37'4 Phasor Addition of Waves

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37.4 Phasor Addition of Waves
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Phasor Addition of Waves, E1

• The sinusoidal wave can be represented
  graphically by a Phasor of magnitude Eo rotating
  about the origin counterclockwise with an angular
  frequency ?
• E1 Eo sin ?t
• It makes an angle of ?t with the horizontal axis
• E1 is the projection on the vertical axis

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Phasor Addition of Waves, E2

• The second sinusoidal wave is
• E2 Eo sin (?t ?)
• It has the same amplitude and frequency as E1
• Its phase is ? with respect to E1

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Phasor Addition of Waves, ER

• The resultant is the sum of E1 and E2
• ER rotates with the same angular frequency ?
• The projection of ER along the vertical axis
  equals the sum of the projections of the other
  two vectors

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ER at a Given Time

• From geometry at t 0, ? ? / 2,
• ER 2E0cos? ?
• ER 2Eocos(? / 2)
• The projection of ER along the vertical axis at
  any time t is

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Finding the Resultant of Several Waves

• Represent the waves by phasors
• Remember to maintain the proper phase
  relationship between one phasor and the next
• The resultant phasor ER is the vector sum of the
  individual phasors

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Finding the Resultant of Several Waves, cont.

• At each instant, the projection of ER along the
  vertical axis represents the time variation of
  the resultant wave
• The phase angle ? is between ER and the first
  phasor
• The resultant is given by the expression
• EP ER sin (?t ?/2)

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Phasor Diagrams for Two Coherent Sources, Comments

• ER is a maximum at ? 0, 2p, 4p,
• The intensity is also a maximum at these points
• ER is zero at ? p, 3p,
• The intensity is also zero at these points
• These results agree with the results obtained
  from other procedures

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Phasor Diagrams for Two Coherent Sources, Diagrams
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Active Figure 37.11
(SLIDESHOW MODE ONLY)
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Three-Slit Interference Pattern

• Assume three equally spaced slits
• The fields are
• E1 Eo sin ?t
• E2 Eo sin (?t ?)
• E3 Eo sin (?t 2?)
• Phasor diagrams can be used to find the resultant
  magnitude of the electric field

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Three Slits Primary Maxima

• The phasor diagram shows the electric field
  components and the resultant field
• The field at P has a maximum value of 3Eo at
  ? 0, 2p, 4p ...
• These points are called primary maxima
• The primary maxima occur when the phasors are in
  the same direction

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Three Slits Secondary Maxima

• Secondary maxima occur when the wave from one
  slit exactly cancels the wave from another slit
• The field at P has a value of Eo
• These points occur at
• ? 0, p, 3p ...

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Three Slits, Minima

• Total destructive interference occurs when the
  wave from all the slits form a closed triangle
• The field at P has a value of 0
• These points occur at
• ? 0, 2p/3, 4p/3 ...

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Three Slits, Phasor Diagrams
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Active Figure 37.13
(SLIDESHOW MODE ONLY)
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Three Slits, Intensity Graphs

• The primary maxima are nine times more intense
  than the secondary maxima
• The intensity varies as ER2
• For N slits, the primary maxima is N2 times
  greater than that due to a single slit

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Three Slits, Final Comments

• As the number of slits increases, the primary
  maxima increase in intensity and become narrower
• As the number of slits increases, the secondary
  maxima decrease in intensity with respect to the
  primary maxima
• As the number of slits increases, the number of
  secondary maxima also increases
• The number of secondary maxima is always
• N 2 where N is the number of slits

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Example 37.3 Six slits (Quiz 37.5)

• Sketch the interference patter from six slits
• The number of secondary maxima is always
• N 2 where N is the number of slits
• Because N 6, the secondary maxima are 1/36 as
  intense as the primary maxima.

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37.5 Phase Changes Due To Reflection - Lloyds
Mirror

• An arrangement for producing an interference
  pattern with a single light source
• Waves reach point P either by a direct path or by
  reflection
• The reflected ray can be treated as a ray from
  the source S behind the mirror

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Interference Pattern from a Lloyds Mirror

• This arrangement can be thought of as a
  double-slit source with the distance between
  points S and S comparable to length d
• An interference pattern is formed
• The positions of the dark and bright fringes are
  reversed relative to the pattern of two real
  sources
• This is because there is a 180 phase change
  produced by the reflection

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Phase Changes Due To Reflection

• An electromagnetic wave undergoes a phase change
  of 180 upon reflection from a medium of higher
  index of refraction than the one in which it was
  traveling
• Analogous to a pulse on a string reflected from a
  rigid support

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Phase Changes Due To Reflection, cont.

• There is no phase change when the wave is
  reflected from a boundary leading to a medium of
  lower index of refraction
• Analogous to a pulse on a string reflecting from
  a free support

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Material for the Final Exam

• Examples to Read!!!
• NONE
• Homework to be solved in Class!!!
• Problems 23, 25