INTERFERENCE & PHYSICS LS13/AS 13th,Nov,09 7.30p.m : INTERFERENCE & PHYSICS LS13/AS 13th,Nov,09 7.30p.m HUYGENS PRINCIPLE INTERFERENCE WAVE FRONT
WAVE OPTICS : WAVE OPTICS INTRODUCTION
The ray optics uses the geometry of straight lines to account for the macroscopic phenomena like rectilinear propagation of light, reflection of light and refraction of light etc.
WAVE OPTICS : WAVE OPTICS However, the microscopic phenomena like interference, diffraction and polarization could not be accounted for by ray optics. To explain these phenomena, concept of waves was introduced.
WAVE OPTICS : WAVE OPTICS The wave theory of light was put forward first of all by Huygens and later on modified by Fresnel.
According to wave theory of light; the light is a form of energy which travels through a medium in the form of transverse wave motion.
The speed of light in a medium depends upon the nature of medium.
WAVE OPTICS : WAVE OPTICS Huygens supposed the existence of a hypothetical medium called ‘luminiferous ether’ which filled the entire space. This medium was supposed to be mass less with extremely high elasticity and very low density.
WAVE FRONT : WAVE FRONT WAVE FRONT
A wave front is defined as the continuous locus of all the particles of medium, which are vibrating in the same phase.
WAVE FRONT : WAVE FRONT A source of light sends out disturbance in all the directions.
In a homogeneous medium, velocity of light waves in all the directions is the same.
Therefore,. disturbance reaches at the same time, at all such particles which are at the same distance from the source.
These particles will naturally vibrate in phase with one another. The locus of all such particles is being called the wave front.
TYPES OF WAVE FRONT : TYPES OF WAVE FRONT Depending on the shape of source of light, wave front can be of three types:
(i) SPHERICAL WAVE FRONT: When the source of light is a point source, the wave front is a sphere with centre at the source.
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TYPES OF WAVE FRONT : TYPES OF WAVE FRONT (ii) CYLINDRICAL WAVE FRONT: When the source of light is linear, e.g. a slit, all the points equidistant from the source lie on a cylinder. Therefore, the wave front is cylindrical
WAVE FRONT : WAVE FRONT (iii) PLANE WAVE FRONT:
When the point source or linear source of light is at very large distance, a small portion of spherical or cylindrical wave front appears to be plane. Such a wave front is called a plane wave front.
RAYS & WAVE FRONT : RAYS & WAVE FRONT According to Huygens, rays are always normal in the wave front and light energy flows along rays.
CHRISTIAAN HUYGENES : CHRISTIAAN HUYGENES
HYGENES PRINCIPLE : HYGENES PRINCIPLE 1.Every point on the given wave front (called primary wave front) acts as a fresh source of new disturbance, called secondary wavelets, which travel in all directions with the velocity of light in the medium.
HYGENES PRINCIPLE : HYGENES PRINCIPLE 2. A surface touching these secondary wavelets, tangentially in the forward direction at any instant gives the new wave front at that instant. This is called secondary wave front.
REFLECTION ON THE BASIS OF WAVE THEORY : REFLECTION ON THE BASIS OF WAVE THEORY AB is a plane wave front incident on a plane mirror M1,M2 at ?BAA´ = ?i.
1,2,3 are the corresponding incident rays perpendicular to AB.
According to Huygens principle, every point on AB is a source of secondary wavelets.
Let the secondary wavelets from B strike M1,M2 at A´ in t seconds.
REFLECTION ON THE BASIS OF WAVE THEORY : REFLECTION ON THE BASIS OF WAVE THEORY ? BA´= c x t …(1)
Where c is the velocity of light in the medium.
The secondary wavelets from A will travel the same distance c x t in the same time. Therefore, with A as centre and c x t as radius, draw an are B´,
so that
AB´= c x t ….(2)
REFLECTION ON THE BASIS OF WAVE THEORY : REFLECTION ON THE BASIS OF WAVE THEORY From A', draw a tangent plane A´B´touching the arc tangentially at B´.
Therefore, A´B´ is the secondary wave front .
REFLECTION ON THE BASIS OF WAVE THEORY : REFLECTION ON THE BASIS OF WAVE THEORY Angle of incidence, i = ?BAA´
and angle of reflection, r = ?B´A´A
REFLECTION ON THE BASIS OF WAVE THEORY : REFLECTION ON THE BASIS OF WAVE THEORY In ?s AA´B and AA´B´,
AA´ is common,
BA´=AB´= c x t, and
?B= ?B´=90o
? ?s are congruent
? ?BAA´= ?B´A´A i.e., ?i= ?r ….(3)
Which is the first law of reflection.
REFRACTION ON THE BASIS OF WAVE THEORY : REFRACTION ON THE BASIS OF WAVE THEORY XY is a plane surface that separates a denser medium of refractive index, ? from a rarer medium.
If c1is velocity of light in rarer medium and c2 is velocity of light in denser medium, then by definition,
REFRACTION ON THE BASIS OF WAVE THEORY : REFRACTION ON THE BASIS OF WAVE THEORY AB is a plane wave front incident on XY at A.
?BAA´= ?i.
1,2,3 are the corresponding incident rays normal to AB.
According to Huygens principle, every point on AB is a source of secondary wavelets
REFRACTION ON THE BASIS OF WAVE THEORY : REFRACTION ON THE BASIS OF WAVE THEORY . Let the secondary wavelets from B strike XY at A´ in t seconds.
? BA´= c1 x t …(5)
REFRACTION ON THE BASIS OF WAVE THEORY : REFRACTION ON THE BASIS OF WAVE THEORY The secondary wavelets from A travel in the denser medium with a velocity c2 and would cover a distance (c2 x t) in t seconds. Therefore, with A as centre and radius equal to (c2 x t), draw an arc B´.
REFRACTION ON THE BASIS OF WAVE THEORY : REFRACTION ON THE BASIS OF WAVE THEORY From A´, draw a tangent plane touching the spherical arc tangentially at B´.
Therefore,A´B´ is the secondary wave front after t sec.
This would advance in the direction of rays 1´,2´,3´,which are the corresponding refracted rays, perpendicular to A´B´
REFRACTION ON THE BASIS OF WAVE THEORY : REFRACTION ON THE BASIS OF WAVE THEORY Let r be the angle of refraction. As angle of refraction is equal to the angle which the refracted plane wave front A´B´ makes with the refracting surface AA´,therefore,
? AA´B´ =r.
REFRACTION ON THE BASIS OF WAVE THEORY : REFRACTION ON THE BASIS OF WAVE THEORY
REFRACTION ON THE BASIS OF WAVE THEORY : REFRACTION ON THE BASIS OF WAVE THEORY It is clear from Fig. That the incident rays, normal to the interface XY and refracted ray, all lie in the same plane. (i.e. in the plane of the paper). This is the second law of refraction.
Hence laws of refraction are established on the basis of wave theory.
BEHAVIOUR OF A PRISM TOWARDS PLANE WAVEFRONT : BEHAVIOUR OF A PRISM TOWARDS PLANE WAVEFRONT Behavior of a prism.
BEHAVIOUR OF A LENS TOWARDS PLANE WAVEFRONT : BEHAVIOUR OF A LENS TOWARDS PLANE WAVEFRONT Behavior of a lens.
BEHAVIOUR OF A SPHERICAL MIRROR TOWARDS PLANE WAVEFRONT : BEHAVIOUR OF A SPHERICAL MIRROR TOWARDS PLANE WAVEFRONT Behavior of a spherical mirror.
SUPERPOSITION PRINCIPLE : SUPERPOSITION PRINCIPLE When two or more wave motions travelling through a medium superimpose one another, a new wave is formed in which resultant displacement at any instant is equal to the vector sum of the displacement due to individual waves
at that instant,
COHERENT SOURCES : COHERENT SOURCES The sources of light, which emit continuous light waves of the same wavelength, same frequency and in same phase or having a constant phase difference are called coherent sources.
For observing interference, coherence of waves is a must. Interference cannot be observed unless the two sources of light are coherent.
COHERENT SOURCES : COHERENT SOURCES Two independent sources of light cannot be coherent. This is because light is emitted from individual atoms, when they return to ground state after being excited by heat or electric discharge. Even the smallest source of light contains billions and billions of such atoms which obviously cannot emit light waves in the same phase. Such independent sources of light whose light waves do not possess a constant phase difference are called non-coherent sources or incoherent sources.
COHERENT SOURCES : COHERENT SOURCES Conditions for Obtaining Two Coherent Sources of light
1. Coherent sources of light should be obtained from a single source by some device.
2.The two sources should give monochromatic light.
3. The path difference between light waves from two sources should be small.
INTERFERENCE OF LIGHT : INTERFERENCE OF LIGHT Interference of light is the phenomenon of redistribution of light energy in a medium on account of superposition of light waves from two coherent sources.
YOUNG’S DOUBLE SLIT EXPERIMENT : YOUNG’S DOUBLE SLIT EXPERIMENT
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE Let the waves from two coherent sources of light be represented as
where a and b are the respective amplitudes of the two waves and ? is the constant phase angle by which second wave leads the first wave.
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE According to superposition principle, the displacement(y) of the resultant wave at time (t) would be given by
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE Thus the resultant wave is a harmonic wave of amplitude R.
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE We get
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE As resultant intensity I is directly proportional to the square of the amplitude of the resultant wave
For constructive interference
I should be maximum, for which
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE Where n = 0, 1, 2……..
If x is the path difference between the two waves reaching point P, corresponding to phase difference ?,then
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE Hence condition for constructive interference at a point is that phase difference between the two waves reaching the point should be zero or an even integral multiple of ?. Equivalently, path difference between the two waves reaching the point should be zero or an integral multiple of full wavelength.
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE For destructive interference,
I should be minimum
Cos ?=minimum = -1
??=?,3?,5?,……
Or ?=(2n – 1)?
Where n = 1, 2……..
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE The corresponding path difference between the two waves:
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE Hence, the condition for destructive interference at a point is that phase difference between the two waves reaching the point should be an odd integral multiple of ? or path difference between the two waves reaching the point should be an odd integral multiple of half the wavelength.
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE Important Notes
1. From the above discussion, we find that interference is constructive, when cos ?=1
i.e.
Amplitudes of two waves, which would be
Maximum.
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE Again, interference is destructive, when
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE the amplitudes of two waves, which would be minimum.
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE We find that
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE When b = a
=0 i.e. dark bands will be perfectly dark and the contrast between bright and dark interference bands will be the best.
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE the resultant intensity as a result of superimposition of two waves, may be pus as
CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE : CONDITIONS FOR CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE 2. If w1 and w2 are widths of two slits from which intensities of light I1 and I2 emanate,then
INTERFERENCE AND ENERGY CONSERVATION : INTERFERENCE AND ENERGY CONSERVATION In the interference pattern, if we take
then average intensity of light in the interference pattern.
INTERFERENCE AND ENERGY CONSERVATION : INTERFERENCE AND ENERGY CONSERVATION If there were no interference, intensity of light from two sources at every point on the screen would be
Which is the same as in the interference pattern.
INTERFERENCE AND ENERGY CONSERVATION : INTERFERENCE AND ENERGY CONSERVATION This establishes that in the interference pattern, intensity of light is simply being redistributed i.e. energy is being transferred from regions of destructive interference to the regions of constructive interference. No energy is being created or destroyed in the process. Thus the principle of energy conservation is being obeyed in the process of interference of light.
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT Suppose A and B are two fine slits, a small distance d apart.
Let them be illuminated by a strong source of monochromatic light of wavelength ?.
MN is a screen at a distance D from the slits.
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT The two waves starting from A and B superimpose upon each other, resulting in interference pattern on the screen, placed parallel to slits A and B.
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT O is centre of distance between the slits A and B. Draw AE, BF and OC perpendicular to MN.
The intensity of light at a point on the screen will depend upon the path difference between the two waves arriving at that point.
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT The point C on the screen is at equal distance from A and B. Therefore, the path difference between two waves reaching C is zero and the point C is of maximum intensity. It is called central maximum.
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT Consider a point P at a distance x from C. The path difference between two waves arriving at P,
= BP – AP
Let O be the mid-point of AB, and
AB = EF + d, AE = BF = D
In Fig.
PE = PC – EC = x – d/2
and PF = PC + CF = x + d/2
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT In ? BPF
BP=[BF2 + PF2]1/2
= [D2 + (x + d/2)2]1/2
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT Expanding Binomially, we get
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT we get, path difference
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT Now the intensity at point P is maximum or minimum according as the path difference (BP – AP) is an integral multiple of wavelength or an odd integral multiple of half wavelength.
Thus for bright fringes ( maxima),
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT Hence bright interference fringes are formed as detailed below:
for n = 0, xo=0 i.e.at
…central bright fringe
for n = 1, x1= …Ist bright fringe
for n = 2, x2 = …2nd bright fringe,
and so on
for n = n, xn = …nth bright fringe
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT Similarly, for dark fringes(minima)
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT Hence dark interference fringes are formed as detailed below:
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT Comparison with the above shows that dark interference fringes are situated in between bright interference fringes and vice-versa.
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT The separation between the centers of two consecutive bright fringes is the width of a dark fringe.
EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT : EXPRESSION FOR FRINGE WIDTH IN INTERFERENCE OF LIGHT Similarly, the separation between the centers of two consecutive dark fringes is the width of a bright fringe.
INTERFERENCE THROUGH THIN FILMS : INTERFERENCE THROUGH THIN FILMS
INTERFERENCE THROUGH THIN FILMS : INTERFERENCE THROUGH THIN FILMS
Slide 74 : For all your Physics Problems
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