Lenses and Images
The basic equations governing image formation by thin lenses are:
1
𝑑𝑜
+
1
𝑑𝑖
𝑚=−
and
=
1
(1)
𝑓
𝑑𝑖
𝑑𝑜
(2)
and the associated sign conventions. As shown in the figure below, 𝑑𝑜 is the distance from the lens to the
object (in this case a light bulb); 𝑑𝑖 is the distance from the lens to the image (the screen); f is the focal
length of the lens; and m is the magnification, i.e. the ratio of the image size to the object size (with sign
convention).
screen
lens
di
Light bulb
do
<x >
2-meter stick
Part I
A quick way to estimate the focal length of a lens is to form an image of a distant object. For a distant
1
object, 𝑑 ≈ 0 so that formula (1) takes the form
𝑜
1
𝑑i
=
1
𝑓
or 𝑑i ≈ 𝑓. This means that when the distance
between the object and the lens is large, the image distance is approximately equal to the focal length. It
is also true in this case that the magnification m in formula (2) above will be small. An image formed of a
distant object may in fact be almost a pinpoint of light.
1.
Estimate the focal lengths of lenses A, B, and C by forming an image of a distant object. Use the
two-meter stick at your lab station as a scale, and record the location of the screen, the lens, and
the front surface of the light enclosure. The lens location should be recorded with an uncertainty,
and you should describe how you chose a value for that uncertainty. Calculate and record the
focal length along with an uncertainty.
2.
When you measure focal length in this way, you are treating the object distance as a large
number. How large must it be? The accuracy of your value of focal length will greater if ratio
𝑑𝑜 /𝑑𝑖 is much larger than one. Review your data, and calculate the ratio 𝑑𝑜 /𝑑𝑖 for each of the
lenses A, B, and C. Based solely on those results, which of the focal lengths do you have most
confidence in? Explain.
3.
The method described above will not work with lens D. Lens D is a diverging lens (it is thinner
at the middle), and it does not focus light. Mathematically, diverging lenses are treated as if they
have a negative focal length. One way to measure the (negative) focal length of a diverging lens
is to combine it with a converging lens. When two lenses, whose focal lengths are f1 and f2, are
placed right next to each other, the combination acts like a single lens whose effective focal
length feff is given by
1
𝑓eff
=
1
𝑓1
+
1
(3)
𝑓2
a)
Hold lenses C and D right next to each other and measure the effective focal length 𝑓eff of the
combination by focusing a distant object as you did before. You have already measured the focal
length of lens C (f1 in the formula above), so you can calculate the focal length of fD (f2 in the
formula) by solving equation (3). Record the focal length of lens D; it should be negative.
b)
This method has some practical limitations. Formula (3) is derived on the assumption that the
two lenses are “right next to each other,” and for your lenses, the holders prevent them being put
right next to each other. Even without the holder, the thickness of the lens itself limits how close
they can be. Measure the center-to-center distance between your two lenses in their holders.
In order to get accurate results for the focal length of the combination, the center-to-center
distance should be much smaller than either of the two individual focal lengths. Comment on
how well is that condition satisfied in your case? Be quantitative.
Part II
4a)
There is a more systematic way to determine the focal length of a lens. Equation (1) can be
1
1
1
1
1
rewritten in the form, 𝑑 = − 𝑑 + 𝑓 so that a graph of 𝑑 (on the horizontal axis) versus 𝑑 (on the
𝑖
𝑜
𝑜
vertical axis) should give a straight line with a slope of -1 and an intercept of
𝑖
1
.
𝑓
Here is a systematic procedure for collecting data. Start with the light source and screen far apart
along the scale. Then find two different locations where the lens can be placed so that a focused
image appears on the screen. Record the location of all three in the Excel spreadsheet. Now
move the screen closer to the light source in increments of your choice. Each time you move the
screen, again find two lens locations that produce a focused image.
1
1
Use Excel to plot a graph of 𝑑 vs. 𝑑 for lens A, then add a trend-line and record the slope and
𝑜
𝑖
intercept. What value do you get for the focal length of the lens? Print out a copy of your graph
and your data table. Get help with this.
b)
You may find that the slope of your graph is significantly different from -1. The reason is that
your measurement of do does not include the distance x that is seen in the figure. We can correct
that. First make and record in the spread sheet a rough estimate of the distance x in the box
provided. Then modify your Excel spreadsheet so that it adds that value to each do. Keep
changing the value of x until the slope of your graph is as close as possible to -1. Use the new
value of the intercept to find and record the focal length of the lens. Print out a copy of the graph
and the data table.
c)
Repeat this process for lens B using your best value of x. Record the slope, intercept, and focal
length of B . Print out a copy of the graph and the data table. Explain why the same value of x
should work for both graphs.
Part III
5.
How large is the light bulb filament? It is not easy to measure the size directly. Here is an
indirect way to measure it. Set up a lens and screen to produce an enlarged image of the filament.
Measure do and di and calculate the magnification. Be sure to include your value of x. Now
measure the size of the image on the screen (just the curly part). By knowing the size of the
image and the magnification, you should be able to find the size of the object (that is, the size of
the actual filament).
Part IV
There is another method that is sometimes used to measure focal length. From your work in part II, you
found that for each position of the screen, there are two different positions of the lens that will bring an
image into focus. Let D be the light bulb to screen distance, and let s be the distance between the two
different lens positions. See figure below. Working from equation (1) it is possible to find a relation
between D, s and f.
𝑓=
6.
𝐷
𝑠 2
[1 − ( ) ]
4
𝐷
You already have the data you need to calculate f in this way. Modify your Excel spreadsheet by
adding new columns for s and D then calculate a value for f for each row. Calculate the average f
for each of your two lenses and print out your table for lens A and lens B.
D
screen
Lens pos. 2
Lens pos. 2
Light bulb
s
meter stick
x
Checklist of materials.
1.
2.
3.
4.
5.
6.
Filled in data sheet
Table and graph for lens A without the x-correction
Table and graph for lens A with the x-correction
Table and graph for lens B with the x-correction
Table for Part IV lens A
Table for Part IV lens B