Optics-Image Formation
Pre-Lab Question
UM Physics Demo Lab 07/2013
What physical process changes the direction of a ray of light as it passes through a
thin lens?
EXPLORATION
Exploration Materials
40-Watt “candle” lamp on wooden base
Power strip
Mesh stand for candle lamp
Cookie sheet pinhole projector
White cardboard projection screen
Hand-held magnifying glass
2 Meter sticks
Clear plastic ruler
4 Large binder clips (support legs for pinhole projector and screen)
Small rubber band
Definitions
A converging lens is thicker in the center than at its edges.
A diverging lens is thinner at the center than at its edges.
You will find diagrams illustrating converging and diverging lenses and
mirrors at the end of this worksheet.
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Pinhole Optics
1. In the diagram below, the arrow represents an object (candle lamp) at some
distance from a small hole in a screen, referred to as a “pinhole”. Behind the pinhole
is a second screen where light passing through the pinhole will be intercepted and
displayed as an image (cardboard sheet). The pinhole is represented as a small gap
in an opaque pinhole screen (aluminum foil baking pan). Using your clear plastic
ruler, trace a ray of light (straight line) from the tip of the arrow (A) through the
pinhole until it intersects the image screen. Repeat this for light rays originating
from the bottom of the arrow (C) and the middle of the arrow (B). Label the points
corresponding to A, B and C on the image screen.
Based on this “ray tracing” analysis, draw the arrow head corresponding to the top of
the object (A) for the image you have constructed on the image screen.
Can a pinhole produce an image of the object? Will the image be upright (“rightside-up”) or inverted (“up-side-down”)? How do you know this?
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2. Set up your pinhole projector (aluminum foil pan) image screen (cardboard
screen) and candle lamp to project an image of the lamp onto the screen as
shown in the diagram for part 1. A wire mesh pedestal is provided to raise
your lamp to the proper height and the large binder clips serve as the support
legs for your projector and screen. Before you turn on your lamp (using
the switch on the power strip) observe the orientation of the wire lamp
filament so you can determine if the images produced by the three pinholes
(small, medium and large) in the projector plate are upright or inverted. Now
turn on the lamp and observe the three images.
A) Are the images upright or inverted?
B) Which pinhole (small, medium, large) produces the brightest image?
C) Which pinhole (small, medium, large) produces the clearest (best focused)
image?
D) Which pinhole (small, medium, large) produces the dimmest image?
E) Which pinhole (small, medium, large) produces the most poorly focused
image?
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F) State the relationship between the quality of focus and brightness for the
image produced by a pinhole.
3.
Now trace three rays from top of the arrow “object” (point A) through the
pin hole. One ray should pass through the top edge of the pinhole, one
through the center of the pinhole and one through the bottom edge of the
pinhole.
A) Do all the rays you have drawn intersect the image screen at the same place?
Can the pinhole produce an image of the point at the tip of the object (arrow) at one
exact point on the image screen?
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B) An ideal pinhole is a geometric point with no physical size. Real pinholes must
have some size (diameter) associated with them since no light can pass through a
true geometric point, which has zero area. Based on your ray tracing above, how
does the diameter of the pinhole affect the quality of the focus for a real pinhole?
C) Is this result (B) consistent with the pinhole images you observe with your
projector? Explain the relationship between pinhole size, image brightness and
quality of focus.
4. The depth of field is defined as the range of distances from a lens or pinhole
where all objects are brought to a clear focus at the image screen (focal
plane). Move the lamp back and forth and observe the clarity of focus for the
images projected by the pinholes on the screen.
A) Based on your observations, how would you characterize the depth of field for a
pinhole (large or small)? Explain how your observations support this
characterization.
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5. The magnification of a lens or pinhole is defined as the ratio of the image
height to the object height, which by similar triangles can easily be shown to
be equivalent to the ratio of the image distance (distance from pinhole to
the image screen) and the object distance (distance from the pinhole to the
object-in this case the lamp filament). In optics this is written as M= -(i/o)
where the minus sign is included to signify that the image is inverted (upside-down) relative to the object.
Start with the screen and the lamp at equal distances (15 cm ~ 6”) from the
pinhole screen.
A) How do the sizes of the images compare with the size of the glowing filament in
the lamp (the object)?
B) Decrease the object distance o by moving the lamp toward the pinhole screen.
What happens to the size of the images? How is this consistent with the
magnification equation?
C) Now increase the object distance o by moving the lamp away from the pinhole
screen. What happens to the size of the images? How is this consistent with the
magnification equation?
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Optics of a Thin Converging Lens
Real Images and Focal Length
1. Focus an image of a distant lamp onto the wall with the magnifying lens and
measure the distance from the image (wall) to the center of the lens with a
meter stick. Parallel light rays from the distant lamp are being brought to a
focus at this distance, which we will call the focal length of the lens.
Record the measured focal length below.
Magnifying Lens
Focal Length (cm)
2. After discussing it with your group, predict what you will observe if you shine
a lamp located at the focal point of the lens through the lens onto a distant
wall.
3. Perform the experiment described in (2): place the candle lamp at the focal
point of the lens and shine it on a distant wall. What do you observe?
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4. Give an example of a practical application of the optical effect you
demonstrated in (3). (Hint: We live near the world’s largest fresh water
ocean, the Great Lakes. This technique had been used since the 19th century
to guard the safety of ships on the Great Lakes and is still in limited use
today).
5. A ray tracing analysis similar to the one you did for the pinhole yields the
following relationship (known as the Thin Lens Equation) between the
image distance i, the object distance o and the focal length f of a thin lens:
1/i + 1/o = 1/f
A) Lay two meter sticks end to end. Place the candle lamp at the zero of the first
meter stick and the cardboard screen at a distance of 1.5 meters from the candle
lamp. Starting with the magnifying lens at the screen, move it toward the lamp
until you project an image of the lamp onto the screen. Record the image and object
distances below. Now calculate the focal length of the lens using the thin lens
equation and compare it to the focal length you have already measured. Show
your calculation.
Image
Distance (cm)
Object
Distance (cm)
Calculated Focal
Length (cm)
Previously Measured
Focal Length (cm)
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B) Now continue to move the lens toward the lamp until you find a second point
where you can project an image of the lamp on the screen. Record the image and
object distances below and again calculate the focal length.
Image
Distance (cm)
Object
Distance (cm)
Calculated Focal
Length (cm)
Previously Measured
Focal Length (cm)
C) How do the image and object distances for the second image you found (lens
near the lamp) compare with the image and object distances you found for the first
image (near the screen)? Does the thin lens equation predict this? (Hint: What
happens to the focal length in the thin lens equation if the image and object
distances are interchanged?)
D) The f/number of a lens is defined as the ratio of its focal length to its diameter.
Measure the diameter of your lens in centimeters and calculate the f/number of the
magnifying below (show your calculation):
f/number = Focal Length/Diameter = f/______
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Virtual Images
1. Mark the location of the focal length of the magnifying lens on a meter stick
by putting a small rubber band around it. Lay a printed page on the table and
stand the meter stick up with the zero end of the meter stick on the page so
you can measure the distance of the magnifying lens from the page as you
move the lens up and down.
A) View the printing through the magnifying glass as you move the lens toward and
away from the page. Compare the quality of focus for the image you observe for
distances less than the focal length to the quality of focus for distances greater
than the focal length. For what range of distances do you see a well-focused,
magnified image of the printing?
B) Is the image upright or inverted?
C) The image distance for this virtual image is defined as negative, since the
image appears to be on the same side of the lens as the object. Does the
magnification equation M= -(i/o) predict the correct orientation for the image
(upright=positive magnification, inverted=negative magnification)? How?
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Everyday Applications
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Household mirrors (plane and magnifying)
Lighthouses
Indoor lighting
Flashlights (converging mirrors)
Slide projectors
Reading glasses
Cameras
APPLICATION
Materials
Desk lamp with 100 watt soft white bulb (“slide projector” light source)
Transparency-film “slide”
Hand-held magnifying glass
Meter stick
Reading glasses
1. The next experiment uses the desk lamp with a 100 watt light bulb as
a bright light source. The lamp will get very hot in use, so please be
careful to avoid touching the hot steel lamp shade and turn the lamp
off when not in use.
A) Using the 100 watt desk lamp as a light source, project an image of your
transparency film slide onto a distant wall with your magnifying lens. Sketch the
arrangement of the light, slide and lens below. At what specific distance must
the slide be from the lens to be projected clearly onto the wall? Is the image of
the slide upright or inverted?
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2. As you did for your magnifying lens, focus an image of a distant lamp onto
the wall with the reading glasses and measure the distance from the image
(wall) to the center of the lens with a meter stick. Record this measured
focal length below.
Reading Glasses
Focal Length (cm)
Opticians define the “Magnifying Power” of a lens in units of Diopters:
Magnifying Power (Diopters) = 1/(focal length in meters).
Note that this “Magnifying Power” is different than the magnification since it does
not depend on image or object distances and in fact is just an alternate way of
specifying the focal length of a lens. It has the advantage that magnifying power
increases with decreasing focal length and therefore correctly predicts a stronger
correction to vision with shorter focal length lenses.
A) Using your measured focal length, calculate the magnifying power of the reading
glasses and compare it to the number printed on the glasses. Show your
calculation and be sure to convert the measured focal length to meters! How well
does your measured magnifying power compare with the value printed on the
glasses?
Calculated Reading
Magnifying Power
For Reading
Glasses
(Diopters)
_______________
Labeled
Magnifying
Power
For Reading
Glasses
(Diopters)
_____________
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Challenge Work:
1. The top diagram shows a typical camera. The shutter, which opens and
closes the lens to light from the outside, is not shown. The bottom diagram
depicts two nearly identical cameras. The only difference between the two
bottom cameras is that their lenses differ in f/number. Since they both have
the same focal length f, the difference between the two cameras is that the
f/3 lens has a larger diameter than the f/22 lens. For your answers refer to
the cameras as the f/3 camera and the f/22 camera.
A) Which has the greater depth of field? (Hint: Which lens is more like a
pinhole?)
B) Which camera allows more light to enter and expose the film when the
shutter is open?
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C) The exposure time is defined as the amount of time that the shutter of the
camera remains open to expose the film to light. Based on your answer to (B) and
assuming the same available light, which camera can photograph the object with the
shortest exposure time?
D) Photographers refer to lenses with low f/numbers such as the f/3 lens as “fast”
lenses. Explain why this makes sense in light of your answers to (B-C) above.
2. The focal length for a pinhole is defined as the distance from the pinhole to
the image screen. Calculate the f/number for a 1 millimeter diameter pinhole
located a distance of 5 centimeters from an image screen. Show your work
and be sure to use consistent distance units (focal length and diameter both
expressed either in millimeters or centimeters).
f/number = Focal Length /Diameter = f/______
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Summary:
1. The Law of Reflection states that the angle of reflection is equal to the
angle of incidence for a reflected ray of light. The angles are defined with
respect to the direction perpendicular to the reflecting surface called the
normal.
2. Images where the light actually emanates from the location of the image
are called real images. Images where the light only appears to emanate
from the image location but does not actually do so are called virtual images.
The image of a lamp filament projected by a thin converging lens is real. The
image in a plane mirror is virtual.
3. Ray tracing demonstrates that pinholes can produce an inverted, real image.
Small pinholes produce well focused, dim images. Large pinholes produce
bright, poorly focused images. The depth of field for a point pinhole is
infinite, and real pinholes have a very large depth of field. The
f/number of a pinhole or lens is defined as the ratio of its focal length to
its diameter. The focal length of a pinhole is the distance from the pinhole
to the image screen. A true pinhole (geometric point) therefore has an
“infinite” f/number (f/∞) (owing to a zero diameter) and real pinholes have
very large f/numbers.
4. Convex lenses can focus parallel light rays to a common focal point and can
therefore produce real images. Concave lenses always produce diverging
rays and can only produce virtual images.
5. The speed of light in a material such as glass, water or plastic is slower
than it is in vacuum. The ratio of (speed of light in vacuum)/(speed of light
in matter) is called the refractive index of the material.
6. The direction of a light ray changes as it enters a material with a higher
refractive index such that its angle with the normal decreases. This is
how lenses change the direction of light rays and is described by Snell’s
Law.
7. The Thin Lens Equation relates the image distance (i), object distance
(o) and focal length (f) for a thin lens and is stated as: 1/i + 1/o = 1/f.
This equation predicts that light from very distant objects (0→∞) will
converge to form a real image at the focal point of a converging lens and that
interchanging the image and object distances will again result in a focused
image. It also predicts that an object placed at the focal point of the lens
(o=f) will project an image “to infinity” (1/i→0 so i→∞), that is, to a very
distant screen (slide projector). Real images produced by a thin converging
lens are inverted (“upside-down”) Objects viewed with image distances less
than the focal length of the lens will produce an upright (“right-side-up”)
virtual image of the object—this how a converging lens is used as a
“magnifying glass”. The thin lens equation also works for diverging lenses
which have a negative focal length.
8. The magnification for any thin lens or pinhole is defined as –(i/o) where the
minus sign predicts that real images are inverted and virtual images
(which have negative image distances) are upright.
9. The speed of light, and hence the index of refraction, is different in optical
materials such as glass or plastic for different wavelengths of light. This
effect is called dispersion. Blue light (short wavelength) is refracted (bent
toward the normal) more than red light (long wavelength). This is how a
prism separates white light into its component colors.
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Reflection
A wave is reflected when it is incident upon a material that redirects it outward.
Reflected waves are redirected according to the law of reflection: the angle of
incidence is equal to the angle of reflection. The angles of incidence and reflection
are defined with respect to the direction normal (perpendicular) to the mirror surface
as shown below.
The law of reflection holds for each of type of mirror, but the result it produces is
different for each. The plane mirror creates an actual-size virtual image of the object
which appears to be behind the mirror. Concave mirrors make incident plane waves
diverge, and convex mirrors cause incident plane waves to converge. Both of these
contribute to the distorted images of faces and bodies you see in carnival mirrors
which often have both convex and concave regions on the same mirror.
Figure 2: Plane Mirror
Figure 3: Concave (Converging) Mirror
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Figure 4: Convex (Diverging) Mirror
Figure 5: Convex (Converging) Lens
Figure 6: Concave (Diverging) Lens
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