Massachusetts Institute of Technology Physics 8 03 Fall 2004 Problem Set 10 Due Friday December 3 2004 at 4 PM Reading Assignment Beke Barrett pages 519 522 525 526 532 548 Problem 10 1 Thin lm interference Do problem 8 1 from Bekefi and Barrett Electromagnetic Vibrations Waves and Radiation Cambridge MA The MIT Press September 15 1977 ISBN 0262520478 White light is incident normally on an air lm of thickness d formed between two glass plates What must be the smallest lm thickness d if only blue light of wavelength 4000 A 4 10 7 m is to be re ected strongly Problem 10 2 Newton rings Do problem 8 4 from Bekefi and Barrett Electromagnetic Vibrations Waves and Radiation Cambridge MA The MIT Press September 15 1977 ISBN 0262520478 This problem was on the 8 03 Final Exam in the spring of 2004 A plano convex piece of glass index of refraction n rests on a plane parallel piece of glass as shown The radius of the spherical surface is R and it is much greater than rm Light of wavelength is incident normally and re ected at the spherical glass air interface and at the air glass interface of the glass plate The two re ected beams then interfere to produce a series of alternately bright and dark concentric circles when viewed from above 1 a Find the radial distance r from the point of contact at which the separation between the spherical surface and the plate upon which it rests is d i e nd the relation between r and d b Derive an expression for the radial distances rm at which bright rings will be observed c Same as b for dark rings Let R 2 m and 640 nm d What then is the spacing di erence in radii between the rst 2 dark rings and what between dark ring 25 and 26 Problem 10 3 Take home experiment 6 Thin lm interference The experiments are very nice and easy to do and they will give you a good insight into this common and intriguing interference phenomenon also observed in soap bubbles and oil spills on the road I strongly encourage you to do the experiments as you will see Problem 10 1 and 10 2 in action You do not have to write up your ndings to get full credit for this Problem Set However in preparing for the Final Exam I will assume that you have done these experiments Problem 10 4 Rainbows A very narrow beam of unpolarized red light of intensity I0 is incident at A on a spherical water drop see gure The angle of incidence is 60 At A some of the light is re ected and some enters the water drop The refracted light reaches the surface of the drop at B where some of the light is re ected back into the water and some emerges into the air The light that is re ected back into the water reaches the surface of the drop at C where some of the light is re ected back into the drop and some emerges into the air The index of refraction n of water for the red light is 1 331 a What is the intensity and what the degree of polarization of the light that refracts into the drop at A 2 b What is the intensity and what the degree of polarization of the light that re ects at B c What is the intensity and what the degree of polarization of the light that emerges into the air at C d Let the angle of incidence at A be 1 and the angle of refraction 2 Express see gure as a function of only 1 and 2 e Calculate the angle in case 1 is 60 Do this for the red light and also for blue light the index of refraction for blue light is 1 343 The speed of blue light in water is about 1 slower than that of red light f For a given wavelength there is one and only one value of 1 for which is a maximum 2 max Prove that this is the case when cos 1 2 n 3 1 Here n is the index of refraction g Using the equation under f calculate the values of 1 for both the red and the blue light that give rise to maximum values for Using your result under d calculate the maximum values for each wavelength will have its own set of values for 1 and associated max h In a world far far away rain comes down as small drops of glass with index of refraction of about 1 5 The living souls there talk about a glass bow What is the maximum value of for these glass bows Compare this with our rainbows Formation of Rainbows The 60 angle of incidence see the gure is very close to the values you found in g Thus the value of as shown in the gure is also very close to your max values in g The fact that max is di erent for the red light than for the blue is key in the formation of the rainbow The geometry shown in the gure will play a central role in the lecture on rainbows on December 7 A rainbow will be made It is advisable to bring an umbrella Problem 10 5 Superposition of N oscillators Do problem 8 5 from Bekefi and Barrett Electromagnetic Vibrations Waves and Radiation Cambridge MA The MIT Press September 15 1977 ISBN 0262520478 We desire to superpose the oscillations of several simple harmonic oscillators having the same frequency and amplitude A but di ering from one another by constant phase increments that is E t A cos t A cos t A cos t 2 A cos t 3 a Using graphical phasor addition nd E t that is writing E t A0 cos t nd A0 and for the case when there are ve oscillators with A 3 units and 9 rad b Study the polygon you obtained in part a and using purely geometrical considerations show that for N oscillators N 1 sin N 2 cos t E t N A N sin 2 2 c Sketch the amplitude of E t as a function of The above calculation is the basis of nding radiation from antenna arrays and di raction gratings 3