LENSES
In this experiment we study focusing of light by lenses. The predictions
of the thin-lens equation are tested in detail for a single lens. A rapid
method for accurate measurement of the focal length is demonstrated.
The effective focal length of two combined lenses is measured.
I. Theory
For image formation by a single lens, the theoretical relationship between object distance do,
image distance di, and focal length f is
1
1 1
 
d o di
f
(1)
(Note: sometimes one uses p for the object distance, and q for the image distance.) The
magnification m is given by
m
hi
d
 i
ho
do
(2)
where hi and ho are the heights of image and object, respectively. Note that an image which is
inverted with respect to the object yields a negative value for the magnification. These relations
are valid for a lens of thickness much smaller than the focal length (a “thin” lens).
II. Experimental Procedure
A. Detailed measurements on one thin converging lens.
For one lens, we will vary the object position over a wide range of values, recording the
image position and the magnification.
As a preliminary, here’s a rough and ready way to determine the approximate focal length of
a lens. Choose one of the convex (converging) lenses. Focus the overhead lights on your lab
table or hand. Measure the distance from the lens to the image. This is the approximate focal
length.
Q1. Why is this distance equal to the focal length (approximately)? (Consider equation (1), in
the limit where do is very large.)
Now set up the light source, 10-cm lens and screen along the optical bench. Be sure that all
three are centered over the bench and at the same height. Place the lens in the center of the
bench, with the light source about two focal lengths away, on one side. Place the screen on the
other side and move it until you see a sharp image of the object focused on it.
The easiest way to do record and analyze your measurements is to enter the data into an
EXCEL spreadsheet; see table 1 for a suggested data-table format. Record the positions yo, yl,
and yi of object, lens, and image, respectively, as shown in the sample data table. Calculate the
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object and image distances do and di (do = abs(yl-yo), di = abs(yi-yl), where “abs” stands for the
absolute value). Also record the measured sizes of the object and the image (ho and hi), and
calculate the experimental magnification, hi/ho, and the “theoretical” magnification, -di/do.
object
yo
(cm)
lens
yl
(cm)
39.2
36.8
33.2
28.1
.
.
Table 1.
61.3
61.3
61.3
61.3
image
yi
(cm)
do
(cm)
di
(cm)
ho
(cm)
79.6
22.1
18.3
8.3
78.3
24.5
16.8
8.3
.
28.1
.
.
.
33.2
.
.
.
.
.
.
.
.
Sample data table for detailed study of lens.
hi
(cm)
hi/ho
-di/do
x=1/do
y=1/di
x
-7.8
-6.85
.
.
-.940
.
.
.
-.828
.
.
.
.0452
.
.
.
.0546
.
.
.
.0452
.
.
.
Now increase do (the distance between object and lens) by moving the object an additional 2
or 3 cm away from the lens, re-focus the screen, and repeat the set of measurements and
calculations. Continue to increase d0 about 6 more times, in suitable steps (i.e., the steps can
grow longer the farther from the lens you go), recording data each time. Choose lens positions
such that you cover the largest possible range of values for do and di. Omit the magnification
measurements when the image becomes less than 0.5 cm high.
When you reach the end of the bench, return the object to its original position, and then
decrease do by about 0.5 cm, by moving the object towards the lens. Repeat all measurements
of do, di, and sizes, recording them in the same table, and continue stepping the object towards
the lens with ever smaller do distances until the image lies beyond the end of the bench. Take
about 4-5 more measurements.
Q2. What happens if the object distance is made less than the focal length (look directly through
the lens to see)?
Compare the theoretical magnification with the experimental magnification by comparing
your hi/ho values to the –di/do values.
First plot: plot the image distance di as a function of the object distance do. (Practical note:
if you have a curve connecting the points, it will look strange unless you sort the entries in
increasing order of do. It may be best just to not connect the points.)
Q3. What happens to di as do becomes large? As do becomes small?
There is a better way to display these data points. The thin-lens equation can be written in
the form
x y 
1
f
where x = 1/do and y = 1/di .
Q4. What should the slope and y-intercept be if you were to plot equation (3)?
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(3)
Second plot: Tabulate x and y (in the same table), and make a graph of y as a function of
x. Use TRENDLINE to find the best-fit straight line.
Q5. Compare the experimental slope with your answer to question 3.
Third plot: Equation (3) says that both the x-intercept and the y-intercept should be equal
to 1/f. Make another plot, of x as a function of y. (Select the last two columns in the data table.)
Use the two intercepts to determine two values of f. Use them to obtain a best estimate f , and
an error,  f , as in the table below. (See the first experiment, "Data Analysis on the IBM PC,"
for methods of calculating the error from two repeated measurements.) Compare this result with
the expected value (the label on the bag the lens came in) in the usual way, giving the
discrepancy, number of sigmas, and quality of agreement. [See Appendix on "Theory of
Statistics," Section IV.] The EXCEL table for doing this calculation should look like this:
f1
(cm)
f2
(cm)
fbar
(cm)
sigf
sigfbar
discrep
# of sigs Q
(cm)
(cm)
(cm)
11.5
9.2
10.35 1.626346
1.15
0.35 0.304348 Very Good
Table 2. Spreadsheet for calculating average value, error, and quality of agreement, for two values of the focal
length f. The expected value for f was taken to be 10.0 cm.
Remember to write down your final result as f   f and discuss possible sources of error.
B. A quick measurement of the focal length.
Here is a simple and accurate way to determine f with just a single measurement.
Place the screen at one end of the optical bench and the light source at the other. Using the
10 cm lens, move it along the bench until you have found both locations which give good images
of the source on the screen. (These two positions are called conjugate positions.) Use equation
(1) to calculate two values for f, one for each of the two positions of good focus. Find the
average value ( f ), and the error on the average value (  f ), as in part A. Compare with your
result from part A, and with the labeled value.
Now use this streamlined procedure to obtain the focal length of the other converging lens.
Q6. What kinds of lenses can be used in this method and why?
C. Combinations of lenses.
The power of a lens to converge (or diverge) light rays is the reciprocal of the focal length: P
= 1/f. For two thin lenses placed close together, the total (effective) power is approximately the
sum of the powers of each lens (Peff = P1 + P2). Thus, the effective focal length for two lenses is
given by
1
1
1
 
(4)
f eff
f1 f 2
Place two lenses as close together as possible, using masking tape to hold them together, and
measure the effective focal length by the method of part B. This will work best if you use the
more powerful of the two converging lenses (f = 5 cm) with the diverging (concave) lens.
Compare your experimental result with feff calculated from equation (4) in the usual way
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(standard deviation, etc). When using equation (4) remember that the focal length of a diverging
lens is negative
III. Equipment
Optical bench
Illuminated source
Diffusing screen
Tensor lamp (for when lights are off)
2 different converging lenses (nominal 5 cm and 10 cm)
1 diverging lens (nominal -10 cm)
3 mounts
1 lens holder
Meter stick
Computer with Excel
lenses - 4