28Refraction through prism and Dispersion of Light

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REFRACTION THROUGH PRISM AND DISPERSION OF LIGHT
▢ Refraction Through Prism:
❖ Definition: Refraction of monochromatic light through two refracting faces of a prism

Table 1: Prism Formulae

Quantity
Formula
Angle of prism
\(A=r_1+r_2\)
Angle of deviation
\(\delta+A=i_1+i_2\)
Refractive index
\(\mu=\frac{\sin i_1}{\sin r_1}=\frac{\sin i_2}{\sin r_2}\)
Mean deviation for yellow colour
\(\delta_y=\frac{\delta_V+\delta_R}{2}\)
Mean deviation
\(\delta_y=A(\mu-1)\)
❖ Symbols:
  • •
    \(i_1\) = angle of incidence at first face
  • •
    \(i_2\) = angle of emergence at second face
  • •
    \(r_1,r_2\) = angles of refraction
  • •
    \(A\) = angle of prism
  • •
    \(\delta\) = angle of deviation
▢ Minimum Deviation:
❖ Condition: For minimum deviation, path of ray inside prism is symmetrical

Table 1: Conditions for Minimum Deviation

Quantity
Condition / Formula
Angles of incidence and emergence
\(i_1=i_2=i=\frac{\delta_m+A}{2}\)
Angles of refraction
\(r_1=r_2=r=\frac{A}{2}\)
Refractive index
\(\mu=\frac{\sin\left(\frac{\delta_m+A}{2}\right)}{\sin\left(\frac{A}{2}\right)}\)
For small angle prism
\(\delta_m=A(\mu-1)\)
Ray inside prism
Parallel to base of prism
❖ Important Points:
  • •
    Angle of deviation first decreases, becomes minimum, then increases with increase in angle of incidence
  • •
    Minimum deviation is characteristic of a prism for particular colour
  • •
    Angle of minimum deviation is independent of angle of incidence
  • •
    Violet deviates most: \(\delta_V>\delta_R\)
❖ Depends On:
  • •
    Refracting angle of prism
  • •
    Nature of material of prism
  • •
    Refractive index of outside medium
  • •
    Colour / wavelength of incident light
❖ Independent Of:
  • •
    Size of prism
  • •
    Angle of incidence at minimum deviation
  • •
    Intensity of incident light
▢ Maximum Deviation:
❖ Condition: Maximum deviation occurs when either incidence or emergence is grazing

Table 1: Maximum Deviation

Case
Condition
First face grazing incidence
\(i_1=90^\circ\)
Second face grazing emergence
\(i_2=90^\circ\)
▢ Grazing Emergence and Limiting Angle:

Table 1: Condition for Grazing Emergence

Condition
Formula / Point
Grazing at first face
\(i_1=90^\circ,\ r_1=C\)
Grazing at second face
\(i_2=90^\circ,\ r_2=C\)
Limiting angle of prism
\(A=r_1+r_2=2C\)
Total reflecting prism
\(A>2C\)
No emergent ray
Light is totally internally reflected from second face
Condition for no emergence
\(\mu>\csc\left(\frac{A}{2}\right)\)
❖ Equilateral Prism:
◉ Condition: \(A=60^\circ\)
◉ No Emergence: \(\mu>\csc30^\circ=2\)
▢ Dispersion of Light:
❖ Definition: Splitting of white light into constituent colours due to unequal deviation of different colours by a prism
❖ Cause: Different colours have different wavelengths, so refractive index and deviation are different
❖ Cauchy's Relation: \(\mu=A+\frac{B}{\lambda^2}\)
❖ Colour Relation:
  • •
    \(\lambda_R>\lambda_V\)
  • •
    \(\mu_V>\mu_R\)
  • •
    \(\delta_V>\delta_R\)
  • •
    Red deviates least
  • •
    Violet deviates most
❖ Formula: \(\delta=A(\mu-1)\)
▢ Angular Dispersion:
❖ Definition: Difference in angle of deviation between most deviated colour and least deviated colour

Table 1: Angular Dispersion

Quantity
Formula / Point
Angular dispersion
\(\theta=\delta_V-\delta_R\)
Formula
\(\theta=A(\mu_V-\mu_R)\)
Most deviated colour
Violet
Least deviated colour
Red
❖ Depends On:
  • •
    Refracting angle of prism
  • •
    Material of prism
  • •
    Nature of chromatic light
▢ Dispersive Power:
❖ Definition: Ratio of angular dispersion to mean deviation

Table 1: Dispersive Power

Quantity
Formula / Point
Dispersive power
\(\omega=\frac{\theta}{\delta_y}\)
Formula
\(\omega=\frac{\delta_V-\delta_R}{\delta_y}\)
Using refractive indices
\(\omega=\frac{\mu_V-\mu_R}{\mu-1}\)
Depends on
Nature of chromatic light and material of prism
Independent of
Refracting angle of prism
Value
Zero or positive; never negative
▢ Combination of Prisms:
❖ Deviation Without Dispersion:
◉ Meaning: Two prisms produce no net dispersion but produce net deviation
◉ **table:
    ❖ Dispersion Without Deviation:
    ◉ Meaning: Two prisms produce dispersion but no net deviation
    ◉ **table:
      ◉ Use: Direct vision spectroscope
      ❖ Identical Opposite Prisms: If two identical thin prisms are placed oppositely, there is no deviation and no dispersion
      ▢ Rainbow:
      ❖ Definition: Rainbow is formed due to dispersion of sunlight by water droplets and total internal reflection

      Table 1: Primary vs Secondary Rainbow

      Feature
      Primary rainbow
      Secondary rainbow
      Formation
      2 refractions + 1 total internal reflection
      2 refractions + 2 total internal reflections
      Brightness
      Brighter
      Fainter
      Sun altitude condition
      Visible when Sun altitude < 42°
      Visible when Sun altitude < 53°
      Innermost arc
      Violet
      Red
      Outermost arc
      Red
      Violet
      ▢ Scattering of Light:
      ❖ Rayleigh Law: Amount of scattering is inversely proportional to fourth power of wavelength
      ❖ Formula: \(Scattering\propto\frac{1}{\lambda^4}\)

      Table 1: Scattering Effects

      Phenomenon
      Reason
      Blue colour of sky
      Blue light has shorter wavelength and scatters more
      Red colour of Sun at sunrise and sunset
      Red light scatters least and reaches observer
      Red used as danger signal
      Least scattering; travels longer distance
      White colour of clouds
      Cloud droplets scatter all colours nearly equally
      Sky black from Moon / space
      No atmosphere, so no scattering
      Cannot see through fog
      Light scattered by droplets in fog
      ❖ Important Point: Wavelength of light is not changed during scattering
      ❖ Clouds: Clouds do not obey Rayleigh law; all colours scatter equally
      ▢ Colour of Objects:
      ❖ Opaque Object: Colour depends on incident light and selective absorption/reflection by object
      ❖ Example:
      • •
        Red rose in white light appears red because it reflects red and absorbs other colours
      • •
        Red rose in light having no red appears black because it reflects no light
      ▢ Colours:

      Table 1: Primary, Secondary and Complementary Colours

      Type / Combination
      Result
      Primary colours
      Red, Green, Blue
      Secondary colours
      Colours formed by mixing primary colours
      Complementary colours
      Two colours which mix to give white
      Red + Green + Blue
      White
      Red + Green
      Yellow
      Green + Blue
      Cyan / Peacock blue
      Red + Blue
      Magenta
      Blue + Yellow
      White
      Green + Magenta
      White
      Red + Cyan / Peacock blue
      White
      ▢ Important Optical Phenomena:

      Table 1: Phenomena and Causes

      Phenomenon
      Cause
      Colour of soap bubble
      Interference
      Colour bands in thin oil film on water
      Interference
      Blue sky
      Scattering
      Red rising/setting Sun
      Scattering
      Air bubble inside water looks silvery white
      Total internal reflection
      Rainbow
      Dispersion + total internal reflection
      Cut diamond sparkles
      Total internal reflection
      Prism
      Deviation + dispersion
      Glass slab
      Neither net deviation nor dispersion
      Hollow prism
      No dispersion
      ▢ Read and Digest:

      Table 1: Important Points

      Fact
      Answer
      Spectrometer
      Measures deviation caused by prism
      Direct vision spectrum
      Uses dispersion without deviation
      Colour of light
      Characteristic of wavelength
      Refrangibility
      Least for red
      Red colour deviation
      Least
      Violet colour deviation
      Most
      Dispersive power
      Independent of angle of prism
      Limiting angle of prism
      \(A=2C\)
      Equilateral prism no emergence
      \(\mu>2\)
      A prism
      Produces both deviation and dispersion
      A glass slab
      Produces neither deviation nor dispersion
      A hollow prism
      Causes no dispersion
      Red glass appears red
      Absorbs all colours except red
      Heated red glass initial glow
      Orange, then yellow, green and so on
      Astronaut sees sky black
      Absence of atmosphere
      ▢ High-Yield Recall:

      Table 1: Prism, Dispersion and Scattering One-Liners

      Fact
      Answer
      Prism deviation formula
      \(\delta+A=i_1+i_2\)
      Prism angle
      \(A=r_1+r_2\)
      Minimum deviation condition
      \(i_1=i_2,\ r_1=r_2\)
      At minimum deviation
      \(r=\frac{A}{2}\)
      Minimum deviation RI
      \(\mu=\frac{\sin\left(\frac{A+\delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}\)
      Thin prism deviation
      \(\delta=A(\mu-1)\)
      Violet deviation
      Maximum
      Red deviation
      Minimum
      Maximum deviation condition
      \(i_1=90^\circ\) or \(i_2=90^\circ\)
      Limiting angle
      \(A=2C\)
      Total reflecting prism
      \(A>2C\)
      No emergence
      \(\mu>\csc\left(\frac{A}{2}\right)\)
      Equilateral prism no emergence
      \(\mu>2\)
      Dispersion
      Splitting of white light into colours
      Cause of dispersion
      \(\mu\) different for different wavelengths
      Cauchy's relation
      \(\mu=A+\frac{B}{\lambda^2}\)
      Angular dispersion
      \(\theta=\delta_V-\delta_R=A(\mu_V-\mu_R)\)
      Dispersive power
      \(\omega=\frac{\mu_V-\mu_R}{\mu-1}\)
      Dispersive power depends on
      Material and chromatic light
      Dispersive power independent of
      Angle of prism
      Deviation without dispersion
      \(\theta+\theta'=0\)
      Dispersion without deviation
      \(\delta+\delta'=0\)
      Primary rainbow
      2 refractions + 1 TIR
      Secondary rainbow
      2 refractions + 2 TIR
      Primary rainbow order
      Violet inside, red outside
      Secondary rainbow order
      Red inside, violet outside
      Rayleigh scattering
      \(\propto\frac{1}{\lambda^4}\)
      Blue sky
      Scattering
      Red danger signal
      Least scattered
      Cloud white
      Equal scattering of all colours
      Primary colours
      Red, Green, Blue
      Red + Green
      Yellow
      Green + Blue
      Cyan
      Red + Blue
      Magenta
      Complementary colours
      Mix to give white
      Q1.
      What is the angle of incident of prism of angle 60° and which has refractive index √2 for which it gives minimum deviation?
      📅BP 2013
      Q2.
      A small angle prism deviates a narrow beam of light by 1.3°. The refractive index is 1.62 and the angle of prism in degree is:
      📅MOE 2014
      Q3.
      Light passing from air to glass at an angle of incidence 60° is deviated to 15°. What is the refractive index?
      📅MOE 2010
      Q4.
      An equilateral prism having angle 60° has refractive index √2. What is the angle of incident for minimum angle of deviation?
      📅IE 2012
      Q5.
      The angle of deviation produced by a small angle prism of angle A and refractive index μ is equal to:
      📅KU 2011
      Q6.
      An achromatic doublet is constructed by lenses of crown glass and flint glass of dispersive powers 0.02 and 0.03. If the focal length of the crown glass is 20 cm, the focal length of the doublet becomes:
      📅IOM 05
      Q7.
      In a right-angled isosceles prism, when the light falls from flat face and for total internal reflection to take place in the second face, the minimum refractive index of the prism is:
      📅IOM 03
      Q8.
      The principle of which of the following is based on deviation without dispersion?
      📅BPKIHS 2000
      Q9.
      For minimum deviation inside the prism, the angle of emergence should be equal to:
      📅IOM 1999
      Q10.
      The power produced by a prism of small angle A:
      📅BPKIHS 2000
      Q11.
      An impure spectrum of Green, Blue, Orange, and Yellow is observed on a screen when light passes through a transparent prism. The refractive index of the prism with respect to the colours will be maximum for:
      📅MOE Curriculum
      Q12.
      To make an achromatic combination, a convex lens of focal length 42 cm having dispersive power 0.14 is placed in contact with a concave lens of dispersive power 0.21. The focal length of the concave lens should be:
      📅MOE
      Q13.
      In the sun's spectrum, if blue & green lights are passed through a transparent prism, then deviation of green light will be:
      📅MOE 2063
      Q14.
      An object PQ is placed facing the hypotenuse of a 45°-90°-45° glass prism. What is the nature of the image formed? (Refractive index of prism is 1.5)
      📅MOE 2056
      Q15.
      In a right-angled glass prism, light is normally incident through the smaller face. Then deviation is:
      📅MOE 2054
      Q16.
      Chromatic aberration is due to:
      📅MOE 2054
      Q17.
      Angle of deviation when passing through a prism is greatest for light:
      📅MOE/KU
      Q18.
      In an equilateral prism, if angle of incidence equals angle of emergence and the angle of incidence is 3/4 of angle of prism. Then the angle of deviation produced is:
      📅KU 2010
      Q19.
      An achromatic lens forms an image of a white object. The colour of image seen will be:
      📅Bangladesh 09
      Q20.
      A yellow object on a blue background appears as:
      📅TE-051
      Q21.
      A prism of refracting angle 60° produces a minimum deviation of 30°. Then the angle of incidence is:
      📅IE-03
      Q22.
      A prism can produce a minimum deviation of 8° in a light beam. If three such prisms are combined, the minimum deviation that can be produced in this beam is:
      📅BPKIHS-95
      Q23.
      The cover of a book appears black when seen through a piece of red glass. Then the cover may be of:
      📅MOE
      Q24.
      A red postage stamp is viewed in yellow light. It appears:
      📅MOE
      Q25.
      A blue object on a white background when seen through a blue filter will:
      📅MOE
      Q26.
      A piece of red glass is heated till it glows in dark. The colour of the glowing piece will be:
      📅MOE
      Q27.
      From two projectors, beams of red light and green light are sent on the same spot of the screen. The screen will look:
      📅MOE
      Q28.
      For which of the following is the dispersive power zero?
      📅MOE
      Q29.
      Dispersion for a medium of wavelength λ is D. Then the dispersion for the wavelength λ/2 will be:
      📅MOE
      Q30.
      White light is passed through a prism of angle 5°. If the refractive indices for red and blue rays are 1.641 and 1.659, the angle of dispersion between them is:
      📅MOE
      Q31.
      A thin prism P1 (angle 4°, μ=1.54) is combined with prism P2 (μ=1.72) to produce dispersion without deviation. The angle of P2 is:
      📅MOE
      Q32.
      The difference in deviation produced by violet and red rays is 2°. The deviation produced by the mean ray is 48°. The dispersive power and deviations are:
      📅MOE
      Q33.
      White light is passed through a prism of angle 5°. The refractive indices for red and blue rays are 1.64 and 1.66. The angle of dispersion is:
      📅MOE
      Q34.
      A glass prism deviates red and blue rays through 10° and 12° respectively. A second prism deviates them through 8° and 10°. The ratio of their dispersive powers is:
      📅MOE
      Q35.
      Find the angle of a prism (dispersive power 0.021, μ=1.52) to form an achromatic combination with a second prism (angle 4.2°, dispersive power 0.045, μ=1.65):
      📅MOE
      Q36.
      A prism (angle 11.25°, dispersive power 0.021, μ=1.52) forms an achromatic combination with a second prism (angle 4.2°, dispersive power 0.045, μ=1.65). The resultant deviation is:
      📅MOE
      Q37.
      The difference between angle of minimum deviation for violet and red rays is 2°. If the angle of mean ray is 48°, the dispersive power of the prism material is:
      📅MOE
      Q38.
      A source emits light of wavelengths 4700Å, 5400Å, and 6500Å. The light passes through red glass before being tested by a spectrometer. Which wavelength is absent in the spectrum?
      📅MOE
      Q39.
      If angle of prism is 60° and angle of minimum deviation is 40°, then the angle of refraction will be:
      📅MOE
      Q40.
      The angle of prism is A and the minimum deviation is (180° - 2A). Then the refractive index of the prism material is:
      📅MOE 2062
      Q41.
      The angle of prism is 6° and its refractive index for green light is 1.5. If a green ray passes through it, the deviation will be:
      📅MOE
      Q42.
      A ray of light suffers minimum deviation through an equilateral prism of refractive index √2. The minimum deviation will be:
      📅MOE
      Q43.
      An equilateral prism has refractive index √2. The angle of incidence which gives minimum deviation is:
      📅MOE
      Q44.
      A ray of light is incident at 60° on a prism (angle 30°). The emerging ray makes 30° with the incident ray. The refractive index of the prism is:
      📅MOE
      Q45.
      A ray passes through an equilateral prism such that the angle of incidence equals the angle of emergence (each 3/4 of the prism angle). The angle of deviation is:
      📅MOE
      Q46.
      A thin prism of glass is placed in air and water successively. If μg = 3/2 and μw = 4/3, the ratio of deviations (air:water) for small angles is:
      📅MOE
      Q47.
      The angle of a prism is 30°. A ray incident at 60° on one face suffers a deviation of 30°. The angle of emergence is:
      📅MOE
      Q48.
      A ray is incident on a prism (angle 75°, μ=√2) and undergoes critical angle at the second face. The angle of incidence at the first face is:
      📅MOE
      Q49.
      A ray is incident on a small-angle prism (angle A, μ=μ) and emerges normally. The angle of incidence is nearly:
      📅MOE
      Q50.
      The angle of prism is 60°. For minimum deviation (refractive index √2), the angle of incidence is:
      📅MOE
      Q51.
      The refracting angle of a small angle prism is A. The relation between δm (minimum deviation) and angle of refraction (r) will be (μ = 1.5):
      📅MOE
      Q52.
      Angle of prism is A and one surface is silvered. Light falls at angle of incidence 2A on the first surface and returns through the same path. The refractive index of the prism is:
      📅MOE
      Q53.
      What should be the refractive index of an equilateral prism for total internal reflection?
      📅MOE
      Q54.
      A ray is incident normally on one face of a right-angled isosceles prism and undergoes total internal reflection. The minimum refractive index is:
      📅MOE
      Q55.
      A ray is incident normally on the first face of an equilateral prism and undergoes total internal reflection at the second face. The refractive index must be:
      📅MOE
      Q56.
      A glass prism (μ=1.5) is immersed in water (μ=4/3). A light beam incident normally on face AB is totally reflected to reach face BC if:
      📅MOE
      Q57.
      The angle of minimum deviation for a prism (angle 60°) is 30°. The refractive index is:
      📅MOE
      Q58.
      A 60° prism (μ=1.414) gives minimum deviation when the angle of incidence is:
      📅MOE
      Q59.
      A prism (angle 45°, μ=1.6) has a minimum angle of incidence to prevent total internal reflection. This angle is:
      📅MOE
      Q60.
      A prism (angle 60°, μ=1.5) produces maximum deviation. The angle of emergence is:
      📅IOM 2017