Lenses and Image Formation
Purpose:
To experimentally determine f , the focal length of a diverging and converging lens using multiple methods.
Introduction:
It is shown in the text that, for a thin lens, the focal length f , which is by definition the image distance
when the object observed is a very great distance away, is related to the object distance do and the image
distance di , corresponding to a relatively close object by the relationship
1
1
1
=
+
f
do
di
(1)
Object h h’ f f Distance of object Image Distance of image The expression, the thin lens equation (equation 1), as written is an approximation. In deriving it we
have neglected the thickness of the lens in comparison to distances comparable to the focal length. However
for relatively long focal length lenses it provides a simple, remarkably accurate way of calculating properties
of images.
In using the above expression one must always be careful to observe the sign conventions used in deriving
it. Here if the object lies to the left of the lens do is taken as positive; if it lies to the right do is taken as
negative. If the image lies to the right of the lens di is taken as positive; if it lies to the left di is taken as
negative hence it is a virtual image. Note that if an image is virtual, the light does not pass through it.
Every object is assumed upright and its height h is taken positive. The height h0 of the image is taken as
positive if the image is upright; h0 is taken as negative if the image is inverted. The focal length f is taken
as positive if the lens will actually focus light from a very distant source. A lens of this type is called a
converging lens and is thicker at the center than at the edges. If the lens will not focus light from a very
distant source it is called a diverging lens.
We can use a converging lens ahead of a diverging lens to converge the rays before they reach the diverging
lens thus using the image of the converging lens as the object for the diverging lens. In this way we can find
the focal length of the diverging lens using the thin lens equation.
Laboratory Procedure:
Part I.I - Taking Direct Measurements: Converging Lens Object at Infinity
1. Make a table in your notebook of values to be measured.
2. Take the converging lens and set it up in such a way as to form an image on a white screen of a very
distant object, i.e., an object well outside the window of the laboratory.
3. Measure the image distance di .
4. Determine δdi based on the precision of the ruler.
i
5. Determine the fractional uncertainty ( δd
di ) for this measurement and record this in your data table.
1
6. Since do is equal to infinity, di is equal to f according to equation 1. Record this result.
Part I.II - Taking Direct Measurements: Converging Lens with An Illuminated Object
1. Now set up an illuminated arrow as an object at an object distance of about four times the focal length
(f =di ) you just measured (i.e set the lamp at 4f from the lens).
2. Record the object distance do
3. Determine δdo based on the precision of the ruler.
o
4. Determine the fractional uncertainty ( δd
do ) for this measurement and record this in your data table.
5. Move the white screen until an image is formed (do not move the lens or arrow).
6. Record the image distance di .
7. Determine δdi based on the precision of the ruler.
i
8. Determine the fractional uncertainty ( δd
di ) for this measurement and record this in your data table.
9. Calculate the focal length from the thin lens equation (equation 1). You may not necessarily get the
same value for the focal length here as you have from Part I.I.
10. Record the length h of the arrow.
11. Determine δh based on the precision of the ruler.
12. Determine the fractional uncertainty ( δh
h ) for this measurement and record this in your data table.
13. Record the length of the arrow’s image, h0 .
14. Determine δh0 based on the precision of the ruler.
0
15. Determine the fractional uncertainty ( δh
h0 ) for this measurement and record this in your data table.
16. Calculate the magnification m,
m=
h0
h
17. Repeat steps 1–16 with an object distance of 2 12 times the focal length and 34 the focal length. Note,
if a real image cannot be formed be sure to explain why one could not see the image in
your results section.
18. Average your three values of the focal length favg1 for the converging lens.
Part II.I - Taking Direct Measurements: Diverging Lens Combination
1. Take the converging lens and set it next to the diverging lens in the same lens holder. The lenses will
touch.
2. Form an image on a white screen of a very distant object, i.e., an object well outside the window of
the laboratory.
3. Measure the image distance di .
4. Determine δdi based on the precision of the ruler.
i
5. Determine the fractional uncertainty ( δd
di ) for this measurement and record this in your data table.
6. Since do is equal to infinity, di is equal to the combined focal length fc according to equation 1. Record
this result.
2
7. Using the average value for the focal length of the converging lens in part I as f1 and the value you just
found for the combined focal length fc , calculate the diverging lens focal length f2 using the following
equation;
1
1
1
=
+
(2)
fc
f1
f2
Part II.II - Taking Direct Measurements: Diverging lens with separated converging lens
1. Place the arrow at one end of the bench and position the converging lens a distance of 4f from the
object (as you’ve done earlier in Part I).
2. Record the object distance do
3. Determine δdo based on the precision of the ruler.
o
4. Determine the fractional uncertainty ( δd
do ) for this measurement and record this in your data table.
5. Using the average value for the focal length of the converging lens in part I as f1 , calculate the image
location (di1 ) using the thin lens equation.
6. Place the diverging lens 8 cm away from the converging lens. This is the separation distance.
7. Calculate the object distance of lens 2, the diverging lens (do2 ), by subtracting d1i from the separation
distance of the lenses.
8. Move the white screen until the image is in focus and record the distance between the screen and lens
2 (di2 ).
9. Calculate the focal length of the diverging lens (f2 ) using the thin lens equation. Be sure to use + and
− signs correctly.
10. Repeat steps 6–9 using separation distances of 4.5 cm and 0 cm between the lenses.
11. Average your three values of the focal length favg2 for the diverging lens.
12. It may help to follow the example outlined below and remember to record all relevant data.
In this example, let f1 =15 cm, then do1 = 60 cm, and do2 is calculated from di1
O1 #1 #2 O2/I1 60cm 8cm 12cm 22.9cm 1
1
1
=
+
f1
do1
d i1
di1 = (
1 −1
1
−
)
f1
do1
di1 = 20.0 cm
3
I2 1
1
1
+
=
f2
do2
d i2
do2 = −12.0 cm = (8.0 cm − di1 )
di2 = 22.9 cm (measured)
f2 = −25.2 cm
13. Repeat steps 6–9 using separation distances of 4.5 cm and 0 cm between the lenses.
14. Average your three values of the focal length favg2 for the diverging lens.
Part III - Determining Uncertainties in Your Final Values
In the results section of your notebook, state the results of part I.II and part II.II of your experiment in
the form favg ±δfavg . Note, δfavg should be equal to the largest fractional uncertainty from your values of
image distance di and object distance do .
δdi δdo
,
δfavg = favg ∗ max
di do
You should also address the following questions:
1. In part I.II, compare your values of magnification m in with the corresponding ratio
they match?
di
do .
How well to
2. Does your result for favg1 in part I.II agree within the uncertainties to the value from part I.I? Be
sure to clearly state the quantitative values you are comparing. If there are any large discrepancies,
quantitatively comment on their possible origin.
3. Does your result for favg2 in part II.II agree within the uncertainties to the value from part II.I? Be
sure to clearly state the quantitative values you are comparing. If there are any large discrepancies,
quantitatively comment on their possible origin.
4