PHY 142!
Assignment 8!
Summer 2013
Reading: Light
Key concepts: Huygens’s principle; reflection; refraction; reflectivity; total reflection; Brewster
angle; polarization by absorption, reflection and Rayleigh scattering.
1.!
Questions about reflected intensity. [Use the formulas on p. 7 of Light.]
a.!
Show that at normal incidence the reflectivity is R =
(n1 − n2 )2
(n1 + n2 )2
, no matter
which index is larger.
2.!
b.!
Show that the sum of the Brewster angles for incidence from medium 1
and from medium 2 is 90°.
c.!
Show that if there is a critical angle it is larger than the Brewster angle.
d.!
Show that R⊥ > R except at normal incidence and when both are 1.
A beam of light in air has a square cross section of width d1 .
d1
It is shown from the side, incident on a glass plate.
a.!
The refracted beam is widened in the dimension
shown, to d2 . (The dimension perpendicular to the
θ1
d2
page is unchanged.) Express d2 /d1 in terms of the
angle of incidence θ1 and the angle of refraction θ 2 .
b.!
Suppose this beam consists of light with E in the plane of incidence, and
let θ1 be the Brewster angle so there is no reflected beam. Express d2 /d1
in terms of n, the refractive index of the glass.
c.!
3.!
With no reflected beam, the total power passing through a cross-section
must be the same in both beams. What is the ratio of the intensities I 2 /I1 ?
A rod made of transparent plastic is immersed in a liquid in a glass
beaker, and is being viewed from the side as shown.
a.!
Despite both the plastic and water being transparent, one can
easily see the rod in the water. How? What light rays allow us
to distinguish the plastic from the water?
b.!
Suppose instead of water the liquid is an oil with the same index of
refraction as that of the plastic. How would this affect our ability to see the
plastic rod? Explain.
PHY 142!
4.!
Assignment 8!
Summer 2013
Explain three things about sunset on a day with only scattered clouds.
a.!
The sun is still visible for several minutes after it has “really” set. That is,
without the atmosphere the sun would disappear earlier than it does with
the atmosphere.
b.!
The sun’s color at sunset is orange-yellow and the intensity of its light is
low enough that we can look directly at it.
c.!
The sky away from the sun has a deep blue color, perhaps with orange
and pink colored clouds.
5.!
You are handed a pair of Polaroid sunglasses on a clear day at the beach. Give
two methods by which you can determine the direction of the transmission axis
of the lenses.
6.!
A beam of unpolarized light is incident at angle θ from a medium with index n1
onto the surface of a medium with index n2 . Comment on the validity of each
statement.
a.!
If n1 < n2 , then at all values of θ other than 0° or 90° the reflected light is at
least partially polarized.
b.!
If n1 > n2 , then there is no value of θ for which the reflected light is totally
polarized.
c.!
If n1 < n2 , then reflection is total only for θ = 90° .
d.!
There is no value of θ for which the reflection is total and the reflected
light is also totally polarized.
7.!
An originally unpolarized beam of light passes through a device and then
through a Polaroid filter. As the filter is rotated about the emerging beam’s axis,
in which case will NO change in transmitted intensity be observed? Explain.
a.!
The device reflected the light from a glass plate (in air) at an angle of
incidence around 40°.
b.!
The device reflected the light from a metal plate (in air) at an angle of
incidence around 40°.
c.!
The device scattered the light through about 90° in a cloudy liquid.
d.!
The device passed the light through a Polaroid filter.
PHY 142!
8.!
Assignment 8!
Summer 2013
Three questions about mirrors.
a.!
The two mirrors used in a clothing store for customers to
see themselves are arranged so that (as seen from above)
they make a right angle as shown. Draw rays from the right
and left shoulders of the person to show that the image seen
is not left-right reversed.
b.!
Draw rays to prove that light entering a corner cube mirror is reflected
straight back.
c.!
Your overall height is h and your eyes are distance d below the top of your
head. Prove that a plane mirror of height 12 h will allow you to see your
entire body in reflection, and determine how far above the floor its bottom
should be placed.
9*.!
An object at the bottom of a pool of
depth d is viewed by a person
looking at the water surface as
shown. The index of refraction of
water is n.
a.!

θ2
d
θ1
Show that the apparent
depth of the object as the
person sees it is given by d′ = (d /n) ⋅ (cosθ 2 /cosθ1 ) . [Assume that the
apparent location is a point on the vertical dotted line above the object.]
b.!
Show that if θ1 is a small angle, then d′ /d ≈ 1/n .
c.!
What value does d′ /d approach as θ1 approaches the critical angle for
total reflection? [What value does θ 2 approach?]
10*.!
Light is incident as shown on a triangular prism made of glass
with refractive index n. It is totally reflected internally. When
the prism is immersed in water (refractive index = 4/3), the
light is no longer totally reflected. Use this information to put
upper and lower limits on the value of n.
30°
PHY 142!
11*.!
Assignment 8!
Summer 2013
Optical fibers are often coated with a
coating
transparent material of a different index of
θ1
refraction. Shown is the surface of such a fiber.
fiber
The fiber has refractive index n and the
coating has index n’. Above the coating is air.
We wish to show that the coating has no effect on total reflection by the fiber.
a.!
Find the angle θ 2 at which the ray strikes the upper surface of the coating.!
b.!
If θ1 is the critical angle for the fiber-air interface, show that θ 2 is the
critical angle for the coating-air interface.
c.!
What happens if θ1 is greater than the critical angle for the fiber-air
interface?
12.!
An object viewed through a plate glass
window appears slightly closer than it is.
Shown is the situation, with the object at
distance x from the window. The ray
y′
y
θ1
indicated emerges from the window
•
x x′
displaced as shown, appearing to have
t
come from the position at distance x’
from the window. You are to find the
distance x − x′ . The glass has thickness t and refractive index n. [Throughout the
problem, use small angle approximations.]
a.!
At what vertical distance y from the horizontal axis does the ray enter the
glass, in terms of x and the angle of incidence θ1 .
b.!
At what vertical distance y’ does it emerge from the glass? [Draw an
enlarged picture of the region where the ray passes through the glass so
the refracted ray can be clearly shown, and find y′ − y .]
c.!
Use the similar triangles to find the answer.
PHY 142!
13.!
Assignment 8!
Summer 2013
A beam of light of intensity I0 impinges at nearly normal
incidence on a film of a transparent material as shown. The
medium above and below the film is air, and the reflectivity at
the surfaces is R. We are interested in the intensities of the
reflected beams a, b and c shown.
a.!
a
b
c
I0
In terms of I0 and R, what are the intensities of beams a,
b and c? [Recall that R + T = 1 , and that (at normal incidence) R for a given
pair of media is the same no matter which medium the light comes from.]
b.!
As an example, let the film be water, for which n ≈ 4 /3 . Find R from the
formula for normal incidence in Question #1. What are the intensities of
the three beams? What can you conclude about the importance of beam c
(and successive reflected beams)?
c.!
Suppose the film is made highly reflecting (R ≈ 1) by evaporating a thin
coating of a metal onto both its surfaces. What can you say about the
relative importance of the three beams in this case?
14.!
15.!
Now we consider the transmitted intensities for light that passes
through the film. Shown are three transmitted beams.
I0
a.!
Find the intensities of the beams in terms of I0 and R.
b.!
If the film is water, what are these intensities? What can
you conclude about the importance of beams b and c?
c.!
As before, let the film be made highly reflecting. What can you
say about the relative importance of these three beams?
a
b
Incident light of intensity I0 moving in the z-direction is polarized along the xaxis. It passes through N filters, at angles π / 2N , 2π / 2N , 3π / 2N , ..., π / 2
relative to the x-axis, so the light emerges polarized along the y-axis.
a.!
Express the final intensity I in terms of I0 and the angle π / 2N .
b.!
For large N this angle is small and we can use the small angle
approximation for the cosine: cosθ ≈ 1 − θ 2 / 2 . Write the formula for I
using this approximation.
c.!
Finally, use the binomial formula (1 + ε )n ≈ 1 + nε (for small ε ) to show
that as N → ∞ , I → I0 .
c