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Notes: Mirrors and Lenses
23.1 Flat Mirrors
Notation for Mirrors and Lenses

The object distance is the distance from the object to the mirror or lens

The image distance is the distance from the image to the mirror or lens

Images are formed at the point where rays actually intersect or appear to originate

Denoted by q

The lateral magnification of the mirror or lens is the ratio of the image height to the object height
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

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
F
lat 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

The image is unmagnified

The image height is the same as the object height

The image is virtual

The image is upright

It has the same orientation as the object

There is an apparent leftright reversal in the image
Example 23.1 A man 1.80 m tall stands in fron of a mirror and sees his full height, no more and no less. If his eyes are 0.14 m from the top of his head, what is the minimum height of the mirror?
23.2 Images formed by Spherical 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

The mirror has a radius of curvature of R

Its center of curvature is the point C

Point V is the center of the spherical segment

A line drawn from C to V is called the principle axis of the mirror

Rays are generally assumed to make small angles with the mirror

W
hen the rays make large angles, they may converge to points other than the image point

This results in a blurred image

This effect is called spherical aberration

Geometry can be used to determine the magnification of the image
Image Formed by a Concave Mirror

Geometry shows the relationship between the image and object distances

This is called the mirror equation

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 ½ the radius of curvature

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
23.3 Convex Mirrors and Sign Conventions

In general, the image formed by a convex mirror is upright, virtual, and smaller than the object
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

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

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

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
R
ay 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
R
ay 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

T
he 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
Example 23.2 Assume that a certain concave spherical mirror has a focal length of 10.0 cm.

Locate the image and find the magnification for an object distance of 25.0 cm. Determine whether the image is real or virtual, inverted of upright, and larger or smaller.

Do the same for the object distances of 10.0 cm and 5.00 cm.
Example 23.3 An object 3.00 cm high is placed 20.0 cm from a convex mirror with a focal length of 8.00 cm.

Find the position of the image

Find the magnification of the mirror

Find the height of the image.
23.6 Thin Lenses
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
F
ocal Length of a Diverging Lens

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

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 equation
Ray Diagrams for Thin Lenses
Ray Diagram for Converging Lens, p > f

The image is real

The image is inverted
R
ay Diagram for Converging Lens, p < f

The image is virtual

The image is upright
R
ay Diagram for Diverging Lens

The image is virtual

The image is upright
Example 23.7 A converging lens of focal length 10.0 cm forms images of an object situated at various distances.

I
f the object is placed 30.0 cm from the lens, locate the image, state whether it’s real or virtual, and find its magnification.

Repeat the problem when the object is at 10.0 cm.

Repeat the problem when the object is 5.00 cm from the lens.
E
xample 23.8 Repeat example 23.7 for a diverging lens with a focal length of 10.0 cm.
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