Solution to Assignment 9 1. LASIK (Laser Assisted in Situ Keratomileusis) is now a common eye surgery in which a laser
is used to remove a thin layer of cornea, correcting vision in the eye so glasses or contacts do not
need to be worn. Typically, a laser known as an excimer laser is used, with a wavelength of 193
nm, which has been shown to produce high-quality incisions in the delicate tissue of the eye.
(a) In what section of the electromagnetic spectrum does a wavelength of 193 nm place this
laser?
mc
Gamma
wave
X-ray
Ultraviolet
Visible
Infrared
Microwave
Radio
Long wave
The wavelength of 193 nm corresponds to Ultraviolet radiation.
(b) If such a laser has a power of 9.00 milliwatts, and is turned on for pulses lasting 34.0 ns each,
how much energy is delivered by the laser to the eye in each pulse? (Note that, for example, the
number 1.23x10^-7 can be entered on WebAssign as 1.23e-7)
The definition of power is energy per unit time. In this problem, 9 mW corresponds to the
amount of energy delivered by laser in 1s. Therefore, energy delivered by the laser in 34 ns
(1ns = 10-9 s) is 9 * 34 * 10-12 J = 3.06 *10-10 J.
2. The power emitted by the Sun in the form of electromagnetic waves is about 4.00 × 1026 W.
The distance from the Earth to the Sun is about 150 million km, and the radius of the Earth is
approximately 6400 km. Using these numbers, determine:
(a) the intensity of sunlight reaching Earth.
The Intensity of light at a distance ‘r’ away from a source of power ‘P’ is given by
. Therefore we get intensity of sunlight at Earth’s surface is 1410W/m2 .
(b) the total energy every second in the sunlight that is incident on the Earth. For reference, on
your electricity bill, you get charged about $1 for every 3.6 x 107 J of electricity you use, so you
might want to think about the value of one second's worth of sunlight hitting the Earth.
Intensity of light is defined as the amount of energy light transfers across a unit area in unit
time. If the Intensity of light is 1410W/m2 then sunlight carries 1410 J of energy across an
area of 1m2. Therefore the amount of energy hitting the earth’s surface is 1410 * crosssectional area of the earth, which is 1410 * π *(radius of earth)2 = 1.82 *1017 J
3. A vertically polarized beam of light with intensity I0 is incident on a polarizing filter. The
transmission axis of the filter is initially vertical, as shown in the figure. Keeping the polarizing
filter perpendicular to the direction of the incident light at all times, the polarizing filter is rotated
a full rotation (360˚).
(a) Which of the six graphs in the figure correctly shows the intensity of the light emerging from
the polarizer, as a function of the angle of the polarizer’s transmission axis?
Graph 1
these graphs
Graph 2
Graph 3
Graph 4
Graph 5
mc
Graph 6
None of the
Since the intensity of light transmitted by the polarizer, when a light of intensity Io is
, where ∆θ is the angle between the transmission axis
incident, is given by
of polarizer and the polarization axis of light. Since, the light is vertically polarized when
the transmission axis is aligned at zero degrees, the angle of the transmission axis is nothing
but
Therefore, graph of Cos2(θ) is graph 6, which is the solution.
(b) If the vertically polarized incident beam is replaced by an unpolarized beam with an intensity
of I0, which of the six graphs correctly shows the intensity of the light emerging from the
polarizer, as a function of the angle of the polarizer’s transmission axis?
Graph 1
these graphs
Graph 2
Graph 3
Graph 4
Graph 5
mc
Graph 6
None of the
The intensity of light transmitted through a polarizer is equal to half of the intensity of the
incident unpolarized light, irrespective of the alignment of transmission axis of the
polarizer. Therefore the Graph 2 is the answer.
4.
The circle above gives you a reference for the angles referred to in this problem.
A particular beam of light, with an intensity of 500 W/m2, is linearly polarized at an angle of
+8.00° to the vertical. You have three polarizers, which have their transmission axes aligned as
follows:
Polarizer
A
the
transmission
axis
is
at
+18.0°
to
the
vertical;
Polarizer
B
the
transmission
axis
is
at
+84.0°
to
the
vertical;
Polarizer C - the transmission axis is at +25.0° to the vertical.
You will select two of these polarizers, and have the light pass through one and then the other.
The two key ideas used in this problem are:
1. The intensity of the transmitted beam has an intensity given by the equation :
, where ∆θ is the angle between the transmission axis of polarizer and the
polarization axis of light and Io is the intensity of the incident beam.
2. The light beam coming out of the polarizer is linearly polarized along the direction of the
transmission axis of the polarizer.
The light transmitted through the polarizer is
(a) If you want to maximize the intensity of the light emerging from the second polarizer, which
two polarizers should you select, and in what order should the light pass through them?
mc
The light should first pass through A, and then through B
through B, and then through A
they are in
The light should first pass
Use polarizers A and B, but it does not matter which order
The light should first pass through A, and then through C
first pass through C, and then through A
which order they are in
The light should
Use polarizers A and C, but it does not matter
The light should first pass through B, and then through C
light should first pass through C, and then through B
matter which order they are in
they are in
The
Use polarizers B and C, but it does not
Use any two polarizers, and it does not matter which order
Since the intensity of light coming out of the polarizer is proportional to square of the
cosine of the difference in angle between the polarization axis of light and transmission axis
of the polarizer, we need to minimize this angle to get maximum intensity. Since we require
maximum intensity coming out of the 2 filters, we would want to minimize the angles for
the both the filters. Since polarizer A has transmission axis at 18o which is closest to 8o.
Then, the light coming out of polarizer A would have its polarization axis at 18o and
polarizer C has the transmission axis that is closest to 18o. Hence the answer.
(b) For the situation in (a), when you maximize the intensity of the light emerging from the
second polarizer, what is that intensity?
Consider the case when the first polarizer is A. The transmitted beam after passing
through filter A has an intensity of
=484.9W/m2. This beam is
linearly polarized at an angle of 180. When this beam is incident on polarizer C then the
intensity of the beam coming out of polarizer C is
478
2
W/m . (c) If, instead, you want to minimize the intensity of the light emerging from the second
polarizer, which two polarizers should you select, and in what order should the light pass through
them?
mc
The light should first pass through A, and then through B
through B, and then through A
they are in
The light should first pass
Use polarizers A and B, but it does not matter which order
The light should first pass through A, and then through C
first pass through C, and then through A
which order they are in
The light should
Use polarizers A and C, but it does not matter
The light should first pass through B, and then through C
light should first pass through C, and then through B
matter which order they are in
they are in
The
Use polarizers B and C, but it does not
Use any two polarizers, and it does not matter which order
Since the intensity of light coming out of the polarizer is proportional to square of the
cosine of the difference in angle between the polarization axis of light and transmission axis
of the polarizer, we need to maximize this angle to get minimum intensity. Since we require
minimize intensity coming out of the 2 filters, we would want to maximize the angles for
both the filters. Since polarizer B has transmission axis at 84o which differs most from 8o.
Then, the light coming out of polarizer B would have its polarization axis at 84o and
polarizer A has the transmission axis that differs most from 84o. Hence the answer.
(d) For the situation in (c), when you minimize the intensity of the light emerging from the
second
polarizer,
what
is
that
intensity?
Consider the case when the first polarizer is B. The transmitted beam after passing
=29.26W/m2. This beam is
through filter B has an intensity of
0
linearly polarized at an angle of 84 . When this beam is incident on polarizer A then the
intensity
of
the
beam
coming
out
of
polarizer
A
is
2
4.84W/m . For the remaining parts of the problem, we now replace the original polarized
unpolarized light of the same intensity. For reference, the intensity is 500 W/m2, and
polarizers
have
their
transmission
axes
aligned
as
Polarizer
A
the
transmission
axis
is
at
+18.0°
to
the
Polarizer
B
the
transmission
axis
is
at
+84.0°
to
the
Polarizer C - the transmission axis is at +25.0° to the vertical.
light by
the three
follows:
vertical;
vertical;
(e) If you want to maximize the intensity of the light emerging from the second polarizer no,
which two polarizers should you select, and in what order should the light pass through them?
mc
The light should first pass through A, and then through B
through B, and then through A
they are in
The light should first pass
Use polarizers A and B, but it does not matter which order
The light should first pass through A, and then through C
first pass through C, and then through A
which order they are in
The light should
Use polarizers A and C, but it does not matter
The light should first pass through B, and then through C
light should first pass through C, and then through B
matter which order they are in
they are in
The
Use polarizers B and C, but it does not
Use any two polarizers, and it does not matter which order
Since the intensity of light coming out of any filter, when unpolarized light is incident on it,
is just half of the intensity of the incident light and the light is polarized along the
transmission axis of the polarizer. Then choosing the polarizer whose transmission axis is
aligned closest to the first polarizer gives the maximum intensity coming out. The order
doesn’t matter since the intensity coming out after passing through the 2 polarizers is
dependent on the difference in angle between the transmission axes of the two polarizers
used. Hence the answer.
(f) For the situation in (e), when you maximize the intensity of the light emerging from the
second polarizer, what is that intensity?
If unpolarized light of intensity 500 W/m2 is incident on the polarizer A, the intensity of the
light coming out is 250W/m2.This beam is linearly polarized at an angle of 180. When this
beam is incident on polarizer C then the intensity of the beam coming out of polarizer C is
246 W/m2. (g) If, instead, you want to minimize the intensity of the light emerging from the second
polarizer, which two polarizers should you select, and in what order should the light pass through
them?
mc
The light should first pass through A, and then through B
through B, and then through A
they are in
The light should first pass
Use polarizers A and B, but it does not matter which order
The light should first pass through A, and then through C
first pass through C, and then through A
which order they are in
The light should
Use polarizers A and C, but it does not matter
The light should first pass through B, and then through C
light should first pass through C, and then through B
matter which order they are in
they are in
The
Use polarizers B and C, but it does not
Use any two polarizers, and it does not matter which order
Since the intensity of light coming out of any filter, when unpolarized light is incident on it,
is just half of the intensity of the incident light and the light is polarized along the
transmission axis of the polarizer. Then choosing the polarizer whose transmission axis is
aligned in a direction away from that of the first polarizer gives the minimum intensity
coming out. The order doesn’t matter since the intensity coming out after passing through
the 2 polarizers is dependent on the difference in angle between the transmission axes of
the two polarizers used. Hence the answer.
(h) For the situation in (g), when you minimize the intensity of the light emerging from the
second polarizer, what is that intensity?
If unpolarized light of intensity 500 W/m2 is incident on the polarizer B, the intensity of the
light coming out is 250W/m2. This beam is linearly polarized at an angle of 840. When this
beam is incident on polarizer A then the intensity of the beam coming out of polarizer A is
41.4W/m2.