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# guide14 - Chapter 14 Interference and Diffraction 14.1...

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Chapter 14 Interference and Diffraction 14.1 Superposition of Waves. ................................................................................... 14-2 14.2 Young’s Double-Slit Experiment . .................................................................... 14-4 Example 14.1: Double-Slit Experiment. ............................................................... 14-7 14.3 Intensity Distribution . ....................................................................................... 14-8 Example 14.2: Intensity of Three-Slit Interference . ........................................... 14-11 14.4 Diffraction. ...................................................................................................... 14-13 14.5 Single-Slit Diffraction. .................................................................................... 14-13 Example 14.3: Single-Slit Diffraction . ............................................................... 14-15 14.6 Intensity of Single-Slit Diffraction . ................................................................ 14-16 14.7 Intensity of Double-Slit Diffraction Patterns. ................................................. 14-19 14.8 Diffraction Grating . ........................................................................................ 14-20 14.9 Summary. ........................................................................................................ 14-22 14.10 Appendix: Computing the Total Electric Field. ............................................ 14-23 14.11 Solved Problems . .......................................................................................... 14-26 14.11.1 Double-Slit Experiment . ........................................................................ 14-26 14.11.2 Phase Difference. ................................................................................... 14-27 14.11.3 Constructive Interference. ...................................................................... 14-28 14.11.4 Intensity in Double-Slit Interference . .................................................... 14-29 14.11.5 Second-Order Bright Fringe . ................................................................. 14-30 14.11.6 Intensity in Double-Slit Diffraction. ...................................................... 14-30 14.12 Conceptual Questions . .................................................................................. 14-33 14.13 Additional Problems . .................................................................................... 14-33 14.13.1 Double-Slit Interference. ........................................................................ 14-33 14.13.2 Interference-Diffraction Pattern. ............................................................ 14-33 14.13.3 Three-Slit Interference. .......................................................................... 14-34 14.13.4 Intensity of Double-Slit Interference . .................................................... 14-34 14.13.5 Secondary Maxima . ............................................................................... 14-34 14.13.6 Interference-Diffraction Pattern. ............................................................ 14-35 14-1

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Interference 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, 11 0 1 1 122 0 2 2 ( , ) sin( ), (,) s i n ( ) xt kx t t ψψ ω φ 2 ψ ±+ == ± + (14.1.1) the resulting wave is simply 10 1 1 1 2 2 2 20 sin( ) sin( ) t t ωφ + + ± + (14.1.2) The interference is constructive if the amplitude of x t is 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 : 0 t = 12 ( ) sin , ( ) 2sin 4 xx x x π ⎝⎠ + (14.1.3) The resultant wave is given by 14-2
() 12 () s i n 2 s i n 1 2s i n 2c o s 4 x xxx x x π x ψ ψψ ⎛⎞ =+=+ + = + + ⎜⎟ ⎝⎠ (14.1.4) where we have used sin( ) sin cos cos sin α βα β + =+ (14.1.5) and sin( / 4) cos( / 4) 2 / 2 ππ == . Further use of the identity [] 22 sin cos sin cos cos sin sin cos sin( ) ab ax bx ab x x x x x φφ φ += + + ++ + + (14.1.6) with 1 tan b a = (14.1.7) then leads to 5 2 2s i n ( ) xx = (14.1.8) where 1 tan ( 2 /(1 2)) 30.4 0.53 rad. = ° = The superposition of the waves is depicted in Figure 14.1.2. Figure 14.1.2 Superposition of two sinusoidal waves. We see that the wave has a maximum amplitude when sin( ) 1 x + = , or / 2 x =− . The interference there is constructive. On the other hand, destructive interference occurs at 2.61 rad x =−= , wheresin( ) 0 = . 14-3

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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 φ π = , this phase difference must not change with time.
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## This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.

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guide14 - Chapter 14 Interference and Diffraction 14.1...

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