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3 Pages

### Chapt21_22_VG

Course: PHYSICS 270, Fall 2010
School: Maryland
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Word Count: 609

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21 Chapters &amp; 22 Interference and Wave Optics Waves that are coherent can add/cancel Patterns of strong and weak intensity Single Spherical Source Approximate Electric Field: E (r, t ) = A(r ) cos( kr t + ) Field depends on distance from source and time. Typically A( r ) 1 / r Most important dependence is in the cosine Two sources that have exactly the same frequency. Coherent E (r, t ) = A(r1 )...

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21 Chapters & 22 Interference and Wave Optics Waves that are coherent can add/cancel Patterns of strong and weak intensity Single Spherical Source Approximate Electric Field: E (r, t ) = A(r ) cos( kr t + ) Field depends on distance from source and time. Typically A( r ) 1 / r Most important dependence is in the cosine Two sources that have exactly the same frequency. Coherent E (r, t ) = A(r1 ) cos( kr1 t + 1 ) r1 + A(r2 ) cos( kr2 t + 2 ) r2 Sources will interfere constructively when ( kr1 + 1 ) ( kr2 + 2 ) = 2 m m = 0, 1, 2, ... Sources will interfere destructively when 1 ( kr1 + 1 ) ( kr2 + 2 ) = 2 m + 2 Field and Intensity far from sources* E (r, t ) = A(r1 ) cos( kr1 t ) + A(r2 ) cos( kr2 t ) Trigonometry suppose A(r1 ) = A(r2 ) E (r, t ) 2 A cos( kr ) cos( kr t ) 2 Field amplitude depends on space *Special case r = r1 r2 r +r r= 1 2 2 Field oscillates in time. 1 = 2 = 0 cos( A) + cos( B) A+ B AB = 2 cos( ) cos( ) 2 2 Average intensity depends in difference in distance to sources, r Field kr E (r, t ) 2 A cos( ) cos( kr t ) 2 r = r1 r2 r +r r= 1 2 2 I ave 1.5 1 1 0 k r = 2 A cos( ) 2 0 2 2 1.2 1 kr ) 2 0.8 Intensity 0.5 E 2 A cos( I= Intensity 0 ! 2 E 0 0 0.6 0.4 -0.5 0.2 -1 0 -1.5 -15 -10 -5 0 t 5 10 15 -15 -10 -5 0 t 5 10 15 Interference of light Coherence because sources are at exactly the same frequency Sources will interfere constructively when ( kr1 + 1 ) ( kr2 + 2 ) = kr = 2 m Phases same because source comes from a single incident plane wave Dark fringes kr = kd sin = 2 m sin m m = m / d m = 0, 1, 2, ... 1 sin m m = m + / d 2 Intensity on a distant screen I ave 1 0 k r = 2 A cos( ) 2 0 2 kr = kd sin kd = Fringe spacing I= 0 2 E 0 2 L 2 y d L y = L d Intensity from a single source 1 0 2 I1 = A 2 0 Maximum Intensity at fringe I = 1 fringe 0 2 2 A = 2 I1 2 0 d Real pattern affected by slit opening width and distance to screen Diffraction Grating N slits, sharpens bright fringes Bright fringes at same angle as for double slit sin m = m / d m = 0, 1, 2, ... Location of Fringes on distant screen sin m = m / d ym = tan m L Intensity on a distant screen Average over time I ave I= 0 2 E 0 1 =I 2 Intensity from a single slit amplitude from a single slit 1 0 2 I1 = A 2 0 At the bright fringe N slits interfere constructively I fringe = 1 0 2 NA = N 2 I1 2 0 Spatial average of intensity must correspond to sum of N slits I SA = NI1 I fringe I SA =N width of fringe fringe width 1 = fringe spacing N sin m = m / d ym = tan m L Measuring Light Spectra Light usually contains a superposition of many frequencies. The amount of each frequency is called its spectrum. Knowing the components of the spectrum tells us about the source of light. Composition of stars is known by measuring the spectrum of their light. sin m = m / d ym = tan m L Accurate resolution of spectrum requires many lines Reflection Grating Propagation of wave fronts from a slit with a nonzero width Sources are not points. How do we describe spreading of waves? Ans. Just solve Maxwells equations. (wave equation) That is not always so easy. In the past, not possible. In the distant past equations were not known. Huygens (1629-1695) Principle 1. Each point on a wave front is the source of a spherical wavelet that spreads out at the wave speed. 2. C. Huygens At a later time, the shape of the wave front is the line tangent to all the wavelets. R. Plant Not the same person. Wikimedia Commons www.guerrillacandy.com/.../ Huygens Principle: When is there perfect destructive interference? Destructive when a r12 = sin = 2 2 1 cancels 2 3 cancels 4 5 cancels 6 Etc. Also: a sin p = 2p 2 p = 1, 2, 3.....
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