Download Chapter 31 The Law of Refraction

Chapter 31 The Law of Refraction
“If in other sciences we should arrive at certainty
without doubt and truth without error, it behooves us
to place the foundations of knowledge in
mathematics.”
Roger Bacon
31.1 Refraction
In chapter 30, we saw that an incident ray of light is reflected from a piece of
reflecting material such that the reflected ray makes the same angle with the
normal as the incident ray. If the reflecting surface is a boundary between two
different transparent mediums, such as air and glass, some of the incident light is
also transmitted into the glass, as shown in figure 31.1. However, it is observed
Figure 31.1 Reflection and refraction of light.
experimentally that this transmitted ray of light is bent as it enters the second
medium. The bending of light as it passes from one medium into another is called
refraction. Refraction of light occurs because light travels at different speeds in
different mediums. Light traveling through a vacuum travels at the speed c = 3.00 ×
108 m/s. But when light enters a medium there is a complex interaction between the
electromagnetic wave (light) and the atomic configuration of the medium. This
interaction causes the electromagnetic wave to slow down in the medium. This
slowing down of the wave as it goes from a vacuum into the medium causes it to
bend. We will use Huygens’ principle to show how this is accomplished in section
31.2.
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Chapter 31 The Law of Refraction
31.2 The Law of Refraction
Let us consider a wave front B1B2 of a plane parallel monochromatic wave
impinging on the boundary of two different mediums, as shown in figure 31.2. The
incident ray makes an angle of incidence i with the normal N. The incident light
Figure 31.2 The law of refraction by Huygens’ principle.
moves at a speed v1 in medium 1 and v2 in medium 2, and we assume that v1 is
greater than v2. The incident wave has just touched the boundary at B1. In a time
∆t, B2, the upper portion of the initial wave front, travels a distance v1∆t, and
impinges at the boundary of the interface at B’2.
In this same time interval ∆t, the wave front at B1 enters the second medium.
By Huygens’ principle, a secondary wavelet can be drawn emanating from the point
B1. This wave moves a radial distance v2∆t in the second medium in the time
interval ∆t, and is shown as the circle of radius v2∆t in the figure. The radial
distance v2∆t is less than the distance v1∆t because v2 is less than v1. By Huygens’
principle, the line drawn from B’2 that is tangent to the secondary wavelet is the
new wave front. The point of tangency is denoted by B’1 and the new wave front in
medium 2 is B’1B’2 .The radius from B1 to B’1, when extended, becomes the refracted
ray B1C. The other refracted rays are drawn parallel to B1C, as shown in figure 31.2.
The angle that the refracted ray makes with the normal is called the angle of
refraction r.
We can obtain the relation between the angles i and r from the geometry of
figure 31.2. Since line B2B’2 makes an angle i with the dashed normal, angle B2B’2B1
is equal to (900 − i), and since the sum of the angles in triangle B1B2B’2 must equal
1800, it follows that angle B2B1B’2 is also equal to the angle i. Using similar
reasoning, angle B1B’2B’1 is equal to r. Hence, from the trigonometry of figure 31.2,
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Chapter 31 The Law of Refraction
sin i = v1∆t
B1B’2
(31.1)
sin r = v2∆t
B1B’2
(31.2)
and
Let us divide equation 31.1 by equation 31.2 and obtain
and
v 1 ✁t
∏
sin i = B 1 B 2
v 2 ✁t
sin r
B 1 B ∏2
sin i = v 1 = constant = n
21
sin r v 2
(31.3)
Equation 31.3 is the law of refraction. It says that the ratio of the sine of the angle
of incidence to the sine of the angle of refraction is equal to the ratio of the speed of
light in medium 1 to the speed of light in medium 2. Because the speed of light in
medium 1 v1 is a constant and the speed of light in medium 2 v2 is a constant, then
their ratio v1/v2, must also be a constant. This constant is called the index of
refraction of medium 2 with respect to medium 1 and is denoted by n21.
If medium 1 is a vacuum, then v1 = c, and the index of refraction of the
medium with respect to a vacuum is
n= c
(31.4)
v
Since the speed v in any medium is always less than c, the index of refraction, n =
c/v, is always greater than 1, except for in a vacuum where it is equal to 1. Indices of
refraction for various substances are given in table 31.1. Notice that the index of
refraction of air is so close to the value 1, the index of refraction of a vacuum, that in
many practical situations, air is used in place of a vacuum.
The law of refraction can be put in a more convenient form by using equation
31.4. We can write the index of refraction of medium 1 with respect to a vacuum as
n1 = c
v1
(31.5)
whereas we can write the index of refraction of medium 2 with respect to a vacuum
as
n2 = c
(31.6)
v2
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Chapter 31 The Law of Refraction
Table 31.1
Index of Refraction for Various Materials (λ = 589.2 nm, the D line of sodium)
Substance
n
Air
1.00029
Benzene
1.5
Diamond
2.42
Glass, crown
1.52
Glass, flint
1.57-1.72
Glass, fused quartz
1.46
Glycerine
1.47
Ice
1.31
Plexiglass
1.49
Quartz crystal
1.54
Water
1.33
Solving for the speeds v1 and v2 from equations 31.5 and 31.6, respectively, and
substituting them into equation 31.3, gives
n21 = v1 = c/n1 = n2
v2
c/n2 n1
(31.7)
Using equation 31.7, we can write the law of refraction, equation 31.3, as
sin i = n21 = n2
sin r
n1
or
n1 sin i = n2 sin r
(31.8)
Equation 31.8 is the form of the law of refraction that we will use in what follows. It
is also called Snell’s law after its discoverer, Willebrord Snell (1591-1626) a Dutch
mathematician who discovered it in 1620, the same year the Pilgrims landed at
Plymouth Rock. Note that if a ray lies along the normal, then the angle of incidence
i is equal to zero, and hence the angle of refraction r must also be zero, and there is
no refraction of this ray.
The fact that the speed of light varies from medium to medium has an
important effect on the wavelength of light. When an initial wave enters a second
medium, its wavelength changes. We can see this from equation 31.3, where
n21 = v1
v2
However, the speed of any wave is given by
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Chapter 31 The Law of Refraction
v = λν
(31.9)
where λ is the wavelength and ν is the frequency of the wave. Hence,
n21 = v1 = λ1ν = λ1
v2 λ2ν λ2
(31.10)
The frequency ν of the wave does not change as it goes across the boundary because
there are the same number of wave fronts passing from medium 1 into medium 2, per
unit time1. Therefore,
λ2 = λ1 = λ1 = n1 λ1
(31.11)
n21 n2/n1 n2
That is, the wavelength of the light in the second medium λ2 is less than the initial
wavelength λ1 by the factor 1/n21 = n1/n2. To summarize, the speed of light varies
from one medium to another but the frequency of the light remains the same in both
mediums. Because of the changing speed of light, the wavelength of the light changes
as it goes into the second medium.
Example 31.1
The refraction of light. A ray of light of 500.0 nm wavelength in air, impinges on a
piece of crown glass at an angle of incidence of 35.00. Find (a) the angle of refraction,
(b) the speed of light in the glass, and (c) the wavelength of light in the glass.
Solution
a. We find the angle of refraction from Snell’s law, equation 31.8, with the indices of
refraction n1 = nair = 1.00 and n2 = nglass = 1.52, found in table 31.1. Therefore,
n1 sin i = n2 sin r
sin r = n1 sin i
n2
= 1.00 sin 35.00 = 0.377
1.52
r = sin−1 0.377
and the angle of refraction is
r = 22.20
That is, you can’t have 5 waves per second incident on the boundary between the two mediums and
only 3 waves per second transmitted through the boundary. Where did the other waves go? You must
have the same number leaving the boundary as you have entering the boundary. Hence the
frequency of the wave must be the same on both sides of the boundary.
1
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Chapter 31 The Law of Refraction
b. The speed of light in the glass, found from equation 31.6, is
v2 = c = 3.00 × 108 m/s
n2
1.52
= 1.97 × 108 m/s
c. The wavelength of light in the glass, found from equation 31.11, is
λ2 = n1 λ1
n2
= (1.00) (500.0 nm)
1.52
= 329 nm
To go to this Interactive Example click on this sentence.
The law of refraction has been derived by Huygens’ principle, treating light
as a wave phenomenon. We will now simplify the analysis of refraction by using
only the ray model of light. Thus, the refraction of waves in figure 31.2 will now be
shown as the equivalent refraction of rays in figure 31.3. The dashed line in figure
31.3(a) is the direction that the incident ray would have followed if it had not
Figure 31.3 The law of refraction by a ray diagram.
entered the second medium. Note that the incident ray was bent away from its
original direction and bent toward the normal. Medium 1 has an index of refraction
that is less than medium 2. The medium with the smaller value of the index of
refraction is called the rarer medium, whereas the medium with the larger value of
n is called the denser medium. Hence, whenever a ray of light goes from a rarer
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Chapter 31 The Law of Refraction
medium to a denser medium the refracted ray is always bent toward the normal. The
larger the value of the index of refraction n2, the greater the amount of bending.
By the principle of reversibility, a light ray that reverses the path in figure
31.3(a), goes from a denser medium to a rarer medium and is bent away from the
normal, as seen in figure 31.3(b). Hence, whenever a ray of light goes from a denser
medium to a rarer medium, the refracted ray is bent away from the normal.
31.3 Apparent Depth of an Object Immersed in Water
An interesting example of the refraction of light is the observation of an object when
it is under water. A stick immersed in water appears to be bent, figure 31.4(a), and
a fish in water is not where it seems to be. Sometimes in a carnival or a bazaar, a
game is played where a glass is placed in water, as in figure 31.4(b), and the
(a)
(b)
Figure 31.4 (a) The stick appears bent because of refraction at the water’s surface.
(b) The apparent depth of an object immersed in water.
patrons try to throw a quarter into the glass, receiving a prize if they are successful.
Of course it is much more difficult than it appears because the glass is not where it
seems to be.
As an example, let the object, the quarter in the glass, be at the bottom of the
water at Q, a distance p from the top of the water. A ray from Q makes an angle of
incidence i with the normal. Because the ray is going from the more dense medium,
water, to the less dense medium, air, the refracted ray is bent away from the
normal, as shown. The observer’s eye, located at R, sees the ray ROQ, but the
person believes that light rays travel in straight lines, and thinks he sees the ray
ROI. The person, therefore, assumes that the quarter in the glass is located at I, a
distance q below the surface of the water. In a sense, you might say that he is
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Chapter 31 The Law of Refraction
looking at the image of the glass. The vertical ray QN’ lies along the normal and
hence is not refracted, as shown. These two rays QOR and QN’ do not intersect
anywhere in medium 2, the air, but when produced backward, the ray ROI
intersects QN’ at I, and in this sense, I appears as the image of Q. The glass appears
to be much closer to the top of the water than it really is. The distance q from the
top of the water to I is called the apparent depth of the object. Let us now solve for
the apparent depth.
From Snell’s law of refraction, equation 31.8, we have
n1 sin i = n2 sin r
We determine the angles i and r from the geometry of figure 31.4 by observing that
tan i = x
p
(31.12)
tan r = x
q
(31.13)
and
To simplify the solution, let us assume that the angles i and r are small enough (less
than 100) to use the small-angle approximation, that is,
sin θ ≈ tan θ
With this assumption, equations 31.12 and 31.13 can be substituted back into
equation 31.8 as
n1 tan i = n2 tan r
n1 x = n2 x
p
q
Solving for the apparent depth,
q = n2 p
(31.14)
n1
Example 31.2
Apparent depth. A glass is placed in the bottom of a tub of water 25.0 cm deep. What
is its apparent depth?
Solution
The apparent depth, found from equation 31.14, is
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Chapter 31 The Law of Refraction
q = n2 p
n1
1
=
(25.0 cm)
1.33
= 18.8 cm
The glass appears to be 18.8 cm below the top of the water when in fact it is really
25.0 cm below the top.
To go to this Interactive Example click on this sentence.
In the solution of the problem of the apparent depth, we made an
assumption, just as is often done in physics. The solution is, of course, only as good
as the assumption, that is, that the angles i and r are less than 100. Suppose they
are not. What effect does this have on the solution? For example, if x in figure 31.4
is 12.5 cm then the angle i, found from equation 31.12, is
i = tan−1 x = tan−1 12.5
p
25.0
0
= 26.6
The angle r, found from Snell’s law, is
sin r = n1 sin i
n2
= 1.33 sin 26.60
1.00
and the angle of refraction is
r = 36.50
This is obviously much larger than the assumed limit of 100. The apparent depth,
now found from equation 31.13, is
q=
x = 12.5 cm
tan r tan 36.50
= 16.9 cm
If the calculation is repeated with x = 25.0 cm, the angle of incidence i is now equal
to 45.00, and the angle of refraction r is now equal to 70.10 and the apparent depth q
is 9.05 cm. These values are significantly different from the values obtained with
the initial assumptions. This is again a reminder that in trying to represent the
physical world in terms of mathematical equations, certain assumptions are made.
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Chapter 31 The Law of Refraction
When these assumptions are valid, the equations are good. When they are not valid
the equations no longer represent the physical world and are essentially useless.
31.4 Refraction through Parallel Faces
A ray of light passing through two parallel boundaries is refracted at each of these
boundaries, but the final ray is in the same direction as the initial ray, as shown in
figure 31.5. An incident ray makes an angle i1 with the normal N. Because medium
Figure 31.5 Refraction through parallel faces.
2 is more dense than medium 1, the refracted ray is bent toward the normal in
medium 2, making an angle of refraction of r2 with the normal. Snell’s law for the
first interface becomes
n1 sin i1 = n2 sin r2
(31.15)
This refracted ray now becomes the incident ray for the interface between medium 2
and medium 1, and makes an angle i2 with the normal N’ .Since normals N and N’
are parallel, angle r2 is equal to angle i2. As the ray goes from the more dense
medium 2 to the less dense medium 1, the ray is refracted away from the normal
through the angle r1, as shown. Snell’s law for the refraction at this interface is
n2 sin i2 = n1 sin r1
(31.16)
Because angle r2 is equal to angle i2, the right-hand side of equation 31.15 is equal
to the left-hand side of equation 31.16. Equating them, we obtain
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Chapter 31 The Law of Refraction
n1 sin i1 = n1 sin r1
and
i1 = r1
Thus, the angle r1 is equal to the initial angle i1, and the final refracted ray comes
out parallel to the direction of the original ray, but slightly displaced.
Example 31.3
Refraction through parallel faces. A ray of light in air makes an angle of incidence of
40.00 with the normal to a plate of glass of n = 1.50, as in figure 31.5. Find the angle
of refraction of the ray in the glass and the angle of refraction of the final ray as it
passes from the glass back into the air.
Solution
The angle of refraction for the first interface, found from Snell’s law, is
n1 sin i1 = n2 sin r2
sin r2 = n1 sin i1
n2
= 1.00 sin 40.00 = 0.4285
1.50
and the angle of refraction for the first interface becomes
r2 = sin−1 (0.4285)
r2 = 25.40
For the second interface, the law of refraction is
n2 sin i2 = n1 sin r1
sin r1 = n2 sin i2 = 1.50 sin 25.40 = 0.6434
n1
1.00
and the angle of refraction for the second interface is
r1 = sin−1 (0.6434)
r1 = 40.00
Note that the final ray makes the same angle as the initial ray, and they are,
therefore, parallel.
31-11
Chapter 31 The Law of Refraction
To go to this Interactive Example click on this sentence.
31.5 Total Internal Reflection
As shown in figure 31.1, when an incident ray falls on a boundary of a transparent
material, part of the ray is reflected back into the first medium, while part of the
ray is refracted through the second medium. If the incident ray is in the dense
medium, then the refracted ray is bent away from the normal, as shown in figure
31.6(a). If the angle of incidence is increased, the angle of refraction is also
increased, as shown in figure 31.6(b). If the angle of incidence is increased still
Figure 31.6 Total internal reflection.
further, the point is reached where the angle of refraction becomes 900, as shown in
figure 31.6(c). The angle of incidence that causes the refracted ray to bend through
900 is called the critical angle of incidence. When the incident angle becomes
greater than the critical angle, no refraction occurs. That is, no light enters the
second medium at all; all the light is reflected. This condition is called total internal
reflection, because refraction has been eliminated entirely. We obtain the condition
for total internal reflection by finding the critical angle of incidence that allows the
angle of refraction to be 900. This is done by applying Snell’s law of refraction:
n1 sin ic = n2 sin 900
or
sin ic = n2 sin 900 = n2
n1
n1
(31.17)
Equation 31.17 gives the condition for total internal reflection.
Total internal reflection can only occur when light travels from a denser
medium to a rarer medium, because it is only then that the refracted ray is bent
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Chapter 31 The Law of Refraction
away from the normal. For a light ray going in the opposite direction, the refracted
ray bends toward the normal and the angle of refraction could never become 900.
Example 31.4
Critical angle for a glass-air interface. What is the critical angle for a ray of light
that goes from glass to air? Take nglass = 1.50.
Solution
The critical angle, found from equation 31.17, is
sin ic = n2 = nair = 1.00 = 0.667
n1 nglass 1.50
ic = sin−1 (0.667)
ic = 41.80
To go to this Interactive Example click on this sentence.
Let us now take an ordinary piece of glass, whose index of refraction is 1.50,
and cut it into a triangle with angles of 45.00, 90.00, and 45.00, as shown in figure
31.7 An incident ray perpendicular to the first face goes directly into the glass
Figure 31.7 Total internal reflection and the prism.
without any refraction, because it is along the normal of the first face. This ray
makes an angle of incidence of 45.00 with the normal as it hits the second face. But
45.00 is greater than the critical angle of 41.80 just found in example 31.4. Hence,
none of this light crosses the interface into the air, but instead it is totally reflected
at this second face. This reflected ray, being normal to the third face, completely
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Chapter 31 The Law of Refraction
passes into the air at the third face without any refraction there. This ordinary piece
of glass, cut into the shape of a triangle, with an angle greater than the critical angle,
is called a prism. One of its functions is to completely reflect light. It is even better
than a plane mirror for reflection because in a plane mirror some light energy is
absorbed by the silver of the mirror, whereas the prism, acting as a device for total
internal reflection, reflects everything. No energy is ever transmitted into the other
medium at the interface. Prisms are used in binoculars, spectrometers, and
reflecting telescopes.
Example 31.5
Critical angle for a water-air interface. What is the critical angle between water and
air? The index of refraction for water is nw = 1.33.
Solution
The critical angle, found from equation 31.17, is
sin ic = n2 = nair = 1.00
n1 nw 1.33
ic = 48.80
The critical angle for a water-air interface is thus greater than the critical angle for
a glass-air interface.
To go to this Interactive Example click on this sentence.
A further example of internal reflection can be found in the area of fiber
optics. An optical fiber consists of a single flexible glass rod, or an array of them, of
high refractive index. Light entering the glass undergoes total internal reflection
from the walls of the glass fiber and the light travels down the length of the fiber
with little or no absorption. Due to their flexibility, the fibers can be curved into
various shapes. But as long as the reflection angle remains greater than the critical
angle, the light travels along the length of the fiber.
31.6 Dispersion
In all the previous discussions of refraction we assumed that the incident light was
monochromatic, that is, it contained light of only one wavelength. If white light, a
mixture of all colors and hence numerous wavelengths, impinges on a transparent
surface, we obtain the refraction pattern shown in figure 31.8. The white light is
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Chapter 31 The Law of Refraction
Figure 31.8 Dispersion of light.
refracted such that each color, or wavelength, is refracted by a different angle.
Hence, the white light is dispersed into its constituent wavelengths. The separation
of white light into its component colors is called dispersion. The band of colors is
known as a spectrum. The spectrum of visible light contains the colors red, orange,
yellow, green, blue, and violet.
Violet light, which has the shortest wavelength, is bent the most, whereas red
light with the longest wavelength, is bent the least. Dispersion occurs because the
index of refraction is not strictly a constant for a particular material, but rather
varies slightly with wavelength. For example, the index of refraction of crown glass
for red light (700.0 nm) is about 1.51, whereas for violet light (400.0 nm) it is about
1.53, which is only a slight difference, yet significant enough to cause the
phenomenon of dispersion.
Example 31.6
Refraction for various wavelengths. White light makes an angle of incidence of 30.00
as it strikes a piece of glass. Find the angle of refraction for (a) red light (700.0 nm)
if n = 1.51 for red light and (b) for violet light (400.0 nm) if n = 1.53.
Solution
a. The angle of refraction for red light, found from Snell’s law, is
n1 sin i = n2 sin rr
sin rr = n1 sin i = 1.00 sin 30.00 = 0.331
n2
1.51
rr = sin−1 (0.331)
rr = 19.30
b. For violet light, the angle of refraction is
n1 sin i = n2 sin rv
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Chapter 31 The Law of Refraction
sin rv = n1 sin i = 1.00 sin 30.00 = 0.3267
n2
1.53
rv = sin−1 (0.3267)
rv = 19.10
Notice that the total angular separation between the red and violet light is only 0.2
degrees for this glass.
To go to this Interactive Example click on this sentence.
Dispersion is most pronounced if a prism is used, as shown in figure 31.9.
Because of the angle between the two faces of the prism, the deviation D of the final
refracted ray is greater.
Figure 31.9 Dispersion by a prism.
Because the index of refraction is a function of the incident wavelength, when
defining the index of refraction of a medium, we need to specify the particular
wavelength used. When the wavelength is not specified in this book it is assumed
that the wavelength is λ = 589.2 nm, corresponding to the yellow D line of the
sodium spectrum.
31.7 Thin Lenses
An optical lens is a piece of transparent material, such as glass or plastic. Because of
its shape, however, light passing through it either converges or diverges to its
principal axes. The entire effect of the lens is due to its shape and the index of
refraction of the lens.
One such shape of a lens is formed from a piece of plane glass by shaping it
into the form of two spherical surfaces with radii of curvature R1 and R2, as shown
in figure 31.10, where C1 and C2 are the centers of the two spherical surfaces. This
shaped piece of glass is called a double convex lens, because the shape of its surfaces
are convex.
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Chapter 31 The Law of Refraction
Figure 31.10 Formation of a lens.
The line going through the center of the lens is called the principal axis and is
shown in figure 31.11. Consider a ray of light AB, parallel and close to the principal
Figure 31.11 Refraction by a lens.
axis and impinging on the first surface at the point B. The normal N1 to the surface
at B is a continuation of the radius of curvature of the first surface, which emanates
from the center of curvature C1 of the first surface. The incident ray AB makes an
angle of incidence i1 with the normal N1. Because the ray is going from the less
dense medium, air, to the more dense medium, glass, the refracted ray is bent
through an angle r1 toward the normal, as shown. This refracted ray impinges on
the second surface at the point D. The normal at D is the continuation of the radius
of curvature of the second surface and is designated N2. Ray BD makes an angle of
incidence i2 with the normal N2. Since the ray now goes from glass to air, it is going
from a more dense medium to a less dense medium. The refracted ray is, therefore,
bent away from the normal N2 through an angle r2 as ray DE. The net effect of the
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Chapter 31 The Law of Refraction
two refractions is to take a ray of light, which is parallel to the principal axis, and
bend it such that it crosses the principal axis. The point where this ray crosses the
principal axis is called the principal focus, and is designated by the letter F. The
distance from the center of the lens to the principal focus is called the focal length f
of the lens. It is important to note that for a particular material, it is the shape of
the lens that determines the normals and hence the refracted rays and thus the
focal length of the particular lens.
Since we have already seen that the converging of a parallel ray to the
principal axis is caused by the refraction at the two surfaces of the lens, it will not
be necessary to go into all this detail every time we consider a lens. Instead, we will
say that a ray of light, parallel to the principal axis, is bent in the middle of the
lens. It then converges to the principal focus, as shown in figure 31.12.
Figure 31.12 Bending of a ray of light by a convex lens.
All rays parallel and close to the principal axis converge to the principal
focus, as seen in figure 31.13. Such a lens is called a converging lens. If the
incident rays are not close to the principal axis, the converging rays do not all
Figure 31.13 A converging lens.
intersect at the same point, a result known as spherical aberration. We will assume
in this book that all rays are sufficiently close to the principal axis so that spherical
aberration can be ignored. If parallel light comes in from the right of the lens it
converges at the same distance to the left of the lens. Note also that, by the
principle of reversibility, light emanating from the principal focus comes out parallel
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Chapter 31 The Law of Refraction
to the principal axis. Figure 31.14 shows some examples of converging lenses. The
actual shapes are determined by the radii of curvature, but an easy way to identify
such a lens is to note that a converging lens is always thicker at the center of the lens
Figure 31.14 Examples of converging lenses.
than it is at the rim. If any of the thin lenses in figure 31.14 are reversed, they still
produce the same effect.
A diverging lens is a lens that takes a bundle of parallel light rays and
diverges them away from the principal axes, as shown in figure 31.15. If the
Figure 31.15 A diverging lens.
diverging rays are produced backward, they all intersect at the same point F, called
the principal focus, or focal point, of the diverging lens. The diverging rays appear to
come from the principal focus. The distance from the center of the lens to the focal
point is called the focal length f of the diverging lens. Some examples of diverging
lenses are shown in figure 31.16. One characteristic of all diverging lenses is that
they are thinner at the center of the lens than they are at the rim.
Figure 31.16 Examples of diverging lenses.
31-19
Chapter 31 The Law of Refraction
31.8 Ray Tracing and the Standard Rays
To determine the location of an image formed by a lens, we use the technique of ray
tracing. An object OP, represented by a brown arrow, is placed a distance p in front
of a convex lens in figure 31.17. A ray, from the bottom of the object at P, travels
Figure 31.17 Ray diagram for a convex lens.
along the principal axis. This ray goes through the lens undeviated. Hence the
bottom of the image must lie somewhere on the principal axis of the lens. To
determine the location of the tip of the image, three rays are drawn from the tip of
the object. These three rays are called the standard rays and are shown in figure
31.17.
The first ray (1), in red, is drawn parallel to the principal axis and after
refraction it passes through the principal focus. A second ray (2), in green, is drawn
from the top of the object and passes through the first focal point. This ray is
refracted through the lens so that it comes out parallel to the principal axis. (Recall
that by the principle of reversibility, light emanating from the principal focus comes
out parallel to the principal axis.) The third of the standard rays (3), in blue, is
drawn from the tip of the arrow and goes through the center of the lens. The
refraction experienced at the first surface is exactly compensated by the refraction
at the second surface, and the ray passes through the lens undeviated. These three
rays intersect at the point I and form the real, inverted image IQ, in yellow. If a
screen were placed at this point a sharp image would be found there. The distance
from the object to the lens is called the object distance p, whereas the distance from
the lens to the image is called the image distance q. Thus, we can find any image by
drawing a ray diagram.
A ray diagram for a diverging lens is shown in figure 31.18. An object OP is
placed a distance p in front of a diverging lens. A ray (1), in red, is drawn from the
object parallel to the principal axis. The diverging lens causes the ray to bend away
from the principal axis, as if the ray had come from the principal focus F. A second
ray (2), in green, is drawn, which would go to the second focal point if there were no
31-20
Chapter 31 The Law of Refraction
Figure 31.18 Ray diagram for a concave lens.
lens present. This ray is caused to bend by the diverging lens so that it comes out
parallel to the principal axis. A third ray (3), in blue, passes through the center of
the lens and is not bent. These three rays do not intersect anywhere to the right of
the lens and hence cannot form a real image there. However, if we follow each of the
three rays backward, they appear to intersect at the point I, and the image is
located at that point. The distance from the lens to the image, the image distance, is
denoted by the letter q in the diagram. We can find an image of any object for any
lens by drawing these standard rays.
31.9 The Lens Equation
A mathematical relation between the object distance p, the image distance q, and
the focal length f, can be obtained from the geometry of figure 31.19.
In triangle OPC,
tan θ1 = OP = ho
PC
p
whereas in triangle QIC,
tan θ1 = QI = hi
QC q
Equating the tangent of θ1 from each equation gives
ho = hi
p
q
Rearranging terms, this becomes
31-21
(31.18)
Chapter 31 The Law of Refraction
Figure 31.19 Determining the lens equation.
hi = q
ho p
(31.19)
We will return to this equation shortly.
In triangle ACF,
tan θ2 = AC = ho
CF
f
whereas in triangle IQF,
tan θ2 = QI = hi
QF q − f
Equating the tangent of θ2 from each equation, gives
Rearranging terms, this becomes
hi = ho
q−f
f
hi = q − f
f
ho
Setting equation 31.20 equal to equation 31.19, gives
q−f = q
f
p
or
q − 1= q
f
p
Dividing each term by q, gives
1 − 1 = 1
f
q
p
Solving for 1/f, we obtain
31-22
(31.20)
Chapter 31 The Law of Refraction
1 = 1 + 1
f
p
q
(31.21)
Equation 31.21 is the lens equation and gives the relation between the object
distance p, the image distance q, and the focal length f of the lens. Notice that it is
the same formula as the mirror equation found in chapter 30.
The linear magnification M of a lens system is defined as the ratio of the
height of the image to the height of the object, that is,
M = height of image = hi
height of object ho
(31.22)
Using equation 31.19, we can also write this as
M = hi = − q
ho
p
(31.23)
The magnification tells how much larger the image is than the object. When we
know M, we find the height of the image hi from equation 31.23, as
hi = Mho
(31.24)
The minus sign in equation 31.23 is placed there by convention. For a single lens,
when the magnification is negative the image is real and inverted, when it is
positive the image is virtual and erect. Notice that the lens equation, the
magnification equation, and the height of the image equation, are the same
equations that were obtained for the spherical mirror in chapter 30.
Example 31.7
A converging lens. An object 5.00 cm high is placed 35.0 cm in front of a converging
lens of 10.0 cm focal length. (a) Where is the image located? (b) What is the
magnification? (c) What is the size of the image?
Solution
a. The image distance, found from the lens equation, equation 31.21, is
1 = 1 + 1
f
p
q
1 = 1 − 1 =
1
−
1
= 0.0714 cm−1
q
f
p 10.0 cm 35.0 cm
q = 14.0 cm
31-23
Chapter 31 The Law of Refraction
b. The magnification, found from equation 31.23, is
M = − q = −14.0 cm = −0.400
p
35.0 cm
Since M is negative, the image is real and inverted.
c. The size of the image, found from equation 31.24, is
hi = Mho = −(0.400)(5.00 cm) = −2.00 cm
The image is thus smaller than the object. The minus sign on hi simply means that
the distance hi is in the negative y direction, and hence, the image is inverted.
To go to this Interactive Example click on this sentence.
Example 31.8
A diverging lens. An object 5.00 cm high is placed 35.0 cm in front of a diverging
lens of −10.0 cm focal length. (a) Where is the image located? (b) What is the
magnification? (c) What is the size of the image?
Solution
a. The image distance, found from the lens equation, equation 31.21, is
1 = 1 + 1
f
p
q
1 = 1 − 1 =
1
−
1
q
f
p −10.0 cm 35.0 cm
−1
= −0.100 cm − 0.0285 cm−1 = −0.1286 cm−1
q = −7.78 cm
Since the image distance q is negative, the image is virtual and is located 7.78 cm in
front of the lens.
b. The magnification, found from equation 31.23, is
q
M = − p = − −7.78 cm = +0.222
35.0 cm
31-24
Chapter 31 The Law of Refraction
Because M is positive the image is virtual and erect.
c. The size of the image, found from equation 31.24, is
hi = Mho = (0.222)(5.00 cm) = 1.11 cm
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31.10 Some Special Cases for the Convex Lens
Case 1: The Object Is Located at Infinity
When the object is very far away from the lens, we can assume that it is at infinity.
Setting p = ∞ in the lens equation, gives, for the location of the image,
1= 1 − 1
q
f
p
= 1 − 1
f
∞
But 1/∞ = 0, and the equation reduces to
q=f
That is, when the object is located at infinity, the image is located at the principal
focus. The magnification becomes
M=− q =−q =0
∞
p
which shows that the image is reduced to a point. This case, which was shown in
figure 31.13, is used as a simple technique to determine the focal length of a
converging lens. A converging lens is held in front of a card and the distance
between the card and the lens is varied until a sharp image is obtained on the card
of a very distant object ( p equal to infinity). The distance between the lens and the
card is the image distance q. But as just seen, if the object is at infinity, then q = f.
Hence, the distance from the lens to the card is the focal length of the converging
lens.
Case 2: The Object Lies between Infinity and 2f, That Is, 2f < p <
∞
This is a rather general case and the ray diagram is shown as the continuous lines
31-25
Chapter 31 The Law of Refraction
in figure 31.20. The actual location of the image is found from the lens equation
when p and f are specified.
Figure 31.20 The object moves closer to the lens.
A very interesting result occurs when the object is moved in toward the lens.
The ray diagram for this case is shown in dashed lines in figure 31.20. Notice that as
the object moves toward the lens the image moves away from the lens and gets bigger.
However, the height of the image hi never gets greater than the height of the object
ho, and thus the magnification is always less than 1. (This will be proved in case 3.)
Case 3: The Object Is Located at a Distance of Twice the Focal
Length from the Lens, That Is, p = 2f
For this case the image, found from the lens equation, is
1 = 1 + 1 = 1 + 1
f
p
q
2f q
or
1 = 1 −1 = 1
q
f 2f 2f
and hence,
q = 2f
That is, when the object is placed a distance of 2f to the left of the lens, the image is
located at the same distance of 2f to the right side of the lens. The magnification for
this case becomes
M = hi = − q = − 2f = −1
ho
p
2f
and
hi = −ho
31-26
Chapter 31 The Law of Refraction
Hence, when p = q = 2f, the height of the image is the same size as the height of the
object. For any value of p greater than 2f the magnification is always less than 1, and
hence the height of the image is less than the height of the object. For any value of p
less than 2f, the magnification is greater than 1, and hence the height of the image is
always greater than the height of the object.
Case 4: The Object Is Located between the Focal Length and
Twice the Focal Length, That Is, f < p < 2f
It is within this region that the lens acts as a magnifier, because if p is less than 2f,
the magnification is greater than 1. The ray diagram is shown in figure 31.21. It is
obvious from the diagram that q is greater than p, and hence, the magnification, M
Figure 31.21 Ray diagram for object located between the focal length and twice the
focal length.
= q/p, is greater than 1, and the image is enlarged. This is also seen in the diagram,
that is, the height of the image is greater than the height of the object.
Case 5: The Object Is Located at the Principal Focus of the Lens,
That is, p = f
When the object is placed at the principal focus, p = f, and the lens equation gives
for the location of the image,
1 = 1 − 1 = 1 − 1 =0
q
f
p
f
f
The only way for 1/q to be equal to zero, is for q to equal infinity, that is,
q=∞
31-27
Chapter 31 The Law of Refraction
The ray diagram for such a case is shown in figure 31.22. The standard ray
(2) cannot be drawn, but rays (1) and (3) can, and as we see from the diagram, they
become parallel on the right-hand side of the lens and hence never intersect to form
a real image there at any finite distance from the lens. The parallel rays only
intersect at infinity and it is there where the image may be found. Note that if rays
(1) and (3) were dashed backward, they would still be parallel, and hence, could not
form a virtual image either.
Figure 31.22 Ray diagram for an object at the principle focus.
Case 6: The Object Is Placed within the Principal Focus, That Is,
p<f
A ray diagram for this case is shown in figure 31.23. Standard ray (1) is drawn
Figure 31.23 Ray diagram for an object placed within the principle focus.
31-28
Chapter 31 The Law of Refraction
parallel to the principal axis and is refracted in the lens such that it passes through
the principal focus F, as shown. Of all the infinite rays that emanate from the tip of
the arrow, standard ray (2) is a ray that is aligned with the dashed straight line
from the principal focus to the tip of the arrow of the object. Therefore, ray (2)
appears to come from the first principal focus, and hence, is refracted in the lens
such that it comes out parallel to the principal axis. The third standard ray (3) is
drawn from the tip of the object and goes through the center of the lens undeviated,
as shown. The three standard rays do not converge anywhere on the right-hand side
of the lens to form a real image there. However, if these standard rays are produced
backward they intersect on the left-hand side of the lens and form a virtual image
there. Thus a person looking through the lens from the right would see the
enlarged, erect, virtual image of the object on the left-hand side of the lens. When a
convex lens is used with the object located within the principal focus, it is called a
simple magnifying glass. In this case, the image distance q is always negative.
Example 31.9
A simple magnifying glass. An object 5.00 cm high is placed 3.00 cm in front of a
convex lens of 5.00-cm focal length. (a) Where is the image located? (b) What is the
magnification? (c) How high is the image?
Solution
a. The image distance, found from the lens equation, is
1 = 1 − 1 =
1
−
1
= −0.1333 cm−1
q
f
p
5.00 cm 3.00 cm
and
Note that q is a negative quantity.
q = −7.50 cm
b. The magnification, found from equation 31.23, is
q
M = − p = − −7.50 cm = +2.50
3.00 cm
The magnification is thus positive and indicates that the image is virtual and erect.
c. The image height, found from equation 31.24, is
hi = Mho
= (2.50)(5.00 cm)
= 12.5 cm
31-29
Chapter 31 The Law of Refraction
Thus the lens used in this way gives an enlarged (magnified 2.5 times), erect,
virtual image of the object. The fact that q is negative is an indication that the
image is virtual.
To go to this Interactive Example click on this sentence.
In summary, for a converging lens, with the object distance p positive, if the
image distance q is positive the image is real and located on the other side of the
lens. If the image distance q is negative, the image is on the same side of the lens as
the object and is a virtual image.
31.11 Combinations of Lenses
Most optical systems consist of a combination of two or more lenses, prisms, or
mirrors. Figure 31.24 shows a combination of two convex lenses separated by a
Figure 31.24 Combination of two convex lenses.
31-30
Chapter 31 The Law of Refraction
distance d. The first lens has a focal length f1 and the second lens has the focal
length f2, as illustrated. An object of height ho1 is placed a distance p1 in front of the
first lens. Let us find the location and size of the resulting image of the lens
combination. We begin the analysis by considering the first lens only, that is, we
temporarily ignore the existence of the second lens. We find the image distance q1 of
the first lens by drawing the standard rays of a ray diagram, as in figure 31.24. This
image of the first lens now acts as the object of the second lens. We now completely
ignore the existence of the first lens and consider the second lens as though an
object has been placed a distance p2 in front of it. We again find the image from this
second lens by drawing the standard rays of a ray diagram, as shown in figure
31.24. Thus, the image distance q2 of the second lens and the image height hi2 are
easily found.
In addition to using the ray diagram, we can also find the final image from
the lens equation. The image distance of the first lens is found from
1 = 1 − 1
q1 f1
p1
(31.25)
whereas the image distance of the second lens is found from
1 = 1 − 1
q2 f2
p2
(31.26)
However, as we can see from the diagram, the object distance p2 is
p2 = d − q1
(31.27)
In most of the cases that we consider in this book, the image distance q1 turns out to
be less than the separation distance d, and the term (d − q1) is positive, giving a
positive object distance p2. There are some cases where the image distance q1 is
greater than the separation distance d. In these cases, the term (d − q1) is a
negative quantity and hence, the object distance p2 is also a negative quantity. An
object having a negative object distance is called a virtual object. For a virtual
object, the rays strike the lens before the object is formed. We will see an example of
this later.
The magnification of the lens combination is defined as the ratio of the height
of the final image to the height of the initial object. That is,
M = height of image 2 = hi2
height of object 1
ho1
But the magnification of the first lens is
31-31
(31.28)
Chapter 31 The Law of Refraction
M1 = height of image 1 = hi1
height of object 1
ho1
(31.29)
whereas the magnification of the second lens is
M2 = height of image 2 = hi2
height of object 2
ho2
(31.30)
Thus, from equation 31.30,
hi2 = ho2M2
(31.31)
and from equation 31.29,
ho1 = hi1
M1
(31.32)
Substituting equations 31.31 and 31.32 back into equation 31.28, and noting that
the height of image 1 hi1 is equal to the height of object 2 ho2, we get
M = hi2 = ho2M2 = ho2M2
ho1
hi1/M1 ho2/M1
or
M = M1M2
(31.33)
Equation 31.33 says that the magnification of a combined optical system is equal to
the product of the magnification of each individual lens. The height of the final
image, obtained from equation 31.28, is
hi2 = Mho1
(31.34)
The location and nature of the image of a particular optical system depends
on the focal lengths f1 and f2, the location of the first object p1, and the separation d
between the two lenses. We will consider different optical systems in section 31.13.
Example 31.10
A combination of lenses. An object 5.00 cm high is placed 15.0 cm in front of a
convex lens of 10.0-cm focal length. A second convex lens, also of 10.0-cm focal
length, is placed 48.0 cm behind the first lens. Find (a) the image of the lens
combination, (b) the magnification of the system, and (c) the height of the final
image.
Solution
a. The image of the first lens, found from equation 31.25, is
31-32
Chapter 31 The Law of Refraction
1 = 1 − 1 =
1 −
1
= 0.0333 cm−1
p1 10.0 cm 15.0 cm
q1 f1
q1 = 30.0 cm
The object distance for the second lens, found from equation 31.27, is
p2 = d − q1 = 48.0 cm − 30.0 cm = 18.0 cm
Hence, the final image, found from equation 31.26, is
1 = 1 − 1 =
1 −
1
= 0.0444 cm−1
q2
f2 p2 10.0 cm 18.0 cm
q2 = 22.5 cm
Thus, the final image of the lens combination is found 22.5 cm behind the second
lens, or 70.5 cm behind the first lens.
b. The magnification of each lens is found as
M1 = − q1 = − 30.0 cm = −2.00
15.0 cm
p1
M2 = − q2 = − 22.5 cm = −1.25
18.0 cm
p2
The final magnification of the system is
M = M1M2 = (−2.00)(−1.25) = 2.50
Note that M1 is negative indicating that the first lens inverts the first image. The
fact that M2 is also negative means that lens 2 inverted the second image from
upside down to erect. The total magnification M is positive indicating that the final
image is right side up. In dealing with combinations of lenses we must be careful of
the convention adopted for the magnification of a single lens. The convention
adopted is that if M is negative, the image is real and inverted, whereas if M is
positive, the image is erect and virtual. In this case of the combined lens, M1 is
negative and image 1 is real and inverted, as expected. Also M2 is negative and
image 2 is real and inverted from its original orientation when in front of lens 2,
also as expected. The inversion of the inverted image, gives a final image that is
erect. Notice however, that the final magnification is positive because it is the
product of two negative numbers. But the final image is real not virtual. Hence, the
convention that a positive magnification indicates that the image must be virtual
cannot be used for a system of multiple lenses.
31-33
Chapter 31 The Law of Refraction
c. The final height of the image of the system, found from equation 31.34, is
hi2 = Mho1 = (2.50)(5.00 cm)
= 12.5 cm
To go to this Interactive Example click on this sentence.
31.12 Thin Lenses in Contact
An interesting case results from the lens combination of section 31.11 when the
distance d between the two lenses is made equal to zero, thereby placing the two
lenses in contact with each other. The lens equation applied to the second lens is
1 = 1 + 1
f2
p2 q2
(31.35)
Setting p2 = d − q1 from equation 31.27 into equation 31.35, yields
1 = 1 + 1
f2
d − q 1 q2
(31.36)
Setting the separation distance d equal to zero in equation 31.36 yields
1 = 1 + 1
f2
− q1
q2
(31.37)
But the lens equation applied to the first lens is given by
1 = 1 − 1
q1 f1 p1
and
1 =−1 + 1
− q1
f1 p1
Substituting equation 31.38 into equation 31.37, yields
1 =−1 + 1 + 1
f2
f1
p 1 q2
and
31-34
(31.38)
Chapter 31 The Law of Refraction
1 +1 = 1 + 1
f1 f2
p1 q2
(31.39)
Let us examine equation 31.39 and see what it is telling us. Here p1 is the object
distance of the first lens, but it is also the object distance of the lens combination,
and we will designate it as pc. Although q2 is the image distance of the second lens,
it is also the image distance of the lens combination, and we can call it qc. Hence, we
can rewrite equation 31.39 as
1 + 1 = 1 + 1
(31.40)
f1 f2
pc qc
The right-hand side of equation 31.40 looks like part of the lens equation, so it is
only natural to define the focal length for the two lenses in contact as fc, and the
lens equation for the combination should then be
1 = 1 + 1
fc pc qc
(31.41)
Combining equations 31.40 and 31.41, gives
1 = 1 + 1
fc
f1
f2
(31.42)
The focal length of the combination is thus obtained from equation 31.42. Hence,
whenever two lenses are in contact, the reciprocal of the focal length of the
combination is equal to the sum of the reciprocals of the focal lengths of the
individual lenses.
Example 31.11
The focal length of two convex lenses in contact. A convex lens of 10.0-cm focal length
is placed in contact with a convex lens of 15.0-cm focal length. What is the focal
length of the combination?
Solution
The focal length of the combination, found from equation 31.42, is
1 = 1 + 1 =
1 +
1
= 0.1666 cm−1
fc
f1 f2 10.0 cm 15.0 cm
and
fc = 6.00 cm
31-35
Chapter 31 The Law of Refraction
To go to this Interactive Example click on this sentence.
Example 31.12
The focal length of a convex and a concave lens in contact. A 10.0-cm convex lens is
placed in contact with a 15.0-cm concave lens. What is the focal length of the
combination?
Solution
The focal length of the combination is found from equation 31.42, but since the
second lens is concave its focal length is negative, that is, f2 = −15.0 cm. Therefore
1 −
1
= 0.0333 cm−1
1 = 1 + 1 =
f1 f2 10.0 cm 15.0 cm
fc
and
fc = 30.0 cm
To go to this Interactive Example click on this sentence.
Equation 31.42, although derived for only two lenses, is completely general
and thus, for any number of lens placed in contact the focal length of the combination
is
1 = 1 + 1 + 1 + 1+…
(31.43)
fc f1
f2
f3
f4
When dealing with many lenses in contact it is convenient to define the
dioptric power of a lens as the reciprocal of the focal length in meters. Thus
P= 1
f
The reciprocal of the focal length in meters is called a diopter.
Example 31.13
The power of a lens. What is the power of a 15.0-cm focal length lens?
Solution
31-36
(31.44)
Chapter 31 The Law of Refraction
The focal length of the lens is f = 15.0 cm = 0.150 m. The power of the lens, found
from equation 31.44, is
P= 1 =
1
= 6.67 diopters
f 0.150 m
To go to this Interactive Example click on this sentence.
The advantage of expressing a lens in terms of its dioptric power is evident
when the lenses are in contact. For example, if there are four lenses of focal lengths
f1, f2, f3, and f4, then the power of each lens is
P1 = 1
f1
P2 = 1
f2
P3 = 1
f3
P4 = 1
f4
The power of these lenses in combination, determined from equation 31.43, is
1 = 1 + 1 + 1 + 1
fc
f1 f 2
f3 f4
and substituting the values of 1/f, we get
1 = P1 + P2 + P3 + P4
fc
The power of the combination is thus
Pc = P1 + P2 + P3 + P4
(31.45)
Expressing a lens in terms of dioptric power is convenient for an optician,
who places many different lenses in contact. When a patient is given an eye exam,
the optician places different lenses in front of the patient’s eyes and asks him to
read a chart. When the patient finds the sharpest image, the optician immediately
gets the power of the combination of lenses used, Pc, from equation 31.45. The
reciprocal of Pc is the focal length of the eyeglasses needed for the patient and a lens
of this focal length can be ground using the lensmaker’s formula, which we now
discuss.
31-37
Chapter 31 The Law of Refraction
31.13 The Lensmaker’s Formula
We showed in figure 31.10 that a spherical lens was made from a piece of plane
glass by shaping it into the form of two spherical surfaces with radii of curvature R1
and R2. We then showed in figure 31.11 that a ray of light parallel to the principal
axis of this lens was refracted and then crossed the principal axis at the focal point
F. Then in figure 31.13 we showed that all rays parallel and close to the principal
axis converge to the principal focus, and we defined the distance from the center of
the lens to the focal point as the focal length f of the lens. We then showed in section
31.10 that the focal length of any lens could be easily determined experimentally by
placing a converging lens in front of a card and the distance between the card and
the lens is varied until a sharp image is obtained on the card of a very distant object
( p equal to infinity). The distance between the lens and the card is the image
distance q. But if the object is at infinity, then q = f. Hence, the distance from the
lens to the card is the focal length of the converging lens. In this way we could
determine the focal length of any lens experimentally. But how do we know how to
grind the lens into the spherical shape that will gives us this focal length?
Recall from chapter 30 that the focal length of a spherical mirror was found
by equation 30.14 to be f = R/2, where R was the radius of curvature of the mirror.
That is, if you want a particular focal length f, just grind the mirror to the radius R
= 2f. In this way you can make a mirror of any focal length. In a similar way, we
would now like to determine the focal length of the lens in terms of its radii of
curvature and the index of refraction of the lens material. The equation that relates
the focal length, index of refraction, and radii of curvature of a lens is called the
lensmaker’s formula. To derive this relation we place an object at the point A, at
the object distance p, in a medium of index of refraction n1, in front of a thick lens as
seen in figure 31.25. We consider the ray of light that leaves the object at A and
t = p2 + q1
n1
n1
n2
B
I
A
F
E
p1
q1
p2
q2
Figure 31.25 Refraction of a ray of light through a thick lens.
31-38
Chapter 31 The Law of Refraction
strikes the spherical surface at the point B. The incident ray AB, upon entering the
second medium of index of refraction n2, is bent toward the principal axis and this
refracted ray crosses the principal axis at the point I, and at this point represents
the image I of the object at A. The distance from the front of the lens to the point I is
the image distance q1. The ray then continues through the second medium until it
arrives at the point E where it is again refracted toward the principal axis and
intersects the principal axis at the point F, which will be the final image of the thick
lens. The distance from the point I to the right surface of the lens is the object
distance for the second spherical surface and is designated as p2. The distance form
the right surface to the final image F is the image distance q2 of the second spherical
surface. Let us now look at what happens at each of the spherical surfaces of the
thick lens.
The incident ray leaves the object at A and strikes the spherical surface at
the point B. The normal at the point B is drawn as the extension of the radius R1 of
the spherical surface whose center of curvature is located at the point C1, figure
31.26. The incident ray AB makes an angle of incidence i1 with the normal N. Upon
entering the second medium of index of refraction n2, the light ray is bent toward
the normal making the angle of refraction r1 as shown in the figure. This refracted
ray crosses the principal axis at the point I, and represents the image I of the object
at A. The law of refraction applied to the interface at point B gives
n1 sin i1 = n2 sin r1
N
n1
A α
p1
i1
B
D
n2
δ r1
R1
(31.46)
θ β
R1 C
γ
I
q1
Figure 31.26 Refraction of a ray of light at the first spherical surface.
We consider only incident rays that are close to the principal axis. Such rays are
called paraxial rays. For paraxial rays, the angle α and hence i1 and r1 will all be
small angles2 and hence the sine of these angles will be equal to the angle itself.
That is, we let sin i1 = i1 and sin r1 = r1, hence the law of refraction becomes
2
The angles in our diagrams are greatly exaggerated to show the details of the refraction.
31-39
Chapter 31 The Law of Refraction
n1 i1 = n2 r1
(31.47)
Recall from geometry that the exterior angle of a triangle is equal to the sum of the
two interior angles. Applying this result to triangle ABC we see that
i1 = α + θ
(31.48)
and for triangle BIC,
θ = r1 + γ
Solving for r1 gives
r1 = θ − γ
(31.49)
Substituting equations 31.48 and 31.49 into equation 31.47 gives
n1 (α + θ) = n2 (θ − γ)
(31.50)
Now recall that the arc s of a circle is given by s = rθ. Applying this relation to arc
BD of the lens we get
BD = R1θ
and hence
θ = BD
(31.51)
R1
Because the ray AB is a paraxial ray, the arc BD is approximately equal to q1γ, that
is
BD ≈ q1γ
and hence
γ = BD
(31.52)
q1
Also arc BD is approximately equal to p1α, that is
and hence
BD ≈ p1α
α = BD
p1
Replacing equations 31.51, 31.52, and 31.53 into equation 31.50 gives
n1 (BD + BD) = n2 (BD − BD)
p1
R1
R1 q1
Canceling out the term BD on both sides of the equation gives
31-40
(31.53)
Chapter 31 The Law of Refraction
n1 + n1 = n2 − n2
p1 R1 R1 q1
Rearranging terms yields
n1 + n2 = n2 − n1
p1 q1
R1 R1
and
n1 + n2 = n2 − n1
p1 q1
R1
For first spherical surface
(31.54)
Let us now consider what happens at the second spherical surface. The ray of
light from the first surface that converged to the point I as the image of the first
surface now becomes the object for the second spherical surface, figure 31.27. The
n1
n2
γ
C2
β θ
R2
i2
p2
G
R2
δ
E
α
F
r2
N
I
q2
Figure 31.27 Refraction of a ray of light at the second spherical surface.
distance from the point I to the right surface of the lens is the object distance for the
second spherical surface and is designated as p2. The distance from the right surface
to the final image F is the image distance q2 of the second spherical surface. Using
the same analysis on the right spherical surface as on the left spherical surface we
obtain the equivalent relation of equation 31.54 as
n2 + n1 = n1 − n2
p2 q2
R2
For second spherical surface
(31.55)
The combined effects of both refractions are found by adding equations 31.54 and
31.55 as
n1 + n2 + n2 + n1 = (n2 − n1) + (n1 − n2)
p1 q1 p2 q2
R1
R2
n1 + n1 + n2 + n2 = (n2 − n1) − (n2 − n1)
p1 q2 p2 q1
R1
R2
n1 n1
1
1
1
1
(31.56)
p 1 + q 2 + n 2 p 2 + q 1 = (n 2 − n 1 ) R 1 − R 2
31-41
Chapter 31 The Law of Refraction
Let us now analyze the third term in equation 31.56.
q1 + p2
n t
n 2 p12 + q11 = n 2 p 2 q 1 = p 22q 1
(31.57)
where we have used the fact that the sum q1 + p2 = t, the thickness of the lens as
seen in figure 31.25. If we make the lens a thin lens then in the limit we can
consider the thickness t of the lens to be small compared to the object and image
distances and hence we let t approach zero. That is, as t → 0, the term in equation
31.57 approaches zero and can be dropped from equation 31.56. Thus equation
31.56 becomes
n1 n1 (
1
1
p1 + q2 = n2 − n1 ) R1 − R2
or
n2 − n1
1
1 − 1
1
(31.58)
n1
p1 + q2 =
R1 R2
The object distance p1 for the first spherical surface is also the object distance of the
entire lens p and so we let p1 = p, and q2 the image distance of the second spherical
surface is the image distance q of the entire lens and so we now let q2 = q. Therefore
equation 31.58 becomes
1 − 1
1 + 1 = n2 − n1
(31.59)
n1
p q
R1 R2
But we showed in equation 31.21, the lens equation, that
1+1 = 1
p q
f
(31.21)
Setting equation 31.21 equal to equation 31.59 gives
1 = n2 − n1
n1
f
1 − 1
R1 R2
(31.60)
Equation 31.60 is called the lensmaker’s equation because it allows the lens maker to
make a lens of a particular focal length f by the appropriate choice of radii of
curvature R1 and R2 of the spherical lens and the index of refraction n1 of the medium
in which the lens will be used and n2 the index of refraction of the lens itself. If the
lens is to be used in air, then n1 = 1 and n2 = n, the index of refraction of the lens,
and equation 31.60 becomes
1 = (n − 1 ) 1 − 1
(31.61)
R1 R2
f
Equation 31.61 is the lensmaker’s formula for a lens used in air, n is the index of
refraction of the glass, R1 is the radius of curvature of the first surface of the lens, R2
is the radius of curvature of the second surface, and f is the focal length of the lens.
31-42
Chapter 31 The Law of Refraction
By convention, if the surface is convex to the incident ray, R is considered
positive. If the surface is concave to the incident ray, R is considered negative. It is
assumed that the ray of light is incident from the left. Thus, for a double convex
lens, R1 would be positive and R2 would be negative. Equation 31.61 says that in
order to make a lens of a particular focal length f for a medium of index of refraction
n, the medium must be ground to the radii of curvature R1 and R2 that satisfies
equation 31.61. Note that in the derivation of equation 31.61, we assumed that the
thickness t of the glass lens is negligible compared to the distance to the principal
focus and to any object or image distance concerned. Such a lens of negligible
thickness is called a thin lens. We have only considered thin lenses in this book.
The focal length of any converging or diverging lens is found from the
lensmaker’s formula, equation 31.61. However, it is important to remember the
convention that R is positive for an incident ray that impinges on a convex surface
and negative for one that impinges on a concave surface. A plane surface is a surface
whose radius of curvature is infinite, and hence, 1/R = 0.
Example 31.14
Finding the focal length of a converging lens. A double convex lens has radii of
curvature of 25.0 cm. The index of refraction is 1.52. What is the focal length of the
lens?
Solution
By our convention, the radius of curvature of the first convex surface is +25.0 cm.
The inside of the second convex surface is concave to the incident light and its
radius of curvature is therefore −25.0 cm. The focal length of the lens, found by the
lensmaker’s formula, equation 31.61, is
1 = (n − 1 ) 1 − 1
R1 R2
f
1
1
(
)
= 1.52 − 1
−
= 0.0416 cm −1
25 cm −25.0 cm
f = 24.0 cm
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Example 31.15
Finding the focal length of a diverging lens. A double concave lens has radii of
curvature of 25.0 cm. The index of refraction is 1.52. What is the focal length of the
lens?
31-43
Chapter 31 The Law of Refraction
Solution
By our convention, the radius of curvature of the first concave surface is −25.0 cm.
The inside of the second concave surface is convex to the incident ray, and hence,
the radius of curvature is +25.0 cm. The focal length of the diverging lens, found
from equation 31.61, is
1 = (n − 1 ) 1 − 1
R1 R2
f
1
1
(
)
= 1.52 − 1
−
= −0.0416 cm −1
−25.0 +25.0
f = −24.0 cm
Note that converging lenses have positive focal lengths while diverging lenses have
negative focal lengths.
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31.14 Optical Instruments
Most optical devices consist of an arrangement or combination of lenses. The focal
lengths of the lenses and their relative positions determines the function of the
optical system. A very brief description of these optical instruments is given here as
an example of lens applications. Most optical instruments consist of many lens
combinations to eliminate various lens defects. We will look at only combinations of
single lenses in order to demonstrate the principle of the device. More detailed
discussions can be found in books on optics. The first optical instrument considered
is the camera.
The Camera
The basic elements of a camera, figure 31.28(a), are a converging lens, contained in
a bellows or tube type mechanism; a roll of photographic film that can be rolled
across a focal plane; an aperture that determines the amount of light that enters
the camera; and a shutter mechanism that determines the time that the lens is
open to light. A schematic diagram of a camera is shown in figure 31.28(b). A ray
diagram shows the location of the image. Because the focal length of the lens is a
constant, increasing or decreasing the object distance p changes the image distance
q, as can easily be determined by the lens equation. Therefore, in order to get a
sharp image of any object, we need to be able to change the distance from the lens to
the film plane. On studio type cameras, this is accomplished by moving the bellows
inward or outward. On small hand cameras, such as a 35 mm camera, there are two
concentric tubes containing the lenses. Rotating the tubes clockwise or
31-44
Chapter 31 The Law of Refraction
counterclockwise causes the lens to move in or out. The total variation of q is
relatively small for a 35 mm camera.
Figure 31.28 A simple camera.
Example 31.16
The change in the image distance for a 35 mm camera. A 35 mm camera has a focal
length of 50.0 mm. By how much will the image distance change when
photographing one object at infinity and another 1.00 m away?
Solution
When the object is at infinity, the image distance q is equal to the focal length f, as
seen in section 31.10, case 1. Therefore,
q1 = f = 50.0 mm = 5.00 cm
When p2 = 1.00 m = 100 cm, the image distance q2 found from the lens equation, is
1 = 1 − 1 =
1 −
1
q2 f
p2 5.00 cm 100 cm
q2 = 5.26 cm
The total variation in the image distance q becomes
∆q = q2 − q1 = 5.26 cm − 5.00 cm = 0.26 cm
31-45
Chapter 31 The Law of Refraction
Thus, the lens does not have to move very far at all to get the two different pictures.
One of the reasons for this is, of course, the relatively small focal length of the lens.
If the focal length of the camera had been 20.0 cm, then the same calculation as
above would have given a variation of the image distance ∆q of 5.00 cm. Therefore,
to obtain a small variation of q a small focal length is used.
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Example 31.17
Getting the entire picture into the camera. At what distance from the 35 mm camera
of example 31.16 should a man, 2.00 m tall, stand in order for his entire length to be
in the picture?
Solution
For 35 mm film, the maximum height that the image hi can be is 35.0 mm.
Therefore, the magnification of the camera is
M = hi = 3.50 cm = 0.0175
ho 200 cm
Since M = q/p, we can solve for the object distance, p, as
p= q
M
Now, as seen above, the image distance q is quite close to the focal length f, so we
will assume that they are equal, that is,
q=f
Hence,
p = q = f = 5.00 cm = 286 cm = 2.86 m
M M 0.0175
That is, for a 2.00-m tall man to have his entire body in the picture he must stand
2.86 m from the camera. Note that the magnification of a camera is quite small; this
is necessary because we usually want to put a large scene onto a small piece of film.
The images must be greatly reduced if they are to fit on the film.
To go to this Interactive Example click on this sentence.
31-46
Chapter 31 The Law of Refraction
The amount of light impinging on the film depends on the speed of the
shutter, that is, 1/50, 1/100, 1/200, 1/500 of a second and so forth, and the size of the
opening at the lens. The size of the opening is determined by an iris diaphragm and
is calibrated in terms of the f-number. The f-number is defined as
f# =
= f
focal length
diameter of aperture d
(31.62)
The standard f-numbers of a camera are: f/2.8, f/4, f/5.6, f/8, f/11, and f/16. Thus, an
f-number of f/4 means that the diameter d of the aperture is 1/4 of the value of the
focal length. If a 50.0-mm focal length is being used, then an f-number of f/4 implies
that the lens opening is
d = f = f = 50.0 mm = 12.5 mm
f#
4
4
The opening for f/16 is
d = f = f = 50.0 mm = 3.13 mm
f# 16
16
Hence, the larger the f-number of a camera, the smaller the lens opening. Thus, a
picture taken in bright sunshine would use a larger f-number than one taken in
subdued lighting, for the same shutter speed.
The Simple Microscope
The simple microscope is the same as the magnifying glass that was treated in
section 31.10, case 6. The object is placed within the principal focus of a converging
lens and an enlarged, erect, and virtual image is formed.
The Compound Microscope
The compound microscope was invented by Galileo in 1610. It consists of two
converging lens. The first lens is called the objective lens and is of short focal length.
The object to be magnified is placed just outside the focal length f1 of the objective,
so that a real, enlarged image is formed at q1, figure 31.29(a). The second lens is
called the eyepiece, and the image of the objective lens becomes the object for the
eyepiece lens. The eyepiece is situated such that the object p2 always lies just within
the principal focus of the eyepiece f2. Thus the eyepiece acts as a simple magnifying
glass, giving an enlarged virtual image at q2. The lens equation is used to find the
location of the final image. Thus, q1 is found from
1 = 1 − 1
q1 f1
p1
Once q1 is determined, the object distance p2 is found from
31-47
Chapter 31 The Law of Refraction
Figure 31.29 The compound microscope.
p2 = d − q1
The final image is found from
1 = 1 − 1
q2
f2 p2
The magnification of the entire system is
M = M1M2
where
M1 = − q 1
p1
and
M2 = − q 2
p2
A typical student microscope is shown in figure 31.29(b).
Example 31.18
The microscope. A bug 2.00 mm in diameter is placed 1.50 cm in front of a 1.00-cm
focal length objective lens of a compound microscope. The eyepiece lens has a focal
length of 23.0 cm and the lenses are separated by a distance of 25.0 cm. Find the
size of the resulting image.
31-48
Chapter 31 The Law of Refraction
Solution
We treat the microscope as a system of two lenses in combination. The image of the
first lens is the object for the second lens. The image of the first lens, found from the
lens equation, equation 31.25, is
1 = 1 − 1
q1
f1 p1
=
1 −
1
1.00 cm 1.50 cm
q1 = 3.00 cm
This image serves as the object of the second lens. The object distance for the second
lens, found from equation 31.27, is
p2 = d − q 1
= 25.0 cm − 3.00 cm
= 22.0 cm
The image of the second lens, found from the lens equation, equation 31.26, is
1 = 1 − 1
q2
f2 p2
=
1 −
1
23.0 cm 22.0 cm
q2 = −506 cm
The magnification of the first lens is found as
M1 = − q1 = −3.00 cm = −2.00
p1
1.50 cm
whereas the magnification of the second lens is found as
q2
M 2 = − p 2 = − −506 cm = +23.0
22.0 cm
The total magnification of the system, found from equation 31.33, is
M = M1M2 = (2.00)(23.0) = 46.0
The size of the final image, found from equation 31.34, is
hi2 = Mho1 = (46.0)(2.00 mm)
= 92.0 mm = 9.20 cm
31-49
Chapter 31 The Law of Refraction
Notice that because M1 is negative, the first image is real, whereas M2 is positive,
indicating that the second image is virtual. This is as expected, since the object p2
lies within the principal focus f2 of the second lens. The final result of the optical
system is to take the initial object, 2.00 mm in size, and make it observable as an
image 92.0 mm in size.
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The Astronomical Telescope
The astronomical telescope is another example of a combination of two converging
lenses. However, this time the objective is a large converging lens of a long focal
length. The astronomical telescope is shown in figure 31.30(a). Because the object is
very far away, the image of the objective is, for all intents and purposes, located at
the principal focus of the objective lens fo. (Recall that when p = ∞, q = f.) It is shown
slightly away from there in figure 31.30 so that the ray diagram is easier to see. The
eyepiece is a converging lens of small focal length, fe, and it is placed such that the
image of the first lens falls just within the principal focus of the second lens, thereby
forming an enlarged, virtual image. That is, the eyepiece acts as a simple
magnifying glass. Since the first image falls approximately at the focal distance of
the first lens, and the second lens is placed so that the image of the first lens falls
just within the focal length of the second lens, the distance separating the lenses is
approximately the sum of the two focal lengths. Hence, the overall length of the
telescope is
L = fo + fe
(31.63)
The final image of the astronomical telescope is, of course, inverted but this is
not serious for analyzing the heavenly bodies. However, the astronomical telescope
is not very good for viewing distant objects on the surface of the earth, because they
are all inverted. (The astronomical telescope can be converted to a terrestrial
telescope by placing a third converging lens between the objective and the eyepiece.)
The magnification of a telescope is usually expressed in terms of the angular
magnification. The angular magnification MA of the lens is defined as the ratio of
the angle subtended at the eye by the object when the lens is used, to the angle
subtended by the unaided eye. We state without proof that the total angular
magnification of an astronomical telescope when the object is at infinity is given by
MA = focal length of objective = fo
focal length of eyepiece
fe
31-50
(31.64)
Chapter 31 The Law of Refraction
(a)
(b)
Figure 31.30 The astronomical telescope.
Example 31.19
An astronomical telescope. Find the angular magnification of an astronomical
telescope whose objective has a focal length of 100 cm, and its eyepiece has a focal
length of 5.00 cm.
Solution
The magnification, found from equation 31.48 is,
31-51
Chapter 31 The Law of Refraction
M = fo = 100 cm = 20.0
fe 5.00 cm
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The Galilean Telescope
The Galilean telescope consists of a converging lens of large focal length for the
objective and a diverging lens of short focal length for the eyepiece. The advantage
of the Galilean telescope is that it gives an erect image. A ray diagram for a
Galilean telescope is shown in figure 31.31. Only two of the standard rays are
Figure 31.31 Ray diagram for a Galilean telescope.
shown to simplify the diagram. The image of the first lens is located at q1. But note
that the second lens is placed before the location of the first image. This is an
example of what is called a virtual object, and the object distance p must be written
as a negative quantity in the lens equation. The effect of the diverging lens for the
eyepiece is to diverge the rays away from the position where they should focus by
the objective lens. For example, when ray (2) hits the diverging lens it is parallel to
the principal axis. However, a ray parallel to the principal axis is diverged away
from the principal axis as though it came from the principal focus of the diverging
lens. This ray is now shown as diverging away from the principal axis. In this
particular example, the second lens is located at the principal focus of the first lens,
hence ray (1) passes through the geometrical center of lens two and is not deviated.
We can see from the figure that these two rays diverge on the right-hand side of the
second lens and therefore can never form a real image there. However if these rays
are produced backward they intersect on the left-hand side of the lens to form an
enlarged, erect, virtual image. Thus the Galilean telescope, also called a terrestrial
telescope, produces enlarged images that are right side up.
31-52
Chapter 31 The Law of Refraction
Example 31.20
A Galilean telescope. A Galilean telescope is made of a + 20.0-cm objective lens and
a −4.50-cm eyepiece lens. The two lenses are separated by a distance d of 15.0 cm.
For an object situated at 1000 cm, find (a) the image distance of the first lens,
(b) the object distance for the second lens, (c) the image distance for the second lens.
Solution
a. The image distance of the first lens, found from the lens equation, equation 31.25,
is
1 = 1 − 1 =
1 −
1
q1 f1
p1 20.0 cm 1000 cm
q1 = 20.4 cm
b. The object distance for the second lens, found from equation 31.27, is
p2 = d − q1 = 15.0 cm −20.4 cm = −5.4 cm
Note that this is a virtual object since p2 is negative.
c. The image distance for the second lens, found from equation 31.26, is
1 = 1 − 1 =
1
−
1
q2
f2 p2 (−4.50 cm) (−5.4 cm)
q2 = −27 cm
Because q2 is negative the final image is virtual.
To go to this Interactive Example click on this sentence.
Have you ever wondered…?
An Essay on the Application of Physics
Nature’s Camera -- the Human Eye
Have you ever wondered how the human eye works? It is one of the greatest
marvels of nature, but one that is usually taken for granted. The human eye is very
much like a camera. It contains a lens, an iris diaphragm to let the light in, and its
film is the retina at the back of the eye, figure 1. The lens focuses incident light onto
31-53
Chapter 31 The Law of Refraction
the retina so a sharp image is observed. The iris diaphragm opens wide in subdued
lighting and closes down in very bright light.
In the eye, the distance from the lens to the retina q, does not change as in a
camera, instead the shape of the lens is changed by the muscles and ligaments of
Figure 1 The human eye.
the eye. Hence, the eye changes its focal length in order to bring objects at different
object distances to a focus at the same image distance q, the distance from the lens to
the retina. This changing of the focal length of the eye is called accommodation. A
schematic of the normal eye is shown in figure 2(a). The greatest amount of bending
Figure 2 Defects of the human eye.
of light in the eye occurs at the cornea, because it is here where the greatest
difference in the index of refraction occurs. The index of refraction of air is of course
equal to one, whereas the index of refraction of the cornea is about 1.35. The index
of refraction of the lens is about 1.44; that of the aqueous and vitreous humor is
about 1.34.
31-54
Chapter 31 The Law of Refraction
Some eyes do not have the ability to change the shape of the lens as
necessary and this is referred to as a defect in vision. One such defect is called
hyperopia, or farsightedness, and is shown in figure 2(b). A farsighted person
can see distant objects clearly, but objects a short distance away are blurred. The lens
is not capable of bending the incoming light of near objects to a focus on the retina.
Instead the light is focused behind the retina, leaving blurred vision. The defect is
easily remedied by placing a converging lens in front of the eye, as shown in figure
2(c). The converging lens brings these incoming rays to a focus at the retina.
Farsightedness is a defect that occurs to most people as they age. The lens hardens
with age and the muscles weaken and are no longer capable of shaping the lens as
needed.
Another defect is myopia, or nearsightedness, as illustrated in figure 2(d).
A nearsighted person can see nearby objects clearly, but distant objects are blurred.
The distant objects come to a focus in front of the retina leaving a blurred image on
the retina. This defect can be eliminated by placing a diverging lens in front of the
eye. The diverging lens causes the rays to converge more slowly and hence the
image is formed farther back at the retina, as shown in figure 2(e).
The minimum distance from the eye at which an object can be seen distinctly
is called the near point of the eye. For the average person, the near point is about
25.0 cm. That is, an object at a distance p of 25.0 cm is the minimum distance that
an object can be placed to give a distinct image on the retina. If the distance is
decreased below this value, the image will be blurred for most people. Remember,
this is only an average value, as we get older, the minimum distance of distinct
vision increases. For example, the near point can be as close as 6.00 cm for a child
and can increase to over a 100 cm for an elderly person.
Example 31H.1
Range of focal length for the eye. The distance from the cornea to the retina in the
human eye is about 20.0 mm. What should the focal length of the human eye be in
order to see clearly an object (a) at infinity and (b) at the near point of most distinct
vision, 25 cm?
Solution
a. For the object at infinity, the image is located at the principal focus. Hence
f = q = 20.0 mm = 2.00 cm
is the focal length of the eye when it views an object at infinity.
b. For the object at p = 25.0 cm, the focal length would have to be
31-55
Chapter 31 The Law of Refraction
1 = 1 + 1 =
1 +
1
= 0.540 cm−1
f
p
q 25.0 cm 2.00 cm
f = 1.85 cm
Thus, the eye only has to change its focal length by (2.00 cm − 1.85 cm = 0.15 cm) 15
mm to see its entire range of vision.
Another common defect of the eye is called astigmatism. Astigmatism occurs
in some people, because their eye is not completely spherical. That is, the radius of
curvature of the eye in the vertical direction is not the same as the radius of
curvature in the horizontal direction. Hence, vertical rays do not converge to the
same position as horizontal rays. This defect is usually corrected with a lens of
cylindrical curvature.
Although the magnification of a lens was defined in equation 31.22 as M =
hi/ho, when viewing an object with the eye it is sometimes more desirable to talk
about the angular magnification of the lens. Figure 3(a) shows an object at three
distances from the unaided eye. The size of the object, as determined by the unaided
Figure 3 Angular magnification and the simple microscope.
eye, depends on the angle subtended by the object at the eye. Thus, when the object
is at position 1, it subtends an angle θ1; when it is at position 2, it subtends the
angle θ2; and when it is at position 3, it subtends the angle θ3. As seen from the
diagram, θ1 < θ2 < θ3. Thus, the closer the object is to the eye, the greater the angle
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Chapter 31 The Law of Refraction
subtended and the easier it is for the eye to see the object. However, the object
cannot be moved closer to the eye than the near point, because that is the closest
point that the eye can still see the object distinctly. In figure 3(b), the object is
placed at the object distance p1 = 25.0 cm in front of a normal eye. The height of the
object ho subtends an angle α at the lens of the eye. The image of the object is
focused on the retina of the eye. This is the largest image that the unaided eye can
make of the object. If the object is placed within the focal point of a converging lens,
the object can be brought even closer to the eye and hence will subtend a larger
angle β, figure 3(c), than with the unaided eye. Therefore, the object is placed within
the focal point, at the position p2, such that the image distance q2 is equal to the
near point of the eye, the closest position of distinct vision. We assume that the
value of the near point q2 = 25.0 cm. The image is virtual, erect, and magnified as
seen earlier. The angular magnification MA of the lens is defined as the ratio of
the angle subtended at the eye by the object when the lens is used, to the angle
subtended by the unaided eye. That is,
MA = β
(31H.1)
α
From figure 3(c), we see that
tan β = ho
p2
whereas from figure 3(b), we see that
tan α = ho = ho
p1 25.0 cm
Let us take the ratio
tan β =
ho/p2
tan α ho/25.0 cm
= 25.0 cm
p2
Because the angles considered are small, we can use the small-angle approximation,
setting the tangent of an angle equal to the angle itself. Hence,
β = 25.0 cm
α
p2
But this ratio is the definition of the angular magnification, equation 31H.1, hence
MA = 25.0 cm
p2
The value of p2, found from the lens equation, is
31-57
(31H.2)
Chapter 31 The Law of Refraction
1 = 1 − 1
p2 f
q2
1 = 1 −
1
= 25.0 cm + f
p2 f −25.0 cm (25.0 cm)f
(31H.3)
And the object distance for the image to be at the near point is
p2 = (25.0 cm)f
25.0 cm + f
(31H.4)
Substituting equation 31H.4 into equation 31H.2, gives
MA = β = 25.0 cm =
25.0 cm
α
p2
(25.0 cm)f /(25.0 cm + f)
MA = 25.0 cm + 1
f
(31H.5)
Equation 31H.5 gives the angular magnification of a converging lens when the image
is located at the near point of the eye, which is assumed to be 25.0 cm for the normal
eye.
Example 31H.2
The angular magnification of a lens when the image is at the near point of the eye. A
converging lens of 10.0-cm focal length is used as a magnifying glass. If the image is
to be at the near point of the eye, find the angular magnification of the lens.
Solution
The angular magnification, found from equation 31H.5, is
MA = 25.0 cm + 1
f
= 25.0 cm + 1
10.0 cm
= 3.5
The eye is more relaxed when viewing an object at infinity than at the near
point. It is, therefore, sometimes more desirable to view an image at infinity than at
the near point. For the image to be at infinity, the object must be placed at the
principal focus. The lens equation, equation 31H.3, shows that if p is at infinity,
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Chapter 31 The Law of Refraction
then the object distance p2 is equal to the focal length f. Hence substituting f for p2
in equation 31H.2, gives
MA = 25.0 cm
(31H.6)
f
Equation 31H.6 gives the angular magnification of a simple microscope when the
image is set for viewing at infinity.
Example 31H.3
The angular magnification of a lens when the image is at infinity. A magnifying
glass of 10.0-cm focal length is used to view an object, such that the image is at
infinity. Find the angular magnification of the lens.
Solution
The angular magnification, found from equation 31H.6, is
MA = 25.0 cm
f
= 25.0 cm = 2.50
10.0 cm
Notice that when the image is viewed at the near point the angular
magnification is 3.5, whereas when viewed at infinity, it is 2.5. If viewing is for a
long time, it might be desirable to view the image at infinity because it is easier on
the eyes. If the magnification is the most important consideration, then the image
should be viewed at the near point of the eye.
The Language of Physics
Refraction
The bending of light as it travels from one medium into another. It occurs because of
the difference in the speed of light in the different mediums. Whenever a ray of light
goes from a rarer medium to a denser medium the refracted ray is always bent
toward the normal. Whenever a ray of light goes from a denser medium to a rarer
medium, the refracted ray is bent away from the normal (p. ).
31-59
Chapter 31 The Law of Refraction
Law of refraction
The ratio of the sine of the angle of incidence to the sine of the angle of refraction is
a constant. The constant is called the relative index of refraction and it is equal to
the ratio of the speed of light in the first medium to the speed of light in the second
medium. Because of the changing speed of light, the wavelength of light changes as
the light passes into the second medium (p. ).
The critical angle of incidence
The angle of incidence that causes the refracted ray to bend through 900. When the
incident angle exceeds the critical angle no refraction occurs. In that case, it is
called total internal reflection because all the light that strikes the interface is
reflected (p. ).
Prism
A triangular piece of transparent material whose angle exceeds the critical angle. A
ray of light falling on one of the smaller sides of the prism enters the prism and is
totally reflected from the longer side of the prism. Prisms are also used for
analyzing the dispersion of white light into its component colors (p. ).
Fiber optics
A flexible glass rod of high refractive index. Light entering the glass undergoes total
internal reflection from the walls of the glass fiber and the light travels down the
length of the fiber with little or no absorption of light (p. ).
Dispersion
The separation of white light into its component colors. It occurs because the index
of refraction of a medium varies slightly with the wavelength of light (p. ).
Lens
A piece of transparent material, such as glass or plastic, that causes light passing
through it to either converge or diverge depending on the shape of the material (p. ).
Lensmaker’s formula
An equation that relates the focal length, index of refraction of the medium and the
lens, and the radii of curvature of the lens (p. ).
Thin lens
A lens whose thickness is negligible compared to the distance to the principal focus
and to any object or image distance (p. ).
Converging lens
A lens that causes light, parallel to its principal axis, to converge to the principal
axis. Converging lenses have positive focal lengths (p. ).
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Chapter 31 The Law of Refraction
Diverging lens
A lens that causes light, parallel to its principal axis to diverge away from the
principal axis. Diverging lenses have negative focal lengths (p. ).
The lens equation
An equation that relates the image distance, the object distance, and the focal
length of a lens. It has the same form as the mirror equation (p. ).
Dioptric power of a lens
The reciprocal of the focal length of a lens. The focal length must be expressed in
meters. For a combination of any number of lenses in contact, the power of the
combination is equal to the sum of the powers of each individual lens (p. ).
Accommodation
The changing of the focal length of the eye in order to focus an image on the retina
of the eye (p. ).
Hyperopia, or farsightedness
A defect of the eye that causes objects far away to be seen clearly while close objects
are blurred. The condition is remedied by placing a converging lens in front of the
eye (p. ).
Myopia, or nearsightedness
A defect of the eye that causes objects close to the eye to be seen clearly, while
objects far away are blurred. The condition is remedied by placing a diverging lens
in front of the eye (p. ).
Near point of the eye
The minimum distance from the eye at which an object can be seen distinctly. For
the average person the near point is about 25 cm (p. ).
Angular magnification
The ratio of the angle subtended at the eye by an object, when a lens is used, to the
angle subtended by the unaided eye (p. ).
Summary of Important Equations
The law of refraction
The index of refraction
sin i = v1 = constant = n21
sin r v2
n1 sin i = n2 sin r
n= c
v
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(31.3)
(31.8)
(31.4)
Chapter 31 The Law of Refraction
v = λν
Speed of a wave in terms of wavelength and frequency
n21 = λ1
λ2
Index of refraction in terms of wavelengths in two mediums
Apparent depth for small angles
Critical angle
Lens equation
Magnification
Height of image
Combination of lenses
Final image
Magnification
P=
Power of combinations of lenses
(31.14)
(31.21)
(31.23)
(31.24)
(31.26)
(31.27)
(31.25)
(31.33)
1 =1 + 1 + 1 + 1+…
fc f1 f2
f3
f4
Power of a lens
(31.10)
(31.17)
1 = 1 − 1
q2 f2 p2
p2 = d − q1
1 = 1 − 1
p1
q1 f1
M = M1M2
where
and
Thin lenses in contact
q = n2 p
n1
sin ic = n2
n1
1 = 1 + 1
f
p
q
M = hi = − q
ho
p
hi = Mho
(31.9)
(31.43)
1
f (in meters)
(31.44)
Pc = P1 + P2 + P3+ P4 +
(31.45)
1 = n2 − n1
n1
f
(31.60)
Lensmaker’s formula, lens in any medium
1 − 1
R1 R2
1 = (n − 1 ) 1 − 1
R1 R2
f
Lensmaker’s formula, lens in air
f-number of a lens
f# = f
d
Angular magnification, astronomical telescope
Angular magnification, viewing at near point
Angular magnification, viewing at infinity
31-62
(31.61)
(31.62)
MA = fo
fe
MA = 25.0 cm + 1
f
MA = 25.0 cm
f
(31.64)
(31H.5)
(31H.6)
Chapter 31 The Law of Refraction
Questions for Chapter 31
*1. Why does a diamond sparkle?
2. When the angle of incidence is equal to zero, the angle of refraction is also
zero. Does the wavelength of the light change when going from one medium to
another under these circumstances?
3. If you are at the bottom of a pool of water and you look upward into the air
at an angle greater than 500 can you see anything in the air?
4. Is it possible to have a ray of light refracted into a medium such that the
new wavelength decreases to the point where it is no longer in the visible spectrum?
What would you see? Does the eye interpret a wavelength or a frequency?
5. What does a wide-angle lens and a telephoto lens do when each is attached
to a camera?
6. Describe the optical system used for (a) a slide projector, (b) a movie
camera, and (c) an overhead projector.
7. A swimmer forgets to take off her glasses as she enters the pool. When she
is under water, wearing the glasses, does the water have any effect on what she can
see with the glasses?
8. When you see yourself in a mirror, your right and left hand are
interchanged. Yet, if you see a picture of yourself made with a camera, containing
lenses, your right and left are not changed. Why?
*9. What are bifocals and why are they used?
*10. What is a mirage and how is it explained by the index of refraction?
*11. How is a rainbow formed?
12. How is a convex lens used, with the help of the sun, to start a fire?
Problems for Chapter 31
31.2 The Law of Refraction
1. A ray of light impinges on a piece of glass (ng = 1.52) at an angle of
incidence of 50.00. Find the angle of refraction.
2. A ray of light passes from water to glass at an angle of incidence of 50.00.
Find the angle of refraction.
3. A ray of light is refracted by an angle of 34.50 as it enters water from glass.
Find the angle of incidence.
4. A ray of light in air makes an angle of incidence of 35.00 as it enters an
unknown liquid. The refracted ray in the fluid is measured to be 22.50. Find the
index of refraction of the unknown liquid. What substance might it be?
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Chapter 31 The Law of Refraction
Diagram for problem 4.
5. Determine the speed of light in (a) water, (b) glycerine, and (c) diamond.
6. If a ray of light of 480.0-nm wavelength in air enters into water, what is
the wavelength of the light in the water? Is this light visible?
7. A ray of light of 580.0-nm wavelength in water, enters into the air. What is
the wavelength of the light in the air?
8. A ray of light of 700.0-nm wavelength in water enters into the air. What is
the wavelength of the light in the air? Will the light be seen in the air?
9. A ray of light of 590.0-nm wavelength in air impinges on a piece of flint
glass at an angle of 30.00 with the vertical. The index of refraction of this flint glass
is 1.57. Find (a) the angle of refraction, (b) the speed of light in the glass, and (c) the
wavelength of light in the glass.
31.3 Apparent Depth of an Object Immersed in Water
10. A rock sits at the bottom of a 3.50-m-deep pool. What is its apparent
depth?
11. A fish appears to be at a depth of 120 cm in water. What is its actual
depth?
Diagram for problem 11.
31.4 Refraction through Parallel Faces
12. A ray of light in air makes an angle of incidence of 30.00 with a sheet of
glass 0.500 cm thick. Find the distance d that the final ray is displaced from its
original direction.
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Chapter 31 The Law of Refraction
Diagram for problem 12.
31.5 Total Internal Reflection
13. What is the critical angle of refraction for a light ray going from water to
glass?
14. Find the critical angle of refraction for a ray of light passing from
glycerine (n = 1.47) into air.
15. The critical angle of a ray of light is measured as 41.80 as it goes from an
unknown liquid into air. Find the index of refraction of the liquid.
31.7 Thin Lenses
16. A double convex glass lens of index of refraction 1.52, has radii of
curvature R1 = +50.0 cm and R2 = −25.0 cm. Find its focal length.
17. A glass lens of index of refraction 1.52, has radii of curvature of R1 =
+40.0 cm and R2 = −15.0 cm. What is its focal length and is it a converging or a
diverging lens?
18. A glass lens of index of refraction 1.52, has radii of curvature of R1 = −40.0
cm and R2 = +15.0 cm. What is its focal length and is it a converging or a diverging
lens?
19. A glass lens of index of refraction 1.52, has radii of curvature of R1 = −40.0
cm and R2 = −15.0 cm. What is its focal length and is it a converging or a diverging
lens?
20. A lens is made of transparent material, with radii of curvature R1 = 15.0
cm and R2 = 90.0 cm. If the focal length of the lens is 5.00 cm, find the index of
refraction of the material.
31.9 The Lens Equation, and Section 31.10 Some Special Cases for the
Convex Lens
*21. An object 3.00 cm high is placed 20.0 cm in front of a converging lens of
15.0 cm focal length. Draw a ray diagram. Find (a) the image distance, (b) the
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Chapter 31 The Law of Refraction
magnification, and (c) the final size of the image. (d) Is the image real or virtual?
(e) Is the image erect or inverted?
*22. An object 3.00 cm high is placed 20.0 cm in front of a diverging lens of
−15.0 cm focal length. Draw a ray diagram. Find (a) the image distance q, (b) the
magnification M, and (c) the height of the image hi. (d) Is the image real or virtual?
(e) Is the image erect or inverted?
23. Where should an object be placed in front of a 20.0-cm lens in order for
the image to be the same size as the object?
24. An object 7.00 cm high is placed 5.00 cm in front of a convex lens of 10.0
cm focal length. (a) Draw a ray diagram. Find (b) the image distance, (c) the
magnification, and (d) the height of the image.
25. How far in front of a 20.0-cm converging lens should an object be placed in
order to produce an image 25.0 cm from the lens on the same side of the lens as the
object?
26. An object is placed 15.0 cm in front of a diverging lens of 5.00 cm focal
length. Where is the image located and what is its magnification?
27. Where should an object be placed in front of a diverging lens of 10.0 cm
focal length in order to give an image a magnification of 0.500?
28. An object 5.00 cm high is placed 20.0 cm in front of a converging lens. The
image is measured to be 7.00 cm high. Where is the image located?
29. An object 5.00 cm high is placed 30.0 cm in front of a converging lens of
7.50 cm focal length. What is the size of the image?
31.11 Combination of Lenses
*30. An object 5.00 cm high is placed 20.0 cm in front of a converging lens of
10.0 cm focal length. A second converging lens of 20.0 cm focal length is placed 30.0
cm behind the first lens. Find (a) the image distance for the first lens, (b) the object
distance for the second lens, (c) the image distance for the second lens, and (d) the
total magnification of the combination. Draw a ray diagram.
*31. An object 5.00 cm high is placed 10.0 cm in front of a 20.0-cm converging
lens. A second converging lens, also of 20.0 cm focal length, is placed 20.0 cm behind
the first lens. Find (a) the location of the final image and (b) its size. Draw a ray
diagram.
*32. A converging lens of +20.0 cm is separated by a distance of 20.0 cm from
a diverging lens of −5.00 cm. An object is located 30.0 cm in front of the first lens.
Find (a) the image distance of the first lens, (b) the object distance for the second
lens, (c) the image distance for the second lens, and (d) the magnification of the
system.
31.12 Thin Lenses in Contact
33. A 20.0-cm convex lens is placed in contact with a diverging lens of
unknown focal length. The lens combination has a focal length of 30.0 cm. Find the
focal length of the diverging lens.
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Chapter 31 The Law of Refraction
Diagram for problem 33.
34. What is the power of (a) a 50.0-cm converging lens, (b) a 20.0-cm
diverging lens, (c) the converging and diverging lenses in contact, and (d) the focal
length of the combination when they are in contact?
35. Ten identical converging thin lenses, each of focal length 6.50 cm, are in
contact. Find the focal length of the composite lens.
Additional Problems
36. A ray of light of 590.0-nm wavelength in glycerine impinges on a piece of
flint glass at an angle of 30.00 with the vertical. The index of refraction of glycerine
is 1.47 and for the flint glass used it is 1.70. Find (a) the angle of refraction, (b) the
speed of light in the glycerine, (c) the speed of light in the glass, and (d) the
wavelength of light in the glass.
37. Light travels in medium 1 at 1.50 × 108 m/s. The light is incident at an
angle of incidence of 40.00 when entering medium 2, where the angle of refraction is
300. Find the speed of light in medium 2.
38. A ray of light of 590.0-nm wavelength makes an angle of incidence of
40.00 on to the cornea of the eye. The index of refraction of the cornea is 1.35. Find
(a) the angle of refraction, (b) the speed of light in the cornea, and (c) the
wavelength of light in the cornea.
39. Light travels from glass of index of refraction 1.50 into water of index
1.33. If there exists a refracted ray, find the angle of refraction when (a) the angle of
incidence is 55.00 and (b) the angle of incidence is 70.00.
*40. If an object is placed a distance d1 in front of the principal focus of a lens
and the image is located a distance d2 beyond the other principal focus, show that
the focal length of the lens is given by
f = d1d2
41. An object is mounted 60.0 cm in front of a screen. At what two positions
will a 12.0 cm focal length lens yield a distinct image on the screen?
42. An optical system creates a virtual object (an object behind the lens) for a
diverging lens of focal length f = −8.00 cm. Find the position of the image and
describe it for the following two object distances: (a) p = −4.00 cm and (b) p = −10.0
cm.
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Chapter 31 The Law of Refraction
43. A farsighted person has a near point of 60.0 cm. What power lens should
be used for eyeglasses such that the person can read this book at a distance of 25.0
cm?
44. A nearsighted person has a far point of 15 cm. (The far point of the eye is
the farthest distance that an object can be seen clearly.) What power lens should be
used to allow this person to view far distant objects?
45. What is the smallest wavelength in air of visible light that can still be
seen on refraction into water?
46. A glass lens, ng = 1.70, has a focal length of 25.0 cm in air. What is its
focal length when it is submerged in water?
Interactive Tutorials
47. The law of refraction. A ray of light of wavelength λ1 = 500.0 nm in
medium 1 (index of refraction of medium one is n1 = 1.00) impinges on a second
medium of index of refraction n2 = 1.52, at an angle of incidence i = 35.00. Find
(a) the angle of refraction r, (b) the speed of light v2 in the second medium, and
(c) the wavelength λ2 in the second medium.
48. The critical angle of incidence. A ray of light in the denser medium (index
of refraction n1 = 1.50) impinges on a less dense medium of index of refraction n2 =
1.00. Find the critical angle of incidence ic for the ray of light going from the more
dense medium to the less dense medium.
49. Lensmaker’s formula. An optical lens has an index of refraction n = 1.52
and radii of curvature of R1 = +35.0 cm and R2 = −15.0 cm. Find the focal length of
this lens.
50. Thin lens. An object of height ho = 8.50 cm is placed at the object distance
p = 35.0 cm of a thin lens of focal length f = 15.0 cm. Find (a) the image distance q,
(b) the magnification M, and (c) the height hi of the resulting image.
51. Thin lens. An object of height ho = 0.500 cm is placed at the object
distance p = 2.00 cm of a thin lens of focal length f = 0.700 cm. Plot (a) the image
distance q and (b) the height hi of the resulting image as the object is moved toward
the lens.
52. A combination of lenses. An object of height ho1 = 5.00 cm is placed at the
object distance p1 = 15.0 cm of a thin lens of focal length f1 = 10.0 cm. A second lens
of focal length f2 = 10.0 cm is placed d = 48.0 cm behind the first lens. Find (a) the
image distance q1 of the first lens, (b) the magnification M1 of the first lens, (c) the
object distance p2 of the second lens, (d) the image distance q2 of the second lens,
(e) the magnification M2 of the second lens, (f) the magnification M of the system,
and (g) the height of the final image hi2.
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clicking on this sentence.
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