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MIT 8 02 - Chapter 14 Interference and Diffraction

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Superposition of WavesYoung’s Double-Slit ExperimentExample 14.1: Double-Slit ExperimentIntensity DistributionExample 14.2: Intensity of Three-Slit InterferenceDiffractionSingle-Slit DiffractionExample 14.3: Single-Slit DiffractionIntensity of Single-Slit DiffractionIntensity of Double-Slit Diffraction PatternsDiffraction GratingSummaryAppendix: Computing the Total Electric FieldSolved ProblemsDouble-Slit ExperimentPhase DifferenceConstructive InterferenceIntensity in Double-Slit InterferenceSecond-Order Bright FringeIntensity in Double-Slit DiffractionConceptual QuestionsAdditional ProblemsDouble-Slit InterferenceInterference-Diffraction PatternThree-Slit InterferenceIntensity of Double-Slit InterferenceSecondary MaximaInterference-Diffraction PatternChapter 14 Interference and Diffraction 14.1 Superposition of Waves..........................................................................................1 14.2 Young’s Double-Slit Experiment ...........................................................................3 Example 14.1: Double-Slit Experiment......................................................................6 14.3 Intensity Distribution..............................................................................................7 Example 14.2: Intensity of Three-Slit Interference ..................................................10 14.4 Diffraction.............................................................................................................12 14.5 Single-Slit Diffraction...........................................................................................12 Example 14.3: Single-Slit Diffraction ......................................................................14 14.6 Intensity of Single-Slit Diffraction.......................................................................15 14.7 Intensity of Double-Slit Diffraction Patterns........................................................18 14.8 Diffraction Grating ...............................................................................................19 14.9 Summary...............................................................................................................21 14.10 Appendix: Computing the Total Electric Field...................................................22 14.11 Solved Problems .................................................................................................25 14.11.1 Double-Slit Experiment ...............................................................................25 14.11.2 Phase Difference..........................................................................................26 14.11.3 Constructive Interference.............................................................................27 14.11.4 Intensity in Double-Slit Interference ...........................................................28 14.11.5 Second-Order Bright Fringe ........................................................................29 14.11.6 Intensity in Double-Slit Diffraction.............................................................29 14.12 Conceptual Questions .........................................................................................32 14.13 Additional Problems ...........................................................................................32 14.13.1 Double-Slit Interference...............................................................................32 14.13.2 Interference-Diffraction Pattern...................................................................32 14.13.3 Three-Slit Interference.................................................................................33 14.13.4 Intensity of Double-Slit Interference ...........................................................33 14.13.5 Secondary Maxima ......................................................................................33 14.13.6 Interference-Diffraction Pattern...................................................................34 0Interference and Diffraction 14.1 Superposition of Waves Consider a region in space where two or more waves pass through at the same time. According to the superposition principle, the net displacement is simply given by the vector or the algebraic sum of the individual displacements. Interference is the combination of two or more waves to form a composite wave, based on such principle. The idea of the superposition principle is illustrated in Figure 14.1.1. (a) (b) (c) (d) Figure 14.1.1 Superposition of waves. (b) Constructive interference, and (c) destructive interference. Suppose we are given two waves, 110111 22022( , ) sin( ), (,) sin( )xt kx t xt kx tψψ ωφ2ψ ψω±+==φ±+ (14.1.1) the resulting wave is simply 10 1 1 1 2 2 220(,) sin( ) sin( )xt kx t kx tψψ ωφ ωφψ+=±+ ±+ (14.1.2) The interference is constructive if the amplitude of (,)xtψis greater than the individual ones (Figure 14.1.1b), and destructive if smaller (Figure 14.1.1c). As an example, consider the superposition of the following two waves at : 0t = 12( ) sin , ( ) 2sin4xx x xπψ ψ⎛==⎜⎝⎠⎞+⎟ (14.1.3) 1The resultant wave is given by ()12() () () sin 2sin 1 2sin 2cos4x xxx x xπxψ ψψ⎛⎞=+=+ +=+ +⎜⎟⎝⎠ (14.1.4) where we have used sin( ) sin cos cos sinα βαβαβ+ =+ (14.1.5) and sin( / 4) cos( / 4) 2 /2ππ==. Further use of the identity []2222 222222sin cos sin coscos sin sin cossin( )abaxbx ab x xab abab x xab xφφφ⎡ ⎤+=+ +⎢ ⎥++⎣ ⎦=+ +=+ + (14.1.6) with 1tanbaφ−⎛⎞=⎜⎟⎝⎠ (14.1.7) then leads to () 5 22sin( )xxψ φ= ++ (14.1.8) where 1tan ( 2 /(1 2)) 30.4 0.53 rad.φ−=+=°= The superposition of the waves is depicted in Figure 14.1.2. Figure 14.1.2 Superposition of two sinusoidal waves. 2We see that the wave has a maximum amplitude when sin( ) 1xφ+=, or /2xπφ=−. The interference there is constructive. On the other hand, destructive interference occurs at 2.61 radxπφ=−=, wheresin( ) 0π=. In order to form an interference pattern, the incident light must satisfy two conditions: (i) The light sources must be coherent. This means that the plane waves from the sources must maintain a constant phase relation. For example, if two waves are completely out of phase with φπ=, this phase difference must not change with time. (ii) The light must be monochromatic. This means that the light consists of just one


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MIT 8 02 - Chapter 14 Interference and Diffraction

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