Ph 122
February 12, 2016
quark%/~bland/docs/manuals/ph122/lenses/lenses.doc
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)
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.
1) 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. Find a distant source, such as a
light at the far end of the room, and focus it onto your hand, your lab book, or the diffusing
screen. Measure the distance from the lens to the image.
Why is this distance equal to the focal length (approximately)? (Consider equation (1), in the
limiting case where do is very large.)
2) 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 1-1/2 focal lengths away, on one side. Place the screen
lenses - 1
on the other side and move it until a sharp image of the object is focused on it. Record the
positions of lens, object, and image as shown in the sample data table below, and calculate the
object and image distances do and di (do = yl-yo, di = yi-yl). (The easiest way to do this is to enter
the data into an EXCEL spreadsheet; see table 1 for a suggested data-table format.) 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.
yo
(cm)
yl
(cm)
39.2
36.8
33.2
28.1
.
.
Table 1.
61.3
61.3
61.3
61.3
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
.
.
.
3) Now increase do (the distance between object and lens) by moving the object 2 or 3
cm away from the lens, re-focus the screen, and repeat the set of measurements and calculations.
Continue to increase d0 in suitable steps (i.e., the steps can grow longer the farther from the lens
you go), recording data each time. Try to choose lens positions such that you cover the largest
possible range of values for do and di. This can probably be done satisfactorily with eight or ten
lens positions in all (parts (3) and (4) combined). Omit the magnification measurements when
the image becomes less than 0.5 cm high.
4) When you reach the end of the bench, return the object to its first 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.
QUESTION: What happens now if the object distance is made even shorter (look through the
lens to see)?
If you want to save your file, put it at c:\labs\122\secn\ .
5) First plot: plot the image distance di as a function of the object distance do. NOTE:
Be sure to make the axes for do and di start at zero; otherwise the asymptotic behavior of the
curve is difficult to interpret. (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.) What happens to di as do becomes large? As do becomes small?
6) Second plot: the thin-lens equation can be written in the form
1
xy 
(2)
f
where x = 1/do and y = 1/di . Tabulate x and y (in the same table), and make a graph of y as a
function of x. Note: once again, make both axes start at zero. The result should be a straight
line. Use TRENDLINE to find the best-fit straight line. What is its slope? What is its yintercept? What do you think it should be, according to equation (2)?
lenses - 2
Equation (2) says that both the x-intercept and the y-intercept should be equal to 1/f. Make
another quick 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.
B. A quick measurement of the focal length. Here is a simple and accurate way to
determine f with just a single measurement.
1)Place the screen at one end of the optical bench and the light source at the other. Using
the lens you have already measured, 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.
2) Now use this streamlined procedure to obtain the focal length of the other converging
lens.
QUESTION: What are the restrictions on the lenses for which this method can be used?
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
(3)

.
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 (3). In using equation (3)
remember that the focal length of a diverging lens is negative!

III. Equipment
optical bench
illuminated source
diffusing screen
lenses - 3
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
Demonstrations
double-reflection real-image lens
lenses - 4