the i>clicker radiation? 1/ 2
f ′ 1014 1 + β = 9 = 105 = f 10 1 − β
1 + β 10 = 1 − β 10
1010 − 1 1 − 10 −10 β = 10 = 10 + 1 1 + 10 −10
Approximation Exercise:
β ≈ 1 − (2 × 10−10 ) Physics 212 Lecture 23, Slide 18
Waves Carry Energy
Physics 212 Lecture 23, Slide 19
Intensity Intensity = Average energy delivered per unit time, per unit area
1 dU I≡ A dt
Length = c dt
Area = A
dU = u ⋅ volume = u Acdt
I =c u
Sunlight on Earth:
I ~ 1000J/s/m2 ~ 1 kW/m2 Physics 212 Lecture 23, Slide 20
Waves Carry Energy
Physics 212 Lecture 23, Slide 21
Comment on Poynting Vector Just another way to keep track of all this - Its magnitude is equal to I – Its direction is the direction of propagation of the wave
Physics 212 Lecture 23, Slide 22
Light has Momentum! Momentum of light is p = U/c → light can exert pressure!! Recall force from mechanics:
Approximate ∆p by p:
But intensity of light is:
∆p F F= ; P= ∆t A
p U P= = A∆t cA∆t
U I= A∆t
Therefore:
I P= c Physics 212 Lecture 23, Slide 23
Light has Momentum! Momentum of light is p = U/c → light can exert pressure!!
Optical Tweezers
Physics 212 Lecture 23, Slide 24
Checkpoint 1b Which of the following actions will increase the energy carried by an electromagnetic wave? A. Increase E keeping ω constant C. Both of the above will increase the energy
B. Increase ω keeping E constant D. Neither of the above will increase the energy
“Only the electric field changing can increase energy. ” “Energy doesn't depend on the amplitude, just the frequency.” “Increasing both will add energy since it has a greater base energy if you increase E and more waves per a period of time if you incease w.” “only velocity of the wave influences energy”
Physics 212 Lecture 23, Slide 25
Checkpoint 1b Which of the following actions will increase the energy carried by an electromagnetic wave? A. Increase E keeping ω constant C. Both of the above will increase the energy
B. Increase ω keeping E constant D. Neither of the above will increase the energy
But then again, what are we keeping constant here? WHAT ABOUT PHOTONS? The energy of one photon is Ephoton = hf = hω/2π Uwave = Nphotons Ephoton = 1/2 ε0E02 Physics 212 Lecture 23, Slide 26
PHOTONS We believe the energy in an e-m wave is carried by photons Question: What are Photons? Answer: Photons are Photons. Photons possess both wave and particle properties Particle: Energy and Momentum localized Wave: They have definite frequency & wavelength (fλ = c) Connections seen in equations: E = hf p = h/λ
Planck’s constant h = 6.63e-34 J-s
Question: How can something be both a particle and a wave? Answer: ~It can’t (when we observe it) What we see depends on how we choose to measure it ! The mystery of quantum mechanics: More on this in PHYS 214 Physics 212 Lecture 23, Slide 27
Calculation 1
y x
An electromagnetic wave is described by: ˆjE0 cos(kz − ωt ) E = where ˆj is the unit vector in the +y direction.
z
Which of the following graphs represents the z-dependence of Bx at t = 0?
X (A)
X (B)
(C)
(D)
E and B are “in phase” (or 180o out of phase)
E = ˆjE0 cos(kz − ωt )
Wave moves in +z direction y E E × B points in direction of propagation x
ˆ 0 cos(kz − ωt ) B = −iB
B
zPhysics 212
Lecture 23, Slide 28
Calculation 2
y
iˆ + ˆj E= E0 cos(kz + ωt ) 2
An electromagnetic wave is described by:
x z
What is the form of B for this wave? −iˆ + ˆj B= ( E0 / c) cos(kz + ωt ) 2
(A)
iˆ + ˆj B= ( E0 / c)cos(kz + ωt ) 2
(C)
(B)
iˆ − ˆj B= ( E0 / c) cos(kz + ωt ) 2
(D) B = −i − j ( E0 / c) cos(kz + ωt )
iˆ + ˆj E= E0 cos(kz + ωt ) 2
ˆ
ˆ
2
Wave moves in –z direction
y E x
+z points out of screen -z points into screen
B E × B points in negative z-direction Physics 212 Lecture 23, Slide 29
Calculation 3 An electromagnetic wave is described by:
E = ˆjE0 sin(kz + ωt )
Which of the following plots represents Bx(z) at time t = π/2ω ω?
(A)
(B)
Wave moves in negative z-direction y +z points out of screen
E
-z points into screen
B
x
E × B points in negative z-direction
(C)
(D)
B = iˆ( E0 / c)sin(kz + ωt ) at ωt = π/2:
Bx = ( E0 / c)sin(kz + π / 2) Bx = ( E0 / c){sin kz cos(π / 2) + cos kz sin(π / 2)}
Bx = ( E0 / c) cos(kz ) Physics 212 Lecture 23, Slide 30
Calculation 4 A certain unnamed physics professor was arrested for running a stoplight. He said the light was green. A pedestian said it was red. The professor then said: “We are both being truthful; you just need to account for the Doppler effect !” Is it possible that the professor’s argument is correct? (λ λgreen = 500 nm, λred = 600 nm)
(A) YES
(B) NO
• As professor approaches stoplight, the frequency of its emitted light will be shifted UP • The speed of light does not change • Therefore, the wavelength (c/f) would be shifted to a smaller value • If he goes fast enough, he could observe a green light !
Physics 212 Lecture 23, Slide 31
Follow-Up A certain unnamed physics professor was arrested for running a stoplight. He said the light was green. A pedestian said it was red. The professor then said: “We are both being truthful; you just need to account for the Doppler effect !” How fast would the professor have to go to see the light as green? (λ λgreen = 500 nm, λred = 600 nm)
(A) 540 m/s
(B) 5.4 X104 m/s
(C) 5.4 X 107 m/s
Relativistic Doppler effect: f ′ = f
f ′ 600 1+ β = = f 500 1− β
1+ β 1− β
36(1 − β ) = 25(1 + β )
Note approximation for small β is not bad: c = 3 X 108 m/s fl v = 5.4 X 107 m/s
(D) 5.4 X 108 m/s
β=
f ′ = f (1 + β )
11 = 0.18 61 1 β = = 0.2 5
Change the charge to SPEEDING! Physics 212 Lecture 23, Slide 32