Assignment 11- Solutions - Boston University...
Transcript of Assignment 11- Solutions - Boston University...
Assignment 11- Solutions
1)
The picture above shows a source of light (the red dot), with the light shining on the origin,
which is marked by the position of the green dot. The horizontal axis is the interface separating
one medium from another. Some of the light reflects back into the first medium, and some
refracts into the second medium.
Choose the three correct statements about this situation from the list below.
The first medium (where the source is) has a higher index of refraction than the second
medium.
FALSE- since The angle inside the blue region is smaller than outside ����� � ����
The first medium has the same index of refraction as the second medium.
FALSE- same.
The first medium has a lower index of refraction than the second medium.
TRUE-same reason.
If the index of refraction of the second medium was increased, the angle of the reflected
beam would change.
FALSE- for the reflected wave the interface has acted like a plane mirror, it is reflected at the
same angle, regardless of the second medium’s refraction index.
Changing the index of refraction of the second medium would have no effect on the
angle of the reflected beam.
TRUE- as the previous one, interfaces acts like a mirror.
If the light source was moved to the position of the purple dot in the second medium, and
the light was incident on the origin again, the light would definitely experience total internal
reflection.
FALSE- Snell’s law doesn’t change if we reverse the direction of the light rays. As long as the
angle in the blue region is the same we have the same equation:
����� sin ����� � ���� sin ���� And so the outgoing ray will be at the same angle. The only difference is that we won’t have the
reflected ray outside (instead we get one inside.)
If the light source was moved to the position of the purple dot in the second medium, and
the light was incident on the origin again, some light would refract out into the upper medium,
traveling toward the position of the red dot.
TRUE- described above.
If the light source was moved to the position of the purple dot in the second medium, and
the light was incident on the origin again, some light would refract out into the upper medium,
but the light would not travel toward the position of the red dot.
FALSE
2)
The picture above shows a source of light in the lower medium (the purple dot), with the light
shining on the origin, which is marked by the position of the green dot. The horizontal axis is the
interface separating one medium from another. In this case, all of the light reflects back into the
lower medium.
Choose all the correct statements about this situation from the list below.
The lower medium (where the source is) has a larger index of refraction than the upper
medium.
TRUE- We are observing total internal reflection while going from the lower medium to the
upper one. This means that there exists a critical angle beyond which this happens and critical
angles are only for going from larger index of refraction to smaller. So lower medium has larger
index.
The lower medium has the same index of refraction as the upper medium.
FALSE- read above
The lower medium has a smaller index of refraction than the upper medium.
FALSE- read above
The angle of incidence in this case is greater than the critical angle for total internal
reflection to occur.
TRUE- read above.
The angle of incidence in this case is smaller than the critical angle for total internal
reflection to occur.
FALSE- read above.
If the light source was moved to the position of the red dot in the upper medium, and the
light was incident on the origin again, the light would definitely experience total internal
reflection.
FALSE- there won’t be total internal reflection in going from lower index to higher and, as we
found before, the lower medium has higher index of refraction.
If the light source was moved to the position of the red dot in the upper medium, and the
light was incident on the origin again, some light would refract into the lower medium, traveling
toward the position of the purple dot.
FALSE- Some light would indeed refract into the lower medium, but it won’t be towards the
purple dot. This we can see from Snell’s law (blue for the lower medium):
�� sin �� � ����� sin �����
But we know that for the ����� in the picture no �� can be found (����� is beyond the critical
angle), and that is why it gets reflected. So if we put some �� in the equation, the ����� that we
get will never be the angle in the diagram.
If the light source was moved to the position of the red dot in the upper medium, and the
light was incident on the origin again, some light would refract into the lower medium, but the
light would not travel toward the position of the purple dot.
TRUE- see above.
3)
This picture shows a light source at point A, at the top left of the picture. The light which refracts
into the lower medium eventually reaches point B, at the bottom right of the picture. Point A is
located at x = 0 m, y = +5 m, and point B is located at x = +10 m, y = -5 m. The upper medium is
air, with an index of refraction of n = 1.00. The interface separating the two media lies along the
line y = 0. Take the speed of light in vacuum to be 3.00 x 108 m/s.
An interesting fact about light is that the path it takes
from A to B always minimizes the total light travel time.
In other words, any other path from A to B, including
the shortest-distance path, would take the light more
time. One can show that this is equivalent to Snell's law.
In the case shown in the picture, the time it takes the
light to travel from point A to the green point, on the
interface, is 26.0 ns. By using Snell's law and
considering the geometry, answer the following.
(a) What is the x-coordinate of the green point?
We can find the distance between A and the green spot
using � � ��:
��
��
��
��
�
� �
�� � �� � 3 � 10� �� � 26.0 ns � 7.8 m
Now this is the hypotenuse of the upper triangle. For �� we thus have:
cos �� � 57.8 ' sin �� � (1 ) cos� �� � 0.768
The horizontal position of the green dot O is therefore:
' *+ � 7.8 m � sin �� � 5.99 m
x = 5.99 m.
(b) What is the index of refraction of the second medium?
To figure out the index we need to use Snell’s law and for that we need the sine of the angles
which is the ratio of opposite over hypotenuse and so:
����
� sin ��sin ��
�*+��
10 ) *+��
We first need to find OB, which is the hypotenuse of the lower triangle:
�� � (5� - .10 ) *+/� � 6.41 m
�� � 1.23
(c) What is the total time, in nanoseconds, it takes the light to travel from point A to point B?
�� � ����
� ��1/��
� 6.41 m � 1.233 � 10� �
�� 26.0 ��
Which is the same as the time in the first region (this is not a coincidence, if the vertical
distance is the same the times are the same.) So the total time will be 2 � 26 �� � 52.2 ��.
4)
A lamp emitting red light is placed at the bottom of a tank of liquid with depth H = 18.0 cm and
index of refraction n = 1.56 (for red light). Above the liquid is air, which has an index of
refraction of 1.00. On the surface of the liquid above the light, an observer sees a bright circle
with radius R.
(a) Why does the observer see a bright circle on the
surface?
same as the Since the lamp is the region of higher index of
refraction, not all of its light-rays will be able to escape
into air. The ones that hit the interface at an angle larger
than the critical angle will undergo total internal
reflection and beyond that point the observer outside
doesn’t see light rays coming out. The angle depends on
the horizontal distance on the interface from the point
right above the lamp.
Because the lamp is circular. False.
Because the pupils in the observer's eyes are circular. False.
Because light striking the surface outside of the circle experiences total internal
reflection. True.
(b) Calculate the radius of the bright circle seen by the observer.
The outermost ray that can come out is at the critical angle, since beyond that the rays will
reflect. First, the ray going at �3 makes the hypotenuse 4:
4 � (5� - 6�
The radius R of the bright circle on the interface is related to �3 by:
�3
�3
5
6
4
But also:
sin1� �
5� -6� �' 5 �
(c) From the list below, choose the two correct statements. The first three statements relate to
light that travels from the lamp toward point A, along the solid line in the figure. The other three
statements relate to what happens when the lamp emits gree
the index of refraction of this liquid decreases as the wavelength of light increases
for most materials.
If a beam of light travels directly from the lamp to the surface at point A, it refracts out into
the air and goes away infinitely far from the lamp.
FALSE- A is beyond �3, total internal reflection.
If a beam of light travels directly from the lamp to
infinitely far from the lamp along the surface of the liquid.
FALSE- A is not at �3, total internal reflection.
If a beam of light travels directly from the lamp to the surface at point A, it reflects back
into the liquid.
If the lamp is adjusted to emit green light instead, the radius of the bright circle
decreases.
If the lamp is adjusted to emit green light instead, the radius of the bright circle stays the
same.
If the lamp is adjusted to emit green lig
sin �3 � 54 � 5
√5� - 6�
sin �3 � 1� � 1
1.56 � 0.641 � 5
√5� - 6� ' (5� - 6� � �5
6� � ��5� � .�� ) 1/5�
� 6√�� ) 1 � 18.0 cm
√1.56� ) 1 � 15.0 cm
(c) From the list below, choose the two correct statements. The first three statements relate to
light that travels from the lamp toward point A, along the solid line in the figure. The other three
statements relate to what happens when the lamp emits green light instead of red light. Note that
the index of refraction of this liquid decreases as the wavelength of light increases
If a beam of light travels directly from the lamp to the surface at point A, it refracts out into
the air and goes away infinitely far from the lamp.
, total internal reflection.
If a beam of light travels directly from the lamp to the surface at point A, it goes away
infinitely far from the lamp along the surface of the liquid.
, total internal reflection.
If a beam of light travels directly from the lamp to the surface at point A, it reflects back
If the lamp is adjusted to emit green light instead, the radius of the bright circle
If the lamp is adjusted to emit green light instead, the radius of the bright circle stays the
If the lamp is adjusted to emit green light instead, the radius of the bright circle increases.
(c) From the list below, choose the two correct statements. The first three statements relate to
light that travels from the lamp toward point A, along the solid line in the figure. The other three
n light instead of red light. Note that
the index of refraction of this liquid decreases as the wavelength of light increases - this is true
If a beam of light travels directly from the lamp to the surface at point A, it refracts out into
the surface at point A, it goes away
If a beam of light travels directly from the lamp to the surface at point A, it reflects back
If the lamp is adjusted to emit green light instead, the radius of the bright circle
If the lamp is adjusted to emit green light instead, the radius of the bright circle stays the
ht instead, the radius of the bright circle increases.
5)
Binoculars generally use pairs of prisms in which the light experiences total internal reflection.
Each prism (in blue on the diagram) is right-angled, with the other two angles being 45°. A
diagram of the path followed by light as it travels through the prisms to your eyes is shown in the
figure above. If the prisms are surrounded by air, determine the minimum index of refraction of
the prism material.
In the figure we see that the vertical light ray incident
on the long edge is perpendicular to the edge and
therefore the angle of incidence with respect to the
normal is �� � 0 (notice it’s not 908!). Therefore from
Snells law �� sin �� � �� sin �� we get �� � 0 as well.
So the light goes through the long edge without
refraction. Now, in order to make sure that the light
emerging from the long edge does not leave the prism
through the shorter edges we need it to undergo total
internal reflection from the shorter edges. If we look
at the figure, this means that 458 has to be larger than
the critical angle of going from the prism to air. You
can observe that if the light reflects off the short edge
to the left at 458 it hits the right one also at 458 and the same thing repeats and the light leaves
the prism through the long edge to the top. So:
sin 458 � sin �3 � �����9:
� 1�
' √22 � 1
� ' � � 2√2 � √2 ; 1.41
So the minimum index is � � 1.41.
458
�
458
458 458
458
6)
As shown in the figure above, a beam of red and violet light is incident along the normal to one
surface of a right-angled triangular glass prism. The glass has an index of refraction of 1.52 for
red light, and 1.54 for violet light. The prism is surrounded by air, which has an index of
refraction of 1.00. Note that you will probably find it helpful to draw a sketch showing how the
red and violet beams travel from the point at which they enter the prism to the side ab of the
prism.
(a) If the angle at vertex a of the prism is = 24.0 degrees, determine the angles of refraction for
the red and violet beams that emerge from the prism from the side ab. Show these refracted
beams on your sketch. The angles of refraction are measured, as usual, from the normal to the
surface.
angle of refraction for the red light is
Just as in previous problem, From Snell’s law �� sin �� ��� sin ��since the light coming from the left impinges on
the vertical edge at right angle, the angle of incidence
relative to the normal to the surface is �� � 0 and not 90.
Therefore �� � 0 too and the light goes through without
refraction.
After this it hits the hypotenuse at angle �9:. To finc this
angle we use the fact that in the figure < - �9: � 90 (the
dashed line is the normal to the interface. Then we notice
that < is also in the small right-angled triangle to the top
and < - 248 � 90. From these two relations we find
�9: � 248. This is the angle for both red and violet light.
To find the angle of refraction of red light we use Snell’s
law with the index for red:
�9:
����
248
���� � 1
�9:
<
�9: sin �9: � ���� sin ���� �9: � �=�> � 1.52, ���� � 1 sin ���� � 1.52 sin 248 � 0.6182 ���� � 38.178
angle of refraction for the violet light is
�9: � �@9���� � 1.54, ���� � 1 sin ���� � 1.54 sin 248 ���� � 38.788
(b) What is the smallest index of refraction the prism could have for light to experience total
internal reflection when it is first incident on the side ab, for the geometry described above?
Assume the light enters the prism as in the diagram.
For this to happen we need �9: � 248 to be the critical angle. Thus:
sin �� � sin 248 � �����9:
� 1�
� � 2.459
7)
Two speakers, which are separated by a distance d = 3.00 m, produce sound waves with the
same amplitude, phase and frequency. You stand a distance of 4.00 m directly in front of the left
speaker, on the dashed line shown in the diagram.
(a) Assume the speed of sound to be 340 m/s. What is the lowest frequency the sound can have
to produce completely constructive interference at your location?
we need Δ4 � �B for constructive interference. But what � should we chose? We want the
lowest frequency. According to CB � � � 340�/� lowest frequency corresponds to the largest
wavelength. Now since Δ4 is also fixed from the picture, �B is a constant and largest B requires
choosing the smallest nonzero � so � � 1. Therefore:
Δ4 � (D� - �� ) D � B B � (3� - 4� ) 4 � 5 ) 4 � 1m C � �
B � 340 6E (b) What is the lowest frequency the sound can have to produce completely destructive
interference at your location?
This time the smallest nonzero � is 1/2. So:
Δ4 � (D� - �� ) D � 12 B
B � 2m C � �
B � 170 6E 8)
Two identical speakers are pointed at one another, along the x-axis. The speakers, which are in
phase with one another, broadcast identical sound waves at a frequency of 170 Hz. Assume the
speed of sound to be 340 m/s. Speaker A is positioned at x = 0, and speaker B is located at x =
+6.20 m.
248
�9:
D
�
(D� - ��
(a) In between the speakers, there are a few locations at which the two waves produce
completely destructive interference. What is the x-coordinate of the point closest to speaker B at
which completely destructive interference occurs?
First let us find the wavelength:
B � �C � 340
170 � 2�
We need to find points of destructive interference which satisfy Δ4 � F� - ��G B
Δ4 � 4� ) 4� � * ) .� ) */ � 2* ) � � H� - 12I B
2* � � - H� - 12I B
We are looking for the point closest to B and in between the two sources. So we have to have:
* J � K 2* J 2� ' H� - 1
2I B J �
Closest to B means the largest x that satisfies the above condition and in turn the largest �.
Now all we need to do is to count from n�0 up and find the largest:
λ�2m
Hn- 12I λK λ
2 �1m , 3λ2 �3m , 5λ
2 �5m , 7λ2 �7m…
Since � � 6.20 m here the largest n that fits is the one that gives 5�. Using that, we find:
4� � *
�
4� � � ) *
� �
2* � � - H� - 12I B � 6.20 m - 5m � 11.2m
* � 5.6m
(b) In between the speakers, there are also a few locations at which the two waves produce
completely constructive interference. How many locations are there, along the line between the
speakers, at which completely constructive interference occurs?
We know that at the point half-way between the two sources the waves interfere
constructively. The next points of constructive interference after that, moving towards 2
happens where Δ4 � �B which are at:
* � 12 .� - �B/ � 1
2 .� - � � 2m/ � �2 - �.1�/ J �
�.1�/ J �2 � 3.10 m
� N 3
The largest � that makes * fit in the distance � this time is � � 3 since � � 6.20 m. From the
other side, approaching source 1, the last � we can have will be )3. So the possible �’s are:
)3, )2, )1,0,1,2,3 and their number is
2�OPQ - 1 � 2 � 6 - 1 � 7
For sources that are in phase, this is always the case. (If they become slightly, or completely
out of phase, we lose the one in the middle and the number will be 2�OPQ. )
9)
The pattern above is produced by interference between identical waves emitted by two sources.
Red denotes positive displacement of the medium, while blue denotes negative displacement.
Black indicates zero displacement. At the orange point near the top, we
interference is occurring. The distance the orange point is from the left source is the length of the
purple line; the distance the orange point is from the right source is the length of the green line.
(a) Select all the true statements about this situation.
For constructive interference to occur at the orange point, the net displacement must be non
zero at all times at that point.
FALSE- only the nodes have the same amplitude (zero) at all times and the anti
will be oscillating between a max and min.
The pattern above is produced by interference between identical waves emitted by two sources.
Red denotes positive displacement of the medium, while blue denotes negative displacement.
Black indicates zero displacement. At the orange point near the top, we say that constructive
interference is occurring. The distance the orange point is from the left source is the length of the
purple line; the distance the orange point is from the right source is the length of the green line.
ments about this situation.
For constructive interference to occur at the orange point, the net displacement must be non
only the nodes have the same amplitude (zero) at all times and the anti-
be oscillating between a max and min.
The pattern above is produced by interference between identical waves emitted by two sources.
Red denotes positive displacement of the medium, while blue denotes negative displacement.
say that constructive
interference is occurring. The distance the orange point is from the left source is the length of the
purple line; the distance the orange point is from the right source is the length of the green line.
For constructive interference to occur at the orange point, the net displacement must be non-
-nodes (peaks)
For constructive interference to occur at the orange point, the distance from each source to
the orange point must be an integer number of wavelengths.
FALSE- not the individual 4� and 4� but 4� ) 4� is the thing that has to be �B. Each of 4� and
4� are free to have values different from �B.
For constructive interference to occur at the orange point, the path-length difference for
the orange point must be an integer number of wavelengths. The path-length difference is the
distance the point is from the left source minus the distance the point is from the right source.
TRUE- see above. The waves must overlap in phase with each other.
In this particular case, the orange point is two wavelengths farther from the left source
than it is from the right source.
TRUE- from the figure, we know that the region around the perpendicular bisector to the line
joining the sources has constructive interference. The region where the orange dot sits is the
second bright region to the right of the bright region in the middle and:
Δ4 � 4� ) 4= � 0B K �R� STURV� Δ4 � 4� ) 4= � 1B K 1�� STURV� SRUD� Δ4 � 4� ) 4= � 2B K 2�� STURV� SRUD�
So the distance of the orange dot from the source to the left (4�) is greater that its distance
from the one to the right (4=) by -2B.
(b) In this case, what is the distance between the two sources, in terms of wavelengths?
The two sources are sitting on bright regions (although it’s somewhat hard to recognize). So
they are on peaks. Let’s say the red regions are maxima and blue ones are minima. From one
maximum to the next is one B. If we count the number of red regions on the line joining the
sources (disregarding the first one the left source is sitting on) we will find their distance in
terms of B: 5B
B B B B B
The pattern above is produced by interference between identical waves emitted by two sources.
Red denotes positive displacement of the medium, while blue denotes negative dis
Black indicates zero displacement. At the orange point near the top, we say that destructive
interference is occurring. The distance the orange point is from the left source is the length of the
purple line; the distance the orange point is from
(a) Select all the true statements about this situation.
Because the orange point is in a region of destructive interference, the net displacement
at the orange point is zero at all times.
TRUE- at points of destructive interference (nodes) the waves are 180 out of phase at all times.
For destructive interference to occur at the orange point, the distance from each source to the
orange point must be an integer number of wavelengths plus half a wav
FALSE- since sources are in phase,
The pattern above is produced by interference between identical waves emitted by two sources.
Red denotes positive displacement of the medium, while blue denotes negative dis
Black indicates zero displacement. At the orange point near the top, we say that destructive
interference is occurring. The distance the orange point is from the left source is the length of the
purple line; the distance the orange point is from the right source is the length of the green line.
(a) Select all the true statements about this situation.
Because the orange point is in a region of destructive interference, the net displacement
at the orange point is zero at all times.
points of destructive interference (nodes) the waves are 180 out of phase at all times.
For destructive interference to occur at the orange point, the distance from each source to the
orange point must be an integer number of wavelengths plus half a wavelength.
since sources are in phase, Δ4 � F� - ��G B gives nodes, not integer multiples of
The pattern above is produced by interference between identical waves emitted by two sources.
Red denotes positive displacement of the medium, while blue denotes negative displacement.
Black indicates zero displacement. At the orange point near the top, we say that destructive
interference is occurring. The distance the orange point is from the left source is the length of the
the right source is the length of the green line.
Because the orange point is in a region of destructive interference, the net displacement
points of destructive interference (nodes) the waves are 180 out of phase at all times.
For destructive interference to occur at the orange point, the distance from each source to the
gives nodes, not integer multiples of B.
For destructive interference to occur at the orange point, the path-length difference for
the orange point must be an integer number of wavelengths plus half a wavelength. The path-
length difference is the distance the point is from one source minus the distance the point is from
the other source.
TRUE- see above.
In this particular case, the orange point is 1.5 wavelengths farther from the left source than it
is from the right source.
FALSE- like the previous problem, the central region has Δ4 � 0. The orange line is on the
second dark region to the left, so: Δ4 � 4� ) 4= � ) 1
2 B K 1�� �WSX STURV� YTC� Δ4 � 4� ) 4= � ) 3
2 B K 2�� �WSX STURV� YTC�
Since it’s closer to the source to the left 4� J 4=.
(b) In this case, the sources are separated by a distance of 15.0 cm, and the distance the orange
point is from the source on the right is 24.6 cm. How far is the orange point from the source on
the left?
We just found that Δ4 � 4� ) 4= � ) Z� B. We are given 4= � 24.6 cm. If we knew B we could
solve for 4�. But the figure is the same as the previous problem and there we found that the
distance between the sources was � � 5B and here we know that � � 15.0 cm. So we get:
B � 15.0 cm5 � 3.0 cm
4� � 4= ) 32 B � 24.6 cm ) 9
2 cm � 20.1 cm