The Wave Nature of Light - Sections 24.4 - 24.6
The Wave Nature of Light - Sections 24.4 - 24.6
The Wave Nature of Light - Sections 24.4 - 24.6
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Announcements Review Diffraction by a Single Slit Final QuestionsReading AssignmentRead section 24.10Homework Assignment 10Homework for Chapter 24 (due at the beginning <strong>of</strong> class on Friday, November 5)Q: 4, 5, 10, 17, 22, 30P: 4, 20, 30, 58PP: 37.2, 38.4<strong>The</strong> <strong>Wave</strong> <strong>Nature</strong> <strong>of</strong> <strong>Light</strong>
Announcements Review Diffraction by a Single Slit Final QuestionsYoung’s double-slit experiment: interference pattern<strong>The</strong> slits S 1 and S 2 are separated by a distance d, and the source is monochromaticAssume that the viewing screen is located a perpendicular distance L ≫ d from the barrier containingthese slitsTo reach any arbitrary point P in the upper half <strong>of</strong> the screen, a wave from the lower slit must travelfarther than a wave from the upper slit by a distance (called the path distance ∆D)∆D = D 1 − D 2 = d sinθ<strong>The</strong> value <strong>of</strong> ∆D determines whether the two waves are in phase when they arrive at point PIf ∆D is zero or some integer multiple <strong>of</strong> the wavelength, the two waves are in phase at point P andconstructive interference occurs<strong>The</strong>refore, the condition for bright fringes (or constructive interference) at point P isd sinθ bright = mλ m = 0,±1,±2,...When ∆D is an odd multiple <strong>of</strong> λ/2, the two waves arriving at point P are 180 ◦ out <strong>of</strong> phase and giverise to destructive interference<strong>The</strong>refore, the condition for dark fringes (or destructive interference) at point P is(d sinθ dark = m + 1 )λ m = 0,±1,±2,...2<strong>The</strong> <strong>Wave</strong> <strong>Nature</strong> <strong>of</strong> <strong>Light</strong>
Announcements Review Diffraction by a Single Slit Final QuestionsQuestionA viewing screen is separated from a double slit by 4.80 m. <strong>The</strong> distance between the two slits is 0.0300 mm.Monochromatic light is directed toward the double slit and forms and interference pattern on the screen. <strong>The</strong> firstdark fringe is 4.50 cm from the center line on the screen. Determine the wavelength <strong>of</strong> the light.<strong>The</strong> <strong>Wave</strong> <strong>Nature</strong> <strong>of</strong> <strong>Light</strong>
Announcements Review Diffraction by a Single Slit Final QuestionsQuestionA viewing screen is separated from a double slit by 4.80 m. <strong>The</strong> distance between the two slits is 0.0300 mm.Monochromatic light is directed toward the double slit and forms and interference pattern on the screen. <strong>The</strong> firstdark fringe is 4.50 cm from the center line on the screen. Determine the wavelength <strong>of</strong> the light.AnswerLet y = 4.50 cm be the distance <strong>of</strong> the first dark fringe from the center line<strong>The</strong>n, tanθ dark = y/L, where L = 4.80 m is the distance to the viewing screenSolving for θ dark , we find thatθ dark = tan −1 ( yL) ( )= tan −1 4.50 × 10−2 m= 0.537 ◦4.80 mFurthermore, dark fringes are caused by destructive interference so(d sinθ dark = m + 1 )λ2Solving for λ (and substituting m = 0), we find thatλ = 2d sinθ dark = 2(0.0300 × 10 −3 m)sin(0.537 ◦ ) = 5.62 × 10 −7 m = 562 nm<strong>The</strong> <strong>Wave</strong> <strong>Nature</strong> <strong>of</strong> <strong>Light</strong>
Announcements Review Diffraction by a Single Slit Final Questions<strong>The</strong> wave theory <strong>of</strong> light: a historical perspectiveNewton’s view <strong>of</strong> light – that it consisted <strong>of</strong> a stream <strong>of</strong> particles – was the prevailing view in mostscientific circles <strong>of</strong> the early 19th centuryDuring Newton’s lifetime, however, there was another theory, advanced by the Dutch physicist ChristianHuygens in 1678, that light was a wave (Huygens’ principle)In 1801 – 123 years after Hugyens first proposed that light was a wave – Thomas Young experimentallydemonstrated the wave nature <strong>of</strong> light (the double slit experiment) by demonstrating interferenceHowever, even with this remarkable demonstration, the scientific community was slow to adopt thewave-theory explanationIn 1819, the French Academy <strong>of</strong> Sciences organized a competition for an essay on diffraction – AugustinFresnel, a French physicist and supporter <strong>of</strong> the wave-theory <strong>of</strong> light, won (Huygens-Fresnel principle)<strong>The</strong> Academy, however, was unconvinced, with the preeminent French mathematician Siméon Poissonpointing out this “strange result”:If Fresnel’s theories were correct, then light waves should diffractinto the shadow region <strong>of</strong> a sphere as they pass its edge, producinga bright spot at the center <strong>of</strong> the shadowHowever, the French physicist and mathematician Dominique Arago soon experimentally verified thisstartling prediction (a Poisson bright spot)<strong>The</strong> <strong>Wave</strong> <strong>Nature</strong> <strong>of</strong> <strong>Light</strong>
Announcements Review Diffraction by a Single Slit Final Questions<strong>The</strong> <strong>Wave</strong> <strong>Nature</strong> <strong>of</strong> <strong>Light</strong>
Announcements Review Diffraction by a Single Slit Final QuestionsSingle-slit diffractionPreviously we saw what happens when monochromatic light from a distant source passes through twonarrow slits (an interference pattern is formed)Now we will examine what happens when monochromatic light from a distant source passes through onenarrow slitRemarkably, the diffraction pattern formed on a viewing screen consists <strong>of</strong> a very bright central maximumand a number <strong>of</strong> narrower and less bright secondary maxima to both sidesThis pattern would be total unexpected in geometrical opticsKeep in mind that the diffraction <strong>of</strong> light is not limited to situations <strong>of</strong> light passing through a narrowopening; it also occurs when light passes an edge<strong>The</strong> <strong>Wave</strong> <strong>Nature</strong> <strong>of</strong> <strong>Light</strong>
Announcements Review Diffraction by a Single Slit Final QuestionsSingle-slit diffraction: locating the minimaWe will now examine how coherent, monochromatic light is diffracted by a single narrow slit <strong>of</strong> width aWhen the diffracted light reaches the viewing screen, waves from different points within the slit undergointerference and produce a diffraction pattern <strong>of</strong> bright and dark fringes<strong>The</strong> single-slit experiment is more mathematically challenging than the double-slit experiment; as a result,we will find equations for the dark fringes onlyTo find the dark fringes, we will pair up all <strong>of</strong> the rays coming through the slit in a clever way, and thendetermine what conditions cause each pair to cancel each otherUsing this procedure, we find that the condition for minima (dark fringes) isasinθ = mλ m = 1,2,3,...In other words, in a single-slit diffraction experiment, dark fringes are produces when the path lengthdifferences (asinθ) between the top and bottom rays are positive integer multiples <strong>of</strong> λThis result may seem counterintuitive, because the waves <strong>of</strong> those two particles rays will be exactly inphase with each otherHowever, the result holds, because each <strong>of</strong> these rays will be part <strong>of</strong> a pair <strong>of</strong> waves that are exactly out <strong>of</strong>phase with each other<strong>The</strong> <strong>Wave</strong> <strong>Nature</strong> <strong>of</strong> <strong>Light</strong>
Announcements Review Diffraction by a Single Slit Final Questions<strong>The</strong> <strong>Wave</strong> <strong>Nature</strong> <strong>of</strong> <strong>Light</strong>
Announcements Review Diffraction by a Single Slit Final QuestionsQuestion<strong>Light</strong> <strong>of</strong> wavelength 633 nm is incident on a narrow slit. <strong>The</strong> angle between the first diffraction minimum on oneside <strong>of</strong> the central maximum and the first minimum on the other side is 1.20 ◦ . What is the width <strong>of</strong> the slit?<strong>The</strong> <strong>Wave</strong> <strong>Nature</strong> <strong>of</strong> <strong>Light</strong>
Announcements Review Diffraction by a Single Slit Final QuestionsQuestion<strong>Light</strong> <strong>of</strong> wavelength 633 nm is incident on a narrow slit. <strong>The</strong> angle between the first diffraction minimum on oneside <strong>of</strong> the central maximum and the first minimum on the other side is 1.20 ◦ . What is the width <strong>of</strong> the slit?Answer<strong>The</strong> width <strong>of</strong> the slit is given bywhere θ is measured from the center linea = mλsinθIn this problem, m = 1 and θ = 0.60 ◦ (why?)<strong>The</strong>refore, solving for a, we find thatm = 1,2,3,...a = (1)(6.33 × 10−7 m)sin(0.60 ◦ )= 6.04 × 10 −5 m = 60.4 µm<strong>The</strong> <strong>Wave</strong> <strong>Nature</strong> <strong>of</strong> <strong>Light</strong>
Announcements Review Diffraction by a Single Slit Final QuestionsReading AssignmentRead section 24.10Homework Assignment 10Homework for Chapter 24 (due at the beginning <strong>of</strong> class on Friday, November 5)Q: 4, 5, 10, 17, 22, 30P: 4, 20, 30, 58PP: 37.2, 38.4<strong>The</strong> <strong>Wave</strong> <strong>Nature</strong> <strong>of</strong> <strong>Light</strong>