Notes: Mirrors and Lenses 23. 1 Flat Mirrors Notation for Mirrors and Lenses



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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

    • Denoted by p

  • 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

    • 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

  • 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

    • q = p

  • The image is unmagnified

    • The image height is the same as the object height

      • h’ = h and M = 1

  • 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

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


  • h

    ’ is negative when the image is inverted with respect to the object



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

  • 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

  • T

    he 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



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.



  1. 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.

  2. 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.



  1. Find the position of the image

  2. Find the magnification of the mirror

  3. Find the height of the image.


23.6 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

  • L

    enses are commonly used to form images by refraction in optical instruments



  • These are examples of converging lenses

  • They have positive focal lengths

  • They are thickest in the middle

  • T

    hese are examples of diverging lenses



  • They have negative focal lengths

  • They are thickest at the edges

  • The focal length, ƒ, is the image distance that corresponds to an infinite object distance

    • This 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

F

ocal 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

  • T

    he geometric derivation of the equations is very similar to that of mirrors





  • 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 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 lengths

    • 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

  • T

    here are an infinite number of rays, these are convenient



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.

  1. 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.

  2. Repeat the problem when the object is at 10.0 cm.

  3. 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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