MSU PHY 184 - Physics for Scientists & Engineers 2

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April 8, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 41April 8, 2005 Physics for Scientists&Engineers 2 2Light as WavesLight as Waves! In the previous chapter we discussed light as rays! These rays traveled in a straight line except whenthey were reflected off a mirror or wererefracted at the boundary between two opticalmedia! Now we will discuss the implications of the wavenature of light! We know that light is an electromagnetic wave! We normally do not think of light as a wave,because its wavelength is so short that we usuallydo not notice this wave behaviorApril 8, 2005 Physics for Scientists&Engineers 2 3Wave OpticsWave Optics! One way to reconcile the wave nature of light with thegeometric optical properties of light is to use Huygens’Principle developed by Christian Huygens, a Dutch physicistwho proposed a wave theory of light in 1678 beforeMaxwell developed his theories of light! This principle states that every point on a propagating wavefront serves as a source of spherical secondary wavelets! At a later time, the envelope of these secondary wavesbecomes a wave front! If the original wave has frequency f and speed v, thesecondary wavelets have the same f and vApril 8, 2005 Physics for Scientists&Engineers 2 4HuygenHuygen’’s s ConstructionsConstructions! Diagrams of phenomena based on Huygens’ Principle arecalled Huygens’ constructions! A Huygens’ construction for a light wave traveling in astraight line is shown belowApril 8, 2005 Physics for Scientists&Engineers 2 5HuygenHuygen’’s s Constructions (2)Constructions (2)! We start with a wave front traveling at the speed of light c! We assume point sources of spherical wavelets along thewave front! These wavelets also travel at c so at a time !t the waveletshave traveled a distance of c!t! If we assume many point sources along the wave front, wecan see that the envelope of these wavelets forms a newwave front parallel to the original wave front! Thus the wave continues to travel in a straight line with theoriginal frequency and speedApril 8, 2005 Physics for Scientists&Engineers 2 6Derivation of SnellDerivation of Snell’’s Laws Law! Now let’s use a Huygens’ construction to derive Snell’s Lawfor refraction between two optical media with differentindices of refraction! Assume that we have a wavewith wave fronts separatedby a wavelength "1 travelingwith speed v1 in an opticallyclear medium incident on theboundary with a secondoptically clear medium asshownApril 8, 2005 Physics for Scientists&Engineers 2 7Derivation of SnellDerivation of Snell’’s Law (2)s Law (2)! The angle of the incident wave front with respect to theboundary is #1, which is also the angle the direction of thewave makes with a normal to the boundary! When the wave enters the second medium it travels withspeed v2! According to Huygens’ Principle the wave fronts are theresult of wavelet propagation at the speed of the originalwave so we can write the separation of the wave fronts inthe second medium in terms of the wavelength in thesecond medium "2April 8, 2005 Physics for Scientists&Engineers 2 8Derivation of SnellDerivation of Snell’’s Law (3)s Law (3)! Thus the time interval between wave fronts for the firstmedium is "1/v1 and the time interval for the secondmedium is "2/v2! This time interval is the same at boundary so we can write! which we can rewrite as! The wavelengths of the light in the two media areproportional to the speed of light in those media!1v1=!2v2!1!2=v1v2April 8, 2005 Physics for Scientists&Engineers 2 9Derivation of SnellDerivation of Snell’’s Law (4)s Law (4)! We can get a relation between the angle of the incidentwave fronts #1 with the boundary and the angle of thetransmitted wave fronts #2 with the boundary by analyzingan expanded region of the Huygens drawingApril 8, 2005 Physics for Scientists&Engineers 2 10Derivation of SnellDerivation of Snell’’s Law (5)s Law (5)! We can see that! Solving for x we get! Remembering that n = c/v we get! Which is Snell’s Law!sin!1="1x and sin!2="2xsin!1sin!2="1"2=v1v2sin!1sin!2=v1v2=c / n1c / n2=n2n1 or n1sin!1= n2sin!2April 8, 2005 Physics for Scientists&Engineers 2 11Light Traveling in an Optical MediumLight Traveling in an Optical Medium! We have seen that the wavelength of light changes whentraveling in an optical medium with index of refractiongreater than one! Taking with case one as a vacuum and case two as a mediumwith index of refraction n we can write! Remembering that v = "f we can write the frequency fn oflight traveling in a medium as!n=!vc=!nfn=v!n=c / n!/ n=c!= fApril 8, 2005 Physics for Scientists&Engineers 2 12Light Traveling in an Optical Medium (2)Light Traveling in an Optical Medium (2)! So the frequency of light traveling in an optical mediumwith n > 1 is the same as the frequency of that lighttraveling in a vacuum! We perceive color by frequency rather than wavelength! Thus placing an object under water does not change ourperception of the color of the object! We can demonstrate that fact by taking a colored objectand putting it in a jar of water! Water has index of refraction n = 1.33! The object appears to have the same color under water asin airApril 8, 2005 Physics for Scientists&Engineers 2 13InterferenceInterference! Sunlight is composed of light containing a broad range offrequencies and corresponding wavelengths! We often see different colorsseparated out of sunlight byrefraction in rainbows! We also sometimes see variouscolors from sunlight due toconstructive and destructiveinterference phenomena on thesurface of DVD’s or CDs or in thinlayers of oil or waterApril 8, 2005 Physics for Scientists&Engineers 2 14Constructive InterferenceConstructive Interference! The geometric optics of theprevious chapter cannot be used toexplain interference! To understand these interferencephenomena we must take intoaccount the wave nature of light! Interference takes place when lightwaves of the same wavelength aresuperimposed! If the light waves are in phase,they interfere constructively, asshown to the rightApril 8, 2005 Physics for Scientists&Engineers 2 15Constructive Interference (2)Constructive Interference (2)! The statement that the two waves interfere constructivelyis the same as saying that the phase difference betweenthe two waves is zero! A phase difference of 2$ radians, 360°, or one wavelengthwill also produce two waves


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MSU PHY 184 - Physics for Scientists & Engineers 2

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