Chapter 23
Mirrors and Lenses
Notation for Mirrors and Lenses
The object distance is the distance from the object to the mirror or lens
Denoted by p
The image distance is the distance from the image to the mirror or lens
Denoted by q
The lateral magnification of the mirror or lens is the ratio of the image height to the object height
Denoted by M
Types of Images for Mirrors and Lenses
A real image is one in which light actually passes through the image point
Real images can be displayed on screens
A virtual image is one in which the light does not pass through the image point
The light appears to diverge from that point
Virtual images cannot be displayed on screens
More About Images
To find where an image is formed, it is always necessary to follow at least two rays of light as they reflect from the mirror
Flat Mirror
Simplest possible mirror
Properties of the image can be determined by geometry
One ray starts at P, follows path PQ and reflects back on itself
A second ray follows path PR and reflects according to the Law of Reflection
Properties of the Image Formed by a Flat Mirror
The image is as far behind the mirror as the object is in front
q = p
The image is unmagnified
The image height is the same as the object height
h
The image is virtual
The image is upright
It has the same orientation as the object
There is an apparent left-right reversal in the image
Application
– Day and Night Settings on Auto MirrorsWith the daytime setting, the bright beam of reflected light is directed into the driver
’s eyesWith the nighttime setting, the dim beam of reflected light is directed into the driver
’s eyes, while the bright beam goes elsewhereSpherical Mirrors
A spherical mirror has the shape of a segment of a sphere
A concave spherical mirror has the silvered surface of the mirror on the inner, or concave, side of the curve
A convex spherical mirror has the silvered surface of the mirror on the outer, or convex, side of the curve
Concave Mirror, Notation
The mirror has a radius of curvature of R
Its center of curvature is the point C
Point V (vertex) is the center of the spherical segment
A line drawn from C to V is called the principle axis of the mirror
Image Formed by a Concave Mirror
Geometry shows the relationship between the image and object distances
This is called the mirror equation
Equivalent Equations
Image Formed by a Concave Mirror
Spherical Aberration
Rays are generally assumed to make small angles with the mirror
When the rays make large angles, they may converge to points other than the image point
This results in a blurred image
Focal Length
If an object is very far away, then p® ¥ and 1/p ® 0
Incoming rays are essentially parallel
In this special case, the image point is called the focal point
The distance from the mirror to the focal point is called the focal length
The focal length is
Focal Point and Focal Length, cont
The focal point is dependent solely on the curvature of the mirror, not by the location of the object
f = R / 2
The mirror equation can be expressed as
Focal Length Shown by Parallel Rays
Convex Mirrors
A convex mirror is sometimes called a diverging mirror
The rays from any point on the object diverge after reflection as though they were coming from some point behind the mirror
The image is virtual because it lies behind the mirror at the point where the reflected rays appear to originate
In general, the image formed by a convex mirror is upright, virtual, and smaller than the object
Image Formed by a Convex Mirror
Ray Diagrams
A ray diagram can be used to determine the position and size of an image
They are graphical constructions which tell the overall nature of the image
They can also be used to check the parameters calculated from the mirror and magnification equations
Drawing A Ray Diagram
To make the ray diagram, you need to know
The position of the object
The position of the center of curvature
Three rays are drawn
They all start from the same position on the object
The intersection of any two of the rays at a point locates the image
The third ray serves as a check of the construction
The Rays in a Ray Diagram
Ray 1 is drawn parallel to the principle axis and is reflected back through the focal point, F
Ray 2 is drawn through the focal point and is reflected parallel to the principle axis
Ray 3 is drawn through the center of curvature and is reflected back on itself
Notes About the Rays
The rays actually go in all directions from the object
The three rays were chosen for their ease of construction
The image point obtained by the ray diagram must agree with the value of q calculated from the mirror equation
Ray Diagram for Concave Mirror, p > R
The object is outside the center of curvature of the mirror
The image is real
The image is inverted
The image is smaller than the object
Ray Diagram for a Concave Mirror, p < f
The object is between the mirror and the focal point
The image is virtual
The image is upright
The image is larger than the object
Ray Diagram for a Convex Mirror
The object is in front of a convex mirror
The image is virtual
The image is upright
The image is smaller than the object
Notes on Images
With a concave mirror, the image may be either real or virtual
When the object is outside the focal point, the image is real
When the object is at the focal point, the image is infinitely far away
When the object is between the mirror and the focal point, the image is virtual
With a convex mirror, the image is always virtual and upright
As the object distance increases, the virtual image gets smaller
Sign Conventions for Mirrors
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Quantity
|
Positive When
|
Negative When
|
|
Object location (p)
|
Object is in front of
the mirror
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Object is behind the
mirror
|
|
Image location (q)
|
Image is in front of
mirror
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Image is behind
mirror
|
|
Image height (h’)
|
Image is upright
|
Image is inverted
|
|
Focal length (f) and
radius (R)
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Mirror is concave
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Mirror is convex
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Magnification (M)
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Image is upright
|
Image is inverted
|
Images Formed by Refraction
Rays originate from the object point, O, and pass through the image point, I
When n2 > n1,
Real images are formed on the side opposite from the object
Sign Conventions for Refracting Surfaces
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Quantity
|
Positive When
|
Negative When
|
|
Object location (p)
|
Object is in front of
surface
|
Object is in back of
surface
|
|
Image location (q)
|
Image is in back of
surface
|
Image is in front of
surface
|
|
Image height (h’)
|
Image is upright
|
Image is inverted
|
|
Radius (R)
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Center of curvature
is in back of surface
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Center of curvature
is in front of surface
|
Flat Refracting Surface
The image formed by a flat refracting surface is on the same side of the surface as the object
The image is virtual
The image forms between the object and the surface
The rays bend away from the normal since n1 > n2
Atmospheric Refraction
There are many interesting results of refraction in the atmosphere
Sunsets
Mirages
Atmospheric Refraction and Sunsets
Light rays from the sun are bent as they pass into the atmosphere
It is a gradual bend because the light passes through layers of the atmosphere
Each layer has a slightly different index of refraction
The Sun is seen to be above the horizon even after it has fallen below it
Atmospheric Refraction and Mirages
A mirage can be observed when the air above the ground is warmer than the air at higher elevations
The rays in path B are directed toward the ground and then bent by refraction
The observer sees both an upright and an inverted image
Thin Lenses
A thin lens consists of a piece of glass or plastic, ground so that each of its two refracting surfaces is a segment of either a sphere or a plane
Lenses are commonly used to form images by refraction in optical instruments
Thin Lens Shapes
These are examples of converging lenses
They have positive focal lengths
They are thickest in the middle
More Thin Lens Shapes
These are examples of diverging lenses
They have negative focal lengths
They are thickest at the edges
Focal Length of Lenses
The focal length,
ƒ, is the image distance that corresponds to an infinite object distanceThis is the same as for mirrors
A thin lens has two focal points, corresponding to parallel rays from the left and from the right
A thin lens is one in which the distance between the surface of the lens and the center of the lens is negligible
Focal Length of a Converging Lens
The parallel rays pass through the lens and converge at the focal point
The parallel rays can come from the left or right of the lens
Focal Length of a Diverging Lens
The parallel rays diverge after passing through the diverging lens
The focal point is the point where the rays appear to have originated
Lens Equations
The geometric derivation of the equations is very similar to that of mirrors
Lens Equations
The equations can be used for both converging and diverging lenses
A converging lens has a positive focal length
A diverging lens has a negative focal length
Sign Conventions for Thin Lenses
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Quantity
|
Positive When
|
Negative When
|
|
Object location (p)
|
Object is in front of
the lens
|
Object is in back of
the lens
|
|
Image location (q)
|
Image is in back of
the lens
|
Image is in front of
the lens
|
|
Image height (h’)
|
Image is upright
|
Image is inverted
|
|
R1 and
R2
|
Center of curvature
is in back of the lens
|
Center of curvature
is in front of the lens
|
|
Focal length (f)
|
Converging lens
|
Diverging lens
|
Focal Length for a Lens
The focal length of a lens is related to the curvature of its front and back surfaces and the index of refraction of the material
This is called the lens maker
’s equationRay Diagrams for Thin Lenses
Ray diagrams are essential for understanding the overall image formation
Three rays are drawn
The first ray is drawn parallel to the first principle axis and then passes through (or appears to come from) one of the focal points
The second ray is drawn through the center of the lens and continues in a straight line
The third ray is drawn from the other focal point and emerges from the lens parallel to the principle axis
There are an infinite number of rays, these are convenient
Ray Diagram for Converging Lens, p > f
The image is real
The image is inverted
Ray Diagram for Converging Lens, p < f
The image is virtual
The image is upright
Ray Diagram for Diverging Lens
The image is virtual
The image is upright
Problem Solving Strategy
Be very careful about sign conventions
Do lots of problems for practice
Draw confirming ray diagrams
Combinations of Thin Lenses
The image produced by the first lens is calculated as though the second lens were not present
The light then approaches the second lens as if it had come from the image of the first lens
The image of the first lens is treated as the object of the second lens
The image formed by the second lens is the final image of the system
Combination of Thin Lenses, 2
If the image formed by the first lens lies on the back side of the second lens, then the image is treated at a virtual object for the second lens
p will be negative
The overall magnification is the product of the magnification of the separate lenses
Combination of Thin Lenses, example
Lens and Mirror Aberrations
One of the basic problems is the imperfect quality of the images
Largely the result of defects in shape and form
Two common types of aberrations exist
Spherical aberration
Chromatic aberration
Spherical Aberration
Results from the focal points of light rays far from the principle axis are different from the focal points of rays passing near the axis
For a mirror, parabolic shapes can be used to correct for spherical aberration
Chromatic Aberration
Different wavelengths of light refracted by by a lens focus at different points
Violet rays are refracted more than red rays
The focal length for red light is greater than the focal length for violet light
Chromatic aberration can be minimized by the use of a combination of converging and diverging lenses
Real vs. Apparent Depth
Real vs. Apparent Depth