Geometrical optic, prisms (A-level physics)
Deviation of light through a prism
The angle of deviation caused by the prism is the angle between the incident ray and the emergent ray.
Consider a ray of light incident in air on a prism of refracting angle A and finally emerges into air as shown.
From the diagram above, MS and NS are normal at the points of incidence and emergence of the ray respectively.
NOTE
Experiments show that as the angle of incidence i is increased from zero, the
deviation D reduces continuously up to a minimum value of deviation Dmin and then increases to a maximum value as the angle of incidence is increased as shown below:
Example 1
A ray of light is incident on a prism of refracting angle 72° and refractive index of 1.3. The ray emerges from the prism at 43°.
Find
(i) the angle of incidence.
(ii) the deviation of the ray.
Solution
Example 2
A prism of refracting angle 67° and refractive index of 1.6 is immersed in a liquid of refractive index 1.2. If a ray of light traveling through the liquid makes an angle of incidence of 53° at the left face of the prism, Determine the total deviation d, of the ray.
Solution
At Q, Snell’s law becomes
1.6 sin 30.2° = 1×2 sin i2
i2 = 42.2°
Substituting for r1, i2, and r2 in equation (i) gives
d = ( 53° – 36.8° ) + ( 42×12° – 30.2° )
d = 28.12°
Minimum deviation of light by a prism
At minimum deviation, light passes symmetrically through the prism. That is to say, the angle of incidence is equal to the angle of emergence.
Consider a ray on one face of the prism at an angle i1and leaves it at an angle i2 to the normal as shown
For minimum deviation, i1 = i2 = i and r1 = r2 = r. From the diagram,
Dmin = d + d
Dmin = 2d where d = i – r
Dmin = 2i – 2r—————————– (a)
But r + r = A
it implies, 2r = A OR r = ————————————- (b)
Combining equation (a) and (b) gives
Dmin = 2i – A
Note that
Equation (e) suggests that if the prism was surrounded by a medium of refractive index nl, then at minimum deviation
Examples 3
Calculate the angle of incidence at minimum deviation for light passing through a prism of refracting angle 70° and refractive index of 1.65.
Solution
Examples 3
An equilateral glass prism of refractive index 1.5 is completely immersed in a liquid of refractive index 1.3. if a ray of light passes symmetrically through the prism, calculate the:
(a) angle of deviation of the ray.
(b) angle of incidence
Analysis:
(a) For an equilateral prism, its refracting angle A =60°
(b) If the ray passes through the prism symmetrically, then the angle of deviation is minimum
Deviation of light by a small angle prism
The small refracting angle of this prism causes the angle i1, r1, r2 and i2 to be small such that sin i1 » i1, sin r1 » r1 sin r2 »r2 and sin i2 » i2.
Examples 4
A ray of light that falls normally upon the first face of a glass prism of a small refracting angle under goes a partial refraction and reflection at the second face of the prism.
The refracted ray is deviated through an angle 1.8° and the reflected ray makes an angle of 9.6° with the incident ray after emerging from the prism through its first face. Calculate the refracting angle of the prism and its refractive index of the glass material.
Solution
Let A be the required refracting angle of the prism as shown
Consider the deviation suffered by the incident light
D = (n – 1) A where D = 1.8°
Dispersion of white light by a transparent medium
Dispersion of whit light is the separation of white light in to its component colors by a transparent medium due to their speed differences in the medium.
When white light falls on a transparent medium, its different component colors travel with different speeds through the medium. They are therefore deviated by different amounts on refraction at the surface of the medium and hence dispersion.
NOTE :
(i) White light is a mixture of various colors. This is called the spectrum of white light.
(ii) The spectrum of white light consists of red, orange, yellow, green, blue, indigo and violet light bands. On refraction, violet is the most refracted colour away from the normal ( violet is the most deviated colour ) while red is least deviated
(ii) When light of two wavelengths say red and blue light is incident at a small angle on a small angle prism of refracting angle A having refractive indices of nr and nb. for the two wave lengths respectively, then the two wave lengths are deviated as shown below.
Examples 5
Light of two-wavelength is incident at a small angle on a thin prism of refracting angle 5° and refractive index of 1.52 and 1.48 for the two wavelengths. Find the angular separation of the two wavelengths after refraction by the prism.
Solution
For a small prism, Angular separation θ = (n1 – n2) A
θ = (1.52 –1.48) ´5°
θ =0×2 °
Examples 6
A glass prism with refracting angle 60° has a refractive index of 1.64 for red light and 1.66 for violet light. Calculate the angular separation of the red and violet rays which emerge from the prism when a ray of white light is incident on the prism at an angle of 45°
Appearance of white light placed in water
Observation
A coloured spectrum is seen inside the water surface with violet on top and red down.
Explanation
The different component colours of white light travel with different speeds through water.
They are deviated by different amounts on refraction at the water surface. Hence different coloured images are formed at different points inside the water surface with a violet coloured image on top.
Grazing property of light rays as applied to prisms.
If a ray of light is either such that the incident angle or the emergent angle is equal to 90° to the normal of the prism, then the ray is said to graze the refracting surface of the prism.
Consider a ray of light incident at an angle i on a glass prism of refracting angle A situated in air with the emergent light grazing the other refracting surface of the prism as shown.
Examples 7
Monochromatic light is incident at an angle of 45° on a glass prism of refracting angle 70° in air. The emergent light grazes the other refracting surface of the prism. Find the refractive index of the glass material.
At P, Snell’s law becomes
na sin 45° = ng sin r ——————————–(a)
From the diagram, r + c = 70°
r = 70° – c ——————————-(b)
Substituting equation (b) in (a) gives
Sin 45° = ng sin ( 70° – c ) ———————————–(c)
At Q, Snell’s law becomes
ng sin c = na sin 90°
Examples 8
A ray of light is incident on one refracting face of a prism of refractive index 1.5 and refracting angle 60°. Calculate the minimum angle of incidence for the ray to emerge through the second refracting face.
Solution
Analysis
For minimum angle of incidence, the emergent ray grazes the second refracting face. At Q, Snell’s law becomes
1.5sin c =na sin 90°
c = 41×8°
But r + c = 60°
r = 60° – c
= 60° – 41.8°
therefore, r = 18.2°
At P, Snell’s law becomes
1.5 sin 18×2° = na sin imin
imin = 28°
NOTE
Limiting angle of the prism
This is the maximum refracting angle of the prism for which the emergent ray grazes the second refracting surface.
Suppose the incident ray grazes the first refracting surface then the limiting angle A is given by A = 2c where c is the critical angle of the glass air interface as shown.
Exercise:
1. (i) Obtain an expression relating the deviation of a ray of light by the prism to the refracting angle and the angles of incidence and emergence.
(ii) The deviation of a ray of light incident on the first face of a 60° glass prism at an angle of 45° is 40°. Calculate the angle of emergence of a ray on the second face of the prism.
[ Ans i2 = 65° ]
(iii) A prism of refractive index 1.64 is immersed in a liquid of refractive index 1.4.
A ray of light is incident on one face of the prism at an angle of 40°. If the ray emerges at an angle of 29°, determine the angle of the prism.
[Answer: 57.7°]
2. (i) For a ray of light passing through the prism, what is the condition for minimum deviation to occur?
(ii) Derive an expression for the refractive index of a prism in terms of the refracting angle, A, and the angle of minimum deviation D.
(iii) A glass prism of refractive index n and refracting angle ,A, is completely immersed in a liquid of refractive index nl. If a ray of light that passes symmetrically through the prism is deviated through an angle θ, Show that
3.(a) A glass prism with refracting angle 60° is made of glass whose refractive indices for red and violet light are respectively 1.514 and 1.530. A ray of white light is set incident on the prism to give a minimum deviation for red light.
Determine the:
(i) angle of incidence of the light on the prism.
(ii) angle of emergence of the violet light.
(iii) angular width of the spectrum.
(b) A certain prism is found to produce a minimum deviation of 51°. While it produces a deviation of 62.8° for a ray of light incident on its first face at an angle of 40.1° and emerges through its second face at an angle of 82.7°.
Determine the:
(i) refracting angle of the prism.
(ii) angle of incidence at minimum deviation.
(iii) refractive index of the material of the prism.
[ Ans (i) 60° (ii) 55.5° (iii) 1.648 ]
4. (i) A ray of monochromatic light is incident at a small angle of incidence on a small angle prism in air. Obtain the expression D = ( n – 1 )A for the deviation of light by the prism.
(ii) A glass prism of small angle, A, and refractive index ng and is completely immersed in a liquid of refractive index nL. Show that a ray of light passing through the prism at a small angle of incidence suffers a deviation given by
5. Explain why white light is dispersed by a transparent medium.
6. Light of two wave length is incident at a small angle on a thin prism of refracting angle 5° and refractive index of 1.52 and 1.48 for the two wave lengths. find the angular separation θ of the two wave lengths after refraction by the prism.
[ Ans θ = 0.2° ]
7. A point source of white light is placed at the bottom of a water tank in a dark room. The light from the source is observed obliquely at the water surface. Explain what is observed.
9. A ray of light is incident at angle of 30° on a prism of refractive index 1.5 calculate the limiting angle of the prism such that the ray does not emerge when it meets the second face.
[ Ans A = 61×3° ]
A spectrometer
It is an instrument used to measure accurate determination of deviation of a parallel beam of light which has passed through a prism. This provides a mean of studying optical spectra and measurement of refractive indices of glass in form of a prism.
It consists of a collimator, a telescope, and a turn table on which the prism is placed as shown.
Before the spectrometer is put in to use, 3 adjustments must be made onto it and these include,
- The collimator is adjusted to produce parallel rays of light.
- The turntable is leveled.
- The telescope is adjusted to receive light from the collimator on its cross wire.
Application of spectrometer
- Measure of refractive angle of a prism
- Measurement of the angle of minimum deviation
- Measurement of refractive index of prism
- Measurement of spectra of light.
Measurement of the refracting angle “A” of the prism
The collimator is adjusted to produce parallel rays of light.
The turn table is leveled.
The telescope is adjusted to receive light from the collimator on its cross wire.
The prism is placed on the turn table with its refracting angle facing the collimator as shown.
With the table fixed, the telescope is moved to position T1 to receive the light from the collimator on its cross wire. This position T1 is noted and the telescope is turned in to a new position T2 to receive light on its cross wire. The angle q between T1 and T2 is measured.
Proof of the relation
Consider a parallel beam of light incident on to a prism of refracting angle A making glancing angles a and b as shown.
NOTE
Method 2: Determining refractive angle of prism using optical pins
A white paper is stuck to the soft board using top-headed pins. Two parallel lines AB and DC are drawn on the paper and the prism is placed with its apex as shown.
Two optical pins P1and P2 are placed along AB and pins P3 and P4 are placed such that they appear to be in line with the images of P1and P2 as seen by reflection from face
PQ .The procedure is repeated for face QR. The prism is removed and angle θ is measured.
The required refracting angle
Measurement of minimum deviation “Dmin” of the prism
The collimator is adjusted to produce parallel rays of light.
The turn table is leveled.
The telescope is adjusted to receive light from the collimator on its cross wire The prism is placed on a turn table with its refracting angle facing away from the collimator as shown.
The telescope is turned in the direction of the base of the prism until light can be seen. With light kept in view, both the telescope and the table are turned until light movesin the opposite direction. Position T3 of the telescope is noted.
The table is then fixed and the prism is removed so that the telescope is turned to a new position T4 to receive the un deviated light. The angle between T3 and T4 is determined and this is the angle of minimum deviation Dmin.
NOTE
(i) Position T3 is noted because, in the position of minimum deviation light viewed through the telescope moves in the opposite direction.
(ii)The refractive index of a glass prism of known refracting angle A can be determined using a spectrometer from the relation
Method 2: measurement of refractive index of prism using optical pins
Two optical pins P1 and P2 are placed along the lines that make different angles of incidence i. Pins P3 and P4 are placed such that they appear to be in line with the images of P1 and P2 as seen through the prism. The angles of deviation d are measured for different angles of incidence. A graph of d against i is plotted to give a curve whose angle of deviation at its turning point is the angle of minimum deviation Dmin of the prism.
Uses of a glass prism
1. They enable the refractive index of a glass material to be measured accurately.
2.They are used in the dispersion of light emitted by glowing objects.
3. They are used as reflecting surfaces with minimal energy loss.
4. They are used in prism binoculars.
Examples 9
A ray of monochromatic light is incident on one face of a glass prism of refracting angle 60° and is totally internally reflected at the next face.
(i) Draw a diagram to show the path of light through the prism.
(ii) Calculate the angle of incidence at the first face of the prism if its refractive index is 1.53 and the angle of incidence at the second face is 42°.
Examples 10
A ray of light is incident on the face AD of a glass block of refractive index 1.52 as shown.
If the ray emerges normally through face BC after total internal reflection, Calculate the angle of incidence, i.
Breakdown
(i) Its after a total internal reflection at Q that the ray emerges through face BC
(ii) At R, there is no refraction. Therefore, Snell’s law does not hold at this point.
Examples 11
A ray of light is incident at 45° on a glass prism of refractive index 1.5 as shown.
Calculate the angle of emergence and sketch the ray diagram.
Solution
Examples 12
A ray of light is incident at 45° on a glass prism of refractive index 1.5 as shown.
Calculate the angle of emergence and sketch the ray diagram.
Solution
Examples 13
The diagram in the figure below shows a cross section of an isosceles right angled prism sides PQ and QR are coated with a reflecting substance. A ray of light is incident on PR at an angle θ as shown
(i) Draw a diagram to show the path of light through the prism.
(ii) Show that the ray leaving the prism is parallel to the incident ray.
From geometry of the figure above the angle of emergence at D is the same as the incident angle q at A. Hence the emergent ray is parallel to the incident ray.
Example 14
(a) Define the term refraction and refractive index.
(b) Derive an expression for apparent displacement of an object when viewed through a parallel- sided block.
(c) A ray of light is incident on a glass-water interface at an angle of 500. If the refractive indices of water and glass are 1.33 and 1.5 respectively, determine the angle made by the refracted ray with the normal at point of incidence.
Solution
(d) A light ray consisting of blue and red light is incident from air to a glass block. The two colors emerge from the block into air at point O and P respectively as shown below.
The speeds of blue and red light respectively in glass are 1.88 x 108ms-1 and 1.84×108ms-1. Find the distance OP. (Answer 0.17cm)
Example 15
The following diagram shows the direction followed by a ray of monochromatic light through a right angled triangular prism of refractive index 1.52. the light emerges in air. Calculate the refractive index of liquid.
Solution
nL = 1.52 sin 610 = 1.33
EXERCISE
1. Draw a labeled diagram of a spectrometer and State the necessary adjustments that must be made on to it before put in to use.
2. Describe how the refracting angle of the prism can be measured using a spectrometer.
3. You are provided with pins, a white sheet of paper, a drawing board and a triangular prism. Describe how you would determine the refracting angle A of the prism
4. A parallel beam of light is incident on to a prism of refracting angle, A, as shown
Show that θ = 2A
5.Describe how the minimum deviation, D, of a ray of light passing through a glass prism can be measured using a spectrometer.
6.You are provided with pins, a white sheet of paper, a drawing board and a triangular prism. Describe how you would determine the angle of minimum deviation, D, of a ray of light passing through a glass prism.
7. Describe how the refractive index of a material of a glass prism of known refracting angle can be determined using a spectrometer.
8. Describe briefly two uses of glass prisms
9. A ray of light is incident on prism surrounded by a liquid of refractive index 4/3 at an angle 300. If the refractive index of the prism is 1.5. Find the angle at which the ray emerges from the opposite face of the prism if its refractive angle is 600. [Answer 38.50].
10. A ray of light is incident on a prism in air along its face. If the ray from opposite face emerge at an angle 300 and the refractive index of the prism is 1.52, find the refractive angle of the prism. [Answer 60.30]
11. (a) Define limiting angle of a prism.
This is the maximum angle of the prism for which light incident on the prism at 900 emerges from the opposite face.
(b) Light incident to a prism at such an angle that it emerges from the opposite face of the prism grazing the surface. If the angle of the prism is 600 and the refractive index of the prism is 1.52, calculate the angle of incidence of light. [Answer = 29.50]
12. Monochromatic light is incident on the refracting face of a prism of refracting angle 600 made of glass of refractive index 1.5. Calculate the minimum angle of incidence for the ray to emerge through the second refractive face. [Answer 27.90)
13. Monochromatic light is incident at an angle of 450 on a glass prism of refractive angle 700 in air. The emergence light grazes the other surface of the prism. Find the refractive index of glass. [Answer 1.5].
14. A ray of light propagating in a liquid is incident on the prism of refracting angle 500 and refracting index 1.6 as shown below. If the ray passes symmetrically through the prism, find the refractive index of the liquid.
{[Answer n= 1.35] NB when light passes symmetrically through the prism r= A/2}
15. Monochromatic light is incident at an angle of 280 on a glass prism of refractive index 1.5. The emergent light grazes the surface of the prism as shown below
(a) Calculate the angle of refraction, r.
(b) Find the critical angle, c for the glass-air interface.
(c) Find the refracting angle, A, of the prism. [Answer: r= 24.20, C = 41.80, A = 660].
16. A ray of light is incident on a prism of refractive index 1.3 and refractive angle 720. The ray emerges from the prism at angle 430. Find
(a) the angle of incidence,
(b) the deviation of they ray [Answer 57.40, d = 28.40]
17. A glass prism of refractive index 1.5 and refracting angle 600 is completely immersed in a liquid of refractive index 1.3. If a ray of light passes symmetrically through the prism, calculate the
(i) angle of incidence
(ii) angle of deviation [Answer: i = 35.20, d= 10.40]
18. A ray of monochromatic light enters one face of a 600 glass prism and is totally internally reflected at the other face
(i) draw a ray diagram to show the path of light through the prism.
(ii) Calculate the angle of incidence at the first face if the refractive index of glass prism is 1.53 and the angle of incidence at the second face is 420.
- Answer 28.20
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