Lesson 11 Interference of Light

Transcription

1 Physics 30 Lesson 11 Interference of Light I. Light Wave or Particle? The fact that light carries energy is obvious to anyone who has focuse the sun's rays with a magnifying glass on a piece of paper an burne a hole in it. But how oes light travel an in what form is this energy carrie? Energy can be carrie from place to place in basically two ways: by particles or by waves. In the first case, material boies or particles can carry energy, such as a thrown baseball or the particles in rushing water. In the secon case, water waves an soun waves, for example, can carry energy over long istances. In view of this, what can we say about the nature of light? Does light travel as a stream of particles away from its source; or oes it travel in the form of waves that sprea outwar from the source? Historically, this question has turne out to be a ifficult one. For one thing, light oes not reveal itself in any obvious way as being mae up of tiny particles nor o we see tiny light waves passing by as we o water waves. The evience seeme to favour first one sie an then the other until about 1830 when most physicists ha accepte the wave theory. By the en of the nineteenth century, light was consiere to be an electromagnetic wave (see Lesson 24). Although moifications ha to be mae in the twentieth century (Lessons 28 to 30), the wave theory of light has prove very successful. We now investigate the evience for the wave theory an how it has explaine a wie range of phenomena. II. The Wave Theory of Light The Dutch scientist Christian Huygens ( ), a contemporary of Newton, propose a wave theory of light that ha much merit. (Refer to Pearson pages 684 to 685.) Still useful toay is a technique he evelope for preicting the future position of a wave front when an earlier position is known. This is known as Huygens' principle an can be state as follows: Every point on a wavefront can be consiere as a point source of tiny seconary wavelets that sprea out in front of the wave at the same spee as the wave itself. The surface envelope, tangent to all the wavelets, constitutes the new wavefront. As a simple example of the use of Huygens' Principle, consier the circular an straight wavefronts AB at some instant in time as shown to the right. The points on the wavefront represent the centres of the new wavelets, seen as a series of small semi-circles. The common tangent to all these wavelets, the line A'B', is the new position of the wavefront a short time later. R.L. & A.K /11/2013

2 Huygens' principle is particularly useful when waves impinge on an obstacle an the wave fronts are partially interrupte. Huygens' principle preicts that waves ben in behin an obstacle. The bening of waves behin obstacles into the shaow region is known as iffraction. Since iffraction occurs for waves, but not for particles, it can serve as one means for istinguishing the nature of light. Does light exhibit iffraction? In the mi-seventeenth century, the Jesuit priest Francesco Grimali ( ) ha observe that when sunlight entere a arkene room through a tiny hole in a screen, the spot on the opposite wall was larger than woul be expecte from geometric rays. He also observe that the borer of the image was not clear but was surroune by coloure fringes. Grimali attribute this to the iffraction of light. Newton, who favoure a particle theory, was aware of Grimali's result. He felt that Grimali's result was ue to the interaction of light corpuscles (ie. little boies ) with the eges of the hole. If light were a wave, he argue, the light waves shoul ben more than that observe. Newton's argument seeme reasonable, yet iffraction is noticeable only when the size of the obstacle or the hole is on the orer of the wavelength of the wave. Newton i not know that the wavelengths of visible light were increibly tiny, an thus iffraction effects were very small. Inee this is why geometric optics using rays is so successful normal openings an obstacles are much larger than the wavelength of the light, so relatively little iffraction or bening occurs. In 1801, a key experiment was performe by the brilliant Thomas Young ( ). (Refer to Pearson pages 685 to 690.) Young irecte light through two parallel narrow slits a small istance apart. The light was then seen on a screen a few meters away. If light consiste of particles the result woul be two bright fringes on the screen (screen b below), but the actual results were quite ifferent. Instea of two bright fringes, there were a series of alternating bright an ark fringes (screen c below). Young reasone that he was seeing a wave interference pattern cause by the iffraction of light through each of the slits. Light iffracting through one of the slits interfere with the iffracte light from the other slit. R.L. & A.K /11/2013

3 III. Interference of Light Recall from Physics 20 that when two waves meet they interfere with each other in an aitive fashion. The waves combine either constructively or estructively epening on the situation. For twoimensional mechanical waves consier the situation where two sets of waves are being generate at the same time an in the same phase a istance () from each other. When crests meet crests an troughs meet troughs, constructive interference occurs an these are calle antinoes or maxima. When crests meet troughs, complete estructive interference occurs an these are calle noes or minima. Notice the pattern of antinoes with noes in between them. (Refer to pages 425 to 428 in Pearson for a iscussion of twoimensional wave interference.) To unerstan how an interference pattern is prouce, consier the iagrams below. Two waves of wavelength are shown to originate from two vibrating sources (S 1 an S 2 ) a istance apart. While waves sprea out in all irections, we will focus our attention on the two wave trains S 1 shown in each of the following iagrams. In iagram (a) the waves travel the same path length when they meet they are in phase an S 2 constructive interference occurs a) constructive interference resulting in a maximum or antinoe. antinoes noes There will also be constructive interference if the path ifference is one wavelength or a whole number multiple of wavelengths (1, 2, 3, n) as shown in iagram (b). S 1 b) constructive interference S 2 R.L. & A.K /11/2013

4 However if the path ifference is ½ or 3/2 or 5/2 an so on, the result is estructive interference resulting in noes or minima. This is illustrate in iagram (c). S 1 S 2 ½ c) estructive interference IV. Derivation of Double Source Interference Equations R x A Q C B S L The variables are: istance between two coherent wave sources L the istance from the wave sources to the viewing screen or area x the istance from the central maximum (point S) to the noe or antinoe of interest the wavelength the angle from the centre line to the noe or antinoe of interest We are intereste in eriving an equation that relates x (RS) with L (QS) an (AC) an. We note that for constructive interference to occur at point R, the path ifference (BC) between AR an CR must be a whole number multiple (n = 1, 2, 3, ) of a wavelength (). BC = n ABC is similar to QRS while B an S are 90. RS x tan (1) an QS L BC n sin (2) AC rearranging equation (2) we get our first useful formula In orer to relate equations (1) an (2) we note sin n sin tan for small angles of (0 to 10 ) n x L **(This equation can only be use if < 10 or if x<<l) rearranging this equation gives us our secon useful formula x nl R.L. & A.K /11/2013

5 V. Using the Double Source Interference Equations The equations that were erive in the previous section are: sin n 1,2,3,...(constructive interference) n sin n 1,2,3,...(estructive interference) n x n 1,2,3,...(constructive interference) nl x n 1,2,3,...(estructive interference) (n)l where: wavelength (m) angle from central line to fringe n orer of fringe L istance from slits to screen (m) x istance from central bright fringe to nth fringe (m) istance between slits (m) These equations are on your formula sheet. These equations may be mae from the formula sheet by subtracting ½ from n. Example 1 A stuent oing Young s experiment fins that the istance between the central bright fringe an the seventh noal line is 6.0 cm. If the screen is locate 3.0 m from the two slits, whose separation is 220 m, what is the wavelength of the light? x = m L = 3.0 m = 220 x 10-6 m n = 7 =? x (n)l m(0.060m) (7)(3.0m) m 676nm Example 2 An interference pattern is prouce through two slits which are mm apart using 550 nm light. What is the angle to the 4 th bright fringe from the central line? = x 10-3 m n = 4 = 550 x 10-9 m sin n n 1 sin 9 1 4(55010 m) sin 3 o m R.L. & A.K /11/2013

6 Example 3 An interference pattern is forme on a screen when a helium-neon laser light ( = nm) is irecte through two slits. If the slits are 43 m apart an the screen is 2.5 m away, what will be the separation between ajacent noal lines? When a question asks for the separation between noes or antinoes, set n = 1 an use the constructive interference equation. The fringes are all the same istance apart, whether they are bright or ark. x =? L = 2.5 m = 43 x 10-6 m n = 1 = x 10-9 m VI. Practice Problems x nl nl x x 9 1( m)(2.5m) m x 3.68cm 1. A slie containing two slits 0.10 mm apart is 3.20 m from the viewing screen. Light with a wavelength of 500 nm falls on the slits from a istant source. How far from the centre line will the 9th bright fringe be? How many bright fringes are possible? (0.144 m, 401) 2. In Young's experiment, the two slits are 0.04 mm apart an the screen is locate 2.0 m away. The thir orer bright fringe is isplace 8.3 cm from the central fringe. A. What is the frequency of the monochromatic light? (5.4 x Hz) B. Where will the secon ark fringe be locate? (4.15 cm) R.L. & A.K /11/2013

7 VII. Han-in Assignment 1. Light of frequency 6.0 x Hz is incient on a pair of straight parallel slits which are 5.0 x 10-5 m apart. Antinoal lines are prouce in the region beyon the slits. What is the maximum number of antinoal lines in the interference pattern? (Hint, what angle woul the last antinoe occur at?) (201) 2. Yellow light of wavelength 615 nm is incient on a ouble slit where slits are 1.3 mm apart. At what angle will the fifth orer antinoal line appear? (0.14 ) 3. How many antinoal lines exist in an interference pattern when raiation of 555 nm passes through slits which are 0.10 mm apart? (361) 4. Light of frequency 6.09 x Hz is incient on a pair of straight parallel slits an prouces an interference pattern on a screen 7.0 m away. If the fringe spacing on the screen is 2.5 cm, etermine the istance between the slits. (0.138 mm) 5. Light of frequency 4.8 x Hz is incient on a pair of straight parallel slits where the slits are 0.16 mm apart. It creates an interference pattern on a screen 8.0 m away. What is the istance from the centre of the pattern to the fourth bright line? (0.13 m) 6. A flat observation screen is place at a istance of 4.5 m from a pair of slits. The separation on the screen between the central bright fringe an the first orer bright fringe is m. The light illuminating the slits has a wavelength of 490 nm. Determine the slit separation. (6.0 x 10-5 m) 7. In a Young s ouble slit experiment the separation between the central bright fringe an the first orer bright fringe is 2.40 cm for 475 nm light. Assuming that the angles that locate the fringes on the screen are small, fin the separation between fringes when light has a wavelength of 611 nm. (3.09 cm) 8. In a Young s ouble-slit experiment, the angle that locates the secon-orer bright fringe is 2.0. If the slit separation is 3.8 x 10-5 m, what is the wavelength of the light? (6.6 x 10-7 m) 9. Re light of wavelength 600 nm passes through two parallel slits. Noal lines are prouce on a screen 3.0 m away. The istance between the first an tenth noal lines is 5.0 cm. What is the separation of the slits? (3.24 x 10-4 m) 10. In an interference experiment, re light with a wavelength of 6.0 x 10-7 m passes through a ouble slit. On a screen 1.5 m away, the istance between the 1 st an 11 th ark bans is 2.0 cm. a) What was the separation between the slits? (4.5 x 10-4 m) b) What woul the spacing be between ajacent noal lines if blue light (450 nm) were use? (1.5 x 10-3 m) R.L. & A.K /11/2013