36.3 Images Formed by Refraction

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36.3 Images Formed by Refraction
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Images Formed by Refraction

• Consider two transparent media having indices of
  refraction n1 and n2
• The boundary between the two media is a spherical
  surface of radius R
• Rays originate from the object at point O in the
  medium with n n1

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Images by Refraction, 2

• We will consider the paraxial rays leaving O
• All such rays are refracted at the spherical
  surface and focus at the image point, I
• The relationship between object and image
  distances can be given by
• (36.8)

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Images by Refraction, 3

• The side of the surface in which the light rays
  originate is defined as the front side
• The other side is called the back side
• Real images are formed by refraction in the back
  of the surface
• Because of this, the refraction sign conventions
  for q and R are opposite the reflection sign
  conventions

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Sign Conventions for Refracting Surfaces
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Flat Refracting Surfaces

• If a refracting surface is flat, then R ??,
  therefore
• q ?(n2 / n1)p (36.9)
• The image formed by a flat refracting surface is
  on the same side of the surface as the object
• For n1 n2 a virtual image is formed between the
  object and the surface
• For n1 of the object

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Active Figure 36.20
(SLIDESHOW MODE ONLY)
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Example 36.5 Gaze Into the Crystal Ball

• A set of coins is embedded in a spherical plastic
  paper weight of radius 3.0 cm, with n 1.50. One
  coin is located 2.0 cm from the edge of the
  sphere. Find the position of the image of the
  coin.
• Here n1 n2 so a virtual image is formed inside
  the paperweight.
• R is negative

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36.4 Thin Lenses

• Lenses are commonly used to form images by
  refraction
• Lenses are used in optical instruments
• Cameras, telescopes, microscopes
• Images from lenses
• Light passing through a lens experiences
  refraction at two surfaces
• The image formed by one refracting surface serves
  as the object for the second surface

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Image Formed by a Lens

• The lens has an index of refraction n and two
  spherical surfaces with radii of R1 and R2
• R1 is the radius of curvature of the lens surface
  that the light of the object reaches first
• R2 is the radius of curvature of the other
  surface
• The object is placed at point O at a distance of
  p1 in front of the first surface

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Image From Surface 1

• There is an image formed by surface 1
• Since the lens is surrounded by the air, n1 1
  and n2 n ?
• Equation (36.8) becomes
• (36.10)
• If the image due to surface 1 is virtual, q1 is
  negative, and it is positive if the image is real

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Image From Surface 2

• For surface 2, n1 n and n2 1
• The light rays approaching surface 2 are in the
  lens and are refracted into air
• Use p2 for the object distance for surface 2 and
  q2 for the image distance, so equation (36.8)
  becomes
• (36.11)
• From the virtual image at surface 1 p2 q1
  t
• q1 is negative and t is the thickness of the lens
• From the real image at surface 1 p2 q1
  t
• q1 is positive

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Image Formed by a Thin Lens

• A thin lens is one whose thickness t is small
  compared to the radii of curvature
• For a thin lens, the thickness, t, of the lens
  can be neglected
• In this case, p2 q1 for either type of image
• Hence equation (36.11) becomes
• (36.12)

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Image Formed by a Thin Lens, 2

• Adding equations (36.10) and (36.12) we obtain
• (36.13)
• Then the subscripts on p1 and q2 can be omitted
  as in the figure and rewrite equation (36.13) as
• (36.14)

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Lens Makers Equation

• The focal length f of a thin lens is the image
  distance q that corresponds to an infinite object
  distance
• This is the same as for a mirror
• Making p ? ? and q ? f, so equation (36.14) will
  become the lens makers equation
• (36.15)
• Given n and f the lens maker can determine the
  values of R1 and R2
• Given R1, R2 and n lens maker can calculate the
  value of f

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Thin Lens Equation

• Using equation (36.15) we can write equation
  (36.14) in a for identical to equation (36.6) for
  mirrors.
• The relationship among the focal length, the
  object distance and the image distance is the
  same as for a mirror
• (36.16)

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Notes on Focal Length and Focal Point of a Thin
Lens

• Because light can travel in either direction
  through a lens, each lens has two focal points
• One focal point is for light passing in one
  direction through the lens
• The other is for light traveling in the opposite
  direction
• However, there is only one focal length
• Each focal point is located at the same distance
  from the lens

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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

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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

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Determining Signs for Thin Lenses

• The front side of the thin lens is the side of
  the incident light
• The back side of the lens is where the light is
  refracted into
• This is also valid for a refracting surface

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Sign Conventions for Thin Lenses
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Magnification of Images Through a Thin Lens

• The lateral magnification of the image is
• When M is positive, the image is upright and on
  the same side of the lens as the object
• When M is negative, the image is inverted and on
  the side of the lens opposite the object

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Thin Lens Shapes

• These are examples of converging lenses
• They have positive focal lengths
• They are thickest in the middle

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More Thin Lens Shapes

• These are examples of diverging lenses
• They have negative focal lengths
• They are thickest at the edges

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Ray Diagrams for Thin Lenses Converging

• Ray diagrams are convenient for locating the
  images formed by thin lenses or systems of lenses
• For a converging lens, the following three rays
  are drawn
• Ray 1 is drawn parallel to the principal axis and
  then passes through the focal point on the back
  side of the lens
• Ray 2 is drawn through the center of the lens and
  continues in a straight line
• Ray 3 is drawn through the focal point on the
  front of the lens (or as if coming from the focal
  point if p
  parallel to the principal axis

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Ray Diagram for Converging Lens, p f

• The image is real
• The image is inverted
• The image is on the back side of the lens

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Ray Diagram for Converging Lens, p

The image is virtual

The image is upright

The image is larger than the object

The image is on the front side of the lens

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Ray Diagrams for Thin Lenses Diverging

• For a diverging lens, the following three rays
  are drawn
• Ray 1 is drawn parallel to the principal axis and
  emerges directed away from the focal point on the
  front side of the lens
• Ray 2 is drawn through the center of the lens and
  continues in a straight line
• Ray 3 is drawn in the direction toward the focal
  point on the back side of the lens and emerges
  from the lens parallel to the principal axis

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Ray Diagram for Diverging Lens

• The image is virtual
• The image is upright
• The image is smaller
• The image is on the front side of the lens

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Active Figure 36.28
(SLIDESHOW MODE ONLY)
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Image Summary

• For a converging lens, when the object distance
  is greater than the focal length
• (p )
• The image is real and inverted
• For a converging lens, when the object is between
  the focal point and the lens, (p
• The image is virtual and upright
• For a diverging lens, the image is always virtual
  and upright
• This is regardless of where the object is placed

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Fresnal Lens

• Refraction occurs only at the surfaces of the
  lens
• A Fresnal lens is designed to take advantage of
  this fact
• It produces a powerful lens without great
  thickness

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Fresnal Lens, cont.

• Only the surface curvature is important in the
  refracting qualities of the lens
• The material in the middle of the Fresnal lens is
  removed
• Because the edges of the curved segments cause
  some distortion, Fresnal lenses are usually used
  only in situations where image quality is less
  important than reduction of weight

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Two Thin Lenses (Combination)

• If two thin lenses are used to form an image
• The image formed by the first lens is located as
  if the second lens were not present
• Then a ray diagram is drawn for the second 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

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Two 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 as a virtual object for the second lens
• p will be negative
• The same procedure can be extended to a system of
  three or more lenses
• The overall magnification is the product of the
  magnification of the separate lenses

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Two Thin Lenses, 3

• Consider a case of two lenses in contact with
  each other
• The lenses have focal lengths of 1 and 2
• If p1 p is the object distance for the
  combination, equation (36.16) becomes
• Since the lenses are in contact, p2 q1

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Two Thin Lenses, final

• For the second lens q2 q ,
• Adding the two previous equations for the
  combination of the two lenses
• (36.17)
• Two thin lenses in contact with each other are
  equivalent to a single thin lens having a focal
  length given by the above equation

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Example 36.6 Where is the Final Image?
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Example 36.6 Where is the Final Image? , 2

• The location of the image formed by lens 1
• The image of lens 1 becomes object for lens 2,
  with p2 20cm 15cm 5cm ?
• Then, the total magnification will be

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Material for the Midterm

• Examples to Read!!!
• Example 36.11 (page 1149)
• Examples in Class!!!
• Example 36.9 (page 1146)
• Example 36.10 (page 1147)
• Homework to be solved in Class!!!
• Question 9
• Problems 21, 28