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Geometric Optics
We can see that light travels in straight lines by observing the behaviour of shadows.
A.
Shadows from a large source.
large
light
source
object
screen
penumbra:
umbra:
Ray diagrams are used to determine an image’s characteristics. Characteristics of an image are:
•
_____________: larger, smaller, or same size (magnification is an expression of relative size).
•
____________________: orientation. i.e., is it inverted or upright?
•
________________: real or virtual.
•
A real image is an image that can be projected onto a screen (light rays converge at one point).
Real images are always inverted.
•
A virtual image cannot be projected onto a screen. You must look through the lens, or in the
mirror to see a virtual image.
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B.
Shadow from a point source.
or
point light
source
object
screen
1.
In the book about Peter Pan by James Barrie, Peter lost his shadow and needs Wendy to help
find it. Peter is 1.25 m tall and is 0.80 m away from a light source. He then sees his shadow on a
screen which is 1.50 m away from him. What is the height of his shadow? (Hint - use similar
triangles.)
2.
A square object has an area of 0.66 m2. It is placed 1.00 m from a point source of light. Calculate
the area of the shadow produced on a screen 4.00 m from the light source.
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The Pinhole Camera
The pinhole camera, or the camera obscura, was invented in the 11th century by the Arabian scholar,
Alhazen. Its construction is shown below.
The light rays from various parts of the object travel in straight lines through the pinhole to form an
inverted image on the screen. A relation between the object and its image is given through similar
triangles.
Demo of a simple pinhole camera
You can make your own pinhole camera and project an image on your retina. Use a pin and a piece of
stiff paper with a small hole in it. Hold the pin between the paper and your eye. Place a few
centimetres between each. Look beyond the pin through the pin hole and you should see an image of
the pin inverted and enlarged.
4.
A pinhole camera shows a 30 mm image of a tree that is 60 m from the camera. The screen is
400 mm from the pinhole. Determine the height of the tree and the magnification factor.
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Spherical Thin Lenses
There are two types of lenses: concave and convex.
N.B., in practice, this works well only with thin lenses. Note that as a ray passes through a lens it is
diffracted twice. Once when entering the lens and once when leaving. We shall use a simplification.
Straight block of glass (angle
doesn’t change).
Actual
Simplification
Convex lens (a.k.a., converging or positive lens):
A lens that is thicker in the middle and thinner at the edges. They converge light rays to a focal point
(i.e., focus; foci is plural of focus). Therefore, light rays that diverge from the focal point will be
rendered parallel by the lens. The following is a cross sectional diagram of a convex lens.
Terms
optical centre (O):
The ___________ of the lens.
principal axis:
A line drawn through the __________ and the _________ centre.
focal point (F):
The point where all the light rays __________.
focal length (f):
The _____________ from the lens to the focal point. It is dependent on
the refractive index of the material and the shape of the lens.
chromatic aberration:
The unwanted spreading out of the colours. It occurs because each
frequency refracts differently through a given medium.
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Ray Diagrams
To draw the ray diagrams:
i.
draw the lens showing the principal axis and the principal focal points.
ii.
place a vertical arrow on the principal axis to indicate the position, size and attitude of the
object.
iii. draw two rays from the tip of the arrow to the lens. Where these two rays meet (real image) or
appear to diverge from (virtual image) is the position of the image. Any two of the following
three rays may be used: (Draw three as a check.)
• A ray coming in parallel to the principal axis emerges towards the focus.
• A ray coming through the focus emerges parallel to the principal axis.
• A ray passing through the optical centre is left unchanged.
Ray Diagrams for Convex (converging or positive) Lenses
1.
Object beyond 2F
Characteristics
attitude
2F
2.
F
F
2F
size
type
Object at 2F
Characteristics
attitude
2F
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F
F
5
2F
size
type
3.
Object between F and 2F
Characteristics
attitude
2F
4.
F
F
2F
size
type
Object at F
Characteristics
attitude
2F
5.
F
F
2F
size
type
Object inside F
Characteristics
attitude
2F
F
F
2F
size
type
Summary Chart for Convex Lens
1.
2.
3.
4.
5.
Position of Object
Position of Image
Beyond 2F
At 2F
Between F and 2F
At F
Inside F
Between F and 2F
At 2F
Beyond 2F
No image (at infinity)
Same side of lens as object (-)
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Characteristics of Image
Attitude Type
Size
NA
NA
NA
1.
Use a ray diagram to determine the size and position of an image projected by a 10.0 cm object,
30.0 cm from a thin positive lens with a 20.0 cm focal length. (Always indicate scale selected.)
Characteristics
di
2F
F
F
2F
hi
M
Ray Diagrams for Concave (diverging or negative) Lenses
E.g., the rays are bent outwards
The characteristics of images produced by a concave lens are always upright, smaller and virtual, and
in-between the lens and the object.
2.
A glowing object 10.0 cm tall is placed 22.0 cm from a concave lens. If the focal length is 12.0 cm
determine the object’s characteristics and dimensions. (Always indicate scale selected.)
Characteristics Dimensions
2F
F
F
2F
•
Real images, formed by convex lenses may be viewed either by looking directly through the lens or
by projecting the real image on a screen.
•
To see a virtual image, you must look directly into the lens.
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Thin Lens Formula
Where: (Any units may be used as long as they are consistent.)
do is the distance from the object to the mirror.
di is the distance from the image to the mirror.
f is the focal length of the mirror.
Sign Convention:
Distance (d)
real focal points or images
virtual focal points or images
Height (h)
upright images
inverted images
+
-
+
-
N.B., All real images are inverted and all virtual images are upright.
Magnification
The magnification of an object can be found from the following formulae:
or
Where: (Any units may be used as long as they are consistent.)
M is magnification
ho is the object’s height
hi is the image height
do is the object distance
di is the image distance
Therefore,
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3.
An object, 3.0 cm high, is 15 cm from a concave lens of 10.0 cm focal length. Determine the
characteristics and the dimensions of the image both graphically and algebraically. Complete the
chart.
Image Dimensions
Graphically
Algebraically
di
hi
M
Characteristics
4.
An object is 4.0 cm from a concave lens of 6.0 cm focal length. Locate its image. What kind of
image is formed? (Hint, ho is not needed.)
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Optics and the Eye
Optics originally referred to the study of vision and the eye. It is now meant to mean the study of all
phenomena related to light. We see objects when light from the object (either reflected or emitted)
enters our pupil, is bent by the lens and falls on the light sensitive tissue on the back of our eye called
the retina. The cells that compose this tissue, rods and cones, are sensitive to light. These cells convert
the light energy into electrical energy. The electrical signals are sent to the brain via the optic nerve.
The brain then interprets these electrical signals.
The eye
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Corrective lenses help our natural lenses to bend the light correctly to ensure it focuses on the retina.
Farsightedness can be corrected using a convex lens. Near-sightedness can be corrected using a
concave lens. The English scientist, Roger Bacon, is often credited with inventing eye glasses in the
13th century.
Normal Vision
image
object
lens
retina
___________ (hyperopia) Can’t see
near objects well.
Far-sighted corrected using a convex
lens
___________ (myopia) Can’t see far
objects well.
Near-sighted corrected using a
concave lens
You can tell if someone is near or farsighted by examining their glasses. Hold the glasses at arm’s
length and look at a distant object. The object will appear smaller if the lens is concave,
(nearsightedness) but larger if it is convex (farsightedness). Surgical techniques (e.g., laser surgery) have
been developed to compensate for many eye problems.
An eyeglass lens will form a virtual image at the person’s near-point where it can be seen clearly. The
eye is not focusing on the object; it is focusing on the virtual image of the object.
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Spherical Reflectors
There are two types of spherical reflectors we shall study.
concave:
Uses:
Reflecting telescope (invented by Newton because
of chromatic aberration in lens telescopes)
convex:
Uses:
principal axis
C
F
V
Terms
concave:
A mirror which causes parallel light rays to __________ on each other.
Therefore it may be called a ____________ mirror.
convex:
A mirror which causes parallel light rays to ___________, therefore it may be
called a ____________ mirror.
vertex (V):
The __________ of the mirror.
centre of curvature (C):
The ___________ of the sphere.
focal point (F):
____________ between the mirror and the centre of curvature. It is the place
where all the light rays meet.
focal length (f):
The ____________ from the mirror to the focal point.
principal axis:
A line drawn through the __________ and the __________ of curvature.
spherical aberration:
Rays reflecting off the far edges of the mirror do not reflect through the
principal focal point. To correct this, the mirror’s shape is changed from
spherical to parabolic. This problem is minor and for our purposes it will be
ignored.
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Ray Diagrams
An object in front of a mirror emits light in all directions, but only some of its light rays will reach the
mirror and be reflected. It is helpful to have a system for locating an image formed by rays reflected
form a curved surface. Any reflected ray follows the law of reflection but certain rays have easily
defined paths that make finding the image easier.
How to draw a ray diagram
i.
ii.
iii.
Draw the mirror showing the principal axis, the centre of curvature, and the principal
focus.
Place a vertical arrow on the principal axis to indicate the position, size and orientation of
the object.
Draw two rays from the tip of the arrow to the lens. Where these two rays meet (real
image) or appear to diverge from (virtual image) is the position of the image. Any two of
the following three rays may be used:
• The ray directly parallel to the principal axis will reflect through or appear to have
come from F.
• The ray directed through F will reflect parallel to the principal axis.
• A ray passing through the centre of curvature is reflected back along the same path.
An object may be at four general locations relative to the mirror. The following examples show how
to find the image and its characteristics for each location.
N.B., This requires a sharp pencil and a good straight edge. (Pens show up better on some grids, but
lots of erasing is often necessary.)
1.
Object is beyond C
Characteristics
attitude
C
2.
size
F
type
Object is at C
Characteristics
attitude
C
size
F
type
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3.
Object is between F and C
Characteristics
attitude
C
size
F
type
4.
Object is at F
Characteristics
attitude
C
size
F
type
5.
Object is between F and V
Characteristics
attitude
size
type
Summary Chart for Concave Mirrors
1.
2.
3.
4.
5.
Position of object
Position of image
Beyond C
At C
Between F and C
At F
Between F and V
Between C and F
At C
Beyond C
no image formed (at infinity)
behind the mirror
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Characteristics of image
Attitude
Type
Size
NA
NA
NA
Curved Mirror Equations
The characteristics of an image in a mirror can also be found algebraically with the following equation.
Where: (Any units may be used as long as they are consistent.)
do is the distance from the object to the mirror
di is the distance from the image to the mirror
f is the focal length of the mirror
Sign Convention:
Distance (d)
real focal points or images
virtual focal points or images
Height (h)
upright images
inverted images
+
-
+
-
N.B., All real images are inverted and all virtual images are upright.
Magnification
The magnification of an object can be found from the following formulae:
or
Where: (Any units may be used as long as they are consistent.)
M is magnification
ho is the object height
hi is the image height
do is the object distance
di is the image distance
Therefore,
The negative sign is needed due to sign convention.
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6.
A 2.0 cm tall candle, is placed 12 cm in front of a concave mirror with a focal length of 8.0 cm.
Use a scale diagram to determine its dimensions and characteristics.
Characteristics Dimensions
Graphical Method for Determination of an Image in Convex Mirrors
7.
By means of a scale diagram determine characteristics of an image 20 cm in height in a convex
mirror (f = 10 cm) if it is 25 cm from the mirror. (Leave room behind convex mirrors for the
image.) Convex mirrors will always give images that will be smaller and virtual.
Characteristics Dimensions
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Summary
Converging (positive)
Lens
Refraction
Mirror
Reflection
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Diverging (negative)