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Chapter 25
Lecture 12April 28, 2005
Electromagnetic waves are transverse
0
0
0 0
0 0 0
cos( )
cos( )
B
P S
E kx t
B kx t
d dd d
dt dt
kE B
E B cBk
E x
B z
E s B A
^
^y
PoyntingVector
0
1S E B
S
2
0 0
~ / : E=cBEB E
Power area notec
S
Poynting Vector and Intensity
• Points in the direction of the wave• the magnitude is the rate of
energy transfer per unit area carried by the wave
20
02avg
EPowerI S
Area c
Average{ [cos(kx-wt)]2 } = 1/2
Traveling Waves
Traveling Waves
Standing Waves between two flat mirrors
EB
My total energy output per unit time is constant
My energy output per unit time and area drops as the
distance2
RNear Earth:P~0.00001 N/m2
Radiation pressureThe theory of relativity: anything moving at the speed of light will carry momentum p=E/c
Light can push stuff!
2
( ) 1
4Sun
Sail Sail Sail Sail
PowerI RF PA A A
c c R
c
I
A
Power
ct
cE
At
p
AArea
Force
1/11
Pressure
Polarization
When E points in one direction the wave is
linearly polarized
There are materials that absorb waves when E points in one direction
E
… but not in the other
●
E
Points out of the screen
There are also other polarizations for which
E changes direction
Polarizers at angles reduce the intensity
Selects one polarization
Only the projection onto the transmission
axis gets through I=I1 cos2
I=I1=I0/2
I=I0
Crossed polarizers transmit approximately what fraction of an electromagnetic wave?
1. 0%
2. 25%
3. 50%
4. 75%
5. 100%
What is light? I believe light is a stream of fast moving particles. This explains why and how light
reflects and refracts.
I can also understand how and why light reflects and refracts if I assume
it is a wave.HUYGENS’ PRINCIPLE
If light is a wave it can should be able to go around small obstacles…and it
does!YOUNG’S INTERFERENCE EXPT.
My equations predicted that light is a high frequency electromagnetic wave in1865.
So, is light a wave or a particle?
Since it sometimes behaves like one and sometimes like the other it is neither.
Instead of trying to force it into some label convenient to us we should find out its
properties.
In many cases light behaves like a wave, but sometimes (when quantum effects are
important) it behaves like a particle.
The ray approximation
When light propagates its wave nature is hidden if
• We never look at distances of the order of (or smaller)• All obstacles have typical sizes much larger than
d
The wave nature of light is not important
for d >>
Light behaves as a ray. In uniform media it travels in a straight
line
The ray approximation
d ~d d
For smaller distances (d ~ ) the wave nature begins to show
up
For d << the wave nature is central in
understanding light’s behavior
When looking at features smaller than the interference of
light waves shows up
The shortest time principle – FERMAT’S PRINCIPLE
When light behaves as a ray and travels from point A to point B it follows the path that gets it to B in the shortest time possible
In a uniform medium where the light speed is c1 …
A
B
This path is longer
This is the shortest path
… for constant speed the shortest path takes the least amount of time
In uniform media light rays travel in straight lines
A B
L
When light behaves as a ray and travels from point A to point B it follows the path that gets it to B in the shortest time possible
Let us look at a reflected ray
mirror
h
x
L - x
x is such that it takes the least amount of time to go from A to B
22 hx 22 hxL
2222 11)( :time hx
vhxL
vxt
110 0
2222 hx
x
vhxL
xL
vdx
dt(x)
'
sin sin
sin1
sin1
0
'
'
'vv
Speed of light in the medium
I. The law of reflection
II. The path of a light ray is reversible.
III. The path of a light ray in vacuum defines what is meant by “a straight line”.
1
25.17
• The reflecting surfaces of two intersecting flat mirrors are at an angle of θ. If a light ray strikes the horizontal mirror, show that the emerging ray will intersect the incident ray at an angle of β=180º-2
Exercise 25.17
This looks like an application to the reflection formula and a bit of geometry
When light behaves as a ray and travels from point A to point B it follows the path that gets it to B in the shortest time possible
Look at a ray going form one medium with v1 to another with v2
v1
v2
h
h
A
B
x
L - x
22 hx
22 hxL
22
2
22
1
11)( hxL
vhx
vxt
22
11
222
221
sin1
sin1
1
1
: 0
vv
hxL
xL
vhx
x
v
dx
dt(x)
1
2
Define the index of refraction:
v
cn
mediumin light of speed
in vacuumlight of speed
2211
22
11
sin sin
sin sin
nn
v
c
v
c
v1
v2
A
B
1
2
Then under refraction,
Index of Refraction
Snell’s law of refraction
n2 sinn1sin1
n2
n1
Given two slabs of transparent material of equal thickness, Fermat’s principle means that the part of a ray passing through the medium with the higher index of refraction is ______ the part passing through the lower index medium.
1. longer than
2. equal to
3. shorter than
4. perpendicular to
5. parallel to
25.13
• When the light in the figure passes through the glass block, it is shifted laterally by the distance d. If n = 1.76, find the value of d.
Exercise 25.13
This looks like an application to the refraction formula and a bit of geometry
2
21
2
21
212
cos
sin
cos
sin
sin cos
θ
θθhd
h
dL
d
L
h
h
L
1122 sin sin nn
d
cm 39.09428.0
1827.02
34.0
5.01sin 5.1
6/ ,51 ,1
2
2
121
d
.nn
Larger than 1
When light goes from medium 1 to medium 2 with n1 > n2
12
12 sin sin
n
n
If we increase 1 the right hand side grows
Eventually, when sin1 = n2/n1 we get sin2=1
If we increase 1 beyond that the wave in medium 2 disappears
The ray suffers total internal reflection
θc is when sinθ2=1 i.e. n1/n2 sin θc =1
Total internal reflection
Find the critical angle for a ray of light in glass
Critical Anglen1 sin1 n1 sinC
n2 sin2 n2 sin(90o)
sinc n2
n1
for n1 n2
Air: n2 = 1 Glass: n1 = 1.5 --> c = 42o
Two Rt Angle PrismsNo Loss of Light; use inoptical instruments
Fiber Optics
core, n1
clad, n2 < n1
n1
n2
Huygen’s principle (1678)
Each point on a wave front is a source of secondary spherical wavelets.
Constructive interference creates the new wave front.
Endoscope
"Foreign Body" in the Stomach Swallowed QuarterHere is a quarter which a young man swallowed and which is lying in the stomach. These are easily removed with a wire snare or device for grasping a coin.
Why is the sky blue?
why are sunsets red?
If n depends on we get dispersion
Dispersion
blue bent more than red
Iscattered=λ-4
longer wavelengthless scattered more scattered
Rainbows
400
420
Primary
520
Secondary
Colors Reversed
Examples
24.22
• At what distance from a 100W electromagnetic wave point source does Emax=15V/m
24.22
24.26
• A possible means of space flight is to place an absorbing sheet into orbit around the Earth and then use the light from the Sun to push this “solar sail.” Suppose a sail of area 6105 m2 and mass 6000kg is placed in orbit facing the Sun.
a) what is the force exerted on the sail? b) What is the sail’s acceleration?c) How long does it take the sail to reach the Moon,
3.84108 m away? Ignore all gravitational effects, assume that the acceleration calculated in part b) remains constant, and assume a solar intensity of 1340 W/m2
24.26
24.35
• An important news announcement is transmitted by radio waves to people sitting next to their radios, 100 km from the station, and by sound waves to people sitting across the news room, 3M from the newscaster. Who receives the news first? Explain. Take the speed of sound in air to be 343 m/s.
Exercise 24.35
• the sound and radio waves start at the same time• sound covers a distance d in a time d/v• radio waves cover a distance L in a time L/c
3sound
34
light 8
3 m s 8.75 10 s
343 m/s
100 103.33 10 s
3 10 /
t
mt
m s
Light wins
24.41
In the figure, suppose that the transmission axes of the left and right polarizing disks are perpendicular to each other.
Also, let the center disk be rotated on a the common axis with angular speed . Show that if unpolarized light is incident on the left disk with an intensity Imax, the intensity of the beam emerging from the right disk is
This means that the intensity of the emerging beam is modulated at a rate four times the rate of rotation of the center disk. Hint: Use the trig. identities
cos2=(1+ cos2)/2 and sin2=(1- cos2)/2
max
1(1 cos 4 )
16I I t
Exercise 24.41
• c is so large that the polarizers appear frozen to a “bit” of light ... At time t the rotation angle will bet• the intensity is decreased by (cos )2
Polarization after the 1st polarizer
Polarization after the 2nd polarizer
Polarization after the 3rd polarizer
0 max
1
2I I 2
01 cos II
Imax
2
2 1
2
0
cos / 2
cos cos / 2
I I
I
24.21
2
12
F=-kx x=Acos( t+ )
1
2spring
f fT
kSHO
m
gPendulum
L
PE kx Energy PE KE
total 0
0
2 2 2 20
2 20
sin( )
/
( / ) ( )
/tan
ma F kx bv F t
F mA
b m
b m
2
)2/(
2
)cos(
m
b
m
k
tAex tmb
/0
t mE E e where
b
( , ) sin( )
2
D x t A kx t
k vk
dampled SHO
driven and damped SHO
traveling waves
2 2
2 2 2
1d d
dt c dx
E E 8
0 0
13 10 m/sc
0
0
0 0
cos( )
cos( )
E kx t
B kx t
E cB
E x
B z
^
^2
av 0 0
1
2u E
0
1S E B
2
0 0
~ /EB E
Power areac
S2
0
02avg
EPowerI S
Area c
p=E/c
1 1 / 1 PowerP
F p E c I
A A t A t A c c
I=I0 cos2
LL S
S
v vf f
v v
positive is from L to Spositive v will use (+), negative v (-)
sin(a+a) + sin(a-a)=2 sin(a) cos(a)
string
Tv