Two-Source Constructive and Destructive Interference Conditions

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1 Two-Source Constructive and Destructive Interference Conditions

2 Crests

3 Crests

4 Crests

5 Crests ath length difference: l = - Travel time difference: t = t 2 - t 1 t = ( - )/v

6 Crests

7 Crests

8 h 1 g 1 f 1 e 1 d 1 c 1 b 1 a 1 Crests a 2 c 2 b 2 d 2 g 2 f 2 e 2 h 2

9 Q1 h 1 g 1 f 1 e 1 d 1 c 1 b 1 a 1 Crests d 2 c 2 b 2 a 2 (A) How long ago, before this snapshot was taken, did a 1, b 1, c 1, d 1, e 1, f 1, g 1, h 1 leave source? How long ago did a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 leave source? Express all your results, here and in the following in terms of the period of oscillation, T! Tabulate the results! g 2 f 2 e 2 Reminder: It takes 1 period for a crest or trough to trave wavelength h 2 (B) Tabulate all pairs of crests and/or troughs which left their resp. sources simultaneously. (C) Do the results in (A) depend on or?

10 Q2 h 1 g 1 f 1 e 1 d 1 c 1 b 1 a 1 Crests (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector,? Tabulate! h 2 g 2 f 2 e 2 d 2 c 2 b 2 a 2 (B) Assume is positioned so that and are equal: = How long after a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector,? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at. (D) Is there constructive or destructive interference at? Or neither? Explain!

11 Q3 h 1 g 1 f 1 e 1 d 1 c 1 b 1 a 1 Crests (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector,? Tabulate! h 2 g 2 f 2 e 2 d 2 c 2 b 2 a 2 (B) Assume is positioned so that exceeds by one wavelength, λ: = + λ How long after a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector,? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at. (D) Is there constructive or destructive interference at? Or neither? Explain!

12 Q4 h 1 g 1 f 1 e 1 d 1 c 1 b 1 a 1 Crests (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector,? Tabulate! h 2 g 2 f 2 e 2 d 2 c 2 b 2 a 2 (B) Assume is positioned so that exceeds by two wavelengths, 2λ: = + 2λ How long after a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector,? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at. (D) Is there constructive or destructive interference at? Or neither? Explain!

13 Q5 h 1 g 1 f 1 e 1 d 1 c 1 b 1 a 1 Crests (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector,? Tabulate! h 2 g 2 f 2 e 2 d 2 c 2 b 2 a 2 (B) Assume is positioned so that is shorter than by two wavelengths, 2λ: = - 2λ How long after (+) or before (-) a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector,? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at. (D) Is there constructive or destructive interference at? Or neither? Explain!

14 Q6 h 1 g 1 f 1 e 1 d 1 c 1 b 1 a 1 Crests (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector,? Tabulate! h 2 g 2 f 2 e 2 d 2 c 2 b 2 a 2 (B) Assume is positioned so that exceeds by one half-wavelengths, λ/2: = + λ/2 How long after (+) or before (-) a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector,? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at. (D) Is there constructive or destructive interference at? Or neither? Explain!

15 Q7 h 1 g 1 f 1 e 1 d 1 c 1 b 1 a 1 Crests (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector,? Tabulate! h 2 g 2 f 2 e 2 d 2 c 2 b 2 a 2 (B) Assume is positioned so that is shorter than by three half-wavelengths, 3λ/2: = - 3λ/2 How long after (+) or before (-) a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector,? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at. (D) Is there constructive or destructive interference at? Or neither? Explain!

16 Q8 h 1 g 1 f 1 e 1 d 1 c 1 b 1 a 1 Crests (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector,? Tabulate! h 2 g 2 f 2 e 2 d 2 c 2 b 2 a 2 (B) Assume is positioned so that is shorter then by one quarter-wavelength, λ/4: = - λ/4 How long after (+) or before (-) a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector,? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at. (D) Is there constructive or destructive interference at? Or neither? Explain!

17 Q9 h 1 g 1 f 1 e 1 d 1 c 1 b 1 a 1 Crests (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector,? Tabulate! h 2 g 2 f 2 e 2 d 2 c 2 b 2 a 2 (B) Assume is positioned so that is exceeds by two third-wavelengths, 2λ/3: = + 2λ/3 How long after (+) or before (-) a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector,? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at. (D) Is there constructive or destructive interference at? Or neither? Explain!

18 Q10 h 1 g 1 f 1 e 1 d 1 c 1 b 1 a 1 Crests Summarize your results for constructive and destructive interference at in terms of two simple mathematical conditions for the ath length difference: h 2 g 2 f 2 e 2 d 2 c 2 b 2 a 2 l = - and, equivalently, for the Travel time difference: t = t 2 - t 1 t = ( - )/v

Constructive Interference =Intensity Maximum: ath length difference: l = - = m λ Travel time difference: t = t 2 - t 1 = m T Destructive Interference =Intensity Minimum: ath

19 Constructive Interference =Intensity Maximum: ath length difference: l = - = m λ Travel time difference: t = t 2 - t 1 = m T Destructive Interference =Intensity Minimum: ath length difference: l = - = (m+1/2) λ Travel time difference: t = t 2 - t 1 = (m+1/2) T m= 0, +1, -1, +2, -2, (m+1/2) = +1/2, -1/2, +3/2, -3/2, where the period T is: T = λ / v

20 Interference athlength Geometry

21 Q11.1 d d = source-to-source spacing, = distance from to, = distance from to. Suppose and are two small loudspeakers, placed 6.8m apart and you can move to any location. What is the largest possible absolute value of the path length difference, Δl =. Explain your reasoning!

22 Q11.2 d Suppose the two small loudspeakers, and, spaced 6.8m apart, oscillate in phase, sending out sound waves of wavelength λ=2.2m. Constructive interference occurs at any location of where Δl = m λ. Here m can be any integer: 0, +1, -1, +2, -2, ; and m is called the order of the interference maximum. What is the largest possible order of interference, m, that can be observed, for any location of?

23 Q11.3 y d L O Lengths and coordinates needed to describe the positioning of sources, and, and detector, : d = source-to-source spacing, L = distance from observation screen to line of sources. y = y-coordinate of, with y-axis along the observation screen and origin O on midline between, and

24 Q11.3 (contd.) d/2 y d d/2 L O (A) Derive exact equations for and, each expressed in terms of d, L, and the y-coordinate of. Hint: ythagoras! (B) From this, obtain an exact equation for the pathlength difference, Δl, in terms of d, L and y (C) At home: Solve the equation from (B) for y, to express y in terms of of d, L, and Δl. Very difficult!

25 Q12.1 y d Θ L O The result in Q11.3 (B) is greatly simplified if d << L, by the so-called Fraunhofer approximation: Δl d sin Θ where tan Θ = y/l. Test this approximation against Q11.3 (B), for fixed d=5cm, fixed Θ=65deg, increasing values of L and y: Tabulate! Hint: Keep enough signif. digits! You re subtracting 2 large numbers with a very small difference.

26 Q12.2 y d Θ L O and, the two loudspeakers, spaced 6.8m apart, oscillate in phase, sending out sound waves of wavelength λ=2.2m. The detector is moved along the y-axis from y=- to y=+, at L = 150m. (A) Find the angles Θ and y-locations of all intensity maxima on the y-axis. How many are there? (B) Find the angles Θ and y-locations of all intensity minima on the y-axis. How many are there?

27 Q12.3 y d Θ L O and, the two loudspeakers, spaced distance d apart, oscillate in phase, sending out sound waves of wavelength λ. Detector is moved along the y-axis from y=- to y=+, at fixed distance L from the source line. The 1 st intensity maximum above midpoint O is found at Θ 1 =24.8 o. (A) How many intensity maxima are there, altogether, on the y-axis from y=- to y=+? (B) How many intensity minima are there, altogether, on the y-axis from y=- to y=+?

28 Q12.4 y d Θ L O and, the two loudspeakers, spaced distance d apart, oscillate in phase, sending out sound waves of wavelength λ. Detector is moved along the y-axis from y=- to y=+, at fixed distance L from the source line. The 3 rd intensity minimum above midpoint O is found at Θ 5/2 =53.8 o. (A) Find Θ m for all intensity maxima (m=0,+1,-1, ) on the y-axis from y=- to y=+? (B) Find Θ m+1/2 for all intensity minima (m+1/2=+1/2,-1/2,+3/2,-3/2, ) on the y-axis from y=- to y=+?

29 Multi-Slit Constructive Interference athlength Geometry Intensity lots

30 Multi-Slit (N-Slit) Interference and Diffraction Grating (N>>1) Notation: Δl = l k+1 l k same for k=1, 2,,N-1. O y Maximally constructive interference occurs when Δl = m λ with m integer Δl 2 Δl 3 Δl Again, by geometry: Δl d sin(θ) assuming L>> Nd; and tan(θ) = y/l Δl Δl

31 rincipal Maxima: sin(θ) = m λ/d with m integer Secondary Maxima

32 Q13 A diffraction grating placed parallel to an observation screen, 40cm from the screen, Is illuminated at normal incidence by coherent, monochromatic light (a laser beam). Assume Fraunhofer conditions (L>> Nd) are satisfied. (a) If the 1 st order principal maximum is observed on the screen 30cm above the central maximum, how many principal maxima altogether, incl. central maximum, are observable? (b) If the 2nd order principal maximum is observed on the screen 30cm above the central maximum, how many principal maxima altogether, incl. central maximum, are observable? Find the angles, Θ, and y-coordinates of all principal maxima on the screen: Tabulate! (c) How would your answers change if the device had been a double-slit (N=2) or a quintuple-slit (N=5) instead of a diffraction grating?

Solutions to Conceptual Practice Problems PHYS 1112 In-Class Exam #2A+2B

Solutions to Conceptual Practice Problems PHYS 1112 In-Class Exam #2A+2B Solutions to Conceptual ractice roblems HYS 1112 In-Class xam #2A+2B Thu. Mar. 19, 2009, 11:00am-12:15pm and 2:00pm-3:15pm C 2.01: In a two-source interference experiment two sources are oscillating in