Physics INTERFERENCE OF LIGHT Q.1 State the principle of superposition of waves an explain the concept of interference of light. Ans. Principle of superposition of waves : When two or more waves, traveling through a meium, arrive at a point simultaneously, then each wave prouces its own isplacement at that point inepenently. The resultant isplacement at that point is the vector sum of the isplacements ue to all the waves. Interference : The phenomenon of changes of isplacements at point ue to superposition of waves is calle as interference of light. 1) When the two waves arriving at a point are in phase, (i.e. the crest of one wave falls on the crest of other wave or the trough of one wave falls on the trough of other wave), the resultant isplacement at that point is maximum. Therefore, the resultant intensity of light at that point is maximum, an that point appears bright. This type of interference is calle a constructive interference. 2) When the two waves arriving at a point are out of phase, (i.e., the crest of one wave falls on the through of the other wave, an vice-versa), the resultant isplacement at that point is minimum. The resultant intensity of light at that point is minimum, an that point appears ark. This type of interference is calle a estructive interference. Steay interference pattern : When the pattern containing alternate bright an ark bans remains unaltere with respect to time, it is calle a steay interference pattern. Q.2 A] State the conitions necessary for obtaining a steay interference pattern. Ans. Conitions for steay interference pattern : 1) The two sources of light must be coherent : The two sources are sai to be coherent if the waves prouce by them are in phase or having constant phase ifference. 2) The two sources of light must be monochromatic : A source of light is sai to be monochromatic, if it emits light waves of only one wavelength. 3) The two sources must be equally bright, i.e., the waves emitte by the sources must have the same amplitue. 4) The two sources must be narrow. 5) The sources shoul be close to each other, their separation being of the orer of the wavelength of light. B] What are the coherent sources? How are they prouce? Ans. The two sources are sai to be coherent, if the waves prouce by them are in phase or having constant phase ifference are erive from the same original source. A steay interference pattern can be prouce ue to interference between the light waves emitte by the coherent sources. A steay interference pattern is that in which the intensity at a point oes not change with time. 1) In Young s experiment, a slit is illuminate by a monochromatic light. The light from the illuminate slit is mae incient on two parallel slits. These slit act as coherent sources, sening out waves having constant phase ifference. 2) In Fresnel s biprism experiment, the biprism prouces two virtual images of the slit by refraction. These virtual images act as coherent sources. Prof. eepak B. Jawale (9326981036) 1
Q.3 State the conitions for constructive an estructive interference. Ans. Conitions for constructive interference (Brightness) : 1) If the two waves interfering at a point are in phase, that point appears bright. 2) Hence the conition for brightness can be written as, Phase ifference = 0, 2, 4, 6, - - - -, etc. 3) Phase ifference 2 correspons to a path ifference ; therefore, the above conition can be written as, Path ifference = 0,, 2, 3, - - -, etc. Thus, a point is bright if the path ifference between the two waves arriving at a point is an integral multiple of the wavelength. Conition for estructive interference (arkness) : 1) If the two waves interfering at a point are in opposite phase, that point appears ark. 2) Hence, the conition for arkness can be written as, Pattern ifference =, 3, 5, - - - -, etc. 3) Path ifference = /2, 3 /2, 5 /2, - - -, etc. Path ifference = (2n 1), 2 where n = 1, 2, 3, 4, -----, Thus, a point is ark if the path ifference between the two waves arriving at a point is an o integral multiple of half a wavelength. Q.4 Give the theory of interference of bans an obtain an expression for the ban with. Ans : Consier two coherent, monochromatic sources A an B separate by a small istance, proucing an interference pattern on a screen place at a istance from the sources. Let be the wavelength of the light. The perpenicular bisector of AB meets the screen at the point P. In figure., AM an BN are the perpeniculars rawn to the screen from A an B respectively. Let Q be a point on the screen at a istance x from the point P. Join AQ an BQ. In BNQ, BQ 2 = BN 2 + NQ 2 In AMQ, AQ 2 = AM 2 + MQ 2 By equation (1) (2), BQ 2 AQ 2 = 2 + (x + /2) 2. (1) = 2 + x 2 + x + 2 /4 = 2 + (x /2) 2. (2) = 2 + x 2 x + 2 /4 = ( 2 + x 2 + x + 2 /4) ( 2 + x 2 x + 2 /4) = 2x (BQ + AQ) (BQ AQ) = 2x Prof. eepak B. Jawale (9326981036) 2
2x Path ifference = BQ AQ = BQ + AQ Now, the istance is very large as compare to x an. AQ + BQ = 2 2x x Path ifference = BQ AQ = = 2 x The point Q will be bright, if = n, where n = 0, 1, 2, 3, ----- x an the point Q will be ark if = (2n 1), 2 Where n = 1, 2, 3, ----- Ban with or fringe with : The istance between the two consecutive bright bans or ark bans in an interference pattern is calle ban with or fringe with (X). Expression for ban with X : Case I : Consiering the istance between two consecutive bright bans : Suppose x n an x n+1 be the istance of n th an (n + 1) th bright bans from the centre of the interference pattern. For n th bright ban, x n n = n, x n =, an for (n + 1) th bright ban, x n+1 (n + 1) = (n + 1) x n+1 = (n + 1) n x n+1 x n = = Ban with, X = Case II : Consiering the istance between two consecutive ark bans : Suppose x m an x m+1 be the istance of m th an (m + 1) th ark bans from the centre of the interference pattern. For m th ark ban, x m = (2m 1), 2 (2m 1) x m =, Prof. eepak B. Jawale (9326981036) 3
x m = (2m 1) an for (m + 1) th ark ban, x m+1 = [2(m + 1) 1] 2 2m + 1 x m+1 = 2 x m+1 = (2m + 1) x m+1 x m = (2m + 1) (2m 1) 2 X = X = Hence, the istance between two consecutive bright bans is the same as that for the ark bans. Thus, bright bans an ark bans are alternate an equispace in an interference pattern. Q.5 escribe Young s ouble slit experiment an state its importance. Ans : Young s ouble slit experiment : 1) This experiment was performe by Sir Thomas Young in 1800. 2) The experimental arrangement consists of a pinhole S illuminate by sunlight. The light S is allowe to fall on two more pinholes S 1 an S 2 separate by a small istance an equiistant from S. 3) S 1 an S 2 emit the seconary light waves an act as coherent sources. 4) The seconary waves from S 1 an S 2 interfere with each other. At the points where the crests (or troughs) of waves from S 1 fall on the crests (or troughs) of waves S 2, the points become bright. At the points where the crests of the waves from S 1 fall on the troughs of the waves from S 2 an vice versa, the points become ark. 5) If a screen is kept parallel to the plane of the slit, alternate bright an ark bans are obtaine on the screen. 6) The pattern is coloure. When the monochromatic light is use to illuminate the slits, the colouration efect is remove. Hence, a stable interference pattern can be obtaine by the ouble slit experiment. 1) It was the first experiment to emonstrate interference of light. 2) It prove conclusively that light is propagate in the from of waves. 3) From this experiment, the wavelength of monochromatic light can be etermine. Prof. eepak B. Jawale (9326981036) 4
Q.6 escribe the biprism experiment for etermining the wavelength of monochromatic light. Ans : Fresnel s biprism experiment : Eyepiece Experimental arrangement : Slit Biprism Ray iagram 1) The apparatus use in the experiment is calle optical bench. 2) It consists of a heavy metal base about 2 m long an 15 cm broa. Four stans, with ajustable heights, are mounte on the optical bench. 3) The stans can be move along the axis of the bench. 4) The stans, carry a slit holer, a holer for biprism, a holer for lens, an an eyepiece with micrometer screw, in that orer. The biprism consists of two small angle prisms cemente together base to base. Ajustments : 1) The optical bench is levele an the slit, the biprism an the eyepiece are brought at the same height. 2) The slit is mae narrow an illuminate by monochromatic light. 3) Refraction through the biprism takes place an the refracte rays appear to come from the two virtual images S 1 an S 2 of the slit. 4) The images S 1 an S 2 are in the same plane of the slit S an act as the two coherent sources. 5) The istance between the slit an the biprism is ajuste to be about 20 cm, an the istance between the biprism an the eyepiece about one meter. 6) The eges of biprism are kept parallel to the slit. 7) The eyepiece is ajuste to observe sharp interference pattern. Measurements : 1) Measurement of : The istance between the slit an eyepiece is measures irectly on the scale marke on the optical bench. 2) Measurement of X : The vertical cross wire of eyepiece is mae to coincie with the bright bans in succession an corresponing micrometer reaings are note. The average istance X between the successive bright bans is etermine. 3) Measurement of : A convex lens is mounte on the stan between the biprism an the eyepiece. The eyepiece stan is ajuste so that the istance between the slit an the eyepiece is more than four times the focal length of the lens. The lens stan is move towars the eyepiece an two iminishe images of the slit are obtaine. The istance 1 between these two images is measure. The lens stan is then move towars the biprism an two magnifie images of the slit are obtaine. The istance 2 between these two images is measure. The istance between the two coherent sources is given by, = 1 2 4) The wavelength of the monochromatic light is etermine by the formula, X X 1 2 = i.e., = Prof. eepak B. Jawale (9326981036) 5
Our Toppers Meical Engineering Q.7 escribe the metho to etermine the istance between the two virtual sources in the biprism experiment. Obtain the necessary formula. Ans : 1) A convex lens is mounte between the biprism an the eyepiece. 2) The istance between the slit an the eyepiece is ajuste to be more than four times the focal length of the lens. 3) The convex lens is move towars the eyepiece an its position is ajuste so that the iminishe an sharp images are obtaine. 4) The istance 1 between the two iminishe images is etermine by using the micrometer screw eyepiece. 5) Then, the lens is shifte towars the biprism an its position is ajuste so that magnifie an sharp images of S 1 an S 2 are obtaine. 6) The istance 2 between the two magnifie images is measure by using the micrometer screw eyepiece. 7) By the property of conjugate foci, the image istance an object istance gets interchange uring the two ajustments. S 1 S 2 Prof. eepak B. Jawale (9326981036) 6
We know that, Image size Image istance = Object size Object istance For the first ajustment, 1 y =.(1) x an for the secon ajustment, 2 x =.(2) y From equation (1) an (2) 1 2 y x = x y 1 2 = 1 2 2 = 1 2 = 1 2 Important Formulae 1) The conition for a bright ban, P.. = n, (n = 0, 1, 2, 3, 4, ---------) 2) The conition for a ark ban, P.. = (2n 1), (n = 1, 2, 3, 4, ---------) 2 x 3) P.. = (refer to the figure in Q.4) 4) The istance of n th bright ban from the central bright ban, n x n = = nx, where n = 0, 1, 2, 3 -------- an x = banwith 5) The istance of n th ark ban from the central bright ban, (2n 1) x n = where n = 1, 2, 3 ------- an a = banwith 6) Ban with or fringe with, X = Prof. eepak B. Jawale (9326981036) 7
7) In Biprism experiment, the wavelength of monochromatic light is, X =, where = 1 2 Image size Image istance 8) = Object size Object istance 1 y = for magnifie images x 2 x = for iminishe images y Prof. eepak B. Jawale (9326981036) 8