Chapter 6: Waves Part 2

1
Chapter 6 Waves Part 2

• Alyssa Jean-Mary

2
Basic Wave Formula

• The speed of a wave is given by the number of
  waves that pass a point per second multiplied by
  the length of each wave
• In equation form, the speed of a wave is given
  by
• v f?
• where v is the speed of the wave, f is the
  frequency of the wave, and ? is the wavelength of
  the wave.

3
Example Calculation of Wave Speed

• Example What is the speed of a wave if it has a
  frequency of 34 Hz and a wavelength of 2.5 m?
• Answer
• Given 34 Hz, 2.5m
• Looking for speed
• Equation v f?
• Solution v (34 Hz)(2.5m) 85 m/s

4
Standing Waves

• When a rope that is attached at one end and free
  at the other end is shaken on the free end, and
  one wave is sent down this rope to the attached
  end, the wave will reform itself and travel back
  along the rope.
• Now, if there is a series of waves that is sent
  down this rope, the reflected waves (i.e. those
  waves coming from the attached end) will meet the
  forward-moving waves (i.e. those waves coming
  from the free end) head on. Thus, each point on
  the rope must respond to two different impulses
  (one from each direction) on the rope at the same
  time. The two impulses will add together
• If the point on the rope is being pushed in the
  same direction by both waves, it will move in
  that direction with an amplitude equal to the sum
  of the amplitudes of the two waves.
• If the point on the rope is being pushed by the
  waves in opposite directions, it will have an
  amplitude equal to the difference of the two wave
  amplitudes.
• If the timing is right, at some points on the
  rope, the two motions will completely cancel out,
  and other points on the rope will be moving with
  twice the normal amplitude. In a situation like
  this, the waves appear not to travel at all
  because some parts of the wave only move up and
  down and other parts of the wave remain at rest.
  This is a standing wave.
• An example of standing waves are the vibrating
  strings in musical instruments.

5
Sound 1

• Most sounds are produced by a vibrating object
• For example, the cone of a loudspeaker is a
  vibrating object that produces sound. When sound
  moves outward from the speaker, the cone pushes
  the air molecules that are in front of it
  together, which forms a region of high pressure
  that spreads outward. Then the cone moves
  backward, which expands the space that is
  available for nearby air molecules. Some of the
  molecules actually flow towards the cone, which
  leaves a region of low pressure that spreads
  outward behind the region of high pressure that
  was previously created. Thus, the repeated
  vibrations of the cone send out a series of
  compressions and rarefactions, which are sound
  waves.
• Sound waves are longitudinal waves because the
  molecules in their paths move back and forth in
  the same direction as the waves (i.e. not
  perpendicular like in transverse waves). The
  molecules that are in the path of a sound wave
  become alternately denser and rarer. The pressure
  change that occurs causes our eardrums to
  vibrate, which is what produces the sensation of
  sound.
• Most sounds are waves i.e. they have a series
  of compressions and rarefactions. Some sounds,
  however, have only one single compression.
  Examples of these types of sounds are the crack
  of a rifle and the first sharp sound of a
  thunderclap.

6
Sound 2

• At ordinary temperatures, in the air at sea
  level, the speed of sound is about 343 m/s (767
  mi/h)
• Sounds travel faster in liquids and solids than
  in gases because since the molecules in liquids
  and solids are closer together than those in
  gases, they can respond faster to one anothers
  motions. Thus, in water, the speed of sound is
  about 1500 m/s and in iron, the speed of sound is
  5000 m/s.
• Our ears are the most sensitive to sounds that
  have frequencies between 3000 Hz and 4000 Hz.
  Sounds with frequencies below 20 Hz, which is
  called infrasound, and above 20,000 Hz, which is
  called ultrasound, are heard by almost no one.
  Most animals, however, have a upper limit that is
  higher than our upper limit. With age, hearing
  deteriorates, especially at the higher
  frequencies.
• Ultrasound, those sounds with frequencies above
  20,000 Hz, have many applications. They are used
  in medical imaging and in determining water
  depths. The technique to determine water depths
  is called sonar. Sonar is also used to detect
  submarines, and it is used by bats in the air to
  detect prey.

7
The Decibel

• The more energy a sound wave carries, the louder
  it sounds, but our ears respond to sound waves in
  a particular way if the rate of energy flow of a
  sound is doubled, the sound doesnt sound twice
  as loud in our ears it is only slightly louder.
  This is why a single instrument can be heard in
  concerto even when an entire orchestra is playing
  at the same time. This is also why you can carry
  on a conversation at a party even when many other
  people are talking at the same time.
• The unit of sound is the decibel (dB). There is a
  special scale that uses the decibel to describe
  how powerful a sound is. If a sound can barely be
  heard by a normal person, it has 0 dB. Every 10
  dB corresponds to a 10-fold change in sound
  energy. Thus, a 50 dB sound is 10 times stronger
  than a 40 dB sound, but is 100 times stronger
  than a 30 dB sound. The sound of ordinary
  conversation is usually about 60 dB, which is 106
  (i.e. a million) times more intense than the
  faintest heard sound.
• Permanent hearing damage can happen if you are
  exposed to sounds that are 85 dB or higher. Rock
  concerts, for instance, can have sounds as high
  as 125 dB, which is why many people have
  significant hearing loss from rock concerts.
  Three-quarters of the hearing loss of a typical
  older person in the United States is due to
  exposure to such loud sounds.

8
The Doppler Effect

• When vehicles are moving towards us, the sounds
  they produce seem to be higher pitched than
  normal, AND when vehicles are moving away from
  us, the sounds that they produce seem to be lower
  pitched than normal. This difference in frequency
  is known as the Doppler effect.
• The Doppler effect is due to the relative motion
  of the listener and the source of the sound.
  Either one or both need to be moving there is
  no Doppler effect when both are not moving. When
  the motion reduces the distance between the
  source of the sound and the listener (i.e. when a
  vehicle is moving towards us), the wavelength of
  the sound decreases, which makes the frequency of
  the sound higher. When the motion increases the
  distance between the source of the sound and the
  listener (i.e. when a vehicle is moving away from
  us), the wavelength of the sound increases, which
  makes the frequency of the sound lower.
• The amount of frequency difference can be seen in
  the following example if a fire engine has a 500
  Hz siren and it moving at 60 km/h (37 mi/h), when
  it is approaching you, you will hear a sound of
  526 Hz, and when it is moving away from you, you
  will hear a sound of 477 Hz. Any such frequency
  change is easy to pick up by our ears.

9
Uses of the Doppler Effect

• The Doppler effect can be used to measure the
  speed of blood in an artery. When an ultrasound
  beam is directed at an artery, the moving blood
  cells with reflect waves that will exhibit a
  Doppler shift in frequency because the cells are
  acting as moving wave sources. The speed of the
  blood can be calculated from this shift in
  frequency. In the main arteries, the speed of
  blood is a few centimeters per second, and in the
  smaller ones, it is less.
• The Doppler effect occurs in light waves. Thus,
  astronomers can use the Doppler effect to detect
  and measure the motions of stars. Stars emit
  light that has only certain characteristic
  wavelengths. If a star is moving toward the
  earth, these wavelengths appear shorter than is
  characteristic, and if a star is moving away from
  the earth, these wavelengths appear longer than
  is characteristic. The amount of difference in
  the frequency from the characteristic frequency
  is used to calculate the speed at which the star
  is moving, whether it is approaching or receding.
  This is how the expansion of the universe was
  discovered.
• There are many other uses for the Doppler effect.

10
Musical Sounds

• Just like other sounds, musical sounds are
  produced by vibrating objects
• The vibrating objects that produce musical sounds
  vary. Some examples are stretched wire in
  stringed instruments, vocal cords in the throat,
  membranes in drums, and air columns in wind
  instruments.
• The simplest vibration of a stretched string is
  when a single standing wave takes up the entire
  length of the string.
• The frequency of a sound may be changed by
  changing the tension in the string. The more
  tension the string has (i.e. the tighter it is),
  the higher the frequency is. The frequency can
  also be changed for a given amount of tension by
  changing the length of the string.
• Depending on where the string is plucked, bowed,
  or struck, more complex vibrations can occur.
  Thus, instead of a standing wave with a single
  crest, the standing wave might have two, three,
  or even more crests. A sound wave that has more
  crests because it has a shorter standing wave has
  a higher frequency. The frequencies are related
  to the frequency of the longest wave (i.e. the
  single standing wave) by simple ratios 21, 31,
  etc. The tone that is produced by a single
  standing wave is called the fundamental tone. The
  tones that occur at higher frequencies when the
  spring vibrates in segments are called overtones.

11
Resonance

• The motion of the string and thus the form of the
  sound wave may be very complex
• The strings of a musical instrument never just
  give a fundamental tone or a single overtone
  they give a combination of the fundamental tone
  and several overtones
• The fundamental tone sounds flat and
  uninteresting to the ear, but as overtones are
  added, the sound becomes richer. The quality, or
  timbre, of the tone depends on which overtones
  are emphasized in the sound. Which overtones are
  emphasized depends mostly on the shape of the
  instrument, which enables it to resonate at
  particular frequencies. The sound part of the
  instrument (i.e. the belly of the violin or the
  soundboard of the piano) has certain natural
  frequencies of vibration. The instrument is more
  readily set to vibrate at these natural
  frequencies than at any other frequencies. A
  resulting sound may have a large number of
  overtones, but the musical quality characteristic
  of an instrument is due to which overtones get
  greater emphasis.

12
Other Musical Sounds

• Wind instruments produce sounds from vibrating
  air columns.
• An organ has a separate pipe for each note. The
  shorter the pipe, the higher the pitch.
• Woodwinds (i.e. flutes and clarinets) have a
  single tube with holes. The length of the air
  column is controlled by the opening and closing
  of these holes.
• Most brass instruments have valves that are
  connected to loops of tubing. If a valve is
  opened, the length of the air column is
  increased, and thus, a note of lower pitch is
  produced.
• A slide trombone varies the length of its air
  column by sliding in or out its telescoping U
  tube.
• Since a bugle has neither holes nor valves,
  different notes are obtained using the lips of
  the bugler.
• The fundamental frequencies of the speaking voice
  on an average 145 Hz for men and 230 Hz for
  women. Even when the overtones that are present
  are considered, the frequencies in ordinary speed
  are for the most part below 1000 Hz. In singing,
  the first and second overtones may be louder than
  their fundamental tones, and even higher
  overtones add to the beauty of the sound.

13
Electromagnetic Waves

• In 1864, British physicist James Clerk Maxwell
  said that an accelerated electric charge
  generates combined electrical and magnetic
  disturbances that are able to travel indefinitely
  though empty space. These combined electrical and
  magnetic disturbances are called electromagnetic
  waves.
• Some examples of electromagnetic waves are light
  waves, radio waves, and x-rays.
• Through electromagnetic induction, a changing
  magnetic field gives rise to an electric current
  in a nearby wire. Thus, we can conclude that a
  changing magnetic field has an electric field
  associated with it. Maxwell said that the
  opposite is also true i.e. that a changing
  electric field has a magnetic field associated
  with it. To prove this, it was easy to measure
  the electric fields that are produced by
  electromagnetic induction because metals have
  very little resistance to the flow of electrons,
  but in Maxwells time, there was no way to detect
  the weak magnetic fields that were produced by
  the electric field. Thus, another way was used to
  check his idea.
• If Maxwell is right, there must be
  electromagnetic (EM) waves that occur in which
  changing electric and magnetic fields are coupled
  together by both electromagnetic induction and
  the mechanism he proposed. The linked electric
  and magnetic fields spread out in space just like
  a ripple spreads out from a stone after it has
  been dropped into a body of water. The energy
  that an EM wave carries is constantly being
  exchanged between its fluctuating electric and
  magnetic fields. In empty space, the speed of an
  EM wave should have the same value as the speed
  of light, 3 x 108 m/s (186,000 mi/s), regardless
  of the frequency or the amplitude of the wave.

14
Creating an EM Wave

• An alternating-current generator is connected to
  a pair of metal rods. Each rod has only a single
  charge at any time, with one rod having a
  positive charge and the other rod having a
  negative charge.
• First, the charges are moving apart from one end
  to the other end of their rod. Since they are
  charges, they have an electric field, and since
  they are moving, they have a magnetic field,
  which is perpendicular to the electric field.
• When the charges have reached the end of their
  rods, they stop, and thus, there is no longer a
  magnetic field being produced. The magnetic field
  that was present before the charges stopped
  continues to spread out along with the electric
  field, though, and both the magnetic field and
  the electric field move with the speed of light.
• The charges then start to move towards each
  other, towards the first end of their rod. Since
  the charges are moving again, a magnetic field is
  produced, but this magnetic field is in the
  opposite direction of the magnetic field that was
  produced when the charges were first moving
  apart. Unlike the magnetic field, however, the
  electric field is in the same direction.
• Once the charges have reached the first end of
  their rods, the polarity of the rods change
  i.e. the rod that had a positive charge now has a
  negative charge, and the rod that had a negative
  charge now has a positive charge.
• The new charges begin to move apart again, to the
  other end of their rods. Since they are moving,
  there is a magnetic field that is produced that
  is in the same direction as the last magnetic
  field that was produced. An electric field is
  also present, but since the polarity of the rods
  changed, the electric field is in the opposite
  direction as before.
• The result of this sequence is that the outermost
  electric and magnetic field lines form closed
  loops that are no longer joined to the
  oscillating charges. Thus, the loops move freely
  though space and make up an EM wave. Closed loops
  of electric and magnetic filed lines continue to
  form and expand outward as the charges in the two
  rods continue this sequence of moving back and
  forth.

15
The Relationship Between Electric Fields and
Magnetic Fields in Electromagnetic Waves

• The relationship between an electric field and a
  magnetic field in an electromagnetic wave is
  shown in the below figure. The fields are
  represented by vectors instead of field lines so
  that the magnitude and the direction of the
  fields in the path of the wave can be shown. The
  fields are perpendicular to each other and to the
  direction of the wave. The fields remain in step
  as they periodically reverse their directions.

16
Types of EM Waves Radios

• The existence of EM waves wasnt proven in
  Maxwells lifetime. Their existence and the fact
  that they behave exactly as Maxwell thought was
  shown through experiments performed by the German
  physicist Heinrich Hertz in 1887.
• Even though Hertz was not concerned about
  commercial uses of EM waves, other scientists and
  engineers were, and thus, the radio was created.
  A radio signal is sent by EM waves that are
  produced by electrons that move back and forth
  millions of times per second in the antenna at
  the sending station. The radio signal resides in
  one of two variations
• In amplitude modulation (AM), it resides in
  variations in the strength of the waves.
• In frequency modulation (FM), it resides in
  variations in the frequency of the waves.
• Frequency modulation is less subject to random
  disturbances (i.e. static) than amplitude
  modulation.
• When the EM waves reach the antenna of a
  receiver, the electrons that are in the antenna
  vibrate in step with the waves. A receiver can be
  tuned to respond only to a narrow frequency band.
  Thus, because transmitters operate on different
  frequencies, a receiver can pick up only the sent
  out signals it wishes. The currents that are set
  up in the antenna of a receiver are very weak,
  but they are strong enough for electronic
  circuits in the receiver to extract the signal
  from the current and turn it into sounds from a
  loudspeaker.

17
Other Types of EM Waves

• The maximum frequency of ordinary radio waves is
  up to about 2 MHz (1 MHz 1 megahertz 106 Hz).
• The maximum frequency of long-range short-wave
  communication is up to about 30 MHz.
• Television and radar use a maximum frequency that
  is higher than 30 MHz. Since waves with
  frequencies this high are short, they cannot be
  reflected by the ionosphere as radio waves are.
  Thus, they need to be obtained by direct
  reception, which is limited by the horizon,
  unless rebroadcast by a satellite station.
• Frequencies that are around 10 GHz (1 GHz 1
  gigahetz 109 Hz) correspond to waves that have
  wavelengths that are only a few centimeters.
  These waves can be readily formed into narrow
  beams. These beams can be reflected by solid
  objects, such as airplanes and ships, which forms
  the basis of radar, which stands for RAdio
  Detection And Ranging. In radar, a rotating
  antenna is used to send out the pulsed beam. The
  distance of a particular target is found from the
  time that is needed for the echo of the pulsed
  beam to return to the rotating antenna. The
  direction of the particular target is the same
  direction in which the rotating antenna is
  pointing at that time.
• The spectrum (i.e. range) of EM waves is shown in
  the figure below. The human eye is only able to
  detect light waves in a very short frequency
  band. This band is from about 4.3 x 1014 Hz,
  which is the frequency for red light, to 7.5 x
  1014 Hz, which is the frequency for violet light.
  This range is referred to as visible light. At
  frequencies lower than visible light, there is
  infrared radiation, and at frequencies higher
  than visible light, there is ultraviolet
  radiation. X-rays and the gamma radiation from
  atomic nuclei are at even higher frequencies than
  ultraviolet radiation.

18
Light Rays

• Much of the behavior of light waves can be
  understood without referencing its
  electromagnetic nature. Also, light can be
  thought of as a ray instead of a wave. This
  simpler model of light can be applied to many
  situations, although not all.
• We become aware that light travels in a straight
  line when we are young. This can be seen by the
  beam of a flashlight on a foggy night. Our entire
  orientation to the world around us, our sense of
  the location of things in space, depend on the
  assumption that light travels in a straight line.
• But, we also are aware that light does not always
  travel in a straight line. The way we see most
  objects is by reflected light, which is light
  that has been turned sharply once it has hit a
  surface. When objects are seen through water or
  though the heated air that is rising above a
  flame, they have a distorted appearance, which
  further shows that light has the ability to be
  bent from a straight-line path. The light in
  these cases (i.e. when objects are seen through
  water or though the heated air that is rising
  above a flame) is said to be refracted. Light is
  refracted when it moves from one transparent
  material to another one.
• Our conscious mind recognizes that light can be
  both reflected and refracted, but it is still
  easy to be deceived about the true positions of
  things. For instance, when we are looking to a
  mirror, our eyes seem to tell us that the light
  comes from an image that is behind the mirror,
  when what we are really seeing is that light
  travels to the mirror, and then from the mirror
  to our eyes. Another example is when we observe
  the legs of someone standing in shallow water.
  The legs of this person appear shorter than they
  do in air because the light that is going from
  the water to the air is bent. Since our eyes and
  our brains have no way to account for this, we
  see the illusion instead of the reality.
• Much about the behavior of light can be learned
  by studying the paths that light follows under
  various circumstances. When the medium is
  uniform, light appears to travel in a straight
  path. The straight lines of the motion of light
  in a uniform medium are called rays.

19
Reflection

• When we look in the mirror, light from all parts
  of all the objects we see are reflected from the
  mirror back to our eyes.
• The light from the foot in the figure follows the
  path CCE. Our eyes see the ray CE and project
  the ray in a straight line. Thus, our eyes
  register the foot at the proper distance, but it
  appears as if it is behind the mirror, at the
  point C. All the light rays from other parts of
  our body are reflected similarly, and thus, a
  complete image is formed that appears to be
  behind the mirror.
• In all mirror images, left and right are
  interchange because front and back have been
  reversed by the reflection. Thus, a printed page
  appears backward in a mirror, and what appears to
  be your right hand in a mirror is actually your
  left hand.
• Images are not seen in walls and furniture as
  they are in mirrors due to the relative roughness
  of surfaces. Rays of light are reflected from
  walls in the same way as they are in mirrors, but
  since there are surface irregularities, the
  reflected rays of light are scattered in all
  directions. The wall is seen by the scattered
  light that is reflected from it.

20
Refraction

• No matter from where the wind is blowing, waves
  always approach a beach almost exactly at right
  angles to the shore. Further out in the ocean,
  however, the direction of a wave many be at a
  slating angle, which is referred to as oblique,
  to the shore. As the waves move towards the
  shore, they swing round so that their crests
  become approximately parallel to the shoreline.
  This is an example of refraction.
• As the obliquely moving wave is traveling towards
  shore, the end that is nearer to the shore
  encounters shallow water before its outer end
  does. The friction between the end that is nearer
  to the shore and the sea bottom slows down this
  part of the wave. As more and more of the wave
  encounters shallow water, more and more of the
  wave is slowed down. As the water gets even
  shallower, the slowing of the water becomes more
  pronounced. Thus, the whole wave turns until it
  is moving almost directly shoreward. The wave
  turns because part of the wave was forced to move
  more slowly than the rest of the wave. Thus, this
  refraction is caused by differences in speed
  across the wave.
• Light will be bent at a sharp angle if it crosses
  a definite boundary between regions in which it
  moves at different speeds. This effect can be
  shown as ripples that move obliquely from deep
  water to shallow water in a tank. If the waves
  approach the boundary of the regions at right
  angles, there is no refraction because the change
  in the speed of the wave takes place across the
  entire wave at the same time.

21
Bending Light 1

• Water is always deeper than it seems to be,
  whether it is in a bathtub, a swimming pool, or a
  puddle. The reason for this is because when light
  goes from one medium to another medium in which
  the speed of light is different (i.e. from water
  to air in this case), light is reflected i.e.
  it changes direction. This effect is similar to
  the refraction of water waves.
• A ray of light from the stone in the figure
  follows the bent path ABE to our eyes, but what
  our brain registers is that the segment BE is
  part of the straight-line path that starts at A.
• The index of refraction (n) of a medium is the
  ratio between the speed of light (c) in free
  space to the speed of light in the medium (v) n
  c/v
• This equation shows that the greater the index of
  refraction, the more a light ray is deflected
  when it enters a medium at an oblique angle from
  the air. Some examples of indexes of reflection
  water 1.33, ordinary glass and clear plastics
  1.52, diamond 2.42. The index of reflection of
  air is assumed to be 1 because the speed of light
  in air is so close to the speed of light in free
  space.
• The apparent depth (h) of an object in a medium
  that is seen from a different medium is related
  to its actually depth (h) and to the index of
  refraction (n) of the medium in which the object
  is in h h/n

22
Example Calculation of Bending Light

• Example What is the index of refraction of a
  medium if the speed of light in the medium is 8.7
  x 108 m/s? What is the apparent depth of an
  object in this medium if it has an actual depth
  of 6.7m?
• Answer 1
• Given 8.7 x 108 m/s
• Looking for index of refraction
• Equation n c/v
• Solution n (3.0 x 108 m/s)/(8.7 x 108 m/s)
  0.345
• Answer 2
• Given 6.7m, 0.345
• Looking for apparent depth
• Equation h h/n
• Solution h 6.7m/0.345 19.4m

23
Bending Light 2

• If light travels more slowly in the second medium
  (i.e. the medium it is entering) than it did in
  the first medium (i.e. the medium it is exiting),
  the light rays that enter obliquely are bent
  toward a perpendicular to the surface between the
  two different mediums. If light travels faster in
  the second medium than it did in the first
  medium, the rays are bent away from the
  perpendicular. If light enters another medium
  perpendicular to the surface between the two
  different mediums, it does not change its
  direction.

24
Internal Reflection

• Because light travels more slowly in glass than
  in air, when light goes from glass to air at an
  oblique angle, it is refracted away from the
  perpendicular to the surface of the glass. If the
  angle is shallow enough, the light will be bent
  back into the glass. This is referred to as
  internal reflection.
• The path that a light ray takes is always
  reversible. To see what the eyes of a fish see,
  the path of the light rays needs to be reversed.
  When the path is reversed, it is shown that the
  light from above the waters surface can reach
  the eyes of a fish only though a circle on the
  surface of the water. The light rays from above
  the surface of the water are all brought together
  in a cone with an angular width of about 98.
  Anything outside this cone is a reflection of the
  underwater scene.

25
Lenses

• A lens is a piece of glass or other transparent
  material that has been shaped so that it can
  produce an image by refracting the light that
  comes from an object.
• Lenses are used for many purposes
• They are used in eyeglasses to improve vision.
• They are use din cameras to record scenes.
• They are used in projectors to show images on a
  screen.
• They are used in microscopes to enable small
  things to be seen.
• They are used in telescopes to enable distance
  things to be seen.
• There are two kinds of lenses
• A converging lens is thicker in the middle than
  at its rim. A converging lens brings a parallel
  beam of light to a single focal point, F. In this
  case, F is called a real focal point because the
  light rays pass though it. If sunlight goes
  though a converging lens, the concentration of
  radiant energy in a single focal point many be
  enough to burn something that is placed on the
  other side of the lens. The distance from the
  lens to its focal point is called the focal
  length of the lens.
• A diverging lens is thinner in the middle than at
  its rim. A diverging lens spreads out a parallel
  beam of light so that the rays of light seem to
  have come from some focal point, F, behind the
  lens. In this case, F is called a virtual focal
  point because the light rays dont actually pass
  though it they only appear to.

26
Ray Tracing A Converging Lens

• A scale drawing of a lens can be used to find the
  properties of the image of an object that is
  formed by the lens. The paths of two different
  light rays from a point of interest on the object
  are traced to where they (or their extensions, if
  it is a virtual image) come together again after
  passing though the lens. A lens has two focal
  points, one on each side. Both focal points are
  the same distance, f, from the center of the
  lens.
• For a converging lens, the three simplest rays to
  trace are
• A ray that leaves the object parallel to the axis
  of the lens. The lens deviates this ray so that
  it passes though the focal point on the other
  side (i.e. the far side) of the lens.
• A ray that passes though the center of the lens.
  The lens does not deviate this ray.
• A ray that passes though the focal point on that
  side of the lens (i.e. the near side). The lens
  deviates the ray so that it continues parallel to
  the axis of the lens.
• If an object is 2f from the center of the lens,
  its image will also be 2f from the center of the
  lens on the other side of the lens. The image is
  a real image because it is passing though a
  converging lens that is the same size as the
  object, but is inverted.

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Ray Tracing A Slide Projector

• When a slide projector uses a converging lens, it
  uses it to produce an enlarged image on a screen.
  Here, the object is in between f and 2f away from
  the center of the lens. The image is a real image
  that is more than 2f away from the center of the
  lens and is thus larger is size than the object.
  The image is also inverted.

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Ray Tracing A Camera

• When a camera uses a converging lens, it uses it
  to produce a reduced image on the film. Here, the
  object is more than 2f away from the center of
  the lens. The image is a real image that is in
  between f and 2f away from the center of the lens
  and is thus smaller is size than the object. The
  image is also inverted.

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Ray Tracing A Magnifying Glass

• When a magnifying glass uses a converging lens,
  it uses it to produce an enlarged image. Here,
  the object is closer than f from the center of
  the lens. The image is a virtual image because
  even though it can be seen by the eye, it cannot
  appear on a screen because no light rays actually
  pass though it. The image also seems to be on the
  same side of the lens as the object because the
  refracted light rays diverge as though coming
  from a point behind it. The image is between f
  and 2f away from the center of the lens and is
  larger is size than the object. The image is not
  inverted as it was in the other cases.

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Ray Tracing A Diverging Lens

• A ray that enters a diverging lens parallel to
  the axis of the lens is deviated away from the
  axis, instead of towards the axis as a converging
  lens does. Thus, an object always produces a
  virtual image when it is seen though a diverging
  lens. The image is not inverted, it is smaller
  than the object, and it is closer to the lens
  than the object is.

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The Eye

• The human eye uses the following parts
• The cornea is the transparent outer membrane of
  the eye.
• The lens is a jellylike substance that, with the
  cornea, is used to focus incoming light onto the
  sensitive retina.
• The retina converts what is seen into nerve
  impulses that are carried to the brain by the
  optic nerve.
• The ciliary muscle focuses on objects that are
  different distances away by changing the shape,
  and thus the focal length of the lens.
• The iris is the colored part of the eye that acts
  like a diaphragm of a camera to control the
  amount of light that enters the pupil.
• The pupil is the opening of the iris. In bright
  light, the pupil is small, and in dim light, it
  is large. A fully opened pupil lets in about 16
  times more light that an fully contracted one.
  The retina also can cope with a wide range of
  brightness.
• The retina has about a million tiny structures
  called cones and rods that are sensitive to
  light. The cones are specialized for color
  vision. They occur in three types one that
  responds to red light, one that responds to green
  light, and one that responds to blue light. Rods
  need a lot less light than cones do to be
  activated, but rods cannot distinguish colors.
  When the illumination is poor, we see shades of
  gray, like a black-and-white photograph. Because
  the central region of the retina contains only
  cones, in dim light, it is easier to see
  something by looking a bit to one side of it,
  instead of looking directly at it.

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Defects of Vision

• There are two common defects of vision
• Farsightedness is when the eyeball is too short,
  and thus light from nearby objects come to focus
  behind the retina. Here, close objects cannot be
  seen, but distant ones can be seen clearly.
  Farsightedness is corrected by a converging
  eyeglass lens.
• Nearsightedness is when the eyeball is too long,
  and thus light from distant objects come to focus
  in front of the retina. Here, far objects cannot
  be seen, but close objects can be seen clearly.
  Nearsightedness is corrected by a diverging
  eyeglass lens.
• Sometimes the cornea or the lens of the eye has
  different curvatures in different planes when
  light rays that are in one plane are in focus on
  the retina of the eye, light rays that are in
  other planes are either in focus in front of the
  retina or behind the retina. Thus, only one bar
  of a cross will be in focus at any time. This
  condition is called astigmatism. Astigmatism
  causes eyestrain because the eye is constantly
  changing the focus of the lens as it tries to
  produce a completely sharp image of what it sees.
  A cylindrical corrective lens is used to correct
  astigmatism.