CERTIFICATE This is to certify Mr. Kshitij Mittal Mittal Of Class XII Roll No.___________ No.___________ has worked under my supervision on The Project_____________________ Project________________________ ___ ______________________ ______________________ and completed completed it to my total satisfaction.
Teacher ______________ (Vivek Misra)
ACKNOWLEDGEMENT ACKNOWLEDGEMENT It is my duty to record my sincere thanks and deep sense of gratitude to my respected teacher______________. teacher______________. For his/her valuable guidance and constant constant encouragement for the fulfillment of the project. I am also highly obliged to our lab assistant Mr. surinder who provided me the required apparatus and materials along with encouragement.
CONTENTS
Aim
Theory
Experiment
Interference Patterns
Derivation
Observations
Graph
Conditions
Bibliography
AIM
To observe the interference pattern by performing Double Slit Experimenent. To derive the expression for fringe width. To draw the intensity vs. path difference di fference graph.
THEORY
COHERENT SOURCES
The sources of light, which emit continuous light waves of the same wavelength, same same frequency and in the same phase difference are called coherent sources. Conditions to obtain coherent light : 1. Two sources should be single source or by same device. 2. Two sources should give monochromatic light. 3. The path path difference b/w light waves from 2 sources should be small.
INTERFERENCE OF LIGHT It is the phenomenon of redistribution of light energy in a medium on account of superposition of light wave from 2 coherent sources. There are 2 types: 1. Constructive Interference. 2. Destructive Interference
EXPERIMENT Young’s Double Slit Experiment Set up the apparatus as in the Figure 1.
The appearance appearance of bright and dark fringes on screen screen can be explained on the basis of interference of light. According to Huygens principle, the monochromatic source of light li ght illuminating the slit S sends out spherical wavefronts. Let the solid arcs represent the crests ant the t he dotted arcs represents the troughs. These wavefronts reach the slits A and B simultaneously which in turn, become sources of secondary wavelets. Thus the 2 waves on on superposition produce interference. The dots (.) represent the positions of constructive interference, where crests of one wave falls on crests of the other and trough falls on trough. The resultant res ultant Amplitude and hence intensity of light is maximum at these positions. The lines joining the dots lead to points C, E, G on the screen. Similarly, the crosses (x) represent the positions posit ions of destructive interference, where crest of one wave falls on trough of the other and vice-versa. The resultant amplitude and hence intensity of light is minimum at these positions. The lines joining the crosses lead to points D, F on the screen. Thus we have Bright Bright Fringes at C, E, and G and Dark Fringes at D and F. These bright and dark fringes fringes are placed alternatively and and they are equally spaced. These are called INTEFERENCE FRINGES.
INTERFERENCE PATTERNS Let the waves from 2 coherent sources of light be represented as
y1 = a sinωt…………………………(1) ωt + θ)……………………..(2) y2 = b sin ( ωt
where a and b ate the respective amplitudes of 2 waves and θ is the constant phase angle by which second wave leads the first wave. According to superposition principle, the displacement y of the resultant wave at time t would be given by ωt + θ) y=y1 + y2 = a sin ωt + b sin( ωt = a sinωt + b sinωt cosθ + b cosωt sinθ y = sinωt(a+b cosθ) + cosωt.b sinθ……………..(3) put a + b cosθ = A cosФ…………….(4) b sinθ = A sinФ……………..(5) Therefore y = sinωt.AcosФ + cosωt.AsinФ = A(sinωt cosФ + cosωt sinФ ) y = A sin (wt + Ф)………………..(6) Thus the resultant wave is a harmonic wave of amplitude A. Squaring (4) and (5) and adding We get, A = √a2 + b2 + 2ab cosθ ……….(7) As resultant resultant intensity I is is directly directly proportional to the square of the amplitude of the resultant wave Thus I α A2 i.e. I α a2 + b2 + 2ab cosθ…….(8)
Conditions for constructive and Destructive Interference,
Constructive Interference I should be maximum, for which
Cosθ = max = +1 so, θ = 0,2 π,4π,….. π, 4π,….. i.e. θ = 2nπ ; where n = 0, 1, 2…… if x is the path difference, then x=λθ/2π x = λ (2n π )/2π i.e. x=nλ Hence, condition for constructive interference at a point is that phase difference b/w the 2 waves reaching re aching the point should be zero or an even integral multiple of π. Or Path difference b/w the 2 waves reaching the point should be 0 or an integral multiple of full wavelength. So Amax=(a+b)
Destructive Interference I should be minimum, for which Cosθ = min = -1 so, θ = π ,3π,5π,….. i.e. θ = (2n-1)π ; where n = 1,2,3……
if x is the path difference, then x=λθ/2π x = λ ((2n-1) π )/2π x=(2n-1)λ/2 Hence, condition for constructive interference at a point is that phase difference b/w the 2 waves reaching re aching the point should be an odd integral multiple of π. Or Path difference b/w the 2 waves w aves reaching the point should be an odd integral multiple of half the wavelength. So Amin= (a-b)
DERIVATION
Expression for Fringe Width in Inerference Looking at Figure 2, The intensity of light at the point on the screen screen will depend on the path difference b/w the 2 waves arriving at that point.the point C is at equal distance from A and B.therefore, the path difference b/w 2 waves reaching C is 0 and the point C is of maximum maximum intensity. It is called CENTRAL MAXIMUM. Consider a point p at a distance x from C. The path difference d ifference b/w 2 waves arriving at P, = BP – AP…….(9) AP…….(9) Let O be the mid point of AB, and AB = EF = d, AE = BF = D From the figure, PE = PC – EC= EC= x-d/2 and PF = PC + CF = x+d/2 By Pythagoras and Binomially, BP= D[1 + (x + d/2) 2/2D2 ]……..(10) and AP= D[1 + (x - d/2)2/2D2 ]………(11)
Putting these values in (9), we get, path difference BP-AP= D[1 + (x + d/2)2/2D2 - 1 - (x - d/2)2/2D2 ] BP - AP= xd/D……………..(12) xd/D……………..(12) For Bright Fringes Path Difference = xd/D = n λ where n= 0,1,2… or x=nλD/d ……..(13)
For Dark Fringes Path Difference = xd/D = (2n-1) λ/2 where n=1,2,3… or x=(2n-1)λD/2d ……..(13)
Comparison shows that dark interference fringes are situated in b/w bright interference fringes and vice-versa Separation b/w the centers of 2 consecutive bright fringes is the width of dark fringe. xn-1 = nλD/d- (n-1)λD/d β = xn – xn-1
β= λD/d……….(14)
Similarly, separation b/w the centers of 2 consecutive bright fringes is the width of dark fringe.
Β’ = x’n – x’n – x’n-1 = [(2n-1)λD/2d]-[(2(n-1)-1) λD/2d] β = β’= λD/d……….(15) So, all bright and dark fringes are of equal width.
OBSERVATIONS
S.No.
Wavelength
λ (cm)
Distance of screen & source D (cm)
Slit Distance d (cm)
Fringe Width
1.
7 x 10-5
25
0.1
0.175
2.
7 x 10-5
50
0.1
0.35
GRAPH
β (cm)
Intensity Vs Path Difference Angular separation of the fringes is just ( λ/d). It is independent of the position on the screen. Further, at sites of constructive interference, Imaxα R 2max α (a+b) 2 = constant Hence all bright interference bands have same intensity At sites of destructive interference, interference, Imaxα R 2max α (a-b) 2 = constant = 0 (at a=b) Hence all dark interference bands have same (zero) intensity. Figure 3 represents the intensity of double slit interference pattern as a function of path difference θ b/w the waves of the screen.
CONDITIONS (For Sustained Inerference)
1. The The 2 sources of light must be coherent i.e. they continuous light waves of same wavelength or frequency, which have either same phase of constant path difference. 2. The The 2 sources should be strong with least background. 3. The The amplitudes of waves from 2 sources should preferably be equal. 4. The The 2 sources should preferably be monochromatic. 5. The The coherent sources must be very close to each other. 6. The The 2 sources should be point and narrow sources.
Figure 1
Interference Pattern Figure 2 Schematic Diagram
Figure 3
Intensity Vs Path Difference Graph
BIBLIOGRAPHY
SITES
http:// www.google.com http://vsg.quasihome.com/interfer.htm BOOKS Pradeeps Physics Practical Physics N.C.E.R.T textbook. Comprehensive Chemistry Practical. Resnick and Halliday.
