Download Discussion Note #32

Discussion Notes and Class Agenda
Physics 240
April 12, 2006
Problem Assignments
Problem
Groups
32.48
1, 3
32.18
5, 7
32.52
2, 4
32.14
6, 8
Review Sessions for Final Exam
Wednesday April 19, 2006
10:00 am—Noon
Room 296 Dennison
and
Thursday April 20, 2006
Noon—4:00 pm
Physics Help Room
Discussion Notes and Class Agenda
Physics 240
April 17, 2006
Problem Assignments
Problem
Groups
33.18
1, 3
33.40
5, 7
33.28
2, 4
33.34
6, 8
Review Sessions for Final Exam
Wednesday April 19, 2006
10:00 am—Noon
Room 296 Dennison
and
Thursday April 20, 2006
Noon—4:00 pm
Physics Help Room
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Discussion: Reflection, Refraction, Dispersion and
Polarization
We now consider how light interacts with mater. Light can be reflected
from smooth, flat surfaces, refracted as it passes between media with
differing indices of refraction, dispersed into component colors when the
refractive index of a medium, and hence the speed at which light travels
in the material, depends on the wavelength of the light, and polarized
with the electric field restricted to a specific orientation in space.
Reflection and Refraction
Reflection is the simplest of these phenomena. For surfaces that are
smooth and flat on a scale size of the wavelength of light, specular
(mirror) reflection obeys the simple relation that the angle of incidence
equals the angle of reflection for the incident and reflected rays of light,
where both angles are defined relative to the normal of the interface.
Specular reflection is the simplest process that can produce an image of
objects from which the light originates. Diffuse reflection directs the
reflected light in random directions and does not form images.
Most generally, light is both reflected and refracted at the interface
between two optically transparent media. The incident and refracted
rays obey Snell’s Law which states that the index of refraction times the
sine of the incident angle in medium a equals the product of the index of
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refraction and the sine of the refracted angle in medium b. Both
specular reflection and refraction are summarized below
 a   r (Specular Reflection)
na sin  a  nb sin b (Snell's Law of Refraction)
Note that angles are defined from the normal to the reflecting/refracting
surface. A ray of light passing from a lower index of refraction to a
higher index of refraction is bent toward the normal. When passing
from a higher to a lower refractive index, the light ray is bent away from
the normal.
For  b above a critical value  c , light traveling from a higher index
medium to a lower index medium can undergo total internal reflection.
For internally reflected light there is no refracted ray transmitted
across the interface between the two media. At the critical angle the
refracted ray is transmitted at  a


2
(90) along the interface
between the two media. Applying Snell’s Law at the critical angle
na sin

2
 na 

n
 b
 nb sin  c
 c  sin 1 
Snorklers and scuba divers are familiar with total internal reflection,
which makes the surface of the water above appear as a sliver mirror
for glancing lines of sight to the surface. Fiber optic wave guides also
employ this effect. A lower index outer layer of the fiber surrounds the
inner fiber made of a higher index material. Light signals can be
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propagated down the fiber without losses at the surface of the inner
fiber despite many reflections at the boundary between the two
materials.
Dispersion
The refractive index in most materials is actually a function of the
wavelength of the light traveling through the medium. The dependence
of the refractive index on wavelength is called dispersion. Dispersion
can occur for electromagnetic radiation other than visible light. Radio
waves traveling through a plasma of charged particles in the upper
regions of the earth’s atmosphere also undergo dispersion, with
different wavelengths traveling at different speeds.
The most familiar demonstration of dispersion is the separation of white
light into its component colors by a glass prism.
The wavelength of light traveling within a refracting medium is

0
n
where 0 is the wavelength of the light in vacuum. For most media, n
decreases with increasing wavelength (decreasing frequency) and
increases with decreasing wavelength (increasing frequency), so violet
light (small wavelength, high frequency) deviates most (is bent most
toward the normal) when entering a higher index medium. Red light
(long wavelength, low frequency) is deviated the least. Many of the
lovely color effects achieved by cut ornaments and gemstones such as
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diamonds are due to the dispersion of the incident light into its
component colors.
Polarization
The solutions we have studied for traveling electromagnetic waves are
linearly polarized, with the electric and magnetic fields each oriented in
a constant direction as the wave propagates through space
E  x, t   Emax cos(kx  t ) yˆ
B  x, t   Bmax cos(kx  t ) zˆ
This wave solution is said to be polarized in the y direction, since the
electric field points in the ŷ direction. By convention, it is the
orientation of the electric field that determines the polarization direction
for a linearly polarized electromagnetic wave.
It is possible to produce linearly polarized light by passing unpolarized
light through a polarizing filter as shown below
Some natural substances, such as the crystal mineral calcite, act as
polarizing filters. In fact, calcite (crystalline CaCO3) also exhibits
birefringence, with a different index of refraction for different
polarization directions. Artificial polarizing filters can also be made by
depositing special molecules on transparent plastic. Polarized
sunglasses are made in this way. If white light of intensity I is passed
through a polarizer, the emerging linearly polarized light has an
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intensity of I/2. If this light is then passed through a second polarizer
(called an analyzer) with its polarizing axis at an angle φ relative to the
initial polarizer, the emerging light will have intensity
I  I max cos2  (Malus's Law)
where Imax is the intensity when the cosine of φ is one, that is, when the
polarizer and analyzer have their polarizing axes aligned.
Linearly polarized light can also be produced by reflection. When light
is reflected and refracted at the surface of a higher index medium (such
as the surface of a body of water) the reflected light will be linearly
polarized parallel to the reflecting surface when the reflected and
refracted rays at the surface form a 90 degree angle
Applying Snell’s Law to this special case we find
na sin  a  nb sin b
na sin  p  nb sin(
tan  p 
nb
na

2
  p )  nb cos  p
(Brewster's Law)
where θp is called the polarizing angle. The polarizing angle is
sometimes referred to as the Brewster angle or Brewster’s angle.
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The phenomenon of linearly polarized light produced by reflection from
horizontal surfaces was a prime motivation in the development of
polarized sunglasses, where the lenses are polarizing filters. The
unwanted glare from the horizontally polarized light reflected from a
horizontal surface (such as the surface of a body of water) can be
eliminated by orienting the polarizing axis of the lenses vertically, to
eliminate the horizontally polarized component of the light reaching the
eyes.
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