PHY 142!
Assignment 9!
Summer 2013
Reading: Lenses and Mirrors; Applications
Key concepts: Focal points and lengths; real images; virtual images; magnification; angular
magnification.
1.!
2.!
3.!
Questions about objects and images.
a.!
Can a virtual image be seen on a projection screen?
b.!
Can a virtual image be photographed?
c.!
What is a virtual object? Give an example.
Questions about the eye and corrective measures.
a.!
Why is our vision so blurred underwater? How do goggles solve the
problem?
b.!
Most eyeglass lenses have a convex outer surface (the one furthest from
the eye) with a standard radius for cosmetic reasons. Draw a cross section
of such a lens that has a negative focal length.
c.!
If you look closely through the eyeglasses at the eyes of the wearer, you
can tell whether the person is nearsighted or farsighted. How?
Questions about magnification.
a.!
The magnification of instruments such as a magnifying glass increases
when the focal length of the eyepiece is reduced. Why are lenses of focal
length shorter than about 5 cm rarely used?
b.!
What is angular magnification, and why is it the appropriate measure
used to describe the power of optical instruments?
c.!
The telescope and microscope both use two positive lenses. Why is the
microscope’s magnifying power so much larger?
4.!
Make a drawing of an overhead projector of the type used in classrooms. Show
why, if the speaker is facing the audience and projecting onto a screen behind her,
the image of the slide she is reading from is upright and enlarged on the screen..
5.!
Questions about focusing.
a.!
To change the focus of a film camera using ordinary lenses from a distant
object to one closer, how is the distance from lens to film changed?
b.!
How is this done in the human eye?
PHY 142!
Assignment 9!
Summer 2013
6.!
Most optical microscopes have systems to provide bright illumination of the
object, yet the image seen through the instrument is not especially bright. Why is
the extra illumination necessary, and why is the image not so bright?
7.!
A device to spread out the beam from a laser is shown. The incoming beam from
the laser has circular cross-section of radius a. The device spreads the beam so its
cross-section radius becomes b, without changing the direction of the beam.
device
light
!
The device has lenses of positive focal length at
each end, as shown. Indicated is a ray at the top of
the incident beam, therefore at distance a from the
axis, entering lens #1. This ray emerges from lens
#2 at distance b from the axis.
1
d
2
a.!
Sketch the path of the ray as is passes
through the device. [It must emerge from lens #2 parallel to the axis.]
b.!
How are the focal lengths f1 and f 2 of the two lenses related to the beam
radii a and b? [Use your drawing.]
c.!
8*.!
Express the two focal lengths in terms of a, b and d.
A method to determine the focal length of a positive lens is shown. Two screens,
each with an object attached to it (indicated by the arrows), are distance L apart.
The lens is positioned first to form an image on the screen #2 of the object on
screen #1; it is then moved to form an image on screen #1 of the object on screen
#2. The distance d between these two locations of the lens is measured.
a.!
b.!
Write the two equations
relating the object and image
distances to L and d. [Use the
symmetry.]
1
d
Find the focal length in terms
of L and d.
L
2
PHY 142!
9.!
Assignment 9!
Summer 2013
An object and a projector screen are separated by distance L. A lens of positive
focal length f is used to make a real image of the object on the screen. There are
two locations for the lens between the object and the screen that will do this.
a.!
Show that the distance d between these two locations is given by
d = L(L − 4 f ) .
b.!
Show that the ratio of the magnifications of the images produced when the
2
⎛ L − d⎞
lens is at these locations is given by ⎜
.
⎝ L + d ⎟⎠
10*.!
In the situation shown the lens has a focal length –10 cm and the mirror has
curvature radius +20 cm. The distance between adjacent dots on the diagram is
10 cm. An object is placed as shown 15 cm in front of the mirror.
•
!
•
•
•
•
•
a.!
Use the formulas to locate the image formed by the mirror and the image
subsequently formed by the lens. (Ignore the image formed directly by the
lens.) Calculate the magnification of the final image relative to the object.
b.!
Draw principal rays (at least two) to locate the image formed by the
mirror.
c.!
Using this as the object for the lens, locate the final image by drawing at
least two principal rays.
[In drawing rays, use the vertical lines to represent the mirror and the lens in the
paraxial ray approximation.]
PHY 142!
11*.!
Assignment 9!
Summer 2013
Shown is a device that displays a real image.
Two shells, identical parts of spherical
mirrors, are put together as shown. An object
(shown as an arrow) is placed at the bottom
of the lower mirror; light is viewed through
the small hole in the top of the upper mirror.
A real image of the object appears in the hole.
Give answers in terms of d.
a.!
An image is formed by the top mirror
at a height x above the bottom; this is the object for the lower mirror,
which produces the image seen in the hole. What is x?
b.!
What is the focal length of the mirrors?
c.!
What is the magnification of the first image?
d.!
What is the overall magnification?
12.!
Questions about focusing power.
!
The “power” of a lens in diopters (D) is the reciprocal of
the focal length in meters. Consider the total power of the
two lenses shown, which have focal lengths f1 and f 2 .
d
Let parallel rays be incident from the left. The point to the
right of the lenses where the rays converge is the focal
point of the combination.
a.!
If the second lens were not present, where would the light focus?
b.!
This image acts as the “virtual” object for the second lens, with a negative
object distance. Locate the final image, showing that for the combination
!
13.!
!
1 1
1
= +
, or Dtot = D1 + D2 .
f
f1 f 2
In the next three questions, take the diameter of the eye to be 2.5 cm.
a.!
In focusing on distant objects, the normal eye does not use the extra power
of the lens. What must the power of this eye in order to give a clear image
of an object far away?
b.!
What additional power must the lens provide in order to focus clearly on
an object at the standard near point, 0.25 m?
c.!
Weakening of the lens has increased a person’s near point distance to
N > 0.25 m . A corrective lens of what power will restore the near point to
0.25 m?
PHY 142!
13.!
14.!
15.!
Assignment 9!
Summer 2013
A nearsighted person cannot reduce the power of her eye (by relaxing the lens)
below 42 D. The lens can add 4 D extra power for near vision.
a.!
How far can an object be from this person and still allow her to focus on it
clearly (the far point distance)?
b.!
What focal length of corrective lens should this person use to make the far
point distance infinite?
c.!
Without corrective lenses, what is this person’s near point distance?
A slightly farsighted person can focus clearly on distant objects by using 3 D of
the 4 D extra power of the lens to augment the insufficient power of his vision.
a.!
Without corrective lenses, between what distances can this person focus
clearly?
b.!
What power corrective lens is needed to let this person focus on an object
at the standard near point of 25 cm?
c.!
With this correction, can the person see clearly at all distances from 25 cm
to infinity? Explain.
A Galilean telescope uses a negative lens as its eyepiece. The arrangement is as
shown. The right focal points of the two lenses coincide.
•
fo
fe
•
f o , fe
•
a.!
Draw a ray from the top of a distant object, passing through the left focal
point of the objective lens, making small angle α with the axis.
b.!
Continue this ray between the lenses, and through the eyepiece.
c.!
The emerging ray makes angle α ′ with the axis. Determine the angular
magnification of the instrument in terms of the focal lengths.
d.!
What are the advantages of this telescope over ones that use two positive
lenses?