Lenses

1
Lenses

• Physics 202
• Professor Lee Carkner
• Lecture 23

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Refraction

• Mirrors can be used to magnify an object or to
  gather light
• Lenses can be used for the same purposes
• Lenses gather light and magnify objects through
  refraction instead of reflection
• Lenses have focal lengths and real and virtual
  images, but their properties also depend on the
  index of refraction
• Unlike a mirror, you look through a lens
• It has two sides we have to account for

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Lenses

• Light incident on a lens is refracted twice, once
  when entering and once when leaving
• We will consider only thin lenses, i.e. thickness
  much smaller than i, p or f
• Our thin lenses are composed of two refracting
  surfaces placed back to back
• If the two surfaces are the same, the lens is
  symmetric

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Lenses and Mirrors

• Mirrors produce virtual images on the opposite
  side from the object
• Lenses produce virtual images on the same side as
  the object
• i is negative in both cases
• Mirrors produce real images on the same side as
  the object
• Lenses produce real images on the opposite side
  as the object
• i is positive in both cases
• If a mirror curves towards the object, f and r
  are positive (real focus)
• If a lens curves towards the object, f and r are
  negative (virtual focus)
• Real is positive, virtual is negative

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Converging and Diverging
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Converging Lens

• A lens consisting of two convex lenses back to
  back is called a converging lens
• Rays initially parallel to the central axis are
  focused to the focal point after refraction
• The focal point is on the opposite side from the
  incoming rays
• f is real and positive
• Converging lenses produce images larger than the
  object
• Magnification is same as for mirrors
• m -i/p

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Diverging Lens

• A lens consisting of two concave lenses back to
  back is called a diverging lens
• Rays initially parallel to the central axis
  diverge after refraction, but can be traced back
  to a virtual focus
• f is virtual and negative
• Th focal point is on the same side as the
  incoming rays
• Diverging lenses produce images smaller than the
  object

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Converging and Diverging
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Lens Equations

• A thin lens follows the same equation as a
  mirror, namely
• 1/f 1/p 1/i
• We can also relate f to the index of refraction
  of the lens n
• 1/f (n-1) (1/r1 -1/r2)
• Where r1 and r2 are the radii of curvature of
  each side of the lens (r1 is the side nearest the
  object)
• If the lens curves towards the object, r is
  negative, if the lens curves away from the
  object, r is positive
• For symmetric lenses r1 and r2 have opposite sign

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Three Types of Images
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Converging Lenses and Images

• The type of image produced by a converging lens
  depends on the distance of the object from the
  focal point
• Objects in front of the focal point (nearer to
  the lens) produce virtual images on the same side
  as the object
• The image is not inverted
• Image is virtual so i is negative
• Objects behind the focal point (further from the
  lens) produce real images on the opposite side of
  the lens
• The image is inverted
• Image is real so i is positive

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Diverging Lenses and Images

• No matter where the object is, a diverging lens
  produces an upright, virtual image on the same
  side as the object
• For either lens the location of images is the
  reverse of that for mirrors
• Virtual images form on the same side as the
  object, real images form on the opposite side
• Real images have positive i, virtual images have
  negative i

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Three Types of Images
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1) Rays that are initially parallel to the
central axis will pass through the focal point
after refraction and vice versa
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2) Rays that pass through the center of the lens
will not be refracted
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Two Lenses

• Many common optical instruments are formed from
  more than one lens (microscope, telescope)
• To find the final image we find the image
  produced by the first lens and use that as the
  object for the second lens
• For a two lens system the magnification is
• M m1m2

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DualLenses
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Optical Instruments

• We can approximate several common optical
  instruments as being composed of a simple
  arrangement of thin lenses
• In reality the lenses are not thin and may be
  arranged in a complex fashion

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Near Point

• You can increase an objects angular size by
  moving it closer to your eye
• The largest clear (unlensed) image of an object
  is obtained when it is at the near point (about
  25 cm for most people)
• If you move the object any closer it will not be
  in focus
• A converging lens will increase the angular
  diameter of an object
• mq q/q

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Magnifying Lens

• You can use a magnifying lens to overcome the
  limitation of your eyes near point
• If the object is inside the near point you can
  view it through a lens which will produce a
  virtual image outside of the near point
• The magnification is
• mq 25 cm /f
• This is the size of the object seen through the
  lens compared to its size at the near point

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Magnifying Glass
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Compound Microscope

• A simple compound microscope consists of an
  objective and eyepiece
• The objective creates a real image focused at the
  focal point of the eyepiece
• The eyepiece acts as a magnifying glass
• The magnification of the objective is m -i/p
• i is very close to the distance between the
  lenses, s
• p is very close to the focal length of the
  objective, fob
• The total magnification is the product of the
  magnification of each
• M (-s/fob)(25 cm/fey)
• where s is the distance between the focal point
  of the lenses (the tube length) and f is the
  focal length

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Microscope
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Refracting Telescope

• In a telescope the two lenses are placed so that
  the two inner focal points are in the same place
• The rays coming in from infinity are refracted by
  the objective to create a real image at the
  common focal point
• The eyepiece then magnifies the real image
• The total angular magnification of the telescope
  depends on the ratio of the eyepieces
• mq -fob/fey

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Refracting Telescope
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Telescopes

• The magnification of the telescope can be altered
  by changing eyepieces
• Short focal length means more magnification
• Magnification is not the most important property
  of a telescope
• Limited by blurring effects of atmosphere
• The true purpose of the objective lens is to
  gather more light than your eye can and focus it
  so that it can be viewed
• The largest practical refracting telescope has an
  objective with a diameter of about 1m
• The objective becomes so large it is hard to
  build and support
• Most large telescopes are reflectors

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Giant 40 inch Refractor at Yerkes
Observatory,Williams BayWisconsin
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Newtonian Telescope