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RAY OPTICS - I1. Refraction of Light2. Laws of Refraction3. Principle of Reversibility of Light4. Refraction through a Parallel Slab5. Refraction through a Compound Slab6. Apparent Depth of a Liquid7. Total Internal Reflection8. Refraction at Spherical Surfaces - Introduction9. Assumptions and Sign Conventions10. Refraction at Convex and Concave Surfaces11. Lens Maker’s Formula12. First and Second Principal Focus13. Thin Lens Equation (Gaussian Form)14. Linear Magnification
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Refraction of Light:Refraction is the phenomenon of change in the path of light as it travelsfrom one medium to another (when the ray of light is incident obliquely).It can also be defined as the phenomenon of change in speed of lightfrom one medium to another.Laws of Refraction: i RarerI Law: The incident ray, the normal tothe refracting surface at the point of Nincidence and the refracted ray all lie in r Denserthe same plane. r N µII Law: For a given pair of media and forlight of a given wavelength, the ratio ofthe sine of the angle of incidence to the i Rarersine of the angle of refraction is aconstant. (Snell’s Law) sin i (The constant µ is called refractive index of the medium, µ= sin r i is the angle of incidence and r is the angle of refraction.)
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TIPS: 1. µ of optically rarer medium is lower and that of a denser medium is higher. 2. µ of denser medium w.r.t. rarer medium is more than 1 and that of rarer medium w.r.t. denser medium is less than 1. (µair = µvacuum = 1) 3. In refraction, the velocity and wavelength of light change. 4. In refraction, the frequency and phase of light do not change. 5. aµm = ca / cm and aµm = λa / λmPrinciple of Reversibility of Light: sin i sin r i Rarer a µb = bµa = sin r sin i (a) a µb x bµa = 1 or a µb = 1 / bµa Denser r (b) If a ray of light, after suffering any number of reflections and/or refractions has its path N reversed at any stage, it travels back to the µ source along the same path in the opposite direction. A natural consequence of the principle of reversibility is that the image and object positions can be interchanged. These positions are called conjugate positions.
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Refraction through a Parallel Slab: N sin i1 sin i2 i1a µb = bµa = Rarer (a) sin r1 sin r2 But aµb x bµa = 1 N Denser r1 δ (b) sin i1 sin i2 t x =1 i2 sin r1 sin r2 M y µ It implies that i1 = r2 and i2 = r1 since i1 ≠ r1 and i2 ≠ r2. r2 Rarer (a)Lateral Shift: t sin δ t sin(i1- r1) y= or y= cos r1 cos r1Special Case:If i1 is very small, then r1 is also very small.i.e. sin(i1 – r1) = i1 – r1 and cos r1 = 1 y = t (i1 – r1) or y = t i1(1 – 1 /aµb)
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Refraction through a Compound Slab: sin i1 a µb = N µa sin r1 i1 Rarer (a) sin r1 bµc = sin r2 N Denser r1 (b) sin r2 r1 cµa = sin i1 µb a µb x bµc x cµa = 1 Denser N (c) r2 r2 or a µb x bµc = aµc µc or bµc = aµc / aµb Rarer (a) i1 µc > µb
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Apparent Depth of a Liquid: N sin i sin rbµa = or a µb = sin r sin i Rarer (a) hr Real depth µaa µb = = r ha Apparent depthApparent Depth of a Number ofImmiscible Liquids: ha r i n hr µb h a = ∑ h i / µi O’ i=1 i Denser (b)Apparent Shift: OApparent shift = hr - ha = hr – (hr / µ) = hr [ 1 - 1/µ]TIPS:1. If the observer is in rarer medium and the object is in denser medium then ha < hr. (To a bird, the fish appears to be nearer than actual depth.)2. If the observer is in denser medium and the object is in rarer medium then ha > hr. (To a fish, the bird appears to be farther than actual height.)
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Total Internal Reflection:Total Internal Reflection (TIR) is the phenomenon of complete reflection oflight back into the same medium for angles of incidence greater than thecritical angle of that medium. N N N N Rarer µa r = 90° (air) ic i > ic i Denser µg O (glass) Conditions for TIR: 1. The incident ray must be in optically denser medium. 2. The angle of incidence in the denser medium must be greater than the critical angle for the pair of media in contact.
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Relation between Critical Angle and Refractive Index:Critical angle is the angle of incidence in the denser medium for which theangle of refraction in the rarer medium is 90° . sin i sin ic gµa = = = sin ic sin r sin 90° 1 1 1 λg or aµg = a µg = or sin ic = Also sin ic = λa gµa sin ic a µgRed colour has maximum value of critical angle and Violet colour hasminimum value of critical angle since, 1 1 Applications of T I R: sin ic = = a µg a + (b/ λ2) 1. Mirage formation 2. Looming 3. Totally reflecting Prisms 4. Optical Fibres 5. Sparkling of Diamonds
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Spherical Refracting Surfaces:A spherical refracting surface is a part of a sphere of refracting material.A refracting surface which is convex towards the rarer medium is calledconvex refracting surface.A refracting surface which is concave towards the rarer medium iscalled concave refracting surface. Rarer Medium Denser Medium Rarer Medium Denser Medium A P • • B B • •P A C C R R APCB – Principal Axis C – Centre of Curvature P – Pole R – Radius of Curvature
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Assumptions:1. Object is the point object lying on the principal axis.2. The incident and the refracted rays make small angles with the principal axis.3. The aperture (diameter of the curved surface) is small.New Cartesian Sign Conventions:1. The incident ray is taken from left to right.2. All the distances are measured from the pole of the refracting surface.3. The distances measured along the direction of the incident ray are taken positive and against the incident ray are taken negative.4. The vertical distances measured from principal axis in the upward direction are taken positive and in the downward direction are taken negative.
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Refraction at Convex Surface:(From Rarer Medium to Denser Medium - Real Image) Ni=α+γ Aγ=r+β or r=γ-β i MA MA rtan α = or α = α γ β MO MO • P M • • C • O R I MA MAtan β = or β = u v MI MI µ1 µ2 Rarer Medium Denser Medium MA MAtan γ = or γ = MC MCAccording to Snell’s law, sin i µ2 i µ2 = or = or µ1 i = µ2 r sin r µ1 r µ1Substituting for i, r, α, β and γ, replacing M by P and rearranging, µ1 µ2 µ2 - µ1 Applying sign conventions with values, + = PO = - u, PI = + v and PC = + RPO PI PC µ1 µ2 µ2 - µ1 + = -u v R
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Refraction at Convex Surface:(From Rarer Medium to Denser Medium - Virtual Image) N A i r µ1 µ2 µ2 - µ1 β α γ + = • • • • -u v R I O uP M R C v µ1 µ2 Rarer Medium Denser MediumRefraction at Concave Surface:(From Rarer Medium to Denser Medium - Virtual Image) N r A µ1 µ2 µ2 - µ1 i = α γ + • • β• R • M P -u v R O I C u µ1 v µ2 Rarer Medium Denser Medium
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Refraction at Convex Surface:(From Denser Medium to Rarer Medium - Real Image) N A r µ2 µ1 µ1 - µ2 i α γ β + = • C • • M P • -u v R O R I u v Denser Medium µ2 Rarer Medium µ1Refraction at Convex Surface:(From Denser Medium to Rarer Medium - Virtual Image) µ2 µ1 µ1 - µ2 + = -u v RRefraction at Concave Surface:(From Denser Medium to Rarer Medium - Virtual Image) µ2 µ1 µ1 - µ2 + = -u v R
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Note:1. Expression for ‘object in rarer medium’ is same for whether it is real or virtual image or convex or concave surface. µ1 µ2 µ2 - µ1 + = -u v R2. Expression for ‘object in denser medium’ is same for whether it is real or virtual image or convex or concave surface. µ2 µ1 µ1 - µ2 + = -u v R3. However the values of u, v, R, etc. must be taken with proper sign conventions while solving the numerical problems.4. The refractive indices µ1 and µ2 get interchanged in the expressions.
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Lens Maker’s Formula:For refraction at LLP1N, µ1 µ1 µ1 µ2 µ2 - µ1 N1 N2 + = A CO CI1 CC1 i(as if the image isformed in the denser • • • C •P P1 • • •medium) O C2 2 I C1 I1For refraction at R2 R1 µ2LP2N, µ2 µ1 -(µ1 - µ2) u v + = -CI1 CI CC2 N(as if the object is in the denser medium and the image is formed in the rarermedium)Combining the refractions at both the surfaces, Substituting the values µ1 µ1 with sign conventions, 1 1 + = (µ2 - µ1)( + ) CO CI CC1 CC2 1 1 (µ - µ1) 1 1 + = 2 ( - ) -u v µ1 R1 R2
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Since µ2 / µ1 = µ 1 1 µ2 1 1 + =( - 1) ( - ) -u v µ1 R1 R2 or 1 1 1 1 + = (µ – 1) ( - ) -u v R1 R2When the object is kept at infinity, the image is formed at the principal focus.i.e. u = - ∞, v = + f. 1 1 1So, = (µ – 1) ( - ) f R1 R2This equation is called ‘Lens Maker’s Formula’. 1 1 1Also, from the above equations we get, + = -u v f
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First Principal Focus:First Principal Focus is the point on the principal axis of the lens at which ifan object is placed, the image would be formed at infinity. F1 F1 f1 f1Second Principal Focus:Second Principal Focus is the point on the principal axis of the lens atwhich the image is formed when the object is kept at infinity. F2 F2 f2 f2
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Thin Lens Formula (Gaussian Form of Lens Equation):For Convex Lens: A M 2F1 F1 F2 2F2 B’ • • C • • • B u v R fTriangles ABC and A’B’C are similar. CB’ B’F2 = CB CF2 A’ A’B’ CB’ = AB CB CB’ CB’ - CF2 =Triangles MCF2 and A’B’F2 are similar. CB CF2 According to new Cartesian sign A’B’ B’F2 conventions, = MC CF2 CB = - u, CB’ = + v and CF2 = + f. A’B’ B’F2 or = 1 1 1 AB CF2 - = v u f
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Linear Magnification:Linear magnification produced by a lens is defined as the ratio of the size ofthe image to the size of the object. I m = Magnification in terms of v and f: O A’B’ CB’ = f-v AB CB m = fAccording to new Cartesian signconventions, Magnification in terms of v and f:A’B’ = + I, AB = - O, CB’ = + v andCB = - u. f m =+I +v I v f-u = or m= =-O -u O uPower of a Lens:Power of a lens is its ability to bend a ray of light falling on it and is reciprocalof its focal length. When f is in metre, power is measured in Dioptre (D). 1 P = f
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RAY OPTICS - II1. Refraction through a Prism2. Expression for Refractive Index of Prism3. Dispersion4. Angular Dispersion and Dispersive Power5. Blue Colour of the Sky and Red Colour of the Sun6. Compound Microscope7. Astronomical Telescope (Normal Adjustment)8. Astronomical Telescope (Image at LDDV)9. Newtonian Telescope (Reflecting Type)10. Resolving Power of Microscope and Telescope
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Refraction of Light through Prism: A A N1 N2 P D δ i e Q r1 O r2 µ B C Prism Refracting SurfacesIn quadrilateral APOQ, From (1) and (2),A + O = 180° …… .(1) A = r1 + r2 (since N1 and N2 are normal) From (3),In triangle OPQ, δ = (i + e) – (A)r1 + r2 + O = 180° …… .(2) or i+e=A+δIn triangle DPQ,δ = (i - r1) + (e - r2) Sum of angle of incidence and angle of emergence is equal to the sum ofδ = (i + e) – (r1 + r2) …….(3) angle of prism and angle of deviation.
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Variation of angle of deviation with angle of incidence:When angle of incidence increases, δthe angle of deviation decreases.At a particular value of angle of incidencethe angle of deviation becomes minimumand is called ‘angle of minimum deviation’. δmAt δm, i=e and r1 = r2 = r (say) 0 i=e iAfter minimum deviation, angle of deviationincreases with angle of incidence.Refractive Index of Material of Prism: A = r1 + r2 According to Snell’s law, A = 2r sin i sin i µ= = r=A/2 sin r1 sin r i+e=A+δ (A + δm) sin 2 i = A + δm 2 µ= i = (A + δm) / 2 A sin 2
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Refraction by a Small-angled Prism for Small angle of Incidence: sin i sin e µ= and µ= sin r1 sin r2 If i is assumed to be small, then r1, r2 and e will also be very small. So, replacing sines of the angles by angles themselves, we get i e µ= and µ = r1 r2 i + e = µ (r1 + r2) = µ A But i + e = A + δ So, A + δ = µ A or δ = A (µ – 1)
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Dispersion of White Light through Prism:The phenomenon of splitting a ray of white light into its constituent colours(wavelengths) is called dispersion and the band of colours from violet to redis called spectrum (VIBGYOR). A D δr N δv R O Y G White B light I V B C ScreenCause of Dispersion: sin i sin i Since µv > µr , rr > rv µv = and µr = sin rv sin rr So, the colours are refracted at different angles and hence get separated.
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Dispersion can also be explained on the basis of Cauchy’s equation. b c µ=a + + (where a, b and c are constants for the material) λ2 λ4 Since λv < λ r , µv > µr But δ = A (µ – 1) Therefore, δv > δr So, the colours get separated with different angles of deviation. Violet is most deviated and Red is least deviated.Angular Dispersion:1. The difference in the deviations suffered by two colours in passing through a prism gives the angular dispersion for those colours.2. The angle between the emergent rays of any two colours is called angular dispersion between those colours.3. It is the rate of change of angle of deviation with wavelength. (Φ = dδ / dλ) Φ = δv - δr or Φ = (µv – µr) A
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Dispersive Power:The dispersive power of the material of a prism for any two colours is definedas the ratio of the angular dispersion for those two colours to the meandeviation produced by the prism.It may also be defined as dispersion per unit deviation. Φ δv + δr ω= where δ is the mean deviation and δ = δ 2 δv - δr (µv – µr) A (µv – µr) Also ω = or ω = (µy – 1) A or ω = (µ – 1) δ yScattering of Light – Blue colour of the sky and Reddish appearanceof the Sun at Sun-rise and Sun-set:The molecules of the atmosphere and other particles that are smaller than thelongest wavelength of visible light are more effective in scattering light of shorterwavelengths than light of longer wavelengths. The amount of scattering isinversely proportional to the fourth power of the wavelength. (Rayleigh Effect)Light from the Sun near the horizon passes through a greater distance in the Earth’satmosphere than does the light received when the Sun is overhead. Thecorrespondingly greater scattering of short wavelengths accounts for the reddishappearance of the Sun at rising and at setting.When looking at the sky in a direction away from the Sun, we receive scatteredsunlight in which short wavelengths predominate giving the sky its characteristicbluish colour.
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Compound Microscope: uo vo B A’’’ fe Fo 2Fo 2Fe α A’• • • • • •2Fo A Fo Po A’’ Fe β Pe Eye fo fo Objective B’ L Eyepiece B’’ D Objective: The converging lens nearer to the object. Eyepiece: The converging lens through which the final image is seen. Both are of short focal length. Focal length of eyepiece is slightly greater than that of the objective.
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Angular Magnification or Magnifying Power (M):Angular magnification or magnifying power of a compound microscope isdefined as the ratio of the angle β subtended by the final image at the eye tothe angle α subtended by the object seen directly, when both are placed atthe least distance of distinct vision. β M = Me x Mo M= α ve D (ve = - D Since angles are small, Me = 1 - or Me = 1 + fe fe = - 25 cm) α = tan α and β = tan β tan β vo vo D M= and Mo = M= (1+ ) tan α - uo - uo fe A’’B’’ D M= x Since the object is placed very close to the D A’’A’’’ principal focus of the objective and the A’’B’’ D image is formed very close to the eyepiece, M= x uo ≈ fo and vo ≈ L D AB -L D A’’B’’ M= (1+ ) M= fo fe AB A’’B’’ A’B’ -L D (Normal adjustment M= x or M≈ x A’B’ AB fo fe i.e. image at infinity)
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Astronomical Telescope: (Image formed at infinity –Normal Adjustment) fo + fe = L fo fe Eye Fo Fe α α • Po β Pe I Eyepiece Image at Objective infinityFocal length of the objective is much greater than that of the eyepiece.Aperture of the objective is also large to allow more light to pass through it.
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Angular magnification or Magnifying power of a telescope in normaladjustment is the ratio of the angle subtended by the image at the eye asseen through the telescope to the angle subtended by the object as seendirectly, when both the object and the image are at infinity. β M= α Since angles are small, α = tan α and β = tan β tan β M= tan α Fe I Fe I M= / PeFe PoFe -I -I M= / - fe fo - fo (fo + fe = L is called the length of the M= telescope in normal adjustment). fe
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Astronomical Telescope: (Image formed at LDDV) fo Eye feα A Fe Fo α • • Po β Pe I Eyepiece ue Objective B D
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Angular magnification or magnifying power of a telescope in this case isdefined as the ratio of the angle β subtended at the eye by the final imageformed at the least distance of distinct vision to the angle α subtended atthe eye by the object lying at infinity when seen directly. β 1 1 1 M= - = α -D - ue fe Since angles are small, 1 1 1 or = + α = tan α and β = tan β ue fe D tan β Multiplying by fo on both sides and M= tan α rearranging, we get Fo I Fo I M= / - fo fe PeFo PoFo M= (1+ ) fe D PoFo + fo M= or M = Clearly focal length of objective must be PeFo - ue greater than that of the eyepiece for larger magnifying power. Lens Equation Also, it is to be noted that in this case M is 1 1 1 larger than that in normal adjustment - = becomes v u f position.
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Newtonian Telescope: (Reflecting Type) Plane Mirror Light from star Magnifying Power: Eyepiece fo M= fe Concave Mirror Eye
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Resolving Power of a Microscope:The resolving power of a microscope is defined as the reciprocal of thedistance between two objects which can be just resolved when seenthrough the microscope. Objective 1 2 µ sin θ θ Resolving Power = = •• ∆d λ ∆d Resolving power depends on i) wavelength λ, ii) refractive index of the medium between the object and the objective and iii) half angle of the cone of light from one of the objects θ.Resolving Power of a Telescope:The resolving power of a telescope is defined as the reciprocal of thesmallest angular separation between two distant objects whose images areseen separately. Objective 1 a Resolving Power = = •• dθ 1.22 λ dθ Resolving power depends on i) wavelength λ, ii) diameter of the objective a.
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WAVE OPTICS - I1. Electromagnetic Wave2. Wavefront3. Huygens’ Principle4. Reflection of Light based on Huygens’ Principle5. Refraction of Light based on Huygens’ Principle6. Behaviour of Wavefront in a Mirror, Lens and Prism7. Coherent Sources8. Interference9. Young’s Double Slit Experiment10. Colours in Thin Films
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Electromagnetic Wave: Y E0 0 X B0 Z 1. Variations in both electric and magnetic fields occur simultaneously. Therefore, they attain their maxima and minima at the same place and at the same time. 2. The direction of electric and magnetic fields are mutually perpendicular to each other and as well as to the direction of propagation of wave. 3. The speed of electromagnetic wave depends entirely on the electric and magnetic properties of the medium, in which the wave travels and not on the amplitudes of their variations. Wave is propagating along X – axis with speed c = 1 / √µ0ε0 For discussion of optical property of EM wave, more significance is given to Electric Field, E. Therefore, Electric Field is called ‘light vector’.
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Wavefront:A wavelet is the point of disturbance due to propagation of light.A wavefront is the locus of points (wavelets) having the same phase ofoscillations.A line perpendicular to a wavefront is called a ‘ray’. Spherical Cylindrical Wavefront Wavefront from a point • from a linear source source Plane Pink Dots – Wavelets Wavefront Blue Envelope– Wavefront Red Line – Ray
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Huygens’ Construction or Huygens’ Principle of SecondaryWavelets: . . . . . . . S• . . . . . . New . . New Wavefront (Spherical) . Wave- front . (Plane) (Wavelets - Red dots on the wavefront)1. Each point on a wavefront acts as a fresh source of disturbance of light.2. The new wavefront at any time later is obtained by taking the forward envelope of all the secondary wavelets at that time.Note: Backward wavefront is rejected. Why?Amplitude of secondary wavelet is proportional to ½ (1+cosθ). Obviously,for the backward wavelet θ = 180°and (1+cos θ) is 0.
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Laws of Reflection at a Plane Surface (On Huygens’ Principle):If c be the speed of light, tbe the time taken by light togo from B to C or A to D or N NE to G through F, then EF FG B D t = + E G c c r i X i r AF sin i FC sin r Y t = + A F C c c AB – Incident wavefront AC sin r + AF (sin i – sin r) t = CD – Reflected wavefront c XY – Reflecting surface For rays of light from different parts on the incident wavefront, the values of AF are different. But light from different points of the incident wavefront should take the same time to reach the corresponding points on the reflected wavefront. So, t should not depend upon AF. This is possible only if sin i – sin r = 0. i.e. sin i = sin r or i=r
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Laws of Refraction at a Plane Surface (On Huygens’ Principle):If c be the speed of light, tbe the time taken by light to N Ngo from B to C or A to D or B RarerE to G through F, then E c, µ1 i EF FG X i F C t = + r Y c v A Denser G v, µ2 AF sin i FC sin r r t = + D c v AC sin r sin i sin r t = + AF ( - ) AB – Incident wavefront v c v CD – Refracted wavefront XY – Refracting surfaceFor rays of light from different parts on the incident wavefront, the values ofAF are different. But light from different points of the incident wavefrontshould take the same time to reach the corresponding points on therefracted wavefront.So, t should not depend upon AF. This is possible onlyif sin i - sin r or sin i sin r = or sin i = c = µ c v =0 c v sin r v
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Behaviour of a Plane Wavefront in a Concave Mirror, Convex Mirror,Convex Lens, Concave Lens and Prism: C A A C D B B Concave Mirror D Convex Mirror C A A C D B B Convex Lens Concave Lens D AB – Incident wavefront CD – Reflected / Refracted wavefront
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A C B D Prism PrismAB – Incident wavefront CD –Refracted wavefrontCoherent Sources:Coherent Sources of light are those sources of light which emit light waves ofsame wavelength, same frequency and in same phase or having constantphase difference.Coherent sources can be produced by two methods:1. By division of wavefront (Young’s Double Slit Experiment, Fresnel’s Biprism and Lloyd’s Mirror)2. By division of amplitude (Partial reflection or refraction)
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Interference of Waves: E1 + E2 Bright Band E1 Dark Band E2 S1 • Bright Band S2 • Dark Band Constructive Interference E = E1 + E2 E1 E1 - E2 Bright Band E2 Crest Destructive Interference E = E1 - E2 Trough Bright Band 1st Wave (E1) Dark Band 2nd Wave (E2) Resultant Wave The phenomenon of one wave interfering Reference Line with another and the resulting redistribution of energy in the space around the two sources of disturbance is called interference of waves.
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Theory of Interference of Waves: The waves are with same speed, wavelength, frequency, E1 = a sin ωt time period, nearly equal amplitudes, travelling in the E2 = b sin (ωt + Φ) same direction with constant phase difference of Φ. ω is the angular frequency of the waves, a,b are the amplitudes and E1, E2 are the instantaneous values of Electric displacement. Applying superposition principle, the magnitude of the resultant displacement of the waves is E = E1 + E2 E = a sin ωt + b sin (ωt + Φ) E = (a + b cos Φ) sin ωt + b sin Φ cos ωt Putting a + b cos Φ = A cos θ (where E is the A sin θ resultant b sin Φ = A sin θ displacement, b sin Φ b A is the resultant A We get E = A sin (ωt + θ) amplitude and θ is the resultant phase difference) Φ θ a A = √ (a2 + b2 + 2ab cos Φ) b cos Φ b sin Φ A cos θ tan θ = a + b cos Φ
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A = √ (a2 + b2 + 2ab cos Φ) Intensity I is proportional to the square of the amplitude of the wave. So, I α A2 i.e. I α (a2 + b2 + 2ab cos Φ)Condition for Constructive Interference of Waves:For constructive interference, I should be maximum which is possibleonly if cos Φ = +1. i.e. Φ = 2nπ where n = 0, 1, 2, 3, …….Corresponding path difference is ∆ = (λ / 2 π) x 2nπ ∆=nλ Imax α (a + b)2Condition for Destructive Interference of Waves:For destructive interference, I should be minimum which is possibleonly if cos Φ = - 1. i.e. Φ = (2n + 1)π where n = 0, 1, 2, 3, …….Corresponding path difference is ∆ = (λ / 2 π) x (2n + 1)π ∆ = (2n + 1) λ / 2 Iminα (a - b)2
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Comparison of intensities of maxima and minima: Imax α (a + b)2 Imin α (a - b)2 Imax (a + b)2 (a/b + 1)2 = = Imin (a - b)2 (a/b - 1)2 Imax (r + 1)2 = where r = a / b (ratio of the amplitudes) Imin (r - 1)2Relation between Intensity (I), Amplitude (a) of the wave andWidth (w) of the slit: I α a2 a α √w I1 (a1)2 w1 = = I2 (a2 )2 w2
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Young’s Double Slit Experiment: S • Single Slit Double Slit P S1 y Screen d/2 S • d d/2 O S2 DThe waves from S1 and S2 reach the point P withsome phase difference and hence path difference∆ = S2P – S1PS2P2 – S1P2 = [D2 + {y + (d/2)}2] - [D2 + {y - (d/2)}2](S2P – S1P) (S2P + S1P) = 2 yd ∆ (2D) = 2 yd ∆ = yd / D
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Positions of Bright Fringes: Positions of Dark Fringes:For a bright fringe at P, For a dark fringe at P,∆ = yd / D = nλ ∆ = yd / D = (2n+1)λ/2 where n = 0, 1, 2, 3, … where n = 0, 1, 2, 3, … y=nDλ/d y = (2n+1) D λ / 2dFor n = 0, y0 = 0 For n = 0, y0’ = D λ / 2dFor n = 1, y1 = D λ / d For n = 1, y1’ = 3D λ / 2dFor n = 2, y2 = 2 D λ / d …… For n = 2, y2’ = 5D λ / 2d …..For n = n, yn = n D λ / d For n = n, yn’ = (2n+1)D λ / 2dExpression for Dark Fringe Width: Expression for Bright Fringe Width:βD = yn – yn-1 βB = yn’ – yn-1’ = n D λ / d – (n – 1) D λ / d = (2n+1) D λ / 2d – {2(n-1)+1} D λ / 2d =Dλ/d =Dλ/dThe expressions for fringe width show that the fringes are equally spaced onthe screen.
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Distribution of Intensity: Suppose the two interfering waves Intensity have same amplitude say ‘a’, then Imax α (a+a)2 i.e. Imax α 4a2 All the bright fringes have this same intensity. Imin = 0y 0 y All the dark fringes have zero intensity.Conditions for sustained interference: 1. The two sources producing interference must be coherent. 2. The two interfering wave trains must have the same plane of polarisation. 3. The two sources must be very close to each other and the pattern must be observed at a larger distance to have sufficient width of the fringe. (D λ / d) 4. The sources must be monochromatic. Otherwise, the fringes of different colours will overlap. 5. The two waves must be having same amplitude for better contrast between bright and dark fringes.
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Colours in Thin Films:It can be proved that the pathdifference between the light partially A Creflected from PQ and that from ipartially transmitted and then Qreflected from RS is P O B µ ∆ = 2µt cos r r tSince there is a reflection at O, the R Sray OA suffers an additional phasedifference of π and hence thecorresponding path difference ofλ/2.For the rays OA and BC to interfere For the rays OA and BC to interfereconstructively (Bright fringe), the destructively (Dark fringe), the pathpath difference must be (n + ½) λ difference must be nλSo, 2µt cos r = (n + ½) λ So, 2µt cos r = n λWhen white light from the sun falls on thin layer of oil spread over water in therainy season, beautiful rainbow colours are formed due to interference of light.
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WAVE OPTICS - II1. Electromagnetic Wave2. Diffraction3. Diffraction at a Single Slit4. Theory of Diffraction5. Width of Central Maximum and Fresnel’s Distance6. Difference between Interference and Diffraction7. Polarisation of Mechanical Waves8. Polarisation of Light9. Malus’ Law10. Polarisation by Reflection – Brewster’s Law11. Polaroids and their uses
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Electromagnetic Wave: Y E0 0 X B0 Z 1. Variations in both electric and magnetic fields occur simultaneously. Therefore, they attain their maxima and minima at the same place and at the same time. 2. The direction of electric and magnetic fields are mutually perpendicular to each other and as well as to the direction of propagation of wave. 3. The speed of electromagnetic wave depends entirely on the electric and magnetic properties of the medium, in which the wave travels and not on the amplitudes of their variations. Wave is propagating along X – axis with speed c = 1 / √µ0ε0 For discussion of EM wave, more significance is given to Electric Field, E.
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Diffraction of light:The phenomenon of bending of light around the corners and theencroachment of light within the geometrical shadow of the opaque obstaclesis called diffraction. X XS• S• Slit Y Y Obstacle Screen Diffraction at a slit Diffraction at an obstacle Screen X & Y – Region of diffraction
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Diffraction of light at a single slit:1) At an angle of diffraction θ = 0°: A θ = 0° 0 • 1 • 2 • 3 • 4 •d 5 • 6 • •O Bright 7 • 8 • 9 10 • 11 • 12 • • B D Plane Wavefront Slit Screen The wavelets from the single wavefront reach the centre O on the screen in same phase and hence interfere constructively to give Central or Primary Maximum (Bright fringe).
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2) At an angle of diffraction θ = θ1: The slit is imagined to be divided into 2 equal halves. θ1 A 0 • 1 • 2 • • P1 Dark 3 • 4 • 5 • θ1 6 • •O Bright • 7 λ/2 8 • 9 10 • 11 • • N 12 θ1 • Plane B λ Wavefront SlitThe wavelets from the single wavefront diffract at an angle θ1 such Screenthat BN is λ and reach the point P1. The pairs (0,6), (1,7), (2,8), (3,9),(4,10), (5,11) and (6,12) interfere destructively with path differenceλ/2 and give First Secondary Minimum (Dark fringe).
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3) At an angle of diffraction θ = θ2: The slit is imagined to be divided into 4 equal parts. • P2 Dark • P1’ A θ2 0 • 1 • 2 • • P1 Dark 3 • λ/2 4 • 5 • θ2 6 • •O Bright 7 • λ 8 • 9 10 • 3λ/2 11 • N 12 • θ2 • B 2λ Plane Wavefront Slit ScreenThe wavelets from the single wavefront diffract at an angle θ2 such thatBN is 2λ and reach the point P2. The pairs (0,3), (1,4), (2,5), (3,6), (4,7),(5,8), (6,9), (7,10), (8,11) and (9,12) interfere destructively with pathdifference λ/2 and give Second Secondary Minimum (Dark fringe).
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4) At an angle of diffraction θ = θ1’: The slit is imagined to be divided into 3 equal parts. • P2 θ1’ • P1’ Bright A 0 • 1 • 2 • • P1 Dark 3 • 4 • • 5 λ/2 θ1’ 6 • •O Bright 7 • 8 • λ 9 10 • 11 • 12 • N θ1’ • Plane B 3λ/2 Wavefront SlitThe wavelets from the single wavefront diffract at an angle θ1’ such that ScreenBN is 3λ/2 and reach the point P1’. The pairs (0,8), (1,9), (2,10), (3,11) and(4,12) interfere constructively with path difference λ and (0,4), (1,5), (2,6),…… and (8,12) interfere destructively with path difference λ/2. Howeverdue to a few wavelets interfering constructively First SecondaryMaximum (Bright fringe) is formed.
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Diffraction at various angles: • P2 • θ2 θ0 θ’ • P1’ θ1’ A θ = 11 A θ2 0 • 1 • 2 • • P11 •P θ1 • 3 λ/2 4 • • 5 λ/2 θθ ’ 21 θ1 θ=0 6 • •O 7 λ/2λ • I 8 • λ 9 10 • 3λ/2 11 • N • N θ’ θ2 θ 1 12 N 1 •Plane BB λ 2λ 3λ/2Wavefront Slit Screen Central Maximum is the brightest fringe. Diffraction is not visible after a few order of diffraction.
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Theory:The path difference between the 0th wavelet and 12th wavelet is BN.If ‘θ’ is the angle of diffraction and ‘d’ is the slit width, then BN = d sin θTo establish the condition for secondary minima, the slit is divided into 2, 4,6, … equal parts such that corresponding wavelets from successive regionsinterfere with path difference of λ/2.Or for nth secondary minimum, the slit can be divided into 2n equal parts.For θ1, d sin θ1 = λ Since θn is very small,For θ2, d sin θ2 = 2λ d θn = nλFor θn, d sin θn = nλ θn = nλ / d (n = 1, 2, 3, ……)To establish the condition for secondary maxima, the slit is divided into 3, 5,7, … equal parts such that corresponding wavelets from alternate regionsinterfere with path difference of λ.Or for nth secondary minimum, the slit can be divided into (2n + 1) equalparts.For θ1’, d sin θ1’ = 3λ/2 Since θn’ is very small,For θ2’, d sin θ2’ = 5λ/2 d θn’ = (2n + 1)λ / 2For θn’, d sin θn’ = (2n + 1)λ/2 θn’ = (2n + 1)λ / 2d (n = 1, 2, 3, ……)
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Width of Central Maximum: θ1 A 0 • 1 • 2 • • P1 Dark 3 • 4 • y1d 5 • θ1 6 • •O Bright D 7 λ/2• 8 • 9 10 • 11 • • N 12 θ1 • Plane B λ Wavefront Slit tan θ1 = y1 / D Screen y1 = D λ / d or θ1 = y1 / D (since θ1 is very small) Since the Central Maximum is spread on either side of O, the d sin θ1 = λ width is or θ1 = λ / d (since θ1 is very small) β0 = 2D λ / d
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Fresnel’s Distance:Fresnel’s distance is that distance from the slit at which the spreadingof light due to diffraction becomes equal to the size of the slit. y1 = D λ / d At Fresnel’s distance, y1 = d and D = DFSo, DF λ / d = d or DF = d2 / λIf the distance D between the slit and the screen is less than Fresnel’sdistance DF, then the diffraction effects may be regarded as absent.So, ray optics may be regarded as a limiting case of wave optics.Difference between Interference and Diffraction: Interference Diffraction 1. Interference is due to the 1. Diffraction is due to the superposition of two different superposition of secondary wave trains coming from coherent wavelets from the different parts sources. of the same wavefront. 2. Fringe width is generally constant. 2. Fringes are of varying width. 3. All the maxima have the same 3. The maxima are of varying intensity. intensities. 4. There is a good contrast between 4. There is a poor contrast between the maxima and minima. the maxima and minima.
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Polarisation of Transverse Mechanical Waves: Narrow SlitTransversedisturbance(up and down) Narrow Slit 90°Transversedisturbance Narrow Slit(up and down)
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Polarisation of Light Waves: • • • • • • • • • • Wave • S - Parallel to the plane • - Perpendicular to the plane Natural Light Representation of Natural Light In natural light, millions of transverse vibrations occur in all the directions perpendicular to the direction of propagation of wave. But for convenience, we can assume the rectangular components of the vibrations with one component lying on the plane of the diagram and the other perpendicular to the plane of the diagram.
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Light waves are electromagnetic waves with electric and magnetic fieldsoscillating at right angles to each other and also to the direction ofpropagation of wave. Therefore, the light waves can be polarised. Optic Axis• • • • • • Unpolarised Plane Plane light Polarised Polarised light light Polariser Analyser Tourmaline Tourmaline Crystal Crystal 90°• • • • • • No light Plane Unpolarised Polarised light light
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90° • • • • • • Unpolarised Plane light Polarised Polariser Analyser light Plane of Vibration Plane of PolarisationWhen unpolarised light is incident on the polariser, the vibrations parallel tothe crystallographic axis are transmitted and those perpendicular to the axisare absorbed. Therefore the transmitted light is plane (linearly) polarised.The plane which contains the crystallographic axis and vibrationstransmitted from the polariser is called plane of vibration.The plane which is perpendicular to the plane of vibration is called planeof polarisation.
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Malus’ Law:When a beam of plane polarised light is incident on an analyser, theintensity I of light transmitted from the analyser varies directly as thesquare of the cosine of the angle θ between the planes of transmission ofanalyser and polariser. a I α cos2 θ a sin θ a cos θIf a be the amplitude of the electric Pvector transmitted by the polariser, Athen only the component a cos θwill be transmitted by the analyser. θIntensity of transmitted light fromthe analyser is Case I : When θ = 0°or 180° , I=I I = k (a cos θ)2 0or I = k a2 cos2 θ Case II : When θ = 90° , I=0 I = I0 cos2 θ Case III: When unpolarised light is incident on the analyser the intensity of the(where I0 = k a2 is the transmitted light is one-half of the intensity ofintensity of light transmitted incident light. (Since average value of cos2θfrom the polariser) is ½)
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Polarisation by Reflection and Brewster’s Law:The incident light wave is made ofparallel vibrations (π – components)on the plane of incidence and • • •perpendicular vibrations (σ – • • θP • • • acomponents : perpendicular to planeof incidence). 90° µAt a particular angle θP, the parallel • • rcomponents completely refracted b • •whereas the perpendicular componentspartially get refracted and partially getreflected.i.e. the reflected components are all inperpendicular plane of vibration and θP + r = 90° or r = 90°- θPhence plane polarised. sin θPThe intensity of transmitted light a µb = sin rthrough the medium is greater than thatof plane polarised (reflected) light. sin θP a µb = sin 90°- θP a µb = tan θP
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Polaroids:H – Polaroid is prepared by taking a sheet of polyvinyl alcohol (long chainpolymer molecules) and subjecting to a large strain. The molecules areoriented parallel to the strain and the material becomes doubly refracting.When strained with iodine, the material behaves like a dichroic crystal.K – Polaroid is prepared by heating a stretched polyvinyl alcohol film in thepresence of HCl (an active dehydrating catalyst). When the film becomesslightly darkened, it behaves like a strong dichroic crystal.Uses of Polaroids:1) Polaroid Sun Glasses2) Polaroid Filters3) For Laboratory Purpose4) In Head-light of Automobiles5) In Three – Dimensional Motion Picutres6) In Window Panes7) In Wind Shield in Automobiles
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