27Refraction at Plane surfaces and Total internal reflection

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REFRACTION AT PLANE SURFACES AND TOTAL INTERNAL REFLECTION
▢ Refraction:
❖ Definition: Bending of light when it passes from one medium to another
❖ Cause: Change in speed of light in different media
❖ Key Points:
  • •
    Light travels faster in rarer medium than denser medium
  • •
    Denser → rarer: ray bends away from normal
  • •
    Rarer → denser: ray bends towards normal
▢ Laws of Refraction:

Table 1: Laws of Refraction

Law
Statement
1st law
Incident ray, refracted ray and normal lie in same plane
2nd law / Snell's law
Ratio of sine of angle of incidence to sine of angle of refraction is constant for given pair of media
❖ Snell's Law for Large Angle:
◉ Condition: \(i\geq10^\circ\)
◉ Formulae:
  • •
    \(\frac{\sin i}{\sin r}=\,^{1}\mu_2=\frac{\mu_2}{\mu_1}=\frac{v_1}{v_2}=\frac{\lambda_1}{\lambda_2}\)
  • •
    \(\mu_1\sin i=\mu_2\sin r\)
❖ Snell's Law for Small Angle:
◉ Condition: \(i<10^\circ\)
◉ Formulae:
  • •
    \(\frac{i}{r}=\frac{\mu_2}{\mu_1}\)
  • •
    \(\mu_1 i=\mu_2 r\)
◉ Note: \(i,r\) are in radians
❖ Colour Points:
  • •
    Frequency remains same when light travels from one medium to another
  • •
    Wavelength of red light is maximum
  • •
    Speed of red light is maximum in a medium
  • •
    Refractive index is minimum for red light
  • •
    Frequency remains constant for all colours during refraction
▢ Refractive Index:
❖ Definition: Ratio of speed of light in vacuum to speed of light in medium
❖ Formula: \(\mu=\frac{v_0}{v}=\frac{c}{v}\)

Table 1: Refractive Index Points

Point
Answer
Nature
Scalar quantity
Temperature increases
\(\mu\) decreases
Depends on angle of incidence
No
Depends on intensity
No
Optically denser medium
Higher refractive index
Optically rarer medium
Lower refractive index
Refractive index of air
Nearly 1 and least
❖ Absolute Refractive Index: \(\mu=\frac{c}{v}=\sqrt{\frac{\mu_m\epsilon_m}{\mu_0\epsilon_0}}=\sqrt{\mu_r\epsilon_r}\)
❖ Symbols:
  • •
    \(\mu_r=\frac{\mu_m}{\mu_0}\) = relative magnetic permeability
  • •
    \(\epsilon_r=\frac{\epsilon_m}{\epsilon_0}\) = relative electrical permittivity
▢ Cauchy's Relation:
❖ Formula: \(\mu=A+\frac{B}{\lambda^2}\)
❖ Meaning: Refractive index depends on wavelength of incident light
❖ Conclusion:
  • •
    \(\mu\propto\frac{1}{\lambda^2}\)
  • •
    Red has least refractive index
  • •
    Violet has greatest refractive index
▢ Lateral Shift:
❖ Definition: Perpendicular distance between incident ray and emergent ray after passing through a parallel glass slab
❖ Key Points:
  • •
    Emergent ray is parallel to incident ray
  • •
    Deviation produced by slab is zero
  • •
    Ray is shifted laterally
❖ Formula: \(d=\frac{t\sin(i-r)}{\cos r}\)

Table 1: Lateral Shift

Condition / Factor
Point
Grazing incidence \((i=90^\circ)\)
\(d=t\)
For small angle
\(\sin i\approx i,\ \cos r\approx1\)
Small angle Snell relation
\(\frac{i}{r}=\mu\)
Depends on
Refractive index, thickness of slab, external medium, angle of incidence
▢ Optical Path:
❖ Definition: Distance travelled by light in air in the same time in which it travels distance \(d\) in a medium of refractive index \(\mu\)
❖ Formula: \(L=\mu d\)
❖ Time Through Glass: \(T=\frac{\mu t}{c}\)
▢ Real and Apparent Depth:
❖ Case 1: Object in denser medium and observer in rarer medium
❖ Effect: Object appears raised
❖ _*table:
    ❖ Case 2:
    ◉ Condition: Object outside medium and observer inside medium
    ◉ Example: Fish inside water looking at bird in sky sees bird higher than actual position
    ◉ Apparent height: \(AI=\mu h\)
    ◉ Shift in height: \(IO=h(\mu-1)\)
    ▢ Multiple Transparent Slabs:
    ❖ **table:
      ▢ Critical Angle:
      ❖ Definition: Angle of incidence in denser medium for which angle of refraction in rarer medium becomes 90°
      ❖ Formula: \(\mu=\frac{1}{\sin C}\)

      Table 1: Critical Angle

      Factor
      Effect
      Nature of media
      Critical angle depends on media
      Temperature increases
      Critical angle increases
      Wavelength increases
      Critical angle increases
      Violet light
      Minimum critical angle
      Red light
      Maximum critical angle
      Water-air interface
      \(C\approx49^\circ\)
      Diamond
      \(C\approx24^\circ\)
      ❖ Reason: \(\mu\propto\frac{1}{\lambda^2}\) and \(\mu=\frac{1}{\sin C}\)
      ▢ Total Internal Reflection:
      ❖ Definition: Complete reflection of light back into denser medium when light travels from denser to rarer medium at angle of incidence greater than critical angle
      ❖ Conditions:
      • •
        Light must travel from denser medium to rarer medium
      • •
        Angle of incidence must be greater than critical angle: \(i>C\)
      ❖ Important Point: In total internal reflection, intensity of light is not reduced because light is reflected back into the same medium
      ▢ Phenomena Due to Total Internal Reflection:
      • •
        Mirage in deserts
      • •
        Air bubbles inside water appear silvery white
      • •
        Brilliance of diamond
      • •
        Optical fibres
      • •
        Light pipes in endoscopy
      • •
        Fish-eye view from inside water
      ▢ Sea Diver / Fish View:
      ❖ Condition: Diver inside water at depth \(h\) sees outside world through circular window

      Table 1: Circular Window Seen by Diver

      Quantity
      Formula
      Radius
      \(r=h\tan C\)
      Using \(\sin C=1/\mu\)
      \(r=\frac{h}{\sqrt{\mu^2-1}}\)
      Area
      \(A=\pi r^2=\frac{\pi h^2}{\mu^2-1}\)
      ❖ Sunrise/Sunset: A diver or fish inside water observes rising or setting Sun at about 41° with horizon
      ▢ Optical Fibre:
      ❖ Principle: Total internal reflection

      Table 1: Optical Fibre

      Feature
      Point
      Diameter
      Approximately \(10^{-6}\ m\)
      Core
      Quartz or glass of higher refractive index
      Core refractive index
      ≈ 1.7
      Cladding
      Transparent material of lower refractive index
      Cladding refractive index
      ≈ 1.5
      Use
      Endoscopy, telephone cables, transmitting cables
      Use
      Transmission of laser beam or other light beam
      ▢ Consequences of Refraction:
      • •
        Sun appears before actual sunrise
      • •
        Sun appears after actual sunset
      • •
        Day becomes about 4 minutes longer
      • •
        Twinkling of stars
      • •
        Tank/pond appears shallower than actual depth
      • •
        Sun appears elliptical or oval near horizon
      • •
        Rod partially dipped in liquid appears bent
      ▢ Bird and Fish Problem:
      ❖ Condition: Bird in air at height \(h\), fish in water at depth \(x\), refractive index of water w.r.t. air = \(\mu\)

      Table 1: Apparent Positions

      Observer
      Observed object
      Apparent distance
      Bird
      Fish
      \(h+\frac{x}{\mu}\)
      Fish
      Bird
      \(x+\mu h\)
      ❖ Extra Point: A man standing on bank appears taller to a swimmer under water looking obliquely
      ▢ Read and Digest:

      Table 1: Important Refraction Points

      Fact
      Answer
      Light in two media
      Velocity, wavelength, intensity and amplitude change
      Frequency during refraction
      Remains unchanged
      Colour of light
      Determined by wavelength
      Air vs water for sound
      Air is denser than water for sound wave
      Blue colour of sky
      Scattering of light
      Glass slab on coloured letters
      Red raised least; violet raised most
      Cut diamond sparkles
      High refractive index and total internal reflection
      Sun near horizon
      Appears bigger and flattened due to atmospheric refraction
      Twinkling of stars
      Fluctuation of atmospheric refractive index
      Stick partly dipped in water
      Appears bent due to refraction
      Fish completely under water
      Length does not change
      Red colour of Sun at sunrise/sunset
      Scattering
      Air bubble in water
      Shines due to total internal reflection
      Cannot see through fog
      Light scattered by droplets
      Rainbow
      Dispersion + total internal reflection
      Mirage in desert
      Refractive index of air increases with height
      Looming in cold region
      Refractive index of air decreases with height
      Diver sees green object as blue
      Wavelength decreases in denser medium
      Transparent material invisible in vacuum
      When refractive index is 1
      Multiple media with parallel faces
      \(\mu_1\sin i_1=\mu_2\sin i_2=\mu_3\sin i_3=constant\)
      ▢ High-Yield Recall:

      Table 1: Refraction and TIR One-Liners

      Fact
      Answer
      Refraction
      Bending of light between media
      Cause of refraction
      Change in speed
      Rarer to denser
      Bends towards normal
      Denser to rarer
      Bends away from normal
      Snell's law
      \(\mu_1\sin i=\mu_2\sin r\)
      Small angle Snell's law
      \(\mu_1 i=\mu_2 r\)
      Refractive index
      \(\mu=\frac{c}{v}\)
      Cauchy's relation
      \(\mu=A+\frac{B}{\lambda^2}\)
      RI for red
      Minimum
      RI for violet
      Maximum
      Speed of red light
      Maximum
      Lateral shift
      \(d=\frac{t\sin(i-r)}{\cos r}\)
      Grazing incidence in slab
      \(d=t\)
      Optical path
      \(L=\mu d\)
      Real/apparent depth
      \(\mu=\frac{Real\ depth}{Apparent\ depth}\)
      Apparent shift
      \(d=t\left(1-\frac{1}{\mu}\right)\)
      Critical angle
      \(\mu=\frac{1}{\sin C}\)
      TIR conditions
      Denser to rarer and \(i>C\)
      Critical angle of diamond
      24°
      Critical angle water-air
      49°
      Critical angle for violet
      Minimum
      Critical angle for red
      Maximum
      Optical fibre principle
      Total internal reflection
      Core RI vs cladding RI
      Core higher, cladding lower
      Mirage
      Refraction + total internal reflection
      Diamond brilliance
      Total internal reflection
      Circular window radius for diver
      \(r=\frac{h}{\sqrt{\mu^2-1}}\)
      Circular window area
      \(A=\frac{\pi h^2}{\mu^2-1}\)
      Day length increase due to refraction
      4 minutes
      Twinkling of stars
      Atmospheric refraction
      Rainbow
      Dispersion and TIR
      Q1.
      The light from a luminous point source placed on the lower face of a rectangular glass slab 2.0 cm thick strikes the upper face and the totally reflected rays outline a circle of diameter 3.2 cm on the lower face. Then refractive index of the glass will be:
      📅BP 2014
      Q2.
      The refractive index is
      📅BP 2013
      Q3.
      The colour of light is due to:
      📅BP 2012
      Q4.
      When light passes from air medium to a glass medium, which remains constant
      📅BP 2012
      Q5.
      The blue color of sea is due to
      📅BP 2011
      Q6.
      Looming occurs due to:
      📅BP 2011
      Q7.
      Newton's corpuscular theory couldn't explain
      📅BP 2010
      Q8.
      To a bird flying in sky, a fish appears to be at 30 cm from water surface. If refractive index of water with respect to air is 4/3, then the real distance of fish from the water surface is
      📅BP 2010
      Q9.
      Optical fibre is based on phenomenon
      📅KU 2013/2016•T.E 2011,012
      Q10.
      A glass slab having thickness 't' and refractive index 'n'. 'v' is velocity of light in glass slab and 'c' is velocity of light in vacuum. Then find time taken to pass the light through the glass slab
      📅KU 2013
      Q11.
      When a ray of light passes into a glass slab from air then
      📅KU 2010
      Q12.
      Which of the following is not correct for TIR to occur?
      📅KU 2010
      Q13.
      The ray of light entering into a rectangular glass slab emerges from the slab. The incident ray and emergent ray are:
      📅I.E 2012
      Q14.
      The refractive index of air with respect to water is 1.33. Then refractive index of water with respect to air is:
      📅I.E 2012
      Q15.
      An air bubble in a glass slab (μ = 1.5) is 5 cm deep when viewed through one face and 2 cm deep when viewed through the opposite face, then thickness of slab is:
      📅T.E 2010
      Q16.
      Which wavelength of light falls under visible wavelength?
      📅KU 2010
      Q17.
      The index of refraction of diamond is 2.4, velocity of light in diamond is
      📅IOM 2016
      Q18.
      A diver in water at a depth 1 m sees the whole outside world in a horizontal circle of radius ... when the refractive index is μ
      📅IOM 04
      Q19.
      When light passes through glass slab
      📅IOM 1997
      Q20.
      The refractive index of glass is 1.5. Then velocity of which light is minimum in the glass
      📅MOE 2008
      Q21.
      The refractive index of glass is 1.5 and water is 1.33. Then what is the critical angle for glass water interface
      📅MOE 2008
      Q22.
      A beam of monochromatic blue light of wavelength 420 nm in air travels in water of refractive index 1.33. Its wavelength in nm in water will be
      📅MOE 2065
      Q23.
      The refractive index (μ) and wavelength of light (λ) when light passes from one medium to another medium
      📅MOE 2063
      Q24.
      A monochromatic beam of light passes from a denser medium to a rarer medium. As a result
      📅MOE 2062
      Q25.
      A glass slab has thickness 6 mm and μ = 1.5. Calculate the time in nanosecond for an instant of light to pass through it.
      📅MOE 2061
      Q26.
      A monochromatic beam of light of wavelength 600 nm in air enters a medium of refractive index 1.5. Its wavelength in the medium will be
      📅MOE 2058
      Q27.
      Refractive index of glass with respect to air is 1.5. What is the velocity of light in the material?
      📅MOE 2055
      Q28.
      A person inside water can see the sun setting at an angle of
      📅MOE 2053
      Q29.
      A light ray is passed from one medium of RI(n1) to another medium of RI(n2) as shown in fig. The correct relation between n1 and n2 is
      📅KU 08
      Q30.
      If a point source of light is placed inside water at a depth of √7 m. Then light emerges out from water surface through a horizontal circle of radius: (μ = 4/3)
      📅
      Q31.
      Which of the following is not correct for TIR to occur?
      [KU 2010]
      📅KU 2010
      Q32.
      If the refractive index of water is μw and that of glass slab immersed in it is μg, what is the critical angle for a ray of light going from glass to water?
      [Bangladesh 2009]
      📅Bangladesh 2009
      Q33.
      A vessel of depth t is half filled with water (refractive index μ1) and half with a liquid (refractive index μ2). The apparent depth of vessel as seen from above is:
      Q34.
      When does total internal reflection occur?
      [TE-02]
      📅TE-02
      Q35.
      A ray of light is travelling from one medium to another. The wavelength of the light in the first and second medium is 4000 Å and 6000 Å respectively. Then the value of critical angle is:
      [TE-03]
      📅TE-03
      Q36.
      Light from vacuum enters a medium of μ = 1.5. If it crosses it within a nanosecond, the thickness of the medium is:
      [BPKIHS 2000]
      📅BPKIHS 2000
      Q37.
      A diver inside a pond looks at an object whose natural colour is green. He sees the object as:
      Q38.
      What happens when monochromatic light travels from air to glass?
      Q39.
      Critical angle of light passing from glass to air is minimum for:
      [IOM/BPKIHS/MOE]
      📅IOM•BPKIHS•MOE
      Q40.
      A, B and C are three optical media with critical angles C1, C2 and C3. Total internal reflection can occur from A to B and from B to C, but not from C to A. The correct relation is:
      Q41.
      Which of the following phenomena cannot be explained by total internal reflection?
      [KU 2011]
      📅KU 2011
      Q42.
      A fish looks at a bird in air. The bird appears:
      [KU 2011]
      📅KU 2011
      Q43.
      A light is incident at an angle of 45° on a liquid surface. It is reflected at 30° and refracted at 60°. The refractive index of the liquid is:
      Q44.
      Total internal reflection is not possible in:
      [KU 2012]
      📅KU 2012
      Q45.
      A diamond sparkles due to:
      [KU 2012]
      📅KU 2012
      Q46.
      Total internal reflection can take place when light travels from:
      Q47.
      The velocity of light in a medium is 2 × 108 m/s. The refractive index of the medium is:
      Q48.
      The apparent depth of a tank when viewed vertically is 1.2 m. If the refractive index of water is 4/3, the real depth of the tank is:
      Q49.
      Critical angle of glass is minimum for:
      [IOM 2004]
      📅IOM 2004
      Q50.
      If the refractive index of water and glass are 1.33 and 1.5 respectively, then the critical angle for light from glass to water is:
      Q51.
      Two identical buckets A and B are filled with different liquids. The refractive index of liquid in A is 1.3 and in B is 1.6. The ratio of apparent depths of the bottoms when viewed from above is:
      Q52.
      A layer of water of refractive index 1.3 and thickness 4 cm floats on benzene of refractive index 1.5 and thickness 6 cm. Then the apparent depth of the bottom of the beaker from free surface of water is:
      Q53.
      A half of the beaker is filled with liquid of refractive index 1.5 and the other half with a liquid of refractive index μ. The total apparent depth is equal to 50% of the total real depth. Then μ is equal to:
      Q54.
      A fish sees an observer of height 24 cm. The height of the observer as seen by the fish when in water of refractive index 4/3 is:
      Q55.
      An air bubble in a glass slab of refractive index μ appears 6 cm when viewed from one side and 4 cm when viewed from opposite side. Then thickness of the glass slab is:
      Q56.
      A diver at a depth of 12 cm in water sees the sky in a cone of semi-vertex angle:
      Q57.
      A transparent cube of 12 cm edge contains a small air bubble. Its apparent depth when viewed through one face of the cube is 6 cm and when viewed through opposite face is 2 cm. What is the actual distance of the bubble from the first face?
      Q58.
      A pond of water is 5 m deep. A flame is held 2 m above the surface. A fish is at depth 4 m from surface. If refractive index of water is 4/3, the apparent height of flame from fish is:
      Q59.
      A point source is located 275 cm below the surface of a lake. The area of the surface that transmits all light from the source is:
      Q60.
      Monochromatic light is refracted from air into glass of refractive index μ. The ratio of the wavelength of incident and refracted waves is:
      Q61.
      A fish rising vertically at speed 3 m/s sees a bird diving vertically at speed 9 m/s. If refractive index of water is 4/3, the actual velocity of the bird is:
      Q62.
      A vessel of depth 2d is half filled with a liquid of refractive index μ1 and upper half with a liquid of refractive index μ2. The apparent depth of the vessel seen perpendicularly is:
      Q63.
      A bird in air looks at a fish vertically below inside water. x is the height of bird above surface and y is depth of fish. If refractive index of water is μ, the distance of fish observed by bird is:
      Q64.
      Light from a luminous point source on the lower face of a rectangular glass slab 2.0 cm thick strikes the upper face and the totally reflected rays outline a circle of radius 3.2 cm on the lower face. The refractive index of the glass is:
      [HSEB]
      📅HSEB
      Q65.
      The refractive index of air with respect to glass is 2/3. The refractive index of diamond with respect to air is 12/5. Then the refractive index of glass with respect to diamond will be:
      Q66.
      The optical path of a monochromatic light is same when it goes through a transparent glass slab of thickness 4 cm and 4.5 cm thickness of water. The RI for glass is 1.53, then RI for water is:
      Q67.
      A transparent glass slab of thickness 4 cm contains same number of waves as 5 cm of water when both are traversed by a monochromatic light. If μ for water is 1.33, then μ for glass is:
      Q68.
      What is the time taken by a ray of light to emerge from a glass slab of thickness 2 cm and refractive index 1.5?
      Q69.
      A point source of light is placed at depth of 4 m below the surface of a transparent liquid whose refractive index is 5/3. What will be the minimum radius of the disc which is to be placed on the liquid surface just above the point source to stop the emergence of any light from the liquid surface:
      Q70.
      Total internal reflection occurs when the light passes from:
      [KU 2016]
      📅KU 2016
      Q71.
      When the light passes from air to glass, which of following changes?
      [KU 2017]
      📅KU 2017