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Lecture 24
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Friday, November 13, 15
Reminder: Snell’s law
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Mirrors
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Curved Mirrors (1)
§ When light is reflected from the surface of a curved
mirror, the light rays follow the law of reflection at each
point on the surface
§ However, unlike the plane mirror, the surface of a curved
mirror is not flat
§ Thus light rays that are parallel before they strike the
mirror are reflected in different directions depending on
the part of the mirror that they strike
§ The light rays can be focused or made to diverge
Soar Telescope
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Friday, November 13, 15
Concave Spherical Mirrors (1)
§ We can abstract this sphere to a two-dimensional
semicircle
§ The optical axis of the mirror is a line through the
center of the sphere, represented in this drawing
by a horizontal dashed line
§ A horizontal light ray above the optical
axis is incident on the surface of the mirror
§ At the point the light ray strikes the mirror, the law of
reflection applies
• θi = θr
§ The normal to the surface is a radius line that points to
the center of the sphere marked as C
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Concave Spherical Mirror (2)
§ Now let us suppose that we have many
horizontal light rays incident on this
spherical mirror as shown
§ Each light ray obeys the law of reflection
at each point
§ All the reflected rays cross the optical
axis at the same point called the focal
point F
§ Point F is half way between point C and
the surface of the mirror
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Concave Spherical Mirror (3)
§ C is located at the center of the sphere so the
distance of C from the surface of the mirror is just
the radius of the mirror, R
§ Thus the focal length, f, of a spherical mirror is
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Forming an Image with a Concave Mirror
Start with convex
mirror and optical
axis
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Friday, November 13, 15
Forming an Image with a Concave Mirror
Start with convex
mirror and optical
axis
Add object to be
imaged
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Friday, November 13, 15
Forming an Image with a Concave Mirror
Start with convex
mirror and optical
axis
Add object to be
imaged
Draw focal length
and radius of mirror
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Friday, November 13, 15
Forming an Image with a Concave Mirror
Start with convex
mirror and optical
axis
Add object to be
imaged
Draw focal length
and radius of mirror
Object has height
ho and distance do
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Friday, November 13, 15
Forming an Image with a Concave Mirror
Start with convex
mirror and optical
axis
Add object to be
imaged
Draw focal length
and radius of mirror
Object has height
ho and distance do
Draw first ray through
center of circle,
reflecting on itself
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Friday, November 13, 15
Forming an Image with a Concave Mirror
Start with convex
mirror and optical
axis
Draw second ray
parallel to the
optical axis
Add object to be
imaged
Draw focal length
and radius of mirror
Object has height
ho and distance do
Draw first ray through
center of circle,
reflecting on itself
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Friday, November 13, 15
Forming an Image with a Concave Mirror
Start with convex
mirror and optical
axis
Add object to be
imaged
Draw focal length
and radius of mirror
Draw second ray
parallel to the
optical axis
Reflect the ray
through the focal
point
Object has height
ho and distance do
Draw first ray through
center of circle,
reflecting on itself
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Friday, November 13, 15
Forming an Image with a Concave Mirror
Start with convex
mirror and optical
axis
Add object to be
imaged
Draw focal length
and radius of mirror
Object has height
ho and distance do
Draw second ray
parallel to the
optical axis
Reflect the ray
through the focal
point
Image is formed
at the intersection
of the two rays
Draw first ray through
center of circle,
reflecting on itself
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Friday, November 13, 15
Forming an Image with a Concave Mirror
Start with convex
mirror and optical
axis
Add object to be
imaged
Draw focal length
and radius of mirror
Object has height
ho and distance do
Draw first ray through
center of circle,
reflecting on itself
Draw second ray
parallel to the
optical axis
Reflect the ray
through the focal
point
Image is formed
at the intersection
of the two rays
Image has height hi
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Friday, November 13, 15
Forming an Image with a Concave Mirror
Start with convex
mirror and optical
axis
Add object to be
imaged
Draw focal length
and radius of mirror
Object has height
ho and distance do
Draw first ray through
center of circle,
reflecting on itself
Draw second ray
parallel to the
optical axis
Reflect the ray
through the focal
point
Image is formed
at the intersection
of the two rays
Image has height hi
Image has distance di
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Friday, November 13, 15
Image with Concave Mirror
§ The reconstruction of this special case produces a real image
• Defined by the fact that the image is on the same side of the mirror as the
object
• Not behind the mirror
§ The image has a height hi located a distance di from the surface of
the mirror on the same side of the mirror as the object with height
ho and distance do from the mirror
§ The image is inverted and reduced in size relative to the object
that produced the image
§ The image is called real when you are able to place a screen at the
image location and obtain a sharp projection of the image on the
screen at that point
§ For a virtual image, it is impossible to place a screen at the location
of the image
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Concave Mirror, do < f (2)
§ To determine the location of the image, we must
extrapolate the reflected rays to the other side of the
mirror
§ These two rays intersect a distance di from the surface of
the mirror producing an image with height hi
C
F
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Conventions for Mirror Distances (1)
§ In this case, we have the image formed on the opposite side of
the mirror from the object, a virtual image
§ To an observer, the image appears to be behind the mirror
§ The image is upright and larger than the object. These results
are very different than those for do > f
§ To treat all possible cases for convex mirrors, we must define
some conventions for distances and heights
§ We define
• Distances on the same side of the mirror as the object to be
positive
• Distances on the far side of the mirror from the object to be
negative
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Conventions for Mirror Distances (2)
§ Thus f and do are positive for concave mirrors
§ In the case of real images, di is positive
§ In cases where the image is virtual, we define di to be
negative
§ If the image is upright, then hi is positive and if the image is
inverted, hi is negative
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The mirror Equation
§ Using these sign conventions we can express the mirror
equation in terms of the object distance, do, and the image
distance, di, and the focal length f of the mirror
§ The magnification m of the mirror is defined to be
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Possible Cases for the Mirror Equation
We can summarize the image characteristics of concave mirrors
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Mirror Equation and Concave Mirrors
d>2f
(real, inverted, reduced)
d<f
(virtual, upright
enlarged)
d<f
(real, upright,
image at infinity)
f<d<2f
(real, inverted,
enlarged)
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Spherical Aberration
§ The equations we have derived for spherical mirrors apply only to
light rays that are close to the optical axis
§ If the light rays are far from the optical axis, they will not be
focused through the focal point of the mirror
§ Thus we will see a distorted image
§ In the drawing several light rays are
incident on a spherical concave mirror
§ You can see that the rays far from the
optical axis are reflected such that they
cross the optical axis closer to the mirror
than the focal point
§ As the rays approach the optical axis, they
are reflected through points closer and
closer to the focal point
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Parabolic Mirror
§ Parabolic mirrors have a surface that reflects light to the
focal point from anywhere on the mirror
§ Thus the full size of the mirror can be used to collect light and
form images
§ In the drawing to the right, horizontal light rays are incident on a
parabolic mirror
§ All rays are reflected through the focal
point of the mirror
§ Parabolic mirrors are more difficult to
produce than spherical mirrors and are
accordingly more expensive
§ Most large telescopes use parabolic
mirrors
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Friday, November 13, 15
Convex Spherical Mirrors (1)
§ Suppose we have a spherical mirror where the
reflecting surface is on the outside of the sphere
§ Thus we have a convex reflecting surface and
the reflected rays will diverge
§ The optical axis of the mirror is a line
through the center of the sphere,
represented in this drawing by a
horizontal dashed line
§ Imagine that a horizontal light ray above the
optical axis is incident on the surface of the mirror
§ At the point the light ray strikes the mirror, the law of reflection
applies, θi = θr
§ The normal to the surface is a radius line
that points to the center of the sphere
marked as C
§ Example: car right-hand rearview mirror
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Convex Spherical Mirrors (2)
§ In contrast to the concave mirror, the normal points away
from the center of the sphere
§ When we extrapolate the normal through the surface of the
sphere, it intersects the optical axis of the sphere
at the center, marked C
§ When we observe the
reflected ray, it seems to
be coming from inside the
sphere
R
f
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Friday, November 13, 15
Convex Spherical Mirrors (3)
§ Now let us suppose that we have many
horizontal light rays incident on this
spherical mirror as shown
§ Each light ray obeys the law of
reflection at each point
§ You can see that the rays diverge and
do not seem to form any kind of image
§ However, if we extrapolate the
reflected rays through the surface of
the mirror they all intersect the
optical axis at one point
§ This point is the focal point of this
convex spherical mirror
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Friday, November 13, 15
Images from Convex Mirrors (1)
§ Now let us discuss images formed
by convex mirrors
§ Again we use three rays
• The first ray establishes that the
tail of the arrow lies on the optical
axis
• The second ray starts from the
top of the object traveling parallel
to the optical axis and is reflected
from the surface of the mirror
such that its extrapolation crosses
the optical axis a distance from the surface of the mirror equal to the
focal length of the mirror
• The third ray begins at the top of the object and is directed so that its
extrapolation would intersect the center of the sphere
• This ray is reflected back on itself
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Images from Convex Mirrors (2)
§ We can see that we form an upright, reduced image on
the side of the mirror opposite the object
§ This image is virtual because it cannot be projected
§ These characteristics are valid for all cases of a convex mirror
§ In the case of a convex mirror, we define the focal length f to be
negative because the focal point of the mirror is on the opposite
side of the object
§ We always take the object distance do to be positive
§ We recall the mirror equation
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Images from Convex Mirrors (3)
§ We can rearrange the mirror equation to get
§
do is always positive and f is always negative, we can see that di
will always be negative
§ Applying the equation for the magnification we find that it is
always positive
§ Looking at our diagram of the ray construction for the convex
mirror will also convince you that the image will always be reduced
in size
§ Thus, for a convex mirror, we always will obtain a virtual, upright,
and reduced image
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Lenses
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Lenses
§ When light is refracted crossing a curved boundary between
two different media, the light rays follow the law of refraction at each
point on the boundary
§ The angle at which the light rays cross the boundary is different along
the boundary, so the refracted angle is different at different points
along the boundary
§ A curved boundary between two optically transparent media is called a
lens
§ Light rays that are initially parallel before they
strike the boundary are refracted in different
directions depending on the part of the lens they
strike
§ Depending of the shape of the lens, the light rays can
be focused or caused to diverge
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Focal lengths
converging lens
f
diverging lens
f
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Lens Focal Lengths
§ Unlike mirrors, lenses have a focal length on
both sides
§ Light can pass through lenses while
mirrors reflect the light allowing
no light on the opposite side
Convex (converging) lens
§ The focal length of a
convex (converging) lens is
defined to be positive
§ The focal length of a
concave (diverging) lens is
defined to be negative
positive
negative
Concave (diverging) lens
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Focal lengths
converging lens
f>0
diverging lens
f<0
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Convex Lenses (1)
§ A convex lens is shaped such that parallel rays
will be focused by refraction at the focal
distance f from the center of the lens
§ In the drawing on the right, a light ray is
incident on a convex glass lens
§ At the surface of the lens, the light ray is
refracted toward the normal
§ When the ray leaves the lens, it is
refracted away from the normal
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Thin Lenses (1)
§ The rays enter the lens, get refracted at the surface,
traverse the lens in a straight line, and get refracted when
they exit the lens
§ In the third frame, we represent the thin lens
approximation by drawing a black dotted line at the center
of the lens
§ Instead of following the
detailed trajectory of the
light rays inside the lens, we
draw the incident rays to
the centerline and then on
to the focal point
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Friday, November 13, 15
Images with Convex Lenses (1)
§ Convex lenses can be used to form images
§ We show the geometric construction of the formation of an
image using a convex lens with focal length f
§ We place an object standing on the optical axis represented by
the green arrow
§ This object has a height ho and is located a distance do from the
center of the lens such that do > f
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Friday, November 13, 15
Images with Convex Lenses (2)
§
We start with a ray along the optical axis of the lens that passes straight
through the lens that defines the bottom of the image.
§
A second ray is then drawn from the top of the object parallel to the optical
axis
•
This ray is focused through the focal point on the other side of the lens
§
A third ray is drawn through the center of the lens that is not refracted in the
thin lens approximation
§
A fourth ray is drawn from the top of the object through the focal point on the
same side of the lens that is then directed parallel to the optical axis
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Images with Convex Lenses (4)
§ Now let us consider the image formed by an object with height ho placed a
distance do from the center of the lens such that do < f
§ The first ray again is drawn from the bottom of the object along the optical
axis
§
§
§
The second ray is drawn from the top
of the object parallel to the optical
axis and is focused through the focal
point on the opposite side
A third ray is drawn through the
center of the lens
A fourth line is drawn such that it
originated from the focal point on the
same side of the lens and is then
focused parallel to the optical axis
•
These three rays are diverging
§
A virtual image is located on the same side
of the lens as the object
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do
f
f
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do
f
f
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do
f
f
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do
f
f
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do
f
f
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do
f
f
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do
f
f
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do
f
f
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di
do
f
f
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di
do
f
f
§ A virtual image is located on the same side of the lens as the object
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