Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 34

The Wave Nature of Light; Interference - all with Video Answers

Educators

+ 1 more educators

Chapter Questions

Problem 1

IThe Problems in this Section are ranked I, II, or III according to
estimated difficulty, with (I) Problems being easiest. Level (III)
Problems are meant mainly as a challenge for the best students, for
"extra credit." The Problems are arranged by Sections, meaning that
the reader should have read up to and including that Section, but
this Chapter also has a group of General Problems that are not
arranged by Section and not ranked.
$$\begin{array}{l}{\text { (II) Derive the law of reflection- namely, that the angle of }} \\ {\text { incidence equals the angle of reflection from a flat }} \\ {\text { surface-using Huygens' principle for waves. }}\end{array}$$

Sandeep Desai

Sandeep Desai

Numerade Educator

Problem 2

(I) Monochromatic light falling on two slits 0.018 $\mathrm{mm}$ apart
produces the fifth-order bright fringe at a $9.8^{\circ}$ angle. What is
the wavelength of the light used?

Farhanul Hasan

Numerade Educator

Problem 3

(1) The third-order bright fringe of 610 $\mathrm{nm}$ light is observed
at an angle of $28^{\circ}$ when the light falls on two narrow slits.
How far apart are the slits?

Derek Walkama

Numerade Educator

Problem 4

(II) Monochromatic light falls on two very narrow slits
0.048 $\mathrm{mm}$ apart. Successive fringes on a screen 6.00 $\mathrm{m}$ away
are 8.5 $\mathrm{cm}$ apart near the center of the pattern. Determine
the wavelength and frequency of the light.

Farhanul Hasan

Numerade Educator

Problem 5

(II) If 720 -nm and 660 -nm light passes through two slits
0.68 $\mathrm{mm}$ apart, how far apart are the second-order fringes
for these two wavelengths on a screen 1.0 $\mathrm{m}$ away?

Derek Walkama

Numerade Educator

Problem 6

(II) A red laser from the physics lab is marked as producing
632.8 -nm light. When light from this laser falls on two
closely spaced slits, an interference pattern formed on a wall
several meters away has bright fringes spaced 5.00 $\mathrm{mm}$ apart
near the center of the pattern. When the laser is replaced by
a small laser pointer, the fringes are 5.14 mm apart. What is
the wavelength of light produced by the pointer?

Farhanul Hasan

Numerade Educator

Problem 7

(II) Light of wavelength $\lambda$ passes through a pair of slits sepa-
rated by 0.17 $\mathrm{mm}$ , forming a double-slit interference pattern
On a screen located a
distance 35 $\mathrm{cm}$ away.
Suppose that the
image in Fig. 9 a is an
actual-size reproduc-
tion of this interfer-
ence pattern. Use a
ruler to measure a
pertinent distance on
this image; then utilize
this measured value
to determine $\lambda(\mathrm{nm})$ .

Derek Walkama

Numerade Educator

Problem 8

(II) Light of wavelength 680 $\mathrm{nm}$ falls on two slits and
produces an interference pattern in which the third-order
bright fringe is 38 $\mathrm{mm}$ from the central fringe on a screen
2.6 $\mathrm{m}$ away. What is the separation of the two slits?

Farhanul Hasan

Numerade Educator

Problem 9

(II) A parallel beam of light from a He-Ne laser, with a
wavelength 633 nm, falls on two very narrow slits 0.068 $\mathrm{mm}$
apart. How far apart are the fringes in the center of the
pattern on a screen 3.8 $\mathrm{m}$ away?

Derek Walkama

Numerade Educator

Problem 10

(II) A physics professor wants to perform a lecture demonstra-
tion of Young's double-slit experiment for her class using the
633 -nm light from a He-Ne laser. Because the lecture hall is
very large, the interference pattern will be projected on a wall
that is 5.0 m from the slits, For easy viewing by all students in
the class, the professor wants the distance between the $m=0$
and $m=1$ maxima to be 25 $\mathrm{cm} .$ What slit separation is
required in order to produce the desired interference pattern?

Farhanul Hasan

Numerade Educator

Problem 11

(II) Suppose a thin piece of glass is placed in front of the
lower slit in Fig. 7 so that the two waves enter the slits $180^{\circ}$
out of phase (Fig, $25 ) .$ Describe in detail the interference
pattern on the screen.

Derek Walkama

Numerade Educator

Problem 12

(II) In a double-slit experiment it is
found that blue light of wavelength
480 nm gives a second-order maximum
at a certain location on the screen.
What wavelength of visible light
would have a minimum at the same
location?

Farhanul Hasan

Numerade Educator

Problem 13

(II) Two narrow slits separated by 1.0 $\mathrm{mm}$ are illuminated
by 544 $\mathrm{nm}$ light. Find the distance between adjacent bright
fringes on a screen 5.0 $\mathrm{m}$ from the slits.

Derek Walkama

Numerade Educator

Problem 14

(II) In a double-slit experiment, the third-order maximum
for light of wavelength 500 nm is located 12 $\mathrm{mm}$ from the
central bright spot on a screen 1.6 $\mathrm{m}$ from the slits. Light of
wavelength 650 $\mathrm{nm}$ is then projected through the same slits.
How far from the central bright spot will the second-order
maximum of this light be located?

Farhanul Hasan

Numerade Educator

Problem 15

(II) Light of wavelength 470 $\mathrm{nm}$ in air falls on two slits
$6.00 \times 10^{-2} \mathrm{mm}$ apart. The slits are immersed in water, as is
a viewing screen 50.0 $\mathrm{cm}$ away. How far apart are the fringes
on the screen?

Derek Walkama

Numerade Educator

Problem 16

(II) A very thin sheet of plastic $(n=1.60)$ covers one slit
of a double-slit apparatus illuminated by $680-\mathrm{nm}$ light. The
center point on the screen, instead of being a maximum, is
dark. What is the (minimum) thickness of the plastic?

Farhanul Hasan

Numerade Educator

Problem 17

(I) If one slit in Fig, 12 is covered, by what factor does the
intensity at the center of the screen change?

Derek Walkama

Numerade Educator

Problem 18

(II) Derive an expression similar to Eq. 2 which gives the
angles for which the double-slit intensity is one-half its
maximum value, $I_{\theta}=\frac{1}{2} I_{0} .$
$$\begin{aligned} d \sin \theta &=m \lambda \\ m &=0,1,2, \cdots \end{aligned} $\left[ \begin{array}{c}{\text { constructive }} \\ {\text { interference }} \\ {\text { (bright) }}\end{array}\right] (2a)$$
$$\begin{aligned} d \sin \theta &=\left(m+\frac{1}{2}\right) \lambda \\ m &=0,1,2, \cdots \end{aligned} $\left[ \begin{array}{c}{\text { destructive }} \\ {\text { interference }} \\ {\text { (bright) }}\end{array}\right]$ (2b)$$

Farhanul Hasan

Numerade Educator

Problem 19

(II) Show that the angular full width at half maximum of the
central peak in a double-slit interference pattern is given by
$\Delta \theta=\lambda / 2 d$ if $\lambda \ll d$

Derek Walkama

Numerade Educator

Problem 20

(II) In a two-slit interference experiment, the path length to a
certain point $P$ on the screen differs for one slit in compar-
ison with the other by 1.25$\lambda$ . (a) What is the phase difference
between the two waves arriving at point $P ?$ Determine
the intensity at P, expressed as a fraction of the maximum
intensity $I_{0}$ on the screen.

Farhanul Hasan

Numerade Educator

Problem 21

(III) Suppose that one slit of a double-slit apparatus is
wider than the other so that the intensity of light passing
through it is twice as great. Determine the intensity $I$ as a
function of position ( $\theta$ ) on the screen for coherent light.

Katie Mcalpine

Numerade Educator

Problem 22

(III) (a) Consider three cqually spaced and equal-intensity
coherent sources of light (such as adding a third slit to the
two slits of Fig. 12 ). Use the phasor method to obtain the
intensity as a function of the phase difference $\delta$
$(\mathrm{Eq} .$ 4). $(b)$ Determine the positions of maxima and
minima.
$$ or \begin{array}{r}{\frac{\delta}{2 \pi}=\frac{d \sin \theta}{\lambda}} \\ {\delta=\frac{2 \pi}{\lambda} d \sin \theta}\end{array} (4)$$

Farhanul Hasan

Numerade Educator

Problem 23

(1) If a soap bubble is 120 $\mathrm{nm}$ thick, what wavelength is
most strongly reflected at the center of the outer surface
when illuminated normally by white light? Assume that
$n=1.32 .$

Derek Walkama

Numerade Educator

Problem 24

(I) How far apart are the dark fringes in Example 6 of "The
Wave Nature of Light; Interference" if the glass plates are
cach 28.5 $\mathrm{cm}$ long?

Farhanul Hasan

Numerade Educator

Problem 25

(II) (a) What is the smallest thickness of a soap film $(n=1.33)$
that would appear black if illuminated with $480-$ nm light?
Assume there is air on both sides of the soap film. (b) What
are two other possible thicknesses for the film to appear
black? (c) If the thickness $t$ was much less than $\lambda,$ why
would the film also appear black?

Katie Mcalpine

Numerade Educator

Problem 26

(II) A lens appears greenish yellow $(\lambda=570 \mathrm{nm}$ is
strongest) when white light reflects from it. What minimum
thickness of coating $(n=1.25)$ do you think is used on
such a glass $(n=1.52)$ lens, and why?

Farhanul Hasan

Numerade Educator

Problem 27

(II) A thin film of oil $\left(n_{\mathrm{o}}=1.50\right)$ with varying thickness
floats on water $\left(n_{\mathrm{w}}=1.33\right) .$ When it is illuminated from
above by white light, the reflected colors are as shown in Fig.
26. In air, the wavelength of yellow light is 580 $\mathrm{nm}$ . (a) Why
are there no reflected colors at point $\mathrm{A} ?$ (b) What is the
oil's thickness $t$ at point $\mathrm{B}$ ?

Katie Mcalpine

Numerade Educator

Problem 28

(II) A thin oil slick $\left(n_{\mathrm{o}}=1.50\right)$ floats on water
$\left(n_{\mathrm{w}}=1.33\right) .$ When a beam of white light strikes this film at
normal incidence from air, the only enhanced reflected
colors are red $(650 \mathrm{nm})$ and violet $(390 \mathrm{nm}) .$ From this infor-
mation, deduce the (minimum) thickness t of the oil slick.

Farhanul Hasan

Numerade Educator

Problem 29

(1I) A total of 31 bright and 31 dark Newton's rings (not
counting the dark spot at the center) are observed when
560 -nm light falls normally on a planoconvex lens resting on
a flat glass surface (Fig. 18). How much thicker is the center
than the edges?

Katie Mcalpine

Numerade Educator

Problem 30

(II) A fine metal foil separates one end of two pieces
of optically flat glass, as in Fig. $20 .$ When light of wavelength
670 nm is incident normally, 28 dark lines are observed (with
one at cach end). How thick is the foil?

Farhanul Hasan

Numerade Educator

Problem 31

(II) How thick (minimum) should the air layer be between
two flat glass surfaces if the glass is to appear bright when
$450$ -nm light is incident normally? What if the glass is to
appear dark?

Penny Riley

Numerade Educator

Problem 32

(II) A uniform thin film of alcohol $(n=1.36)$ lies on a flat
glass plate $(n=1.56) .$ When monochromatic light, whose
wavelength can be changed, is incident normally, the
reflected light is a minimum for $\lambda=512 \mathrm{nm}$ and a
maximum for $\lambda=635 \mathrm{nm}$ . What is the minimum thickness
of the film?

Farhanul Hasan

Numerade Educator

Problem 33

(1I) Show that the radius $r$ of the $m^{\text { th }}$ dark Newton's ring, as
viewed from directly above (Fig. $18 ),$ is given by
$r=\sqrt{m \lambda R}$ where $R$ is the radius of curvature of the
curved glass surface and $\lambda$ is the wavelength of light used.
Assume that the thickness of the air gap is much less than
$R$ at all points and that $r \ll R$ . [Hint: Use the binomial
expansion.]

Lainey Roebuck

Numerade Educator

Problem 34

(II) Use the result of Problem 33 to show that the distance
between adjacent dark Newton's rings is
$$\Delta r \approx \sqrt{\frac{\lambda R}{4 m}}$$
for the $m^{\text { th }}$ ring, assuming $m \gg 1$ .

Farhanul Hasan

Numerade Educator

Problem 35

(II) When a Newton's ring apparatus (Fig. 18) is immersed
in a liquid, the diameter of the eighth dark ring decreases
from 2.92 $\mathrm{cm}$ to 2.54 $\mathrm{cm} .$ What is the refractive index of the
liquid? [Hint: see Problem $33 . ]$

Katie Mcalpine

Numerade Educator

Problem 36

(II) A planoconvex lucite lens 3.4 $\mathrm{cm}$ in diameter is placed
on a flat piece of glass as in Fig. 18. When 580 -nm light
is incident normally, 44 bright rings are observed, the last
one right at the edge. What is the radius of curvature of the
lens surface, and the focal length of the lens? [Hint: see
Problem $33 .$]

Farhanul Hasan

Numerade Educator

Problem 37

(II) Let's explore why only "thin" layers exhibit thin-film
interference. Assume a layer of water, sitting atop a flat
glass surface, is illuminated from the air above by
white light (all wavelengths from 400 $\mathrm{nm}$ to 700 $\mathrm{nm}$ ).
Further, assume that the water layer's thickness $t$ is
much greater than a micron $(=1000 \mathrm{nm}) ;$ in particular,
let $t=200 \mu \mathrm{m} .$ Take the index of refraction for
water to be $n=1.33$ for all visible wavelengths. $(a)$ Show
that a visible color will be reflected from the water layer
if its wavelength is $\lambda=2 n t / m,$ where $m$ is an integer.
(b) Show that the two extremes in wavelengths $(400 \mathrm{nm}$
and 700 $\mathrm{nm}$ ) of the incident light are both reflected from
the water layer and determine the $m$ -value associated
with each. $(c)$ How many other visible wavelengths,
besides $\lambda=400 \mathrm{nm}$ and $700 \mathrm{nm},$ are reflected from the
"thick" layer of water? (d) How does this explain why
such a thick layer does not reflect colorfully, but is white
or grey?

Brandy Heflin

Numerade Educator

Problem 38

(III) A single optical coating reduces reflection to zero
for $\lambda=550 \mathrm{nm}$ . By what factor is the intensity reduced by
the coating for $\lambda=430 \mathrm{nm}$ and $\lambda=670 \mathrm{nm}$ as compared
to no coating? Assume normal incidence.

Farhanul Hasan

Numerade Educator

Problem 39

(II) How far must the mirror $M_{1}$ in a Michelson interferometer
be moved if 650 fringes of 589 -nm light are to pass by a
reference line?

Katie Mcalpine

Numerade Educator

Problem 40

(II) What is the wavelength of the light entering an
interferometer if 384 bright fringes are counted when the
movable mirror moves 0.125 $\mathrm{mm}$ ?

Farhanul Hasan

Numerade Educator

Problem 41

(II) A micrometer is connected to the movable mirror of an
interferometer. When the micrometer is tightened down on
a thin metal foil, the net number of bright fringes that move,
compared to the empty micrometer, is $272 .$ What is the
thickness of the foil? The wavelength of light used is 589 $\mathrm{nm}$ .

Mayukh Banik

Numerade Educator

Problem 42

(III) One of the beams of an interferometer (Fig, 27$)$ passes
through a small evacuated glass container 1.155 $\mathrm{cm}$ deep.
When a gas is allowed to slowly fill the container, a total of
176 dark fringes are counted to move past a reference line.
The light used has a wavelength of 632.8 $\mathrm{nm} .$ Calculate the
index of refraction of the gas at its final density, assuming
that the interferometer is in vacuum.

Farhanul Hasan

Numerade Educator

Problem 43

(III) The yellow sodium $\mathrm{D}$ lines have wavelengths of 589.0
and 589.6 $\mathrm{nm}$ . When they are used to illuminate a Michelson
interferometer, it is noted that the interference fringes
disappear and reappear periodically as the mirror $\mathrm{M}_{1}$ is
moved. Why does this happen? How far must the mirror
move between one disappearance and the next?

Eduard Sanchez

Numerade Educator

Problem 44

(II) The illuminance of direct sunlight on Earth is about
$10^{5} \operatorname{lm} / \mathrm{m}^{2}$ . Estimate the luminous flux and luminous intensity
of the Sun.

Farhanul Hasan

Numerade Educator

Problem 45

(II) The luminous efficiency of a lightbulb is the ratio of luminous flux to electric power input. (a) What is the luminous efficiency of a $100-\mathrm{W}, 1700-\mathrm{lm}$ bulb? $(b)$ How many $40-\mathrm{W}, 60-\mathrm{Im} / \mathrm{W}$ fluorescent lamps would be needed to provide an illuminance of $250 \mathrm{Im} / \mathrm{m}^{2}$ on a factory floor of area $25 \mathrm{~m} \times 30 \mathrm{~m} ?$ Assume the lights are $10 \mathrm{~m}$ above the floor and that half their flux reaches the floor.

Katie Mcalpine

Numerade Educator

Problem 46

Light of wavelength $5.0 \times 10^{-7} \mathrm{m}$ passes through two
parallel slits and falls on a screen 4.0 $\mathrm{m}$ away. Adjacent
bright bands of the interference pattern are 2.0 $\mathrm{cm}$ apart.
(a) Find the distance between the slits. (b) The same two
slits are next illuminated by light of a different wavelength,
and the fifth-order minimum for this light occurs at the
same point on the screen as the fourth-order minimum for
the previous light. What is the wavelength of the second
source of light?

Farhanul Hasan

Numerade Educator

Problem 47

Television and radio waves reflecting from mountains or
airplanes can interfere with the direct signal from the
station. $(a)$ What kind of interference will occur when
75-MHz television signals arrive at a receiver directly from
a distant station, and are reflected from a nearby airplane
122 $\mathrm{m}$ directly above the receiver? Assume $\frac{1}{2} \lambda$ change in
phase of the signal upon reflection. (b) What kind of
interference will occur if the plane is 22 $\mathrm{m}$ closer to the
receiver?

Eduard Sanchez

Numerade Educator

Problem 48

A radio station operating at 88.5 $\mathrm{MHz}$ broadcasts from two
identical antennas at the same elevation but separated by a
9.0 -m horizontal distance $d,$ Fig. $28 .$ A maximum signal is
found along the midline, perpendicular to $d$ at its midpoint
and extending horizontally in both directions. If the midline
is taken as $0^{\circ},$ at what other angle(s) $\theta$ is a maximum signal
detected? A minimum signal? Assume all measurements are
made much farther than 9.0 $\mathrm{from}$ the antenna towers.

Farhanul Hasan

Numerade Educator

Problem 49

Light of wavelength 690 $\mathrm{nm}$ passes through two narrow slits
0.66 $\mathrm{mm}$ apart. The screen is 1.60 $\mathrm{m}$ away. A second source
of unknown wavelength produces its second-order fringe
1.23 $\mathrm{mm}$ closer to the central maximum than the $690-\mathrm{nm}$
light. What is the wavelength of the unknown light?

Katie Mcalpine

Numerade Educator

Problem 50

Monochromatic light of variable wavelength is incident
normally on a thin sheet of plastic film in air. The reflected
light is a maximum only for $\lambda=491.4 \mathrm{nm}$ and
$\lambda=688.0 \mathrm{nm}$ in the visible spectrum. What is the thickness of
the film $(n=1.58) ?[$Hint . Assume successive values of $m .]$

Farhanul Hasan

Numerade Educator

Problem 51

Suppose the mirrors in a Michelson interferometer are
perfectly aligned and the path lengths to mirrors $M_{1}$ and $M_{2}$
are identical. With these initial conditions, an observer sees
a bright maximum at the center of the viewing area. Now
one of the mirrors is moved a distance $x$ . Determine a
formula for the intensity at the center of the viewing area as
a function of $x,$ the distance the movable mirror is moved
from the initial position.

Jacob Shpiece

Numerade Educator

Problem 52

A highly reflective mirror can be made for a particular
wavelength at normal incidence by using two thin layers of
transparent materials of indices of refraction $n_{1}$ and
$n_{2}\left(1 < n_{1} < n_{2}\right)$ on the surface of the glass $\left(n > n_{2}\right)$ . What
should be the minimum thicknesses $d_{1}$ and $d_{2}$ in Fig. 29 in
terms of the incident wavelength $\lambda,$ to maximize
reflection?

Farhanul Hasan

Numerade Educator

Problem 53

Calculate the minimum thickness needed for an antireflec-
tive coating $(n=1.38)$ applied to a glass lens in order to
eliminate $(a)$ blue $(450 \mathrm{nm}),$ or $(b)$ red $(720 \mathrm{nm})$ reflections
for light at normal incidence.

Katie Mcalpine

Numerade Educator

Problem 54

Stealth aircraft are designed to not reflect radar, whose
wavelength is typically $2 \mathrm{cm},$ by using an antireflecting
coating. Ignoring any change in wavelength in the coating,
estimate its thickness.

Farhanul Hasan

Numerade Educator

Problem 55

Light of wavelength $\lambda$ strikes a screen containing two slits a
distance $d$ apart at an angle $\theta_{1}$ to the normal. Determine the
angle $\theta_{m}$ at which the $m^{16}$ -order maximum occurs.

Hubert Agamasu

Numerade Educator

Problem 56

Consider two antennas radiating 6.0 $\mathrm{MHz}$ radio waves in
phase with each other. They are located at points $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$
separated by a distance $d=175 \mathrm{m},$ Fig. $30 .$ Determine the
points on the $y$ axis where the signals from the two sources
will be out of phase (crests of one meet troughs of the
other).

Farhanul Hasan

Numerade Educator

Problem 57

What is the minimum (non-zero) thickness for the air layer
between two flat glass surfaces if the glass is to appear dark
when $680-$ nm light is incident normally? What if the glass is
to appear bright?

Katie Mcalpine

Numerade Educator

Problem 58

Lloyd's mirror provides one way of obtaining a double-slit
interference pattern from a single source so the light is
coherent. As shown in Fig. 31 , the light that reflects from the
plane mirror appears to come from the virtual image of the
slit. Describe in detail the interference pattern on the
screen.

Farhanul Hasan

Numerade Educator

Problem 59

Consider the antenna array of Example 5 of "The Wave
Nature of Light; Interference," Fig. 15. Let $d=\lambda / 2,$ and
suppose that the two antennas are now $180^{\circ}$ out of phase
with each other. Find the directions for constructive and
destructive interference, and compare with the case when
the sources are in phase. (These results illustrate the basis
for directional antennas.)

Lainey Roebuck

Numerade Educator

Problem 60

Suppose you viewed the light transmitted through a thin film
layered on a flat piece of glass. Draw a diagram, similar to
Fig. 17 or $23,$ and describe the conditions required for
maxima and minima. Consider all possible values of index
of refraction. Discuss the relative size of the minima
compared to the maxima
and to zero.

Farhanul Hasan

Numerade Educator

Problem 61

A thin film of soap $(n=1.34)$ coats a piece of flat glass
$(n=1.52) .$ How thick is the film if it reflects 643 -nm red
light most strongly when illuminated normally by white
light?

Katie Mcalpine

Numerade Educator

Problem 62

Two identical sources $\mathrm{S}_{1}$ and $\mathrm{S}_{2},$ separated by distance $d,$
coherently emit light of wavelength $\lambda$ uniformly in all direc-
tions. Defining the $x$ axis with its origin at $S_{1}$ as shown in
Fig. $32,$ find the locations (expressed as multiples of $\lambda )$
where the signals from the two sources are out of phase
along this axis for $x>0$ , if $d=3 \lambda.$

Farhanul Hasan

Numerade Educator

Problem 63

A two-slit interference set-up with slit separation $d=$
0.10 $\mathrm{mm}$ produces interference fringes at a particular set of
angles $\theta_{m}$ (where $m=0,1,2, \ldots )$ for red light of frequency
$f=4.6 \times 10^{14} \mathrm{Hz}$ . If one wishes to construct an analogous
two-slit interference set-up that produces interference fringes
at the same set of angles $\theta_{m}$ for room-temperature sound of
midde-C frequency $f_{\mathrm{s}}=262 \mathrm{Hz}$ , what should the slit sepa-
ration $d_{\mathrm{S}}$ be for this analogous set-up?

Katie Mcalpine

Numerade Educator

Problem 64

A radio telescope, whose two antennas are separated by
$55 \mathrm{m},$ is designed to receive 3.0 -MHz radio waves produced
by astronomical objects. The received radio waves create
3.0 -MHz electronic signals in the telescope's left and right
antennas. These signals then travel by equal-length cables to
a centrally located amplifier, where they are added together.
The telescope can be "pointed" to a certain region of the sky
by adding the instantaneous signal from the right antenna to
a "time-delayed" signal received by the left antenna a time
\Deltat ago. (This time delay of the left signal can be easily
accomplished with the proper electronic circuit.) If a radio
astronomer wishes to "view" radio signals arriving from an
object oriented at a $12^{\circ}$ angle to the vertical as in
Fig. 33, what time delay $\Delta t$ is necessary?

Farhanul Hasan

Numerade Educator

Problem 65

In a compact disc (CD), digital information is stored as a
sequence of raised surfaces called "pits" and recessed
surfaces called "lands." Both pits and lands are highly reflec-
tive and are embedded in a thick plastic material with index
of refraction $n=1.55$ (Fig, 34). As a 780 -nm wavelength
(in air) laser scans across the pit-land sequence, the transi-
tion between a neighboring pit and land is sensed by moni-
toring the intensity of reflected laser light from the CD. At
the moment when half the width of the laser beam is
reflected from the pit and the other half from the land, we
want the two reflected halves of the beam to be $180^{\circ}$ out of
phase with each other. What should be the (minimum)
height difference $t$ between a pit and land? [When this light
enters a detector, cancellation of the two out-of-phase
halves of the beam produces a minimum detector output.

Jacob Shpiece

Numerade Educator

Problem 66

(II) A Michelson interferometer can be used to determine the
index of refraction of a glass plate. A glass plate $(thickness t)$
is placed on a platform that can rotate. The plate is placed in
the light's path between the beam splitter and either the
fixed or movable mirror, so that its thickness is in the direc-
tion of the laser beam. The platform is rotated to various
angles, and the number of fringes shifted is counted. It can
be shown that if $N$ is the number of fringes shifted when the
angle of rotation changes by $\theta$ , the index of refraction
is $n=(2 t-N \lambda)(1-\cos \theta) /[2 t(1-\cos \theta)-N \lambda]$ where
$t$ is the thickness of the plate. The accompanying Table shows
the data collected by a student in determining the index of
refraction of a transparent plate by a Michelson interferometer.
$$\begin{array}{|l|l|l|l|l|l|}\hline N & {25} & {50} & {75} & {100} & {125} & {150} \\ \hline \theta(\text { degree) }& {5.5} & {6.9} & {8.6} & {10.0} & {11.3} & {12.5} \\ \hline\end{array}$$
In the experiment $\lambda=632.8 \mathrm{nm}$ and $t=4.0 \mathrm{mm}$ . Deter-
mine $n$ for each $\theta$ and find the average $n .$

Farhanul Hasan

Numerade Educator