LENSES
LENSES
References
Crummett and Western, Physics: Models and Applications, Sec. 36-5,6
Halliday, Resnick, and Walker, Fundamentals of Physics (5th ed.), Sec. 35-6
Tipler, Physics for Scientists and Engineers (3rd ed.), Sec. 31-4
Introduction
In this experiment, you will encounter some basic characteristics of thin converging and diverging
lenses and simple lens systems.
A converging lens bends light incident upon it toward its axis of symmetry. It is capable of forming a
real image, which can be projected on a screen. In Figure 1, an object is at distance s from a converg-
ing lens whose focal length is f. f is the distance from the lens at which light from a very distant source
(s ~ ) is brought to a focus. Nearer objects form real, inverted images at distances s' > f, as shown in
the figure.
Conversely, a diverging lens is one which bends incident light away from its axis. Since it does not
converge light, it cannot form a real image. However, if one looks through a diverging lens at an object,
light from the object comes to the eye as if from an image behind the lens. (See Figure 2 on the next
page.) This image isn't really there, in the sense that there isn't really any point in space from which the
rays of light are actually diverging; it is spoken of as a virtual image. Since the image is on the opposite
side of the lens from the space in which a real image would be formed, the image distance s' is taken
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LENSES
as negative. A virtual image does not represent a real convergence of light rays and
thus cannot be projected on a screen.
Figure 2
Virtual image formation by a negative (diverging) lens
A converging lens will also form a virtual image if the object is inside its focal length.
For either a converging or a diverging lens, it is easy to see that, if the thickness of the lens itself is
negligible,
1 1 1
+ =
s s f
(1)
where f is considered to be negative for a diverging lens. You can also show that, if y and y' are the
heights (measured from the axis) of the object and the image, then
m
y
s
=y
s
(2)
m is called the lateral magnification; negative values of m are interpreted as referring to inverted
images.
Equipment
optical bench
light source
image screen holder, cross-hair
lenses (1 each AA or A, B, C, and D)
Procedure
The lenses' mounting in their holders isn't perfectly
Figure 3 Lens position measurement
symmetric, so the best way to measure a lens position is to
record it with the holder turned one way, repeat it with the holder
turned around, and take the average of the two measurements (see Figure 3).
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Figure 4
Measuring the focal length of a converging lens
(1) Using converging lens AA (or A), set up as shown in Figure 4. The object is a wire mesh
illuminated by a light source behind it. It is the wire mesh you want to form an image of on the
screen. Put the mesh at zero on the scale, the screen at 120.0 cm, and the lens in between.
Carefully adjust the lens position until a clear image of the mesh is formed on the screen. Record
the lens position (as discussed above). Repeat for object-to-screen distances of 150 cm and 180
cm.
NOTE: If you have an A lens instead of an AA, use object-to-screen distances of
150, 170, and 190 cm.
(2) Repeat step (1) to obtain a second independent determination of the lens position for each
object-to-image distance.
(3) Replace lens AA (or A) by lens B. Place it about 5 cm from the light source, at the far end of the
optical bench, facing a wall. Adjust the source-lens separation until you get a clear image of the
wire-mesh on the wall (use a tape measure, if needed). Record the object and image distances (s
and s’) once you get a clear image of the wire mesh on the screen. Trace the image of several
squares from the wall on to a piece of paper (include it in your lab report). Measure the grid
spacing of the wire-mesh object, by counting the number of squares in a 1-cm (or other convenient)
width. Work out the magnification by two different methods. For the first method, directly compare
the image size (say one square) to the corresponding dimensions of the original object by taking
the ratio of the image size and object size. Make sure you have an estimate of uncertainty for your
result. Now, calculate the magnification by using the well known expression m=-s’/s, propagate the
uncertainty, and compare it to the first estimate of magnification. Repeat for at least two
independent trials.
(4) Using diverging lens C and converging lens D, set up the arrangement of Fig. 5. Set the
diverging lens in place from 3 to 8 cm from the object. With an object-to-screen distance of around
80 cm, carefully adjust lens D to form an image on the screen. Record the positions of all system
elements (O, C, D, and I). Don’t forget Figure 3!) Repeat this procedure twice more.
LENSES
Figure 5
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Measuring the focal length of a diverging lens
(5) The focal length of lens D is 10.16 cm. Given this, you have enough information to calculate the
focal length of lens C from your data. (Note that the image formed by lens C is the "object" that is
imaged by lens D.)
(6) Take the object/lamp box off the optical bench and set up the telescope system of Figure 6. The
Figure 6
An elementary
telescope
"eyepiece" lens B should be right at the end of the bench, so you can get your eye close to it
without injuring yourself. To obtain a reasonable estimate of the magnification of the telescope, set
up two meter sticks along the wall at the far end of your lab bench. Place your table lamp so it
illuminates the meter sticks. Looking through the B lens, move the cross-hair back and forth until it
is in focus. Then move the objective lens AA to bring one of the meter stick into focus. (If you have
the cross-hair properly positioned, it and the image will move together when you move your head
from side to side a little.) Measure distances x1 and x2. With one eye focus on the actual meter
stick , and with the other eye focus on the image of the meter stick formed by the telescope. By
comparing markings on the image to those on the actual meter stick you should be able to estimate
the magnification.
Now replace the B lens with the C lens and remove the crosshairs. Adjust the lenses till you get
a clear image. This image should be erect (lens C is a diverging lens). This type of a telescope is
called a Galilean telescope and is also referred to as an “opera glass”. The distance between the
lenses is supposed to be the difference between the focal lengths of the objective and the
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eyepiece. Is that the case? Measure the distance between the two lenses and verify.
Analysis
(1) From your data of steps (1) and (2), use Equation (1) to calculate six values of the focal length of
lens A (or AA). Find their mean, standard deviation, and standard error. From your data of step
(3), calculate three values for the focal length of lens B, and find their mean and standard error.
(2) One source of error in this experiment is that the position O of the object (the wire-mesh screen) is
not very precisely defined. To see how much difference this makes, redo the calculations of (1)
with the object distances s increased by 1 mm. Is the change in the answer significant? Do you
think it is credible that the position of the object screen is really that far off?
(3) From your data on image and object size in procedure (3), calculate the lateral magnification m
from the grid dimensions, m = -y'/y. Compare this to what you get from m = -s’/s. Do the two
results agree within your estimates of experimental errors?
(4) Use your data from procedure (4) and (5), and the given value of fD = 10.16 cm, to calculate the
focal length of lens C (three independent values). Note that s'C + sD = XCD where XCD is the
distance between the two lenses. (Why?) Find an average value and standard error for fC.
(5) What is your estimate of the apparent magnification of the telescope in procedure (6) above? For
distant objects, the apparent magnification should be the ratio of focal lengths of the objective and
the eye lenses. Calculate this number and compare to your estimate. For distant objects, the
image formed by the objective (at the plane of the cross-hairs) should be at the focal point of both
lenses, so that x1  fB and x2  fAA. How do these values compare?
last update 7/03