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REFRACTION
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OPTICAL REFHACTJO~
OnJE(:TI\'ES
. Describe the relationship
between optical refraction and
the wave character of light.
. Show the effect of refraction on
the speed of light.
. Describe the control of light
beams with lenses.
. Analyze the formation of images
by ray diagrams.
. Solve object-image problems.
Study the magnification of
images.
Describe the dispersion of light
.
.
.
by prisms.
Define color as a property of
light.
. Discuss primary and secondary
colors.
U2
14.1 The Nature
of Optical
Refraction
In Section
10.12 we discussed refraction as a property of waves. Now
we shall examine the refractive behavior of light and relate
this behavior to its wavelike nature.
When aiming a rifle at a target, one relies on the common observation that light travels in straight lines. It does
so, however, only if the transmitting medium is of the
same opticaldensity throughout. Optical density is a property of a trallsparellt material that is all i,werse measllre of tlte
speed of light through the material.
Consider a beam of light transmitted through air and
directed onto the surface of a body of water. Some of the
light is refJected at the interface (boundary) between the
air and water; the remainder enters the water and is transmitted through it. Because water has a higher optical density than air, the speed of light is reduced as the light
enters the water. This change in the speed of light at the
air-water interface is diagrammed in Figure 14-2.
A ray of light that strikes the surface of the water at an
oblique angle (less than 900 to the surface) changes direction abruptly as it enters the water because of the change
in speed. The reason for this change in direction with a
change in speed can be illustrated if we redraw the wavefront diagram of Figure 14-2(A) to make the angle of the
incident ray oblique, as in Figure 14-2(B). When interpreting this diagram, you should remember that a light ray
REFRACTION
l
l
I
,
.
.
,
I
.
II
1-
indicates the direction the light travels and is perpendicular to the wave front. We have alreaclv defined refraction
as a bending of a wave disturbance.
"(See Section 10.12.)
This bending of a light ray is called optical refro(tiol/.
Optical refraction is the bmdillg of lisht rays as tlley pass 0/1liquely frolll DlICmcdium into Ill/otlter of d~tfcrCllt oplical dfl/::.ity.
Because of refraction, a fish observed from the bank
appears nearer to the surface of the water than it actually
is. A teaspoon in a tumbler of water appears to be bent at
the surface of the \ at£;'f. A coin in the bottom of an empty
teacup that is out of the line of vision of an observcr may
become visible when the cup is filled with watl'r.
14.2 Refractioll
alld the Speed of Light
Line MN of
Figure 14-3 represents the surface of a body of water. The
line AD represents a ray of light through the air striking
the water at D. Some of the light is reflected along DE.
Instead of continuing in a straight line along OF, the light
ray entering the water is bent as it passes from air into
water, taking the path DB.
The incident ray AO makes the angle AOe with the normal. Angle AOe is the angle of incidence, i. Recall that thl'
angle of incidence is defined as the angle between the incident ray and the normal at the point of incidence. The
refracted ray DB makes the angle DOB with the normal
produced. Till.'angle bt'twet'll the refracted ray and file /loTl/w!
(
-
Figure 14-1. Parallel beams of
light incident upon a rectangular
glass plate. The beams are mainly
refracted at both surfaces. Some
reflection occurs at each surface.
however.
Figure 14.2. Wave-front diagrams
illustrating (A) the difference in
the speed of Ught in air and In
water and (8) the refraction of
light at the air-water boundary.
Pl.n......
PI.ne......
I
I
I
Normal
I
T
"
Air
Boundllry
Boundary
w.~
-
-..
-".'z_
Lightr.y-
Lighlrly~
IAI
181
3.H
CHAPTER 14
,
c
I
I
A
at tilt:' poilit of refractioll i~ called tlte angle of refraction,
E
ro-- Normal
An~le of
I
incidence
,---'
I
I
I
M
N
I
-
-
"0-.
I',
,i!',-..
I
I
I
"'-,,:
"-
Angleof ">,
"-
r.'
I
I...f~ion
~
"b-
'....- F
NOI'fI\OIlprodu.ced--t
I
--
Figure
14.3.
.
I
D
A ray diagram
B
_
of re-
showing the angle of incidence and the angle of refraction.
fraction
r.
In the examples given so far, the light rays have passed
from one medium into another of higher optical density,
with a resulting reduction in speed. When a ray enters the
denser medium normal to the interface, no refraction oc~
curs. When a ray enters the denser medium at an oblique
angle, refraction does occur and the ray is bent tmi'lmi the
normal.
What is the nature of the refraction if the light passes
obliquely from one medium into another of lower optical
density? Suppose the light source were at B in Figure 14-3.
Light ray BO then meets the surface at point 0, and angle
BOD is the angle of incidence. Of course some light is
reflected at this interface. However, on entering the air the
refracted portion of the light takes the path OA. The angle
of refraction in this case is angle COA. It shows that the
light is bent away from the normal. When a light ray enters a medium of lower optical density at an oblique angle,
the ray is bent away from the normal. HJd the light ray
entered this less dense medium normal to the interface, no
refraction would have occurred.
14.3 The lndex of Refraction
The speed of light in a
vacuum is approximately 300 000 kilometers per second.
The speed of light in WJter is approximJtcly 225 000 kilo.
meters per sccond, or just about threc fourths of that in a
vacuum. The spt:'ed of light in ordinary glass is approximately 200 000 kilometers per second or about hvo thirds
of that in a vacuum. The ratioof tltl!spced af 1(..-;l1t iI/a PIlCIIIIIII
to its speed ill a substance is called the iIJdex of refraction for
that sllbstmlce. For example:
.
I n d ex 0 f re f rachon
Figure
14-4. Willebrord Snell bea professor of mathematics
and physics at the University of
came
Leyden at the age of 21. He discovered the law of refraction now
known as Snell's law.
(g Iass)
=
speed of light in vacuum
speed of light in glass
Using the approximate values just given, the index of
refraction for glass would be about 1.5 and for water about
1.3. The index of refraction for a few common substances
is given in Appendix B, Table 18. The speed of light in air
is only slightly different from the speed of light in a vac.
uum. Therefore, with negligible error, we can use the
speed of light in a vacuum for cases where light travels
from air into another medium.
The fundamental principle of refraction was discovered
bv the Dutch mathematician
and astronomer Willebrord
Snell (1580-1626). See Figure 14-4. He did not publish his
discovery, but his work was taught at the University of
Leyden,
where he was a professor
of mathematics
and
1-
,,-
REFRACTION
:U5
physics. The French mathematician and philosopher Rene
Descartes (1596-1650) published Snell's work in 1637.
Sm:!U'sdiscoveries about refraction were not stated in
terms of the speed of light. The speed of light in empty
space was not determined unti1 1676, and the speed in
water was not measured until 1850. From his observations, howe\rer, Snell defined the index of refraction as the
ratio of the sine of the angle of inddence to the sine of the
angle of refraction. This relationship is known as Snell's
law. If II represents the index of refraction, i the angle of
incidence, and r the angJe of refraction,
sin i
n=sin r
The sines of angles from 0° to 90° arc given in Appendix B,
rable 6.
The relationship between Sm.'Il's law and the ratio of the
speeds of light in air and a refracting medium can be recognized from Figure 14-5. Rays of light travel through the
first medium (air) with a speed Vt and enter the refracting
medium in which their speed is V2'
A wave front MP approaches the refracting interface
MN in the first medium at an incident angle i. The wave
front at P travels to N in the time t with a speed VI. Simultaneously the wave front at M travels in the second medium to Q in the same time t but with a speed V2' The new
wave front is NQ, which tra\'els forward in the second
medium at a rdractive angle r. The distances PN and MQ
are respectively VIt and V2t, so that
PN
v,t
VI
- - - MQ
v2t 112
In triangle MNP
..
slnl=-
In triangJe MNQ
.
slnr=-
The ratio
sin
-=
sin
i
r
From Snel1' 5 la w
Therefore
These relationships
example.
PN
MN
MQ
MN
PN/MN
PN
--=MQ/MN
MQ
n=-
v.
v,
sin i
sin r
v,
n=- v,
are illustrated
in the following
"
Figure 14-5. A wave-front
of refraction.
diagram
CHAPTER 14
:l:l6
EX.UIPU;
A ray of light passes from air into water, striking at an
angle of 65.0<' with the boundary between the two media. The index of
refraction of water is 1.33. Calculate the angle of refraction of the ray of
light.
"'"
.
e
~
65.00
i = 90.00
flw
= 1.33
".,!!!!
.
.,
IIasu' t'ljuullun
sin i
n w =- .
smr
lInk",,"n
r
Gin-II
-e
Solution
Working equation: sin r =
sin(90°
- II)
"w
r
= arcsin
(-0.423
1.33 )
.
~18S
PIUCTICE PHOBLE\lS
1. Using the information given in the Example
and the fact that the speed of light in air is 3.00 x let mis, calculate the
speed of light in water.
Ails. 2.26 x lOS m/s
2. A ray of light passes from air into a gemstone at an angle of incidence
of 40.0". The angle of refraction is measured to be 15.4°. Using the information in Appendix B, Table 18, determine whether the gemstone is a
diamond.
Ails. Yes
When a piece of glass is immersed in water, the edges of
the glass are visible because of the difference between the
index of refraction of glass and that of water, even though
both substances are transparent. The difference in the indices of refraction of air and water also make the top surface of the water visible.
But when
a piece
of crown
glass
(n
=
1.5172)
is im-
mersed in benzene (II = 1.501), the crown glass seems to
disappear in the benzene bec..use the nearly identical indices of refraction make the edges of the glass virtu..lIy invisible.
The index of refraction of a homogeneous substance is a
constant quantity that is a definite physical property of the
substance. Consequently, the identity of such a substance
can be determined by measuring its index of refraction
with an instrument known ..sa refractometer, Por example,
"
'-,
,
.ti
,
"
"
REFRACTION
butterfat and margarine have different indexes of refraction. One of the first tests made in a food-testing laboratory to determine whether butter has been mixed with
margarine is the measurement of the index of refraction.
The high index of refruction of a diamond furnishes
one of the most conclusive tests for its identification.
Because Ught travels very slightly faster in outer space
.....
than it does through air, light from the sun or the stars is
refracted when it enters the earth's atmosphere obliquely.
Since the atmosphere is denser near the earth's surface, a
ray of light from the sun or a star striking the atmosphere
Rgure 14-6. The sun is visible
obliquely foHows a path suggested by the curve shown in before actual sunrise and after
Figure 14-6. There is no abrupt refraction such as that actual sunset because of atmowhich occurs at the interface between two media of differ- spheric refraction.
ent optical densities.
Atmospheric refraction prevents the sun and stars from
being seen in their true positions except when they are
directly overhead. In Figure 14-6, the sun appears at S'
E
instead of 5, its true position. Since the index of refraction
from outer space to air is only 1.00029, the diagram is
greatly exaggerated to show the bending. Refraction of
sunlight by the earth's atmosphere causes the sun, when
geometrically on the horizon, to appear about one ..iiame-
.
l
I
L
L
'"
l"
L
,
\
ter (i0) higher
r--,
than it really is. Around
required for the earth to rotate through
iO
2 minutes
"
I~~
ia
,
are
of arc. Thus, we
gain about 4 minutes of additional daylight each day because of atmospheric refraction at sunrise and sunset.
~
14.4 The Laws of Uefractioll If the index of refraction
of a transparent substance is known, it is possible to trace D
the path that a ray of light will take in passing through the
R.v 0' refracted as il
entenendasltl_
substance. In Figure 14-7 rectangle ABCD represents a
the jIIuspllU'
piece of plate glass with parallel surfaces. The line EO
represents a ray of light incident upon the glass at point O.
From point 0 as a center, two arcs are drawn. The radii of
these arcs are in the ratio 3/2 based on the index of refrac- Figure 14-7. A method of tracing
tion of the glass being 1.5 and that of the air being 1. The a light ray through a glass plate.
normal OF is drawn. Then a line is drawn parallel to OF
through the point H where tne inddent ray intersects the
smaller arc. This line intersects the larger arc at point G.
The line OP, determined by points G and 0, marks the
path of the refracted ray through the glass.
If the ray were not refracted as it leaves the glass, it Test the validity of this COIJStruCwould proceed along the line PK. To indicate the refrac- lionusingFigure 14-5. and
tion at point P, this point is used as a center and arcs are S}!eIl'staw.
drawn having the same ratio, 3/2, as before. The normal
PL and a Hne MN parallel to the normal are drawn. The
parallel line is drawn this time through point N where the
r--\
-
c
CHAPTER 14
338
/
/
/
larger arc is intersected by the extension of refracted ray
OP. This line intersects the smaller arc at point M. Points P
/
/'\r
and M detennine the path of the refracted ray as it enters
the air.
In Figure 14-8 line AD represents a light ray incident
upon a triangular glass prism at point D. Passing obliquely
from air into glass, the ray is refracted toward the normal
along line DB. Observe that at point Dangle i in air is
larger than angle r in glass. As light ray DB passes obliquely from glass into air at point B, it is refracted away
from the normal along line Be. At point Bangle i in glass
is smaller than angle r in air.
Refraction of light can be summarized in three laws of
refraction:
1. The incident ray, the refracted rayI and the normal to the
surface at the point of incidence are all in the same plane.
2. The index of refraction for any homogeneous medium is a
constant that is independent of the angle of incidence.
3. When a ray of light passes obliquely from a medium of
lower optical density to one of higher optical density, it is bent
toward the normal to the surface. Conversely, a ray of light passing obliquely from an optically denser medium to an optically
rarer medium is bent away from the normal to the surface.
.,
A
Figure 14-8. The path of a light
ray through a prism. As the ray AD
passes from the optically less
dense to the optically more dense
medium at D, its path is refracted
toward the normal. As the ray
passes from the optically more
dense to the optically less dense
medium at B, its path is refracted
away from the norma\.
refraction of 90°.
14.5 Total Reflection
Suppose an incident ray of light,
AD, passes from water into air and is refracted along the
line DB, as shown in Figure 14-9. As the angle of incidence, i, is increased, the angle of refraction, r, also increases. When this angle approaches the limiting value,
Figure 14-10. (Right) Total reflection at the water-air interface occurs when the angle of incidence
exceeds the critical angle.
path that gets closer to the water surface. As the angle of
incidence continues to increase, the angle of refraction finally equals 90" and the refracted ray takes the path ON
Figure 14-9. (Left) The critical
angle ic is the limiting angle of
incidence in the optically denser
medium that results in an angle of
rL = 90", the refracted ray emerges from the water along a
,.
C'
,
I
I
I
I
I
I
I
I
B
I
Air
I
r-,.-.../
I
/
01/
M
"
\
II
.
N
I
Iii
,
/'"
Water..
I
I
I
I
C
//
--
-
_/ ",~'"
Water
E
..-
"
,," -"-
D
-.-
Nolight
~.
I rL\ ,
M
-
I:------j
I
A
Air
--
-~
0
I
'"
o!,
enters
the air
N
_L__....
I
I
.---i
.
I
'-"i>--f-.
(I
Angleof I
incidence I,
,
Angle of
reflection
-
,
339
REFRACTION
along the water surface. The limiting angle of incidence in the
optically denser medium that results in an angle of refractionof
90° is known as the critical angle, ie:. The critical angle for
water is reached when the incident ray DO makes an
angle of 48.5° with the normal; the critical angle for crown
glass is 42°, while that for diamond is only 24°.
In Section 14.3 we defined the index of refraction of a
material as the ratio of the speed of light in a vacuum (air)
to the speed of light in the material or as the ratio
sin i/sin r. In Figure 14-9 the light passes from the optically denser water to the air. Thus in the form of Snell's
law, the roles of the angles of incidence and refraction are
reversed. Here the angle of refraction, r, is related to the
speed of light in air. The angle of incidence, i, is associated
with the speed of light in water. The index of refraction of
the water in this instance is
n~
Figure 14-11. The meter stick
appears to be bent at the surface
of the water as a result of the refraction of light. Can you account
for the second image of the end
of the meter stick at the lower
left?
sin r (air)
sin i (water)
At the critical angle, ie, r is the limiting value r u which
equals 90". Thus
sin rL
n=-=
sin ie
Therefore,
sin 90"
1
~sin ic
sin ic
in general
sIn 1,,-
1
n
where n is the index of refraction of the optically denser
medium relative to air and ic is the critical angle of this
medium.
If the angle of incidence of a ray of light passing from water
into air is increased beyond the critical angle, no part of the
incident ray enters the air. The incident ray is totally reflected
from the water interface. In Figure 14-10 EO represents a ray
of light whose angle of incidence exceeds the critical angle,
the angle of incidence EOC being greater than the critical
angle DOC. The ray of light is reflected back into the water
along the line OE', a case of simple reflection in which the
Total internal reflection is the
angle of incidence EOC equals the angle of reflection E'0c.
basis for fiber optics, which is
Total reflection always occurs when the angle of incidence ex- used in modern telecommunicaceeds the critical angle.
tions.
A diamond is a brilliant gem because its index of refraction is exceedingly high and its critical angle is therefore
correspondingly
small. Very little of the light that enters
the upper surface of a cut diamond passes through the
CHAPTER 14
3411
diamond; most of the light is reflected internally (total reflection), finally emerging from the top of the diamond.
The faces of the upper surface are cut at such angles as to
ensure that the maximum light entering the upper surface
is reflected back to these faces. See also Plate VI(A).
QUESTIONS:
GROUP A
1. What three conditions must be met
for refraction to occur?
2. Does the fact that light refracts violate
the principle of rectilinear propagation? Explain.
3. Snell's law relates angles of incidence
and refraction. Did he know why refraction occurs? Explain.
4. Why is the index of refraction a characteristic of a material?
5. (a) How could you distinguish between a diamond and a piece of glass
cut the same way? (b) Why does this
happen?
6. What is the relationship between the
velocity of light and the index of refraction of a transparent substance?
GRUUp B
7. Why do we gain about four minutes
of daylight each day?
8. Research "optical fibers" and relate
their operation to this section.
9. (a) Most of what you see outside
your classroom window consists of
images of the actual objects. Explain.
(b) Under what circumstances would
you see the actual object?
10. A friend throws a coin into a pool.
You dive toward the spot where you
saw it from the edge of the pool in
order to retrieve it. Explain what will
happen.
11. (a) In which of the gases listed in
Table 18 in Appendix B, does light
travel the slowest? (b) Which material
has the highest optical density?
12. According to the definition, what is
the index of refraction of air?
PROBU:;i\1S: GROUP A
1. (a) What is the refractive index of a
material in which the speed of light is
1.85 x 10' m/s? (b) Using Table 18,
identify this material.
2. Determine the speed of light in each of
the following materials: (a) a sapphire
(b) ice (c) glycerol.
3. Light passes from air into water at an
angle of incidence of 42.3°. Determine
the angle of refraction in the water.
4. The angles of incidence and refraction
for light going from air into an optically more dense material are 63.5° and
42.9°, respectively. What is the index
of refraction of this material?
GROUP B
5. A man in a boat shines a light at a
friend under the water. If the beam in
the water makes an angle of 36.2° relative to the normal, what was the angle
of incidence?
6. Calculate the critical angle for light
going from glycerol into air.
7. A ray of light passes from water into
Lucite. If the angle relative to the normal in the Lucite is 45.0°, what was
the angle in the water?
8. Light moves from flint glass into water
at an angle of incidence of 28.7°;
(a) What is the angle of refraction?
(b) At what angle would the light have
to be incident to give an angle of refraction of 90.00?
.HI
REFRACTION
LENS OPTICS
14.6 Types of Lenses
A lens is any transparent object
having two nonparallel curved surfaces or one plane surface and one curved surface. The curved surfaces can be
spherical, parabolic, or cylindrical. Lenses are usually
made of glass but can be made of other transparent materials. There are two general classes of lenses based upon
their effects on incident light.
1. Converging
lenses. Cross sections of converging
lenses are shown in Figure 14-12(A). All are thicker in the
middle than at the edge. The concavo-convex
lens is
known as a meniscus lens.
,
c
f
f
f
r
r
r
I
I
I
r"
Double
convex
Concavo.
Double
concave
convex
convex
(A) Converging lenses
Planoconcave
(B) Diverging lenses
Convexoconcave
Figure 14-12.
Common
lens con-
figurations.
Recall that light travels more slowly in glass and other
lens materials than it does in air. Light passing through
the thick middle region of a converging lens is retarded
more than light passing through the thin edge region.
Consequently, a wave front of light transmitted through a
converging lens is bent as shown in Figure 14-13(A). The
plane wave in this diagram is incident on the surface of the
converging lens parallel to the lens plane. The wave is
refracted and converges at the point F beyond the lens.
Lens plane
Lens plane
,,
,,
,,
Tf1Il
Pl."....:
I
(A)
~
(,
(j\II!\\
c<
PrnclP.1
F
'I
/1
I
/1
Plane wave
'
'B)
Figure 14-13. Refraction of a
plane wave (A) by a converging
lens and (B) by a diverging lens.
The incident wave is perpendicular
to the lens axis.
CHAPTER
342
14
2. Diverging lenses. Lenses that are thicker at the
edges than in the middle are diverging lenses. Their cross
sections are shown in Figure 14-12(B). The convexoconcave lens is a meniscus lens.
Light passing through the thin middle region of a diverging lens is retarded less than light passing through the
thick edge region. A wave front of light transmitted
through a diverging lens is bent as shown in Figure
14-13(B). Observe that the plane wave in this diagram is
also incident on the surface of the lens parallel to the lens
plane. In this case, however, the refracted wave diverges
in a way that makes it appear to come from the point F' in
front of the lens.
A spherical lens usually has two centers of curvaturethe centers of the spheres that form the lens surfaces.
These centers of curvature determine the principal axis of
the lens. The radii of curvature of the two surfaces of double convex and double concave lenses are not necessarily
equal, although they are so drawn in the ray diagrams in
this chapter. The geometry of lens surfaces having differ~
ent radii of curvature is shown in Figure 14-14.
Figure 14-14. The geometry of
lens surfaces that have different
radii of curvature. Observe that
the surfaces of the double convex
lens are sections of intersecting
spheres.
14.7 Ray Diagrams
The nature and location of the
image formed by a lens is more easily determined by a ray
diagram than by a wave-front diagram. The plane waves
of Figure 14-13 can be represented
by light rays drawn
perpendicular to the wave fronts. Since these wave fronts
-- - "
Lens
plane
" "-
/
"-
/',
/
/'
//r1
C,
/
/
Principala"i!
--
_/
//
/ /i\ I '
I
/
' "-
--
/
"-
(AI Doublaconvex lens
"-
"
.......__
/
/
/
\
\
,
"-
(8) Double COncave lens
"-
"-
""
-- --
REFRACTION
343
approaching the lens are shown parallel to the lens plane,
their ray lines are parallel to the principal axis of the lens.
Observe that the principal axis passes through the center
of the lens and is perpendicular
to the lens plane.
In Figure 14-15(A) these parallel rays (which are also
parallel to the principal axis) are shown incident on a converging lens. They are refracted as they pass through the
lens, and they converge at a point on the principal axis
that locates the principal focus F of the lens. At point B the
incident ray is refracted toward the normal drawn to the
front surface of the lens. At point D the incident ray is
refracted away from the normal drawn to the back surface
of the lens. Because the rays of light actually pass through
the principal focus F, it is called the real focus. Real images
are formed on the same side of the lens as the real focus.
Had these rays approached the lens from the right parallel
Real images can he projected
screen.
on a
c
>:
f
r
."
o
,
,,
Realfocuo
,
;
/
LenIPlane'//
,,
f
IAI
IBI
Figure 14-15. The converging
lens. (A) A ray diagram that illustrates the refractive convergence
of incident light rays parallel to
the principal axis of the lens.
(8) A photograph using light pencils shows a similar convergence.
CHAPTER
.~44
14
to the principal axis, they \vould have converged at a point
on the principal axis that located the principal focus F'.
Rays parallel to the principal axis are shown incident on
a diverging lens in Figure 14-16(A). The rays of light are
refracted as they pass through the Jens and diverge as if
they had originated at the principal focus F' located in
front of the lens. Because the rays do not actually pass
through this principal focus, it called a virtual focus. Virtual images arc formed on the same side of the lens as the
virtua I focus.
Focallen!Jlh
IAI
Figure 14.16. The diverging lens.
(A) A ray diagram illustrates the
refractive divergence of incident
light rays parallel to the principal
axis of the lens. (8) A photograph
using light pencils shows a similar
divergence.
IBI
In general, lenses refract rays that are parallel to the
principal axis and the refracted rays either converge at the
real focus behind the (converging) lens or diverge as if
they originated at the virtual focus in front of the (diverging) lens. However, these foci are not midway between
the lens and the center of curvature as they are in spherical
mirrors. The positions of the foci on the principal axis depend on
the index of refraction of the lens. A common double convex
lens of crown glass has principal foci and centers of curvature that practically coincide and a focal length that is ap-
r
:~4,5
REFRACTION
proximately
equal to its radius of curvature.
The focal
length, [, of a lens is the distance between the optical center
of the lens and the principal focus. The focal length of any
lens depends
on its index of refraction and the curvature
of its surfaces.
The higher its index of refraction,
the
shorter is its focal length. The longer its radius of curvature, the longer is its focal length.
A "bundle"
of parallel light rays incident on a converging lens parallel to its principal axis is refracted and converges at the principal
(real) focus behind the lens. The
image is ideally a point of light at the real focus. This refraction is shown in Figure 14-17(A).
Bundles of parallel rays are not always incident on a lens
,,
e
f
parallel to its principal axis. If such rays are incident on a
converging lens at a small angle with the principal axis,
they converge at a point in the plane that contains the
principal focus of the lens and is perpendicular to the principal axis. This plane is called the focal plane of the lens. A
bundle of parallel rays incident on a converging lens at a
I
I
r
I
r
small angle a (alpha)
Figure 14-17(B).
with
its principal
axis is shown
The foea/length of a lens is the
same for light traveling in either
directioneven if the two lens surfaces
have different radii of curva-
tUre.
Bundle of rays: the essentially
parallel rays in a beam or a pencil
of light.
The focal plane of a concave mirror is described in Section 13.7.
in
~
,
c--.
'-'
I'r
tI
t~'\
~
,
IAI
by Refraction
The image of a point on an
object is formed by intersecting
refracted rays emanating
from the object point. In Section 13.8 we recognized
that
the task of locating graphically
an image formed by reflection is simplified
if we select incident
rays that have
known paths after reflection. These rays, called principal
rays, are shown with their reflected paths in Figure 13-18.
Their paths after refraction are also known. They are
shown with their refracted paths in Figure 14-18.
The image 5' of point 5 (the object in Figure 14-18) is
formed by the converging
lens when the refracted
rays
from point 5 intersect behind the lens. Ray 1, parallel to the
principal axis, is refracted through
the real focus, F. See
Figure
14-15. Ray 2, along the secondary
Focal plane
181
f
14.8 Images
~
axis, passes
Figure14-17. In (A) parallel rays
are shown incident on a converg-
ing lens parallel to its principal
axis. The refracted rays converge
at its principal focus. In (B) the
rays are incident at a small angle
a with the principal axis. Here the
refracted rays are focused at a
point in the focal plane F'F" of the
lens.
Review Fi:.;ure 13-18.
346
CHAPTER
14
through the optical center of the lens, 0, without being
appreciably refracted. Ray 3 passes through the principal
focus, F', and is refracted parallel to the principal axis. The
three refracted rays intersect at the image point 5'. Observe that any two of the principal rays emanating from the
same object point are able to locate the image of that point.
I_Focal-J
I length
Object
,
1
2
Principal
axis
Figure 14-18. The principal rays
used in ray diagrams. Any two of
tnese three rays from the same
point on the object locate the
image of that point.
I
I
I
I
I
I/Realfocus
oe,
"
Virtual
,'-
f~w
Image
I
I Lens
I plane
Lenses and mirrors differ in several ways.
1. Secondary axes pass through the optical center of a
lens and not through either of its centers of curvature.
2. The principal focus is usually near the center of curvature, depending on the refractive index of the glass from
which the lens is made. Thus the focal length of a double
convex lens is about equal to its radius of curvature.
3. Since the image produced by a lens is formed by rays
of light that actually pass through the lens, a real image is
The lens principles discussed in
this chapter apply to thin lenses.
Ray diagrams become more
plicated when
thick lenses.
com-
they are applied
to
fonned 011the side of the lens opposite the object. Virtual
images formed by lenses appear to be on the same side of
the lens as the object.
4. Convex (converging) lenses form images in almost
the same manner as concave mirrors, while concave (diverging) lenses are like convex mirrors in the manner in
which they form images.
Spherical lenses, like spherical mirrors, have aberration
defects. See Figure 14-19. When such lenses are used with
large apertures, images formed by rays passing through
the central zones of the lens are generally sharp and welldefined, while images formed by rays passing through the
edge zones are fuzzy. This defect of lens images is called
spherical aberration.
Similarly, rays of light coming from an object point not
on the principal axis are not brought to a sharp focus in the
image plane. This defect of lenses is known as lens astigma-
tism. By using a combination of lenses of suitable refractive
indexes and focal lengths, lens makers produce anastigmatic lenses. Such lenses give good definition over the entire image even when used with large apertures.
347
REFRACTION
Rays not teftactad
through princiPilI focus
:rA,.rt""
,
,,
,
,-
,
F
F
:1
,,-
,.,
IAI
A lens of short focal length that can be used with a large
aperture has a large light-gathering capability. It is said to
be a "fast" lens. The light-gathering
power of a camera
lens is given in terms of its f-number. This number is determined by the focal length of the lens and its effective
diameter. The effective diameter is the diameter of the
camera aperture (diaphragm) that determines the useful
lens area. The light-gathering power, or "speed," of a lens
is expressed as the ratio of its focal length to its effective
diameter. If the speed of a lens is given as f/4, it means
that its focal length is 4 times its effective diameter.
Because the useful area of a lens is proportional to the
square of its effective diameter, the light-gathering power
of a lens increases four times when its effective diameter is
doubled. An f/4lens is 4 times as fast as an f/8 lens, and is
16 times as fast as an f/16Iens. It follows thatthe required
time of exposure increases as the square of the f-number.
It should be recognized that all lenses having the same
f-number give the same illumination in the image plane,
regardless of their individual diameters. Therefore, a lens
having twice the diameter of another lens of the same fnumber will have four times the light-gathering
power,
but this light will be spread over an image having four
times the area and will give the same image brightness.
14.9 Images Formed by Converging Lenses
We shall
consider six different cases of image formation. These
cases are illustrated in Figure 14-20.
Case 1. Object at an infinite distance. The use of a small
magnifying glass to focus the sun's rays upon a point approximates this first case. While the sun is not at an infinite distance, it is so far away that its rays reaching the
earth are nearly parallel. When an object is at an infinite
distance and its rays are parallel to the principal axis of the
lens, the image formed is a point at the real focus. See Figure
14-20(A). This principle can be used to find the focal
length of a lens by focusing the sun's rays on a white
Figure 14-19.
(A) Rays
parallel
to
the principal axis but near the
edge of a converging
lens are not
refracted through the principal
focus. (B) An aperture can be
used to block these rays from the
lens.
The area
of a circle:
A = 7Tr2 = -."d'
4
Thus, A 0:;d2.
348
CHAPTER
,
14
..
-
"
---w
"
(C~Case 3.
,Obj"'t~,
2F'
F'
Image
(D! Case 4.
,
,
O"~~
2F'
F"
Image
-
"
"
2>'
(F!Case6.
Figure 14-20. Ray diagrams of
image formation by converging
,
F
2>
"
2>
lenses.
screen. The distance from the screen to the optical center
of the lens is the focal length of the lens.
Case 2. Object at a finite distance beyond twice the focal
length. Case 2 is illustrated in (B). Rays parallel to the principal axis, along the secondary axis, and emanating from a
point on the object are used to locate the corresponding
image point. The image is real, inverted, reduced, and located
between F and 2F on the opposite side of the lens. The lenses of
the eye and the camera, and the objective lens of the refracting telescope are all applications of this case.
Case 3. Object at a distance
equal to twice the focal length.
The
,
~
~
REFRACTION
construction of the image is shown in (C). The image is real,
inverted, the same size as the object, and located at 2F on the
opposite side of the lens. An inverting lens of a field telescope, which inverts an image without changing its size, is
an application of Case 3.
Case 4. Object at a distance between one and two focal lengths
away. This is the converse of Case 2 and is shown in (D).
The image is real, inverted, enlarged, and located beyond 2F on
the opposite side of the lens. The compound microscope, slide
projector, and motion picture projector are all applications
of a lens used in this manner.
Case 5. Object at the principal focus. This case is the converse of Case 1. No image is formed, since the rays of light
are parallel as they leave the lens (E). The lenses used in
lighthouses and searchlights are applications of Case 5.
Case 6. Object at a distance less than one focal length away.
The construction in (F) shows that the rays are divergent
after passing through the lens and cannot form a real
image on the opposite side of the lens. These rays appear
to converge behind the object to produce an image that is
virtual, erect, enlarged, and located on the same side of the lens
as the object. The simple magnifier and the eyepiece lenses
of microscopes, binoculars, and telescopes form images as
shown in Case 6.
14.10 Images Formed by Diverging Lenses
The only
kind of image of a real object that can be formed by a diverging lens is one that is virtual, erect, and reducedin size.
Diverging lenses are used to neutralize the effect of a converging lens, or to reduce its converging effect to some Figure 14.21. The image of an
extent. The image formation is shown in Figure 14-21. object formed by a diverging lens.
A
...-----F,--
B
Object
- --
;;
""
""
""
Virtual
image
F
:J50
CHAPTER 14
14.11 Object-Image
Relationships
For thin lenses,
the ratio of object size to image size equals the ratio of the
object distance to image distance. This rule is the same as
the rule for curved mirrors. Thus
hi
-
The lens equation is valid providing these sign conventions
are followed:
posilive for real objects
d "IS'
negative for virtual objects
positive for real images
d1
'
IS
'
f
A'
....';-"""
'I
::
iI
-~"
-:.,--
"-
::Virtual
Image
,I.
::
"'
..----':" .._;::'-"-L__.....-
"
images
positive for converging,
nega Ive or /Vergzng
'
-A--,
lenses
I enses
,,'
~ t:
F"
"'!---
1
1
1
-=-+f
do
di
where do represents the distance of the object from the
lens, di the distance of the image from the lens, and f the
focal length.
The numerical value of the focal length f is positive for a
converging lens and negative for a diverging lens. For real
objects and images, the object and image distances do and
dj have positive values. For virtual objects and images, do
and d; have negative values.
14.12 The Simple Magnifier
A converging lens of short
focal length can be used as a simple magnifier. The lens is
placed slightly nearer the object than one focal length and
the eye is positioned close to the lens on the opposite side.
This is a practical example of Case 6; the image is virtual,
erect, and enlarged as shown in Figure 14-22. A reading
glass, a simple microscope, and an eyepiece lens of a compound microscope or telescope are applications of simple
magnifiers.
Magnification M is simply the ratio of the image height to the
object height.
~
,;.,
--,-:;
di
ho
do
where ho and hi represent the heights of the object and the
image respectively, and do and dj represent the respective
distances of the object and image from the optical center of
the lens.
The equation used to determine the distances of the object and image in relation to focal length for curved mirrors
applies also to lenses. It can be restated here as
Observe that the lens equations
are the same as the mirror equations of Section 13.12.
[
IS [negative for virtual
[ " fi d
-
".,:
'
'.',
...
But,
-,'
hi
ho
Eye
Object
Figure 14-22. The simple magnifier.
So,
di
do
,
f
I,
351
REFRACTION
Suppose an object is viewed by the unaided eye. As it is
moved closer and closer to the eye, the image formed on
the retina becomes larger and larger. Eventually, a nearest
point is reached for the object at which the eye can still
form a clear image. This minimum distance for distinct
vision is approximately 25 em from the eye. Although this
nearest point varies among individuals, 25 em is taken as
the standard distance for most distinct vision; it is called
the near point. As a person grows older the muscles of the
eye, which thicken the lens and thus increase its convergence (shorten its focal length), gradually weaken. Consequently the near point moves out with aging.
If a converging lens is placed in front of the eye as a
simple magnifier, the object can be brought much closer
and the eye focuses on the virtual image. When the lens is
used in this way, the object is placed just inside the principal focus (do = f). The image is then formed approximately at the near point. Magnification, shown above to
be equal to the ratio dJdo, can now be expressed for a
simple magnifier as the ratio of the distance for most distinct vision to the focal length of the lens. When f is given
in centimeters, the magnification becomes approximately
M~
25 em
f
Magnifiers are labeled to show their magnifying power.
Thus a magnifier with a focal length of 5 em would be
marked 5X. One with a focal length of 2.5 em would be
marked lOX, etc. Observe that the shorter the focal length
of a converging lens, the higher is its magnification.
14.13 The Microscope
The compound
microscope,
thought to be invented in Holland by Zacharias Janssen
about 1590, uses a lens, the objective, to form an enlarged
image as in Case 4. This image is then magnified, as in
Case 6, by a second lens, called the eyepiece.
In Figure 14-23 a converging lens is used as the objective, with the object AB just beyond its focal length. At
A'B', a distance greater than twice the focal length of the
objective lens, an enlarged, real, and inverted image is
formed. The eyepiece lens acts as a simple magnifier to
enlarge this image.
The magnifying power of the objective is approximately
equal to the length of the tube, I, divided by the focal
length, fo, of the objective, or lIfo. The magnifying power
of the eyepiece, acting as a simple magnifier, is approximately 25 cm/fE. The total magnification is the product of
,,
EYePlec~.
_
2
, :.'.,..
m,:'::m.~
/
!::--1<':
</
I
_v-
..:
,.
I
L
I
>'/
Figure 14-23. Image formation
a compound microscope.
by
352
CHAPTER
the two lens magnifications.
approximation.)
M~
(The equation
14
is again an
2Scmxl
t, x to
14.14 Refracting
Telescopes
A refracting astronomical
telescope has two lens systems. The objective lens is of large
diameter so that it will admit a large amount of light. The
objects to be viewed in telescopes are always more distant
than twice the focal length of the objective lens. As a consequence the image formed is smaller than the object. The
eyepiece lens magnifies the real image produced by the objective lens. The magnifying power is approximately equal
to the focal length of the objective, fo, divided by the focal
length of the eyepiece, fE' or fJfh.
The lenses of a terrestrial, or field, telescope form images
just as their counterparts do in the refracting astronomical
telescope. Since it would be confusing to see objects inverted in a field telescope, another lens system is used to
reinvert the real image formed by the objective. This additional inverting lens system makes the final image erect, as
shown in Figure 14-24. The inverting lens system does not
magnify the image because the lens system is placed exactly its own focal length from the image formed by the
objective.
A"
A
....,
V
II
U
Second
Enlarged
image
image
(erect)
Eyepiece
"
Objective
,
-->2fo~I"
>1,
Figure 14-24. Image formation by
a terrestrial telescope.
The prism binocular is actually a double field telescope
that uses two sets of totally reflecting prisms instead of a
third lens system to reinvert the real images formed by the
objective lenses. This method of forming final images that
,
r
,
353
REFRACTION
are upright and correctly oriented also has the effect of
folding the optical path, thus making binoculars more
compact and easier to use than telescopes.
The prism binocular customarily nas descriptive markings stamped on its case, such as 7 x 35, 8 x 50, etc. The
first number gives its magnification, and the second number gives the diameters (in mm) of its objective lenses.
A viewing drt,ice with vile eyepiece and vile objecti!.'e lens is
called
{/ mOl/ocular.
r
,
,
Figure 14-25. A pair of totally reflecting prisms reinvert the image
in binoculars.
QUESTIONS:
1. (a) What is the difference
between a
convex and a concave lens? (b) What
effect does each have on light?
2. How does the relationship between
the focal length and radius of curvalure of a Jens differ hom that of
(/
curved mirror?
a
3.
4.
5.
6.
7.
~
,,
,
GROUP A
8.
(a) What
factors
determine
the focal
length of a lens? (b) How do they affect the focal length?
How many principal rays are needed
to locate the image of an object when
using a lens?
Why is no image formed when the
object is at the focal point of a convex
lens?
What is the relationship between the
focal length of a converging lens and
its magnification?
Which optical instrument most clearly
approximates the human eye?
What property of real images explains
why you have to load a slide projec-
tor with the slide upside down?
9. (a) How many lenses does a refracting telescope have? (b) Name and
describe them.
10. What must be true about the relationship between the index of refraction
of the air and the index of refraction
of the lens material in order for a
convex lens to be a converging lens?
GROUP B
11. What additional property of real images would help you distinguish between a real and a virtual image?
12. Describe how you would determine
the focal length of a converging lens.
13. If there are two converging lenses in
a compound microscope, why is the
image still inverted?
14. (a) What are you doing when you
focus vour camera? (b) How docs this
differ from the way the eye focuses?
15. What is happening to your
granddad's eyes that makes him say
3.54
CHAPTER
"my arms are too short to read the
newspaper"?
16. Why is an extra set of lenses needed
in a terrestrial telescope?
17. A student uses a lens to focus an inverted, reduced image of a candle.
(a) What kind of lens is it? (b) Where
is the candle located?
PROBLEMS:
GROUP /1
1. A converging lens with a £ocallength
of 15.0 em is placed 53.0 em from a
light bulb. Where would you place a
screen to focus an image of the object?
2. An object is 32.5 em from a converging lens with a focal length of
12.0 em. (a) Locate and describe the
image using a ray diagram. (b) Calculate the distance of the image from
the lens.
3. A convex lens of focal length 25.0 em
is placed 5.50 ill from a screen.
(a) Where should you place a candle
to form a sharp image? (b) If the, can~
dIe flame is 1.85 em high, how high
will its image be?
4. An object 30.0 em from a converging
lens forms a real image 60.0 em from
the lens. (a) Find the focal length of
the lens. (b) If the object is 9.75 em
high, how high is the image?
5. What is the magnifying power of a
simple magnifier whose focal length
is 15 em?
6. A camera lens has a S.10-em focal
length. How far must the lens be
from the film to take a clearly focused
picture of your friend, 6.50 m away?
7. The objective lens of a compound
microscope has a focal length of
0.500 em and the eyepiece has a focal
length of 2.00 em. If the tube of the
microscope is 15.0 em long, what is
the magnifying power of this microscope?
14
GROUP B
8. You set up a slide projector 3.50 m
from the screen to get an image
1.3S m high. (a) If the slide is 3.50 em
talt how far from the lens is the
slide? (b) What is the focal length of
this lens?
9. A camera, equipped with a lens of
focal length 4.80 em, is to be focused
on a tree that is 10.0 m away.
(a) How far must the lens be from the
film? (b) How much would the lens
have to be moved to take a picture of
another tree that is onlv 1.75 m
away?
10. The distance from the front to the
back of youreye is approximately
1.90 em. If you are to see a clear
image of your physics book when it
is 35.0 em from your eye, what must
be the focal length of the lens/cornea
system?
11. Suppose you look out the window
and see your friend, who is standing
15.0 m away. To what focal length
must your eye muscles adjust your
lens so that you may see your friend
clearly? (See Problem 10)
12. When a S.O-cm object is placed 12 em
from a converging lens, an image is
produced on the same side of the
lens as the object but 60.0 em away
from the lens. (a) What type of image
is this? (b) Find the focal length of
the lens. (c) Calculate the image size.
-'
PHYSICS ACTIVITY
Borrow a pair of glasses from a nearsighted friend, unless you have a pair
yourself. Hold them about 12 em from
your eye and look at different objects
through them. Describe what you see. If
possible, use lenses with different correction factors. How does the correction factor affect what you see? Try this with the
glasses of a far-sighted person, too.
,
f
f
.'155
REFRACTION
DISPERSION
14.15 Dispersion
by a Prism
Suppose a narrow beam
of sunlight is directed onto a glass prism in a darkened
room. If the light that leaves the prism falls on a white
screen, a band of colors is observed, one shade blending
gradually into another. This band of colors produced when
sunlight is dispersedby a prism is calleda solar spectrum. The
dispersion of sunlight was described by Newton, who
observed that the spectrum was "violet at one end, red at
the other, and showed a continuous gradation of colors in
between."
We can recognize six distinct colors in the visible spectrum. These are red, orange, yellow, green, blue, and violet.
Each color gradually blends into the adjacent colors giving
a continuous spectrum over the range of visible light. A
continuous spectrum is shown in Plate VII of the color
insert between pages 336 and 337. Light consisting of several
colors is called polychromatic light; light consisting of only one
color is called monochromatic light. All colors of the spectrum are present in the incident beam of sunlight. White
light is a mixture of these colors.
The dispersion of light by a prism is shown in Plate I of
the color insert. It is evident that the refraction of red light
by the prism is not as great as that of violet light; the refractions of other colors lie between these two. Thus the
index of refraction of glass is not the same for light of different colors. If we wish to be very precise in measuring
the index of refraction of a substance, monochromatic
light must be used and the monochrome color must be
stated. Some variations in the index of refraction of glass
are given in Table 14-1.
14.16 The Color of Light
A hot solid radiates an appreciable amount of energy that increases as the temperature
is raised. At relatively low temperatures,
a small amount
of energy is radiated in the infrared region. As the temperature of the solid is raised, some of the energy is radiated
at higher frequencies. These frequencies range into the red
portion of the visible spectrum as the body becomes "red
hot." At still higher temperatures, the solid may be "white
hot" as the major portion of the radiated energy shifts
toward the higher frequencies.
Suppose we have a clear-glass tungsten-filament
lamp
connected in an electric circuit so that the current in the
filament, and thus the temperature of the filament, can be
controlled. A small electric current in the filament does not
Table 14-1
VARIATION OF THE INDEX
OF REFRACTION
Color
,ed
yellow
blue
violet
Flint
Crown
glass
glass
1.515
1.517
1.523
1.533
1.622
1.627
1.639
1.663
:l;j6
CHAPTER
14
change the filament's appearance.
As we gradually increase the current, however, the filament begins to glow
with a dark red color. To produce this color, electrons in
the atoms of tungsten must have been excited to suffi~
ciently high-energy levels so that upon de-excitation they
emit energy with frequencies of about 3.9 x 1014 Hz. Even
before the lamp filament glows visibly, experiments show
that it radiates infrared rays that can be detected as heat.
As the current is increased further, the lamp filament
gives off orange light in addition to Ted; then the filament
adds yellow, and finally at higher temperatures
it adds
enough other colors to produce white light.
A photographer's
tungsten-filament
flood lamp operates at a very high temperature. If the white light of such a
lamp is passed through a prism, a band of colors similar to
the solar spectrum is obtained. Figure 14-26 shows the
distribution of radiant energy from an "ideal", or blackbody, radiator for several different temperatures.
,
6000K
5000K
4000K
Figure 14.26. Distribution of
radiant energy from an ideal
radiator.
o
,
Ivbgyor
4000;"
7600;"
Considering the wavelengths of various colors shown in
Plate I, it is evident that our eyes are sensitive to a Tange of
frequencies equivalent to about one octave. The wavelength of the light at the upper limit of visibility (7600 A) is
about twice the wavelength of the light at the lower limit
of visibility (4000 A). We use the word color to describe a
psychological sensation through the visual sense related
to the physical stimulus of light. The color perceived for
monochromatic
light depends on the frequency of the
light. For example, when light of about 4.6 x 1014 Hz
(6500 A) enters the eye, the color perceived is red.
14.17 The Color of Objects
Color is a property of the
light that reaches our eyes. Objects may absorb certain
REFRACTION
II-
1-
frequencies from the light falling upon them and reflect
other frequencies. For example, a cloth that appears blue
in sunlight appears black when held in the red portion of a
solar spectrum in a darkened room. A red cloth held in the
blue portion of the solar spectrum also appears black. The
color of an opaque object depends upon the frequenciesof light it
reflects. If all colors are reflected, we say it is white. It is black
if it absorbs all the light that falls upon it. It is called red if
it absorbs all other colors and reflects only red light. The
energy associated with the colors absorbed is taken up as
heat.
A piece of blue cloth appears black in the red portion of
the spectrum because there is no blue light there for it to
reflect and it absorbs all other colors. For the same reason,
a red cloth appears black in the blue portion of the spectrum. The color of an opaque object depends on the color of the
light incident upon it.
Ordinary window glass, which transmits all colors, is
said to be colorless. Red glass absorbs all colors but red,
which it transmits. The stars of the United States flag
would appear red on a black field if viewed through red
glass. The color of a transparent object depends upon the color of
the light t/wt it transmits.
14.18 Complementary
Colors
Because polychromatic
light can be dispersed into its elementary colors, it is reasonable to suppose that elementary colors can be combined to form polychromatic light. There are three ways in
which this can be done.
1. A prism placed in the path of the solar spectrum
formed by another prism will recombine the different colors to produce white light. Other colors can be compounded in the same manner.
2. A disk that has the spectral colors painted on it can
be rotated rapidly to produce the effect of combining the
colors. The light from one color forms an image that persists on the retina of the eye until each of the other colors
in turn has been reflected to the eye. If pure spectral colors
are used in the proper proportion, they will blend to produce the same color sensation as white light.
3. Colored light from the middle region of the visible
spectrum combined with colored light from the two end
regions produce white light. This method is described in
Section 14.19.
Two prisms are used as described above, but with the
red light from the first prism blocked off from the second
prism. The remaining spectral colors are recombined by
the second prism to produce a blue-green color called
3.'>7
Actually, in order to be reflected
the red light must interact with
the object. The interaction consists
of a resonant absorption by electrons of the object's atoms. The
energy is immediately re-emitted
or "reflected."
In physics, the term "color" refers to light. In art, the term
"c%r" often refers to the shade
of a pigment.
Combining colored lights is an
additive process.
.158
Become familiar with the six elementary colors and their complements illustrated in Plate II.
CHAPTER
14
cyan. Red light and cyan should therefore combine to pro-
duce white light, and a rotating color wheel shows this to
be true. Any two colors that combine to form white light are said
to be complementary.
In similar fashion it can be shown that blue and yellow
are complementary colors. White fabrics acquire a yellowish color after continued laundering. A blue dye added to
laundry detergents neutralizes the yellow color and the
fabrics appear white. Iron compounds in the sand used for
making glass impart a green color to the glass. Manganese
gives glass a magenta, or purplish-red,
color. However, if
both these elements are present in the right proportion,
the resulting glass will be colorless. Green and magenta
are complementary colors. The complements of the six elementary spectral colors are shown in Plate II.
14.19 The Primary Colors
The six regions of color in
the solar spectrum are easily observed by the dispersion of
sunlight. Further dispersion within a color region fails to
reveal any other colors of light. We generally identify the
range of wavelengths
comprising a color region by the
COIOfof light associated with that region. These are the six
elementary colors of the visible spectrum; they combine to
produce white light. However, the complement of an elementarv color is not monochromatic but is a mixture of all
of the elementary colors remaining after the one elementary color has been removed.
Experiments with beams of different colored lights have
shown that most colors and hues can be described in
terms of three different colors. Light from one end of the
visible spectrum combined with light from the middle region in various proportions will yield all of the color hues
in the half of the spectrum that lies in between them. Light
from the opposite end, when combined with light from
the middle region, will also yield all hues in the half of the
spectrum that lies in between them. Colored light from the
wo end regions and the middle region can be combined to
match most of the hues when mixed in the proper proportions. The three colors that can be used most successfullv
in color matching experiments of this sort are red, gree~,
and blue. Consequently these have been called the primary
colors.
Suppose we project the three primary colors onto a
white screen as shown in Plate IIHA). The three beams
can be adjusted to overlap, producing additive mixtures of
these primary colors. Observe that green and blue lights
combine to produce cyan, the complement of red; green
and red lights combine to produce yellow, the comple-
r
r
REFRACTION
ment of blue; and red and blue lights combine to produce
magenta, the complement of green. Thus two primary colOfS combine to produce the complement of the third primary color. VVhere the three primary colors overlap, white
light is produced.
359
Combining primary colors is an
additive process.
f
,
~
~
14.20 Mixing Pigments
When the complements blue
light and yellow light are mixed, white light results by an
additive process. If we mix a blue pigment with a yellow
pigment, a green mixture results. This process is subtractive since each pigment subtracts or absorbs certain colors.
For example, the yellow pigment subtracts blue and violet
lights and reflects red, yellow, and green. The blue pigment subtracts red and yellow lights and reflects green,
blue, and violet. Green light is the only color reflected by
both pigments; thus the mixture of pigments appears
green under white light.
The subtractive process can be demonstrated by the use
of various color filters that absorb certain frequencies and
transmit others from a single white-light source.
When pigments are mixed, each one subtracts certain
colors from white light, and the resulting color depends
on the frequencies that are not absorbed. The primary pigments are the complements of the three primary colors. They are
cyan (the complement of red), magenta (the complement
of green), and yellow (the complement of blue). When the
three primary pigments are mixed in the proper proportions, all the colors are subtracted from white light and the
mixture is black. See Plate III(B).
14.21 Chromatic Aberration
Because a lens configuration has some similarity to that of a prism, some dispersion occurs when light passes through a lens. Violet light
is refracted more than the other colors and is brought to a
focus by a converging lens at a point nearer the lens than
the other colors. Because red is refracted the least, the
focus for the red rays is farthest from the lens. See Figure
14-27(A). Thus images fanned by ordinary spherical
lenses are always fringed with spectral colors. The nvnfocusing oflight of different colors is called chromatic aberration. Sir Isaac Newton developed the reflecting telescope
to avoid the objectionable effects of chromatic aberration
that occur when observations are made through a refracting telescope.
Th.e English optician John Dollond (1706-1761) discovered that the fringe of colors could be eliminated by means
of a combination of lenses. A double convex lens of crown
glass used with a suitable plano-concave lens of flint glass
Combining primary pigments is a
subtractive process.
Compare the additive combinations of primary colors in Plate
lll(A) with the subtractive combinations of primary pigments in
Plate III(B).
,160
CHAPTER
Figure 14-27.
(A) Chromatic
aberration is caused by unequal refraction of the different colors.
(B) A two-lens combination of
crown and flint glass
chromatic aberration.
corrects
14
corrects for chromatic aberration without preventing refraction and image formation. A lens combination of this
type is shown in Figure 14-27(B). It is called an achromatic
(without color) lens.
v
v
QUK5TJONS: CROUP ,1
1. What are the six colors of the spectrum?
2. Use Table ]4.1 to answer these questions: (a) Which of the two materials
is optically less dense? (b) How does
the index of refraction vary with the
wavelength of the light? (c) Which
color changes speed the most?
3. (a) What are the three primary additive colors? (b) What happens when
you mix them?
4. You have a magenta opaque object.
What color would it look under the
following colors of light? (a) white
(b) red (c) cyan (d) green (e) yellow
5. (a) What are complementary
colors
with reference to light? (b) Name the
three pairs of complementary
colors.
SUMMARY
CI
6. Explain what could happen when you
mix the following: (a) cyan and yellow pigment (b) blue and yellow light
(c) spectral blue and yellow pigments.
7. A student, operating a spotlight for
the school show, points out that there
are colored fringes around the edges
of the white spot from the light. Explain what is happening here.
8. (a) What is meant by the term dispersion? (b) Who first explained this phenomenon?
9. How does the range of frequencies
that our eyes perceive as visible light
compare to the range of vibrations
our ears perceive as sound?
10. If a black body absorbs all visible radiation incident on its surface, how
can we see it?
.....
The laws of refraction describe the behavior of light rays that pass obliquely
from one medium into another of different optical density. The index of refraction of any transparent material is defined
in terms of the speed of light in a vacuum and the speed of light in the
material. The index of refraction is also
expressed in terms of Snell's law. Total
reflection is explained on the basis of the
critical angle of a material and the limiting value of the angle of refraction.
Converging lenses have convex surfaces
and form images in a manner similar to
that of concave mirrors. Diverging lenses
have concave surfaces and form images in
REFRACTION
.161
a manner similar to that of convex mirrors. The general lens equations correspond to those for curved mirrors. These
equations relate object and image sizes
with their respective distances from the
lens and object and image distances with
the focal length of the lens.
A converging lens forms either real or
virtual images of real objects depending
on the position of the object relative to
the principal focus of the lens. A diverging lens forms only virtual images of real
objects. Lens functions in common types
of refractive instruments such as microscopes and telescopes can be analyzed by
considering each lens separately.
Sunlight is composed of polychromatic
VOCABUUIIY
<=::!
achromatic lens
angle of refraction
chromatic aberration
complementary
color
converging lens
critical angle
diverging lens
elementary color
eyepiece
f-number
focal length
focal plane
r
(
light that undergoes dispersion when refracted by a prism. Six elementary colors
are recognized in dispersed white light.
The visual perception of color is related
to the frequency of visible light.
The removal of an elementary coJor
from white light leaves a polychromatic
color that is the complement of the color
removed. Addition of complementary
colors produces white light. White light is
produced by adding the primary colors.
Any two of the three primary colors will
combine to produce the complement of
the third. The primary pigments are complements of the primary colors. Their
combination is considered to be a subtractive process.
.
index of refraction
lens equation
monochromatic
near point
objective
optical density
optical refraction
polychromatic
primary color
primary pigment
principal axis
principal focus
principal ray
real focus
real image
refraction
refractometer
secondary axis
Snell's law
solar spectrum
spherical aberration
total reflection
virtual focus
virtual image