Waves Part 3B: Interference

Transcription

1 Waves Part 3B: Interference Last modified: 31/01/2018

2 Contents Links Interference Path Difference & Interference Light Young s Double Slit Experiment What Sort of Wave is Light? Michelson-Morley Experiment Electromagnetic Spectrum Gravity Waves

3 Interference Contents The term interference usually refers to the superposition of waves that are otherwise identical, but have different phases. Consider two waves described by the equations: y 1 = A sin(kx ωt) and y 2 = A sin(kx ωt + φ) The following plots show the superposition y = y 1 + y 2 calculated for several example values of φ: φ = 0 φ = π 3 + = + = φ = 2π 3 + = φ = π + =

4 Contents In each case, the superposition has the same wavelength and frequency as the original waves, but has a different amplitude. The maximum amplitude occurs when φ = 0 (the waves are in phase). This is known as constructive interference. The minimum amplitude occurs when φ = π (the waves are out of phase). This is known as destructive interference. Intermediate values of φ give partially destructive interference. Since adding a phase of 2nπ (where n is an integer) doesn t change the function, more generally we have: φ = 2nπ constructive interference φ = (2n + 1)π destructive interference

5 Path Difference Contents The phase difference between two waves is very often due to a difference in the distances travelled by the two waves. As an example consider the situation shown at right. A point source S is producing sound waves, which are then detected by a microphone M. Two sound waves reach M - one directly from S and the other reflecting off a wall at W. S W M The microphone M measures the superposition of these waves: y M = A sin(kx ωt) + A sin(k(x + ) ωt) ( ) k = 2A cos sin(kx ωt + k 2 2 ) A sin(kx ωt + k 2 ) where = SW + WM SM is the difference in path lengths taken by the two waves.

6 Contents The sound measured at M has the same frequency as emitted at S, but the amplitude A = 2A cos ( ) k 2 depends on the path difference. We should remember from a previous lecture that the loudness or intensity of sound is proportional to amplitude squared, so the intensity I measured at M is a function of (I max is the maximum intensity): ) ) I( ) = I max cos 2 ( k 2 = I max cos 2 ( π λ Looking at this formula, it is clear that the maximum intensity (i.e. constructive interference) will occur when cos ( ) π λ = ±1. max volume is when = nλ (n = 0, ±1, ±2...) Also, the minimum intensity (i.e. destructive interference) occurs when cos ( ) π λ = 0 min volume is when = (n + 1 )λ (n = 0, ±1, ±2...) 2

7 Young s Double Slit Experiment Contents Interference is involved in a very historic experiment performed by Thomas Young in Since the time of Newton (late 1600 s) there had been much controversy about whether light is a wave or instead a beam of particles. (Newton himself believed the latter- he named the individual particles of light corpuscles ). Young was able to (at least temporarily) settle the question with his famous Double Slit experiment. The apparatus is shown at right. A point light source is placed in front of a screen containing two thin, closely spaced slits. Another screen is placed behind the first screen. If Newton s belief that light consists of beams of particles coming from the source is correct, then we would expect to only see light making it through to two points on the second screen as shown. source screen

8 Contents If light is a wave, the behaviour is very different. The properties of the slits are important - a wave transmitted through a thin slit will appear the same as produced by the slit being a point source - i..e circular wavefronts. If the light source is placed equidistant from the two slits, then this ensures that the wavefronts transmitted by the two slits are in phase. Light from the two slits reaches all points on the screen. But the phase difference (and hence the interference) between the two sources will vary across the screen. When the wavefronts arrive together there will be a bright line seen (constructive interference), and when out of phase there will be darkness (destructive interference). dark bright

9 With these two very different predictions, Young performed the experiment and observed a series of alternating bright and dark lines (often called fringes ) as expected in the wave model of light. Light must therefore be a wave! Contents The brightness of the light seen on the screen is given by the intensity of the wave. The calculation is exactly the same as the previous example: ( ) π I( ) = I max cos 2 λ Note that the centre of the screen, halfway between the slits, always has a bright fringe. Other maxima, as before, are found where = nλ, and minima when = (n )λ, for integer values of n. = 3λ = 5 λ 2 = 2λ = 3 λ 2 = λ = 1 λ 2 = 0 = 1 λ 2 = λ = 3 λ 2 = 2λ = 5 λ 2 = 3λ

10 Contents To determine the position on the screen for a particular value of, we need to do a little geometry, as shown below. Let d be the separation of the slits, L be the distance between the slits and the screen and consider the point P a distance y from the centre of the screen. L θ y d θ When L d, then the purple and orange lines become close to parallel and the green and blue triangles will be similar. From the green triangle: y = L tan θ L sin θ (true when θ is small) and from the blue triangle: sin θ = d

11 Contents Combining these two formulas gives the connection between y and : y = L sin θ = L d As discussed earlier, maxima occur when = nλ, so bright fringes are seen on the screens at the positions: y max = nλl d (n = 0, ±1, ±2...) and minima have = (n )λ, so dark fringes are located at: y min = (n )λl d (n = 0, ±1, ±2...) In both cases the fringes are evenly spaced, with separation λl d, and the dark fringes lying midway between the bright. Remember these formulas are approximations, valid when L d

12 Contents Two thin slits, 1.00 mm apart are illuminated by a laser, causing an interference pattern on a screen 2.00 m behind the slits. The distance between two adjacent bright fringes on the screen is measured to be 1.26 mm. What is the wavelength of the laser? From the previous page, we know the separation of two adjacent slits is: δy = λl d λ = δy d L = ( ) ( ) 2.00 = m = 630 nm Note: as done here, it is very common to express the wavelength of light in nanometres (1 nm = m).

13 What Sort of Wave is Light? Contents After Young established that light was indeed a wave, the obvious next question is: A wave in what? Other waves require a medium to travel in - air for sound, water for an ocean wave etc. What is the medium for light waves? This mysterious substance was given the name luminiferous ether (or just ether). Since we can see light coming from distant galaxies, the ether must be everywhere around us. Measuring and understanding the properties of the ether was very important. A key observation is that since the Earth is spinning on its axis, and also moving around the Sun, the Earth must be moving relative to the ether. A famous experiment trying to measure the effects of this relative motion was the 1887 Michelson-Morley experiment.

14 Michelson-Morley Experiment Contents The Michelson interferometer is another apparatus that relies on interference of light waves. The basic idea is shown below right. mirror Light shines on a beam splitter (a half-mirror that reflects 50% of light while the rest passes directly through). The two split d 1 beams are then reflected back by two regular mirrors before recombining at a light beam splitter detector. mirror When they reach the detector, the two beams have a path difference of: = 2(d 1 d 2 ) detector d 2

15 Contents The detector will detect fringes based on the value of - bright when = nλ and dark when = (n )λ. Because of the motion of the apparatus relative to the ether, we expect the speed of light to be different for the two paths (compare to a boat travelling upstream and downstream). This means that the times taken to travel each branch will be constantly changing, and so there will be a phase difference between the two superposed waves. Experimentally we expect this to cause a variation in the brightness of light measured in the detector. In the actual experiment, no such change is observed. This result raises serious doubt about the existence of the ether. Also in 1887, Heinrich Hertz produced and detected electromagnetic waves consisting of vibrating electric and magnetic fields, and requiring no other medium. We now understood visible light to be one of these electromagnetic waves, which also include radio waves, microwaves, X rays etc.

16 Electromagnetic Spectrum Contents Electromagnetic waves consist of vibrating electic and magnetic fields. In a vacuum they travel at a constant speed: c = m/s Different ranges of wavelengths are given different names as shown below (Note that these ranges are only approximately defined, and will vary a little in usage): frequency (Hz) Gamma Rays X Rays Ultra- Violet Infrared Microwaves Radio wavelength (m) wavelength (nm) Some familiar examples: Dental X-ray Hz Microwave oven 2.45 GHz AM radio 1 MHz FM radio/tv 100 MHz WiFi 2.4 GHz Mobile phones 900, 1800 MHz

17 Gravity Waves Contents In 1916, Albert Einstein, as part of his General Theory of Relativity predicted the existence of gravitational waves. These are oscillations in space itself. In 2015, the LIGO (Laser Interferometer Gravitational-Wave Observatory) experiment in the USA detected these waves for the first time. The design of this experiment is essentially the same as the Michelson-Morley experiment, with a passing gravity wave changing one of the path lengths d 1 or d 2, and thus causing a change in the interference pattern seen in the detector. In the original Michelson-Morley experiment these distances d 1 and d 2 were less than a metre, and the whole experiment fit comfortably on a benchtop. LIGO consists of two interferometers, each of which has d i = 4 kilometres! The detector is able to measure a change in path length of only m - about one thousandth the size of a proton.