Lab 9: Light Focusing by Lenses
Objectives: To investigate the formation of images by converging and diverging lenses;
to categorize images as Real or Virtual, Upright or Inverted, Reduced or Enlarged;
to estimate the focal length of lenses by the images they form of a distant object;
to relate the focal length to the radius of curvature of the lens surfaces.
To predict image distances and characteristics using the thin-lens focus equation;
to see how the effective focal length of 2 lenses together relates to individual f ’s;
to see that 2 lenses far apart cannot be described by a single effective lens;
to design and assemble an open-frame telescope or microscope.
Apparatus:
3 different lenses, lens holder stand, rectangular glass/plastic plate, glass/plastic prism
Equipment: optical bench with lens holders, illuminated object lamp, image screens
Background: Light from any very distant source “object” arrives as essentially parallel rays with flat
wave-fronts. A converging lens deflects this light so that it converges to a Focal point F
at some point behind the lens. Usually we want all the light to focus to the same point,
which forms an “image” where the rays converge, so most lenses are symmetric around
an axis of rotation. If the lens’ symmetry axis (the “optic axis”) is along the original
parallel rays, the Focal point will also occur on the optic axis. Then, the location of the
Focal point relative to the lens “center” is called the lens’ focal length f . Since the
light really does converge there, the image formed is called a Real image.
Notice that light thru the lens far from the axis (at large y) must be deflected sharply, or
y
it won’t cross at F ; it needs   tan  
. For small-angle deflections by a prism,
f
  (n  1) ; the apex angle  also widens at large y , and the lens curvature /y is
nearly uniform. In uniform curvature, the surface curves as a sphere around its radius.
Low curvature is nearly flat around a large radius, but a small radius curves tightly:
 1
 n 1

… so each curved surface in a lens has “focusing power”
. A ray

y R
y
R
traveling with small ray-to-axis angles in a thin lens will have almost the same y at
each surface; this means we can add their focusing powers (not just their deflections):
 1
n
1 n 1 1 n
1
1 
 .


… if material no is outside the lens,
 ( lens  1) 
f
R1
R2
f
no
 R1 R2 
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Diverging lenses deflect light to diverge from their axis, appearing to emanate from
some point along it in front of the lens. With their focus on the negative side of the
optical axis, diverging lenses have
negative focal lengths. Since rays
never really converged there, it is
a Virtual image. The lens-maker
equation shows that diverging
lenses are essentially concave.
If the light rays are only small angles from parallel with the optic axis (“paraxial” rays),
and some lenses are very close together, their focusing powers add to act like one lens.
1
1
1
1
The effective focal length of this set of lenses would then be
…



f
f1 f 2 f 3
in this approximation, the order of the lenses (1st, 2nd, 3rd ) does not matter.
What if light is diverging from an object on the optic axis that is NOT “very far away”?
y
Its rays strike the lens at y from the lens center, at incident angle  ob  tan  ob 
.
d ob
y
They’ll be deflected thru almost the same angle  
as the parallel rays at that y ,
f
so they will leave the lens in a direction θob less steep than the parallel rays there. In
y
y
y


 tan  im   im .
small-angle approximation that angle subtraction is:
f d ob d im
1
1
1


The thin-lens focus equation
just cancels the arbitrary y’s .
f d ob d im
Notice that an object only 1f away from a converging lens has infinite image distance;
for objects closer than that, their rays are still diverging even after the lens deflected
them (so the rays only seem to cross after tracing-back to their virtual image).
Interesting objects have height hob from the optic axis; 3 rays from their tip are easy to
trace thru a thin lens, based on the focal length of that lens. Ray 1 goes undeflected thru
the center of the lens (the 2 sides of the lens are parallel there.) Ray 2 starts parallel to
the optic axis, and is deflected thru the lens focal point (like rays from a distant object).
Ray 3 starts toward the optic axis at –F and is deflected to become parallel to the optic
axis (like a zero-height object at dob = f ). These rays all intersect (focus) at the same
point, at some height him from the optic axis at the image distance.
An optical setup’s Lateral Magnification M = him/hob compares the image height to
the object height. M larger than 1 means the image is enlarged (bigger), smaller than 1
means the image is reduced in size; negative M images are inverted (upside-down).
For a single lens, Lateral Magnification should be –dim/dob … (see ray 1’s triangles).
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Notice :(1) rays may be drawn as if all the deflection occurs at one surface (mid-lens).
(2) the lens is presumed to have large enough diameter that rays do not “miss” it.
(3) lines that are virtual ray extensions (trace-backs or forward) may be dashed.
(4) after converging to form an image, rays diverge from it … so it acts like an object.
A second lens in sequence merely uses the image from the first lens as its object;
remember that you need to measure that object distance from the 2nd lens!
Activity 1. Estimates Inspect each lens to estimate the center that its surfaces curve around; hold it
close to one eye to estimate the “angular size factor” with the lens / without the lens.
Deduce whether the object’s light rays were deflected toward the axis (converging) or
away from the optic axis by the lens. Finally, hold them near the table “far” from the
ceiling lights; vary their distance from the table-top – can you form an image of the
ceiling light on the table-top? If so, then the distance from lens to image is a rough
estimate of the lens’ focal length. For the diverging lens, you’ll have to look thru it at a
distant object, and “guess” (using your primate stereo vision) how far (behind the lens)
its image appears to be … where would you reach to, to grab it … don’t confuse
location with the small image size!
lens
R1 radius of R2 radius
angular
converging
curvature
of curvature magnification /diverging
f estimate
concave
thick convex
thin convex
Now that you have a focal length estimate, and radii of curvature estimates, deduce the
refractive index for the glass that the concave lens and one other lens is made from.
concave n ________________
_______ convex n  ______________
Comment: are your deduced refractive indexes plausible for glass?
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Activity 2. Image characteristics Return to a convex lens: form an image from a very distant object.
Is that image Real, or Virtual? Inverted, or Upright? Made Larger, or made Smaller?
Use the same choices to describe the images made by this lens for an object that is
closer than 2f from the lens, but farther than 1f … you might want to set the lens on the
optical rail. Then describe the image this lens forms, for an object that is closer than 1f.
Finally, describe the images formed by the concave lens for various object distances.
Lens type
very distant object
closer than 2f
closer than f
concave
|
|
|
|
|
|
convex
|
|
|
|
|
|
Activity 3. Computing Focal Length from Object Distance & Image Distance
A. Thin Convex Place the lighted object box to touch one end of the meterstick rail. Set
the thin convex lens in a lens holder between 1f and 2f from the object. Slide the screen
along the rail until an image forms clearly. Measure the object distance from the lens
center, and the image distance from the lens center, and record in the table at the top of
the next page. Also measure the image height or width (the size of some image part),
including sign, and the same dimension on the object (use a caliper for small portions).
Then slide the lens toward (or away from) the screen until another image forms clearly.
For this image also, measure the object and image distance and height (including sign).
Use the thin-lens focus equation in each instance to obtain two computed focal lengths.
Use the definition of Lateral Magnification to compute M; compare with –dim/dob .
B. Thick Convex Repeat the object and image measurements (distance and height) for
the thick convex lens – first where the lens is between 1f and 2f from the object, then
when farther than 2f from the object. Obtain two values for this lens’s focal length; if
they agree fairly well, average them. Also compute M, and –dim/dob .
C. Combined Concave + Thick Convex To get a real image from the concave lens, you’ll
need to combine it with the thick convex lens – preferably in the same lens-holder so
they are very close together and well-aligned. Form a Real Inverted Large image and
measure object and image distance and height. Then form a Real Inverted Small image
and measure distances and heights. Use this data in the thin-lens focus equation to
obtain 2 focal length values for the combined lenses; if they agree, average them.
Also compute M, and –dim/dob , for this lens combination.
Finally, use the lens-power addition equation, with the thick convex lens’ focal length,
to calculate the focal length for the concave lens alone.
… is it supposed to be negative?
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Lens
dob
dim
him
hob
f
M
–dim/dob
thin vex
thin vex
thick vex
thick vex
combine
combine
Sample computation of f :
Sample computation of M :
Thin convex f (avg) ____________ thick convex f (avg) _____________
combined lens f (avg) ___________
show computation of concave f :
concave lens f ________________
Activity 4. Microscope
Making a microscope with 2x – 6x objective (lateral magnification), and look at the
image with the other convex lens as “eyepiece”. Measure the lateral magnification of
the objective by comparing image size to object height (is it inverted?): ________
There is an apparent angular magnification due to looking at the first image closely,
through the “magnifying glass”, compared with looking at it with your naked eye.
It is tricky to estimate that angular magnification, and it depends on how well you’re
using the eyepiece … it is best to have your eye very close to the eyepiece; to do this,
the eyepiece will need to be 1 focal length from the first image (or less): ________
Compute the overall magnification of your microscope: m  M = ____________.
Sketch the ray diagram of this set-up below:
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Lab 9 Summary
Name :_______________________
Lab Partner : _____________
1. Under what circumstance does a converging lens form a virtual image?
under what circumstances does a converging lens form a real image?
Specifically describe the situation in which a diverging lens can form a real image.
(hint: didn’t we see a real image with the combined lenses?)
2. With a converging lens, using a very distant object, why is the image at the focal length?
With that converging lens, how does one obtain a very distant image?
A very distant real image would be huge, and hence quite dim.
Why is the very distant virtual image not extremely dim?
3. Modern car rear-view mirrors on each side say “objects in mirror are closer than they appear”
– are their images farther than the object? _____ (it might help to draw a ray diagram)
What do they really mean by that statement?
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