Principles of Technology
CH 13 LIGHT AND GEOMETRIC OPTICS
4
Name____
KEY OBJECTIVES
At the conclusion of this chapter you will be able to:
• Explain how curved surfaces refract light.
• Describe how images are produced by spherical lenses, and use the lens equations to
solve problems relating to these images.
Lenses
One important practical application of refraction is the construction of lenses. As light
passes through a lens, it can either converge or diverge, depending on the shape of the
lens.
Lenses that are thicker in the middle and thinner at the ends are known as Converging
lenses. Samples of converging lenses are shown below. Lenses that are thinner in the
middle and thicker at the ends are called diverging lenses and are illustrated below.
Converging Lenses
Diverging Lenses
In this book we will assume that all of the lenses we discuss are very thin. This
assumption allows us to use much simpler mathematical relationships.
1. Which statement is false?
a. An important practical application of refraction is the construction of lenses.
b. As light passes through a lens, it can either converge or diverge, depending on the shape of
the lens.
c. All lenses are very thin and can only be analyzed through complex mathematical
relationships.
d. Lenses that are thicker in the middle and thinner at the ends are known as converging
lenses. Lenses that are thinner in the middle and thicker at the ends are called diverging
lenses
CONVERGING (CONVEX) LENSES
If parallel monochromatic light rays strike a
converging lens, the light passes through the
lens and converges to a point known as the
focus. The diagram illustrates this situation.
Note that the principal axis is also one of the rays of light. It does not bend because it
strikes the lens along a normal to the surface. The distance between the center of the lens
and the principal focus is known as the focal length of the lens.
2. Which statement is false?
a. If parallel monochromatic light rays strike a converging lens, the light passes through the lens and
converges to a point known as the focus.
b. The principal axis is also one of the rays of light. The principal axis rays of light do not bend because
it strikes the lens along a normal to the surface.
c. The principal axis rays of light refract more than other light rays that are applied to a convex lens.
d. The distance between the center of the lens and the principal focus is known as the focal length of the
lens.
The degree to which a lens refracts light depends on its index of refraction and its
curvature. Large indices of refraction and highly curved surfaces produce more powerful
lenses. (Eye-care professionals use a unit called the diopter to measure the refractive
ability of a lens. The diopter is the reciprocal of the focal length and has the unit m-1
If an object is placed at various distances from a converging lens, the lens will produce
different types of images. Three special rays emerging from the top of the object may be
used to locate the image produced by the lens.
3. Which statement is false?
a. The degree to which a lens refracts light depends on its index of refraction and its
curvature. Large indices of refraction and highly curved surfaces produce more powerful
lenses.
b. Eye-care professionals use a unit called the diopter to measure the refractive ability of a
lens. The diopter is the reciprocal of the focal length and has the unit m-1
c. If an object is placed at various distances from a converging lens, the lens will produce
different types of images. Three special rays emerging from the top of the object may be
used to locate the image produced by the lens.
d. The degree to which a lens refracts light is highly unpredictable in all lenses thus lenses
always produce the same size image, regardless of the distance of the object from the lens.
The diagram below represents an object placed in front of a converging lens.
We substitute a straight line for our converging lens because, as stated above, we consider the lens to be
extremely thin.
A ray that enters the lens parallel to the principal axis will be refracted so that it passes through the
principal focus, as illustrated above.
A ray that strikes the principal axis at one focal length from the lens will emerge parallel to the
principal axis as illustrated above.
Any ray that passes through the optical center of the lens will pass through the lens without being
refracted. We note in the preceding diagram that the principal axis is one such ray and can be used in
locating the base of the object.
Any two rays along with the principal axis can
be used to locate the image.
4. Which statement is false?
a. A ray that enters the lens parallel to the principal axis will be refracted so that it passes
through the principal focus.
b. Any ray that runs along the principal axis and passes through the optical center of the lens
will be refracted greatest.
c. A ray that strikes the principal axis at one focal length from the lens will emerge parallel to
the principal axis.
d. Any ray that passes through the optical center of the lens will pass through the lens without
being refracted.
e. Any two rays along with the principal axis can be used to locate the image.
The table and diagram below summarize the various types of images that can be
produced by a converging lens. We note that real as well as virtual images can be formed
by this type of lens.
In every case, a diverging concave lens
produces an erect, virtual image that is located
in front of the object and on the same side of
the lens as the object. The image is always
smaller than the object. The diagram below
illustrates how such an image is formed.
5. Which statement is false?
a. The principal axis passes through the optical
center of the lens will pass through the lens
without being refracted and can be used in
locating the base of the object.
b. Only one type of image can be produced by a
converging lens, a concave image.
c. In every case, a diverging concave lens
produces an erect, virtual image that is located
in front of the object and on the same side of
the lens as the object.
d. The image is always smaller than the object.
DIVERGING (CONCAVE) LENSES
We note the following:
1. A ray entering the lens parallel to the principal axis is refracted at an angle such that, if we project it
back through the lens, it passes through a virtual focus.
2. A ray entering the lens at an angle such that, if it were projected as a straight line, it would pass
through point F on the diagram, is refracted
parallel to the principal axis.
3. A ray that passes through the optical center of the lens passes straight through and is not refracted.
In all cases involving single lenses, real images are inverted and virtual images are erect.
LENS EQUATIONS
We can calculate the sizes and distances of images formed by lenses by means of the following
equations, which are identical to the equations used for curved mirrors:
6. Which statement is false?
a. A ray entering the lens parallel to the principal axis is refracted at an angle such that, if we project it back
through the lens, it passes through a virtual focus.
b. A ray entering the lens at an angle such that, if it were projected as a straight line, it would pass through
point F on the diagram, is refracted parallel to the principal axis.
c. A ray that passes through the optical center of the lens passes straight through and is not refracted. In all
cases involving single lenses, real images are inverted and virtual images are erect.
d. The sizes and distances of images formed by lenses can not be calculated and must be estimated by high
skilled optometrists.
The following sign conventions are used for these equations: The focal length is considered positive for
converging lenses and negative for diverging lenses. The object distance is considered positive if it is
on the side of the lens from which light is coming, that is, if the object is real, as usually the case.
(Exceptions are some situations involving more than one lens where the object of one lens is the image
of another and is located on the opposite side of the lens from which light is coming, that is, the object
is virtual. In that case the object distance is considered negative.)
7. Which statement is false?
a. The focal length is considered positive for converging lenses and negative for diverging lenses.
b. The object distance is considered positive if it is on the side of the lens from which light is coming, that
is, if the object is real, as usually the case.
c. Images can not be produced by more than one lens simultaneously.
d. The image distance is considered positive if it is on the opposite side of the lens from which light is
coming. If the image distance is on the opposite side of the lens then image is real and the distance is
positive.
The image distance 15 considered positive if it is on the opposite side of the lens from
which light is coming, that is, if the image is real; if the image distance is on the same
side of the lens, that is, if the image is virtual, the distance is negative.
The equation for the magnification of a (m) mirror can be used also to calculate the
magnification of a lens:
The magnification is the ratio of the height of the image to the height of the object.
The negative sign is inserted as a convention. Object and image heights are considered
positive if they are above the axis and negative if below the axis. Thus the magnification
is positive for an erect image and negative for an inverted image.
8. Which statement is false?
a. If the image distance is on the same side of the lens then image is virtual and the distance is
negative.
b. The magnification is the ratio of the height of the image to the height of the object.
c. The negative sign is inserted to indicate very small magnitudes of size of images.
d. Object and image heights are considered positive if they are above the axis and negative if
below the axis. The magnification is positive for an erect image and negative for an
inverted image.
PROBLEM
A 0.70-meter (ho) object is placed 0.50 meter (do) from a converging lens whose focal
length is 0.25 meter (f). Describe the image (hi) and its placement (di). The focal length
is considered positive for converging lenses and negative for diverging lenses.
1/f = 1/do + 1/di
hi/ho = -di/do
The image is located 0.50 meter from the lens and is real. It is the same height as the
object and is below the principal axis, that is, inverted.
PROBLEM
A 0.70-meter (ho) object is placed 0.50 meter (do) from a diverging lens whose focal
length is 0.25 meter (f). Describe the image (hi) and its placement (di). The focal length
is considered positive for converging lenses and negative for diverging lenses.
1/f = 1/do + 1/di
hi/ho = -di/do
SOLUTION
The image is located 0.17 meter
away from the lens and on the
same side of the lens as the object,
and is therefore virtual. It is 0.24
meter in height above the principal
axis; therefore, it is erect.
9.
A 1.90-meter (ho) object is placed 0.60 meter (do) from a converging lens whose
focal length is 0.05 meter (f). Describe the image (hi) and its placement (di). The
focal length is considered positive for converging lenses and negative for
diverging lenses.
1/f = 1/do + 1/di
hi/ho = -di/do
a. The image is located 0.15 meter away from the lens and on the same side of the
lens as the object, and is therefore virtual. It is 2.5 meter in height above the
principal axis; therefore, it is erect.
b. The image is located 0.05 meter from the lens and is real. It is 0.15 meters in
height and is below the principal axis, that is, inverted.
c. The image is located 0.90 meter from the lens and is real. It is the same height as
the object and is parallel to the principal axis, that is, virtual erect.
d. The image is located 4.1 meter away from the lens and on the same side of the
lens as the object, and is therefore virtual. It is 1.1 meter in height above the
principal axis; therefore, it is inverted
10.
A 1.90-meter (ho) object is placed 0.60 meter (do) from a diverging lens whose
focal length is 0.05 meter (f). Describe the image (hi) and its placement (di). The
focal length is considered positive for converging lenses and negative for
diverging lenses.
1/f = 1/do + 1/di
hi/ho = -di/do
a. The image is located 0.055 meter away from the lens and on the same side of the
lens as the object, and is therefore virtual. It is 0.17 meter in height above the
principal axis; therefore, it is erect.
b. The image is located 0.15 meter from the lens and is real. It is 0.26 meters in
height and is below the principal axis, that is, inverted.
c. The image is located 1.90 meter from the lens and is real. It is the same height as
the object and is parallel to the principal axis, that is, virtual erect.
d. The image is located 4.1 meter away from the lens and on the same side of the
lens as the object, and is therefore virtual. It is 1.1 meter in height above the
principal axis; therefore, it is inverted