LIGHT: Geometric Optics Ch 23

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LIGHT Geometric Optics Ch 23
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Reflection off of smooth surfaces such as mirrors
or a calm body of water leads to a type of
reflection known as specular reflection.
(1)
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Reflection off of rough surfaces such as
clothing, paper, and the asphalt roadway leads to
a type of reflection known as diffuse
reflection.
(1)
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There are several interesting applications of
this distinction between specular and diffuse
reflection. One application pertains to the
relative difficulty of night driving on a dry
asphalt roadway compared to a wet asphalt
roadway.
(1)
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The Law of Reflection
In the diagram, the ray of light approaching the
mirror is known as the incident ray
The ray of light which leaves the mirror is known
as the reflected ray
At the point of incidence where the ray strikes
the mirror, a line can be drawn perpendicular to
the surface of the mirror this line is known as
a normal line
(1.)
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This law of reflection can be derived from
Fermat's principle.
Fermat's Principle Light follows the path of
least time
The path-length L from A to B is
Since the speed is constant, the minimum time
path is simply the minimum distance path. This
may be found by setting the derivative of L with
respect to x equal to zero.
(4)
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This derivation makes use of the calculus of
maximum-minimum determination, the derivative of
a square root, and the definitions of the
triangle trig functions.
(4)
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The normal line divides the angle between the
incident ray and the reflected ray into two equal
angles.
The angle between the incident ray and the normal
is known as the angle of incidence.
The angle between the reflected ray and the
normal is known as the angle of reflection.
(1.)
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Object distance Image distance
(1)
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Lets try an experiment!
(1)
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Image Formation in Plane Mirrors
An image is a position in space where all the
reflected light appears to diverge from.
Virtual images are images which are formed in
locations where light does not actually reach.
Whenever a mirror (whether a plane mirror or
otherwise) creates an image which is virtual, it
will be located behind the mirror where light
does not really pass.
(1)
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3. The image is said to be upright, as opposed to
inverted.
Besides the fact that plane mirror images are
virtual, there are several other characteristics
which are worth noting.
1. If you raise your left hand, you will notice
that the image raises its right hand. If you
raise your right hand, the image raises its left
hand. This is termed left-right reversal.
      
2. For plane mirrors, the object distance (often
represented by the symbol do) is equal to the
image distance (often represented by the symbol
di).
4. the dimensions of the image are the same as
the dimensions of the object.
(1)
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Images formed by plane mirrors are virtual,
upright, left-right reversed, the same distance
from the mirror as the object's distance, and the
same size as the object.
The ratio of the image dimensions to the object
dimensions is termed the magnification. Plane
mirrors produce images which have a magnification
of 1.
(1)
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• CURVED OR SPHERICAL MIRRORS
• There are two types
• Concave
• Convex

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Beginning a study of spherical mirrors demands
that you first become acquainted with some
terminology which will be periodically used.
Principal axis Center of Curvature (C) Vertex (A)
Focal Point (F) Radius of Curvature (R) Focal Length (f)
(1)
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The point in the center of sphere from which the
mirror was sliced is known as the center of
curvature
(1)
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The point on the mirror's surface where the
principal axis meets the mirror is known as the
vertex
The vertex is the geometric center of the mirror.
(1)
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Midway between the vertex and the center of
curvature is a point known as the focal point
(1)
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The distance from the vertex to the center of
curvature is known as the radius of curvature.
The radius of curvature is the radius of the
sphere from which the mirror was cut.
(1)
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Finally, the distance from the mirror to the
focal point is known as the focal length
Since the focal point is the midpoint of the line
segment adjoining the vertex and the center of
curvature, the focal length would be one-half the
radius of curvature.
(1)
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The focal point is the point in space at which
light incident towards the mirror and traveling
parallel to the principal axis will meet after
reflection.
(1)
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In our study of concave mirrors, we are going to
assume that all light-rays which strike a mirror
parallel to its principal axis (e.g., all rays
emanating from a distant object) are brought to a
focus at the same point. This is only an
approximation. It turns out that as rays from a
distant object depart further from the principal
axis of a concave mirror they are brought to a
focus ever closer to the mirror. This lack of
perfect focusing of a spherical mirror is called
spherical aberration. The approximation in which
we neglect spherical aberration is called the
paraxial approximation.3 Likewise, the study of
image formation under this approximation is known
as paraxial optics. This field of optics was
first systematically investigated by the famous
German mathematician Karl Friedrich Gauss in
1841. (3)
(4)
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When the problematic portion of the mirror is
disabled so that it can no longer focus (or
mis-focus) light, the image appears more focused
(1)
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The only type of mirror which does not suffer
from spherical aberration is a parabolic mirror
A ray traveling parallel to the principal axis of
a parabolic mirror is brought to a focus at the
same point, no matter how far the ray is from the
axis. Since the path of a light-ray is completely
reversible, it follows that a light source placed
at the focus of a parabolic mirror yields a
perfectly parallel beam of light, after the light
has reflected off the surface of the mirror.
A 40' deep focus parabolic mirror generating 88KW
worth of steam power
(2)
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The receiving dishes of radio telescopes are
generally parabolic they reflect the incoming
radio waves from (very) distant astronomical
sources and bring them to a focus at a single
point, where a detector is placed.
A car headlight consists of a light-bulb placed
at the focus of a parabolic reflector. The use of
a parabolic reflector enables the headlight to
cast a very straight beam of light ahead of the
car.
(1)
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For a concave mirror, the normal at the point of
incidence on the mirror surface is a line which
extends through the center of curvature. Once the
normal is drawn the angle of incidence can be
measured and the reflected ray can be drawn with
the same angle
(1)
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Ray diagrams can be used to determine the image
location, size, orientation and type of image
formed of objects when placed at a given location
in front of a concave mirror.
(1)
From the geometry of the spherical mirror, note
that the focal length is half the radius of
curvature
(4)
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The mirror equation expresses the quantitative
relationship between the object distance (do),
the image distance (di), and the focal length
(f).
The Magnification equation relates the ratio of
the image distance and object distance to the
ratio of the image height (hi) and object height
(ho).
These two equations can be combined to yield
information about the image distance and image
height if the object distance, object height, and
focal length are known.
(1)
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As in the case of lenses, the cartesian sign
convention is used here, and that is the origin
of the negative sign above. The radius r for a
concave mirror is a negative quantity (going left
from the surface), and this gives a positive
focal length, implying convergence.
(4)
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If the object is outside the focal length, a
concave mirror will form a real, inverted image.
                                               
                                                  
                                                
(4)
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If an object is placed inside the focal length of
a concave mirror, and enlarged virtual and erect
image will be formed behind the mirror.
(4)
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Diverging rays
Converging rays
(1)
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When the object is located beyond the center of
curvature (C), the image is located between the
center of curvature (C) and the focal point (F).
On the other hand, when the object is located
between the center of curvature (C) and the focal
point (F), the image is located beyond the center
of curvature (C).
(1)
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 The best means of summarizing this relationship
is to divide the possible object locations into
five general areas or points
Case 1 the object is located beyond the center
of curvature (C)
Case 2 the object is located at the center of
curvature (C)
Case 3 the object is located between the center
of curvature (C) and the focal point (F)
Case 4 the object is located at the focal point
(F)
Case 5 the object is located in front of the
focal point (F)
(1)
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Case 1 The object is located beyond C
Case 2 The object is located at C
Case 3 The object is located between C and F
(1)
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Case 4 The object is located at F
Case 5 The object is located in front of F
(1)
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Nine different object locations are drawn and
labeled with a number the corresponding image
locations are drawn in blue and labeled with the
identical number.
(1)
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A convex mirror forms a virtual image.
The virtual image that is formed will appear
smaller and closer to the mirror than the object
(4)
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The center of that original sphere is known as
the center of curvature (C) and the line which
passes from the mirror's surface through the
sphere's center is known as the principal axis.
The mirror has a focal point (F) which is located
along the principal axis, midway between the
mirror's surface and the center of curvature.
Note that the center of curvature and the focal
point are located on the side of the mirror
opposite the object - behind the mirror. Since
the focal point is located behind the convex
mirror, such a mirror is said to have a negative
focal length value.
(1)
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A convex mirror is sometimes referred to as a
diverging mirror due to its ability to take light
from a point and diverge it.
After reflection, the light rays diverge
subsequently they will never intersect on the
object side of the mirror. For this reason,
convex mirrors produce virtual images which are
located somewhere behind the mirror.
(1)
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Any incident ray traveling parallel to the
principal axis on the way to a convex mirror will
reflect in a manner that its extension will pass
through the focal point.
A
Any incident ray traveling towards a convex
mirror such that its extension passes through the
focal point will reflect and travel parallel to
the principal axis.
B

• Ray Diagrams - Convex Mirrors

(1)
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Image Characteristics for Convex Mirrors
(a) located behind
the convex mirror (b) a virtual image (c)
an upright image (d) reduced in size
Unlike concave mirrors, convex mirrors always
produce images which share these characteristics.
The location of the object does not effect the
characteristics of the image.
(1)
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Another characteristic of the images of objects
formed by convex mirrors pertains to how a
variation in object distance effects the image
distance and size.
(1)
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The Mirror Equation - Convex Mirrors
The Magnification equation
(1)

• Sample Problem

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Lenses that cause an incoming plane wave to bend
towards the axis through its middle are called
converging or convex lenses.
Concave lenses, on the other hand, are thicker at
their edges than in the middle they cause an
incoming plane wave to bend away from its central
axis and are hence also known as a diverging
lenses.
(2.)
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1. http//www.glenbrook.k12.il.us/gbssci/phys/Clas
s/refln/u13l1c.html
2. http//www.sparknotes.com/physics/optics/geom/s
ection3.rhtml
3. http//farside.ph.utexas.edu/teaching/302l/lect
ures/node114.html
4. http//hyperphysics.phy-astr.gsu.edu/Hbase/hfra
me.html