L Destructive interference:

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1 SPH3UW/SPH4UI Unit 9.3 Interference of Light in Two Diensions Page 1 of 11 Notes Physics Tool box A pair of ientical point sources operating in phase prouces a syetrical pattern of constructive an estructive interference. The Phase ifference between two waves can change if the waves travel ifferent lengths fro one point to another. For an interference pattern to appear, the light rays reaching any point P ust have a phase ifference that oes not change in tie. (sai to be coherent) Constructive interference: o P S1 P S ; 0,1,,3.. o sin ; 0,1,,3... y o ; 0,1,,3,... L Destructive interference: 1 o PS 1 PS ; 0,1,,3.. 1 o sin ; 0,1,,3... x 1 o ; 0,1,,3,... L The interference pattern prouce by two coherent sources of water waves can be reaily seen in a ripple tank with a shallow layer of water. Constructive an estructive interference can prouce fixe patterns of interference (sae frequency an thus sae wavelength). As successive crests an troughs travel out fro the two sources, they interfere soeties crest on crest an trough on trough, an soeties crest on trough. The interference fors patterns of lines of estructive interference (noal lines) an areas of constructive interference. The noal lines actually are hyperbolic in shape.

2 SPH3UW/SPH4UI Unit 9.3 Interference of Light in Two Diensions Page of 11 When the frequency of the sources increase, the wavelength ecreases, bringing the noal lines closer together an increasing there nuber. If the istance between the two sources is increase, the nuber of noal lines also increases. Young s Double Slit Experient We can apply the sae principles with light as we i with water, but our light ust have a couple of properties: 1. Must have two (or ore) waves. Must have the sae frequency 3. Must be coherent. Coherent Light: The sources ust aintain a constant phase with respect to each other. If we have two ientical points sources S 1 an S of coherent light, separate by a istance,, then the path ifference between a light ray fro the top an botto slit can prouce interference patterns. If the istance ifference is a ultiple of a coplete wavelength, then the waves will arrive in synch an we obtain constructive interference an thus a bright spot on the screen. If the rays arrive copletely out of synch (by 180 o ), then we have estructive interference an thus a ark spot will appear on the screen. If we take a point P 1 on one the screen an connect it to each of the two sources by the lines PS 1 1 an PS 1 we can eterine if there is a ifference in the two path lengths: Difference is calculate by PS 1 1 PS 1

3 SPH3UW/SPH4UI Unit 9.3 Interference of Light in Two Diensions Page 3 of 11 Constructive Interference For constructive interference (the waves arrive with crest to crest an trough to trough) the path ifference ust be ultiples of the wavelength of the light. PS 1 PS ; 0,1,,3.. Where eterines the constructive interference (bright) ban nuber. If =0 this is the central axiu, if =1 we have the first orer axiu (is on either sie of the central axiu. Destructive Interference For estructive interference (the waves arrive with crest to trough an trough to crest) the path ifference ust be ultiples of the wavelength of the light. 1 PS 1 PS ; 0,1,,3.. Where eterines the estructive interference (ark) ban nuber. If =0 this is the first orer iniu, if =1 we have the secon orer iniu.

4 SPH3UW/SPH4UI Unit 9.3 Interference of Light in Two Diensions Page 4 of 11 Exaple If one source to the screen is 1.x10-5 farther than the other source an re light with wavelength 600 n is use, eterine the orer nuber of the bright spot. Solution Path ifference is 5 PS 1 PS We nee only eterine how any ultiples of our re light wavelengths this is equal to: P S P S 1 PS 1 PS Therefore this woul represent the 0 th orer axiu. Now if we o not know the actual path ifferences the two rays travel, but we o know the angle a specific orer nuber of the interference pattern akes with respect to the line connecting the slits to the screen. Then we can also eterine a forula that relates the orer nuber with this angle. With the help of Figure below. The viewing screen is locate a perpenicular istance L fro the ouble slits, S1 an S. These slits are separate by a istance, an the source is onochroatic.

5 SPH3UW/SPH4UI Unit 9.3 Interference of Light in Two Diensions Page 5 of 11 To reach any point P in the upper half of the screen, a ray fro the lower slit ust travel farther than a ray fro the upper slit by a istance sin. This istance is calle the path ifference. If we assue that r 1 an r are parallel, which is approxiately true if L is uch greater than, then the path ifference is given by The value of point P. If sin sin rr 1 sin eterines whether the two waves are in phase when they arrive at is either zero or soe integer ultiple of the wavelength, then the two waves are in phase at point P an constructive interference results. Therefore, the conition for bright fringes, or constructive interference, at point P is: When sin sin ; 0,1,,3... is an o ultiple of, the two waves arriving at point P are 180 out of phase an give rise to estructive interference. Therefore, the conition for ark fringes, or estructive interference, at point P is: Exaple 1 sin ; 0,1,,3... Light of wavelength 450 n shines through openings 3.0 u apart. At what angle will the first-orer axiu occur? What is the angle of the first orer iniu? Solution For constructive interference sin sin 1 sin sin For Destructive Interference sin sin sin sin

6 SPH3UW/SPH4UI Unit 9.3 Interference of Light in Two Diensions Page 6 of 11 Finally it is useful to obtain expressions for the positions along the screen of the bright an ark fringes easure vertically fro O to P, the y istance. In aition to our assuption that L>>, we also assue that >>. Uner these conitions, the angle is very sall; thus, we can use the sall angle approxiation that sin tan Thus we have sin tan y L. This gives us our final forula for constructive an estructive interference patterns y Constructive: ; 0,1,,3,... L Destructive: y 1 ; 0,1,,3... L Exaple For light of wavelength 650 n, what is the spacing of the bright bans on a screen 1.5 away if the istance between slits is 6.6x10-6? Solution Since we are given the istance the screen is away fro the slits, then we can use y L y L

7 SPH3UW/SPH4UI Unit 9.3 Interference of Light in Two Diensions Page 7 of 11 Exaple Two ientical points sources 4.0 c apart are operating in phase at a frequency of 7.0 Hz. The sources generate interference patterns. A certain point on the first noal line is locate 9 c fro one source an 10 c fro the other. Deterine a) the wavelength of the waves b) the spee of the waves. a) b) 1 PS1 PS 1 10c 9c 0.0c v f 7.0Hz.0c 14 c s Exaple Approxiate the wavelength it the istance between the two sources is 3.0 c an the total nuber of noal lines is 4. sin c 0.86c

8 SPH3UW/SPH4UI Unit 9.3 Interference of Light in Two Diensions Page 8 of 11 Exaple The istance fro the right bisector to the secon noal line in a two-point interference pattern is 6.0 c. The istance fro the ipoint between the two sources to point P is 3 c. what is angle for the secon noal line. y sin L 6.0c sin 3c 11 Exaple A raio station operating at a frequency of 1500 khz has two ientical vertical ipole antennas space 400 apart, oscillating in phases. At istance uch greater than 400, in what irections is the intensity the greatest? Since we want constructive interference we use sin, with 0, 1,,

9 SPH3UW/SPH4UI Unit 9.3 Interference of Light in Two Diensions Page 9 of 11 sin Therefore 0, 30, 90 Exaple A point P is on the secon-orer axiu, 65 fro one source an 45 fro another source. The sources are 40 apart. Fin the wavelength of the wave an the angle the point akes in the ouble slit. Solution First we will locate the wavelength P S P S Now with the wavelength, we will locate the angle sin 1 sin 10 1 sin 30 40

10 SPH3UW/SPH4UI Unit 9.3 Interference of Light in Two Diensions Page 10 of 11 Exaple You are to eterine the wavelength of a source when it is shone through slits that are 0.10 apart an the interference pattern is create on a screen.5 away. You easure the istance between the first an seventh consecutive ark lines to be 9.0 c. what is the wavelength. Now 9.0c y 6 1.5c This istance is the sae as the istance between any two ajacent bans. y L y L Exaple The thir-orer ark fringe of 631 n light is observe at an angle of 14. How Far apart are the slits. sin 1 1 sin sin

11 SPH3UW/SPH4UI Unit 9.3 Interference of Light in Two Diensions Page 11 of 11 Extra Notes an Coents