Optics Experiment 2
Thin Lenses and Microscopes
Safety
Make sure that you have read the safety notes in the introductory section of this manual before
beginning any practical work.
Do not, under any circumstances, attempt to repair any of the equipment should you think it to
be faulty. Rather, turn off the apparatus at the power-point and consult your demonstrator.
In the course of this experiment you should note that the lamps you use will become very hot. Be
careful to make sure you do not burn yourself.
References
121/2: Page 922 (Geometrical Optics); Section 38.2 – 38.4.
141/2: Sections 35.6 – 35.7.
Prelab Questions
Note: Information on the thin lens formula and ray tracing appears in the text on the following
pages. A guide to calculating uncertainties appears in the introductory chapter.
Question (a)
An object is located at a position u = 25 cm from a lens of focal length f = 15 cm.
Use ray-tracing techniques to locate the image of this object below.
Question (b) Use the thin lens formula to confirm the value your diagram shows is the distance v
between the image and the lens.
Question (c)
If the object distance u has an experimental uncertainty of ±0.5 cm and the
focal length, f is known within an accuracy of ±0.1 cm, determine the
uncertainty in the image distance v.
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Introduction
There are many familiar applications of the control of light paths by lenses. In general their
purpose is to provide a clear image of an object, ie. a one-to-one mapping of the points of the
object. These images may be real or virtual, magnified or diminished, upright or inverted,
depending on the purpose of the optical instrument.
In this experiment you will concentrate on the role of lenses as magnifying glasses, in cameras
and microscopes. Initially you will need to remind yourself about various concepts and
techniques involved in the analysis of optical systems. Be prepared – much of what appears on
the next few pages is background review information. RayTrace simulations have also been
prepared to illustrate some of the ideas - ask your demonstrator about these.
Outline of Experiment
Section A
Several techniques for finding the positions of images produced by thin lenses will be
explored - you will lead up to locating the images experimentally by predicting their
location using ray tracing diagrams and the Thin Lens Equation, as well as viewing
the formation of images in more detail courtesy of the RayTrace program.
Section B
In this section you will investigate using a single lens magnification system, the
“magnifying glass”.
You will choose between Sections C and D.
Section C
Now you will build a more complex magnifying system - a model microscope having
two lenses. The image produced by the first lens acts as an image for the second, with
magnification occurring at both stages.
Section D
A lens produces a perfectly focused image in one location for each object position.
However the effectiveness of cameras relies on the fact that for a particular lensimage distance, objects in a range of positions, produce blurry but acceptable images.
This range of acceptable object locations is called depth of field. You will investigate
the effect of aperture diameter on depth of field. A sound understanding of how
lenses produces images will be required to explain this adequately.
Section A:
A Review: Images Produced by Thin Lenses
The Thin Lens Approximation
The diagram shows the path
of a light ray passing through
the centre of a glass lens.
Due to refraction at the glass
surfaces, the ray will be bent
away from its original path
(dotted line).
Figure 1
However, the thin lens approximation assumes that for a sufficiently thin lens, such a ray is not
significantly deflected from its path. This is a good approximation provided that:
(a)
the lens is thin, and
(b)
the rays are paraxial (ie.  is small).
It will be assumed that the thin lens approximation holds in all the work to follow.
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Focal Length of a Lens
Figure 2
The focal point (F) is the point through which incident light parallel to the optic axis then
converges for a convex (converging) lens, or appears to diverge from for a concave (diverging)
lens. The focal length (f) is defined as the distance from the centre of the lens to the point F.
Ray Tracing Techniques
When constructing ray diagrams by tracing light rays through thin lenses, four steps should
always be followed:
(a)
changes of direction of a light ray should be drawn as though they occur
at the lens’s central plane,
(b)
rays passing through the centre of the lens are not deviated,
(c)
rays parallel to the optic axis must
travel in a direction which passes
through the focal point,
F
(d)
parallel rays of light not parallel to
the optic axis meet on the focal
plane .
Figure 3
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Example:
The following sketch is an example of the use of ray tracing to construct the
image formed by an arrow 2 cm high with its base on the optic axis at a
distance of 10 cm from a large thin lens. The lens has a focal length of 4 cm.
Figure 4
The Thin Lens Equation
It can be shown that:
1
f

1 1

u v
where
u = distance between object and lens,
v = distance between image and lens,
f = focal length of lens.
Figure 5
The sign conventions we will use are:
u > 0
v > 0
f > 0
real object
real image
convex lens
u < 0
v < 0
f < 0
virtual object
virtual image
concave lens
Your input starts here …
Question (d) Now interpret the thin lens formula.
What does it predict about the location of the image:
Exercise:
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(i)
as the distance between the object and the lens, u, becomes larger?
(What position does the image approach for extremely large distances
between the lens and object?)
(ii)
as the object approaches the focal point?
(iii)
as the object passes the focal point, so u < f.
A RayTrace file simulates this behaviour, the file is called “thinlns2.ray”.
Double click on the object then drag it towards and away from the lens and
observe the behaviour of the rays. Does this agree with your answers to
Question (d)?
Physics 121/2 and 141/2 Laboratory Manual
Experiment (i)
Uncertainties

Are the optical elements aligned with the holes through which the measurements are
taken? What kind of error would be introduced if the alignment were not perfect?

How accurately can you determine u, the separation of the arrow slide and the lens?

How far can you move the screen and still have the image appear sharp? Hence how
accurately can you determine the image distance v?

Using your experimental uncertainties for u and v, calculate an uncertainty in your value
of f.

In your prelab exercise you predicted the position of the image of an object at a distance of
25 cm from a lens having a 15 cm focal length. Record this in your logbook here together
with the uncertainty determined in answering Question (a).
Using the slide transparency (illuminated) as the object, the optical bench, the ground glass
screen to determine the image position, and one of the (nominally) 15 cm focal length
lenses, check your prediction.
Did you expect the measured image position to be exactly the calculated image position
considering the focal length of the lens is nominally 15 cm? Compare any disagreement
with the uncertainty in the image position you calculate in answering Question (a).

Using this data and the thin lens formula calculate an experimental value for the focal
length of the lens.

Perform two more measurements to complete the following table in your book.
Case
u
1.
u1 = 25cm
v1 = ?
2.
u2 = 2f
v2 = ?
3.
u3 = v 1
v3 = ?
Question (e)
v
Image:
upright/
inverted
Image:
magnified/
diminished
Image
virtual/
real
Comment on the relationship between u1, v1 and u3, v3. Could you have
predicted this given the thin lens formula?
Your brief review of the properties of thin lenses is now complete. Continue now to the application of
these ideas to understanding magnifying glasses – the simplest possible microscope.
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Section B:
The Microscope
What Eyes Do:
When you need to see an object in detail, your instinctive response is to bring it as close to your
eye as possible. This helps, because the size of the image formed on the eye's retina (the retinal
image) depends on the distance of the object from the eye, being proportional to the height of the
object, h, and inversely proportional to its distance from the eye, d.
Figure 5: (a) The eye produces an image of an object a distance d away;
(b) The object is at the near point, and the retinal image is the largest clear image that is possible
The eye focuses on the object by adjusting the radius of curvature of the eye's lens until the image
forms on the retina (at the back surface of the eyeball), instead of behind or in front of it. The lens
is at its most relaxed position when parallel light is entering it - called relaxed vision.
At a certain distance from your eye the optical system of your eye can no longer bring the rays
diverging from each point on the object to focus at a single point your retina For young "normal
eyes" this is taken as 25 cm. The image on your retina at this stage is as large as possible for an
unaided eye, and this arrangement is used as the reference point for the more powerful
magnifying systems like the magnifying glass and microscope.
Magnification
The eye alone is often not sufficient for observing detail, so the next step is to introduce a single
lens or a combination of lenses as a magnifier. It is often useful to refer to a quantity which is a
measure of the magnification produced by an optical system.
Two ways of describing magnification can be used, transverse (or lateral) magnification and
angular magnification.
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Transverse Magnification
Transverse magnification, MT, compares the height of the final image with the height of the
object.
MT 
It is defined as:
hi
,
ho
… (1)
where
hi =
image height
and
ho =
object height.
Exercise:
Set aside a whole page for the following diagram, using it side-on.
(you are probably familiar with this)
Construct a ray diagram for an object that is an upright arrow 2.5 cm high,
with its base on the optic axis 6 cm from the centre of a large thin lens. The
lens has a 10 cm focal length. (Leave plenty of room to the left of your
lens, when drawing the diagram.)
Where do the rays of light emerging from the lens appear to diverge from?
Question (f)
This is called a virtual image.
Comment on the agreement between the object and image locations in your
diagram and the thin lens formula.
Exercise:
Recalling the sign conventions described earlier in this experimental show, by
use of similar triangles, that MT = – v/u. Calculate the transverse
magnification for the case you have just constructed. Does it agree with the
height measurements taken from your diagram?
Using the thin lens formula it can easily be shown that
MT 
f
f v

f u
f
… (2)
Question (g) What does this equation predict for the effect on the transverse magnification of
increasing the lens–object distance from f/2 to f, to 2f, and then increasing still
further? Describe in words how this description is consistent with your
experience of using a convex lens. Look back at Section A for some relevant
results.
Magnifying Power and Angular Magnification
A simple comparison between image and object sizes does not give a realistic idea of what you
actually see. Even when you view an object without the aid of a magnifying glass, there is still
an optical system involved, ie. the eye's lens. Your unaided eye’s view of the object is a more
useful standard than the object itself. So, when defining the ability of a magnifying glass to
magnify, the reference “standard” we use is the retinal image when the object is placed at the
near point of an unaided eye. As indicated above, it is conventional to assign 25 cm to this
normal viewing distance. The magnifying power, MP, of an optical system is then defined as:
MP
=
size of retinal image as seen through the instrument
size of retinal image as seen by unaided eye when object is at near point
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Figure 6
I is image formed by the magnifying glass of the object O,
I' is the image formed by the eye of the image I on the retina,
and
L is the eye to magnifying glass (focal length l) distance.
For the small entry angles common to optical systems this is equivalent to the angular
magnification, MA, which compares the angle subtended by the image at the lens of the eye (ifor
the object on the diagrams above) with the angle subtended by the object at the near point (np).
Figure 7
MP = MA
=
=
angle subtended by image seen through instrument
angle subtended by image as seen by unaided eye when object is at near point
i
 np

hi di
hnp dnp

MT
dnp
… (3)
di
Again we have applied the small angle approximation
 ~ tan 
here.
The equation shows that if the object and images are located in about the same place, ie. dnp ~ di,
the transverse magnification, MT, and the angular magnification, MA, have very similar values.
Re-expressing this equation (3), using both equation (2) and noting that
MP
=
MA
=
d np f  v
f L v
di = L – v gives:
… (4)
Note that dnp is the location of the object when at the reference point, ie. a near point of 25 cm.
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Some special cases
(a)
The case where the magnifying glass is held very close to the eye, ie. L  0:
Equation (4) reduces to:
MA
1 1 
 

v 
f

dnp 
=
As dnp and f are fixed, the greatest angular magnification will correspond to the smallest
possible value of –v for clear vision, which, for a "standard observer", is – v = 25 cm = dnp,
so
MA
=
1 +
25
f
… (5)
where f is in cm.
(b)
The usual case with a magnifying glass, where you place the object at the glass' focal point
(F). Then the light entering the eye is parallel, (ie. v = ), and so vision is relaxed. This is
therefore the case where u = f:
Equation (4) reduces to:
MA
=
d np
f
=
25
f
… (6)
A Simple Microscope (Magnifying Glass)
A single convex lens may be used as a magnifying lens. If the object to be viewed is placed either
inside or at the focal plane of the lens, a magnified image is seen.
Experiment (ii) - Estimation of the Magnifying Power of a Lens
Uncertainties

How accurately can you position the top of one of the magnified squares on the reference
graph paper screen.

How many magnified squares should you count in order to minimise the uncertainty of the
magnification measurement?

For a (nominal) focal length of 10 cm, and image distance v =  25 cm, calculate the object
distance u. Set up the optical bench in this way, and observe the arrow object with your
eye very close to the lens (L  0 case).

To measure the magnification, replace the arrow object with the graph paper slide. Place
the reference screen (also graph paper) as shown in the diagram. Keep the reference
screen in this position for all of your magnifying power measurements.
reference
graph paper
screen
25 cm
right eye
object (graph paper)
image
left eye
L~0
Figure 8
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
Simultaneously focus both eyes, and estimate the magnification by superimposing the two
images. (Consult your demonstrator if you find this difficult.) Compare your result to that
obtained from the appropriate formula for MP (special case (a)).

Least eye fatigue is felt over long periods if the eye is completely relaxed, ie. adjusted to
receive parallel light. Practical optical instruments should be used with the eye in this
state. Adjust your experimental set up to these conditions.
Question (h)

Where must the object be positioned, relative to your left eye?
The basic method used above to determine the MP must be modified somewhat. This is
because the reference paper will still be at 25 cm, but the image of the object graph paper
will be at , and it would be difficult to focus simultaneously on both. Instead, use the
1 mm aperture to increase the depth of field for viewing the reference paper, so that it can
be seen even when the eye is focusing at . Determine the MP, and compare it to that
obtained from the appropriate formula for MP (special case (b)).
You have a choice to make – complete this prac by doing either Section C or Section D. You may complete
the other section as Further Work if you have time!
Section C
Compound Microscopes
We often need a greater magnification than a single lens can provide. The model compound
microscope considered here has the basic features of a practical scientific microscope. It consists
of two lenses, L1 and L2, both of small focal length. These lenses are arranged so that the
objective lens, L1, forms a real image (I1), and the eyepiece lens, L2, forms a virtual image (I2). I2 is
the image inspected by the eye.
Figure 9
The objective lens, L1, produces a real image, and so its magnification is given by MT
(equation(1)). The eyepiece, L2, produces a virtual image viewed by the eye, and so its
magnification is given by MA. Therefore the overall magnification of the compound microscope is:
MP
=
MT,objective x MA,eyepiece
In the situation shown in the diagram above the eye would not be relaxed. For relaxed vision you
would need to ensure that the final image, I2, is at  so light enters your eye in parallel beams.
Then the intermediate image, I1 would be located at the focal point of the eyepiece lens, ie. F2.
When the image produced by the eyepiece lens is at , MA,eyepiece = 25/f2. Combining this
information with the results from Section A, it can be shown that
MP
=
–
25 s
f1 f 2
for the final image at , ie. for relaxed vision, where s is the separation of the two focal points F1'
and F2 (see diagram).
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Experiment (iii)

The compound microscope will be constructed using two 10 cm (nominal) focal length
lenses. Place the object arrow 15 cm away from the objective lens.
For this arrangement, where do you expect the real image, I 1, formed by the
objective lens, to be located?
Question (i)

Check this position, using a sheet of paper.
Note:
Small errors in the object to objective lens distance will result in a large error
in the location of the image I1. If I1 is not in the calculated position, adjust the
location of the object to place it there.
For relaxed vision, what should be the distance between I 1 and lens L2?
Question (j)

Position the eyepiece lens in the location you have just calculated. Observe the magnified
image, qualitatively describe how the magnification compares to the single lens case.

If there is time, experimentally estimate the magnification of the compound microscope,
using the same method as in the simple microscope case. You will need to use the 1 mm
aperture to increase the depth of field.
How does the measured magnification compare to that given by the formula for
MP?
Question (k)
Section D:
The Camera
Complete either Section C or Section D.
The single lens system you have been investigating can be used to simulate the optics of a
camera. First an introductory observational exercise …
Exercise:
(i)
Set up your object, lens and screen to produce a clearly
focused image.
Before trying it, write down your prediction for what
will happen to the image if the lens is covered except
for a small hole? (Will only part of the image appear,
will it be smaller or more blurry or dimmer than the
image produced by the whole lens?)
(ii)
Now, keeping the screen to lens distance constant,
move the object towards and away from the lens,
centred on the position that gives a clear image.
Repeat for an uncovered lens and a lens covered except
for a small hole. No measurements are required here.
Comment on your observations.
Depth of field is a term used to indicate the range of
object positions for which the images appear reasonably
sharp.
Figure 10
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This arrangement of screen, lens and aperture is a simple model of a camera. With a real camera
you have control over the lens to screen (film) distance to focus on the subject of interest. You can
also vary the aperture size to control the amount of light reaching the film and the depth of field.
The amount of light reaching the film can also be controlled by varying the length of time the
shutter is open.
You will now examine this behaviour in more detail.
Depth of Field
A camera made of a simple lens would only produce a perfectly focused image for objects at one
object-lens distance (u), once the lens-film distance (v) was fixed. However, we usually want to
photograph objects in more than one position at the same time.
A large depth of field means that an acceptable image is formed over a large variation of object to
camera distances. When you want only one plane of the scene being viewed to be clearly focused
on film then you need a limited depth of field.
Experiment (iv)
Uncertainties

How reliably can you determine when the image on the screen begins to go fuzzy?
Considering that this uncertainty arises at both ends of the sharp region, what then is the
uncertainty in the depth of field measurement?

With the object positioned approximately 25 cm from the lens, obtain the sharpest possible
image. Fix the lens and screen in those positions. By moving the object toward and away
from the lens, estimate the depth of field of the lens by itself, then with the 15 mm, 10 mm
and 5 mm apertures (place the centre of each aperture in turn over the centre of the lens).
Question (l)
Using the diagrams below as a starting point, draw your own diagrams to
explain how the presence of the aperture enables the object to be moved while
retaining a reasonable image on the ground glass screen.
f
f
aperture
f
f
Figure 11
A real camera uses an infinitely variable aperture called an iris diaphragm, which is made up of
a set of thin, sliding metal leaves.
In a real camera, the image recorded on the film depends on the intensity of light falling on the
film, and the length (in time) of the exposure. The latter is determined by the shutter speed.
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Question (m) If you “stopped down” the camera (ie. reduced the aperture size) to increase the depth
of field, will you need to keep the shutter open for a longer or shorter period of
time? In that case, are you more or less likely to get a blurred image of fast
moving objects?
The "speed" of the film is another variable. This refers to the time it takes for the chemicals on
the film to react to light. A "fast" film is therefore useful for photography in dark conditions, (but
it may give a coarser quality photograph).
Further Work
Complete either Section C or Section D, whichever you have not yet attempted.
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