Polarization of Light Waves
Light is an electromagnetic wave whose electric and magnetic vibrations are perpendicular to each other and to the direction of propagation.
Polarization of light is the process of confining the vibrations of the electric vector of light waves to one direction.
- In unpolarized light, the electric field vibrates in all directions perpendicular to the direction of the wave.
- The commonly used pictorial representation of an unpolarized light wave is shown below:
An unpolarized light beam is equivalent to two equally intense beams whose planes of vibration are perpendicular to each other.
After reflection or transmission through certain substances, the electric field is confined to one direction, and the radiation is said to be plane-polarized light.
Polaroid
This is a device used to produce plane-polarized light.
In a polarizer, there is a characteristic direction called the transmission axis, which is indicated by the dotted line.
If a polarizer is placed in front of an unpolarized light source, the transmitted light is plane-polarized in a specific direction.
Since the human eye is unable to detect polarized light, it is necessary to use an analyzer to detect the direction of polarization.
If the plane of polarization of the polarizer and the plane of the analyzer are perpendicular, then no light is transmitted when the polarizer and the analyzer are combined.
Methods / Ways of Production of Plane Polarized Light
- By Polaroid
- By reflection
- By double refraction
- By using Nicol prism
Polarization by Polaroids
Polaroid is an artificial crystalline material that can be made in thin sheets.
It has the property of allowing light vibrations only of a particular polarization to pass through.
Uses of Polaroid
- They are used in sunglasses to reduce the intensity of light and to eliminate glare.
- They are used to control the intensity of light entering trains and aeroplanes.
- They are used in windshields of automobiles.
Polarization by Reflection
The reflecting surface of a transparent medium can produce plane-polarized light.
This happens when unpolarized light is incident on any transparent medium, e.g., glass.
Where:
- AO – Incident natural light
- OB – Strongly plane-polarized reflected ray
- OC – Partially plane-polarized reflected ray
Brewster’s Angle
- Polarization by reflection occurs at a certain special angle of incidence at which maximum polarization occurs.
Example
- For glass with a refractive index of 1.5, Brewster‘s angle is 57°.
Brewster’s Law
- The law states:
The extent of polarization of light reflected from a transparent surface is maximum when the reflected ray is at right angles to the refracted ray.
By Snell‘s law of refraction of light:
- When η = refractive index of the transparent medium
- From the figure above, we have:
- Equation (1) above becomes:
This equation leads to Brewster’s law.
- The equation shows that the angle of incidence for maximum polarization depends only on the refractive index of the medium.
Polarization by Double Refraction
Double refraction is the property possessed by certain crystals, e.g., calcite (Iceland spar), of forming two refracted rays from a single incident ray.
o-ray – ordinary ray
e-ray – extraordinary ray
- Where a beam of unpolarized light is incident on one face of the crystal, its internal molecular structure produces two beams of polarized light E and O whose vibrations are perpendicular to each other.
Nicol Prism
- Is a device for producing plane-polarized light.
- It consists of two pieces of calcite which are stuck together with Canada balsam (a transparent material used to join the two pieces of calcite).
- The extraordinary ray, E, passes through the prism while the ordinary ray, O, undergoes total internal reflection at the interface between the two crystals when the angle of incidence exceeds the critical angle value.
Note
If the incident ray to the Nicol prism does not produce double refracted rays, it means that the incident ray is a polarized ray of light.
Application of Polarized Light
1. Reducing Glare
Glare caused by light reflected from a smooth surface can be reduced by using polarizing materials since the reflected light is partially or completely polarized.
Example:
- In sunglasses.
- In photography as filters, where they are placed in front of the camera lens.
2. Optical Activity
Certain crystals, e.g., sugar solutions, rotate the plane of vibration of polarized light passing through them and are said to be optically active.
Definition:
Optical activity is the ability of certain substances to rotate the plane of vibration of plane-polarized light as it passes through them.
For a solution, the angle of rotation depends on its concentration, which can be measured by the instrument known as a polarimeter.
3. Stress Analysis
When glass, Perspex, polythene, and some other plastic materials are under stress (e.g., by bending or uneven heating), they become doubly refracting.
The effect is called photoelasticity and is used to analyze stresses in plastic models of various structures.
Problems
Problem 71
A point P is situated at 20.1 cm and 20.28 cm from two coherent sources. Find the nature of illumination at point P if the wavelength of light is 6000 Å.
Problem 72
The path difference between two identical waves arriving at a point is 85.5 μm. Is the point bright or dark? If the path difference is 42.5 μm, calculate the wavelength of light.
Problem 73
In Young’s experiment, the distance between the two slits is 0.8 mm and the distance of the screen from the slits is 80 cm. If the fringe width is 0.6 mm, find the wavelength of light.
Problem 74
In Young’s experiment, interference bands were produced on a screen placed at 1.5 m from two slits 0.15 mm apart and illuminated by light of wavelength 6500 Å. Find:
- Fringe width
- Change in fringe width if the screen is moved away from the slits by 50 cm
Problem 75
Two parallel slits 1.2 mm apart are illuminated with light of wavelength 5200 Å from a single slit. A screen is placed at 1.0 m from the slits. Find the distance between the fifth dark band on one side and the seventh bright band on the other side of the central bright band.
Problem 76
In a biprism experiment, the distance between the slit and the eyepiece is 80 cm and the separation between the two virtual images of the slit is 0.25 mm. If the slit is illuminated by light of wavelength 6000 Å, find the distance of the second bright band from the central bright band.
Problem 77
In a biprism experiment, with the distance between the slit and the screen as 0.5 m and the separation between the two virtual images of the slit as 0.4 cm, an interference pattern is obtained with light of wavelength 5500 Å. Find the distance between the 3rd and the 8th bands on the same side of the central band.
Problem 78
In a biprism experiment, the distance of the 10th bright band from the center of the interference pattern is 6 mm. Find the distance of the 15th bright band from the center.
Problem 79
The fringe separation in a biprism experiment is 3.2 × 10-4 m when red light of wavelength 6.4 × 10-7 m is used. By how much will this change if blue light of wavelength 4 × 10-7 m is used with the same setting?
Problem 80
In a biprism experiment, the fringe width is 0.4 mm when the eyepiece is at a distance of 1 m from the slit. Find the change in fringe width if the eyepiece is moved 25 cm towards the biprism without changing any other arrangement.
Problem 81
In a biprism experiment, the distance between the slit and the screen is 1.0 m and the distance between the images of the slit is 2.7 mm. If the fringe width is 0.2 mm, find the wavelength of light used.
Problem 82
In a biprism experiment, the distance between the slit and the screen is 0.8 m and the two virtual sources formed by the biprism are 0.4 mm apart. The wavelength of light used is 6000 Å. Find the band width.
Problem 83
Calculate the distance between the second dark band and the fifth bright band on the same side of the central bright band of an interference pattern produced by coherent sources separated by 1.2 mm. The screen is placed at 1 m from the coherent sources and the wavelength of light used is 6000 Å.
Problem 84
In a biprism experiment, the slit is illuminated by light of wavelength 5000 Å. The distance between the slit and the biprism is 20 cm and the distance between the biprism and the eyepiece is 80 cm. If the distance between the two virtual sources is 0.25 cm, calculate the distance between the fifth bright band on one side of the central bright band and the sixth dark band on the other side.
Problem 85
In a biprism experiment, the distance between the two virtual images of the slit is 1.5 mm and the distance between the slit and the focal plane of the eyepiece is 1 m. Find the distance between the second and the eighth dark fringe on the same side if the wavelength of light used is 5000 Å.
Problem 86
In a biprism experiment, fringes were obtained with a monochromatic source of light. The eyepiece was kept at a distance of 1.2 m from the slit and fringe width was measured. When another monochromatic source of light was used without disturbing slit and biprism, the same fringe width was obtained when the eyepiece was at 0.8 m from the slit. Find the ratio of wavelengths of light emitted by the two sources.
Problem 87
The distance between two consecutive dark bands in a biprism experiment is 0.32 mm when red light of wavelength 6400 Å is used. By how much will this distance change if yellow light of wavelength 5900 Å is used with the same setting?
Problem 88
Newton’s rings are observed with a plane convex lens in contact with a glass plate. The radius of the first bright ring is 1 mm. If the radius of the convex surface is 4 m, what is the wavelength of light used?
Problem 89
The diameter of the 10th dark ring in a Newton’s ring system viewed normally by reflected light of wavelength 5900 Å is 0.5 cm. Calculate the thickness of the air film and radius of curvature of the lens.
Problem 90
Newton’s rings formed with sodium light between a flat glass plate and a convex lens are viewed normally. What will be the order of the dark ring which will have double the diameter of that of the 40th dark ring?
Problem 91
If the diameters of two consecutive Newton’s rings in reflected light of wavelength 5890 Å are 2.00 cm and 2.20 cm respectively, what is the radius of curvature of the lens surface in contact with the plane glass surface?
Problem 92
Newton’s rings formed with sodium light (5890 Å) between a plane glass plate and convex lens surface. The diameters of two successive dark rings are 2 mm and 2.236 mm. What is the radius of curvature of the lens surface?
Problem 93
Newton’s rings are formed by placing a lens on a glass surface. If the 10th bright ring of sodium light by reflection (5890 Å) is 5 mm in diameter, what is the radius of curvature of the lens?
Problem 94
In a Newton’s ring experiment, the plane convex lens and the glass plate are in optical contact and the thickness of the film at that point is zero. Find the thickness of the air wedge at the fourth bright ring for light of λ = 500 Å.
Problem 95
In Young’s double-slit experiment, sodium light of wavelength 0.59 μm was used to illuminate a double slit with separation 0.36 mm. If the fringes are observed at a distance of 30 cm from the double slits, calculate the fringe separation.
Problem 96
In an experiment using Young’s slit, fringes were found to occupy 3.0 mm when viewed at a distance of 36 mm from the double slits. If the wavelength of the light used is 0.59 μm, calculate the separation of the double slits.
Problem 97
When red monochromatic light of wavelength 0.70 μm is used in a Young’s double-slit arrangement, fringes with separation 0.60 mm are observed. The slit separation is 0.40 mm. Find the fringe spacing if:
- Yellow light of wavelength 0.60 μm is used
- The slit separation becomes 0.30 mm
- The slit separation is 0.30 mm and the fringe distance is doubled
Problem 98
Interference fringes are formed in an air wedge using monochromatic light of wavelength 0.60 μm. The fringes are formed parallel to the line contact, and a dark fringe is observed along the line of contact. Calculate the thickness of the air wedge at positions where:
- The twentieth dark fringe
- The thirtieth bright fringe from the line of contact
Problem 99
When interference fringes are formed using an air wedge, it is found that the twentieth bright fringe is formed at an air thickness of 6.8 μm. Calculate:
- The wavelength of the light used
- The diameter of the wire
Problem 100
Interference fringes of separation 0.40 mm are formed with yellow light of wavelength 0.60 μm. Calculate the fringe spacing if blue light of wavelength 0.45 μm is used.
Problem 101
Interference fringes of spacing 1.0 mm are obtained using light of wavelength 6000 Å incident on an air wedge of angle θ. The angle of the wedge is now doubled and the light replaced by one of wavelength 1.5 μm. Calculate the new fringe separation.
Problem 102
A loudspeaker emits a note which gives a beat frequency of 4 Hz when sounded with a standard tuning fork of frequency 280 Hz. The beat frequency decreases when the fork is loaded by adding a small piece of plastic to its prongs. Calculate the frequency of the note emitted by the loudspeaker.
Problem 103
A note from a loudspeaker gives a beat frequency of 10 Hz when sounded with a tuning fork of frequency 440 Hz. Calculate:
- The beat period
- Two possible values for the frequency of the note emitted by the loudspeaker
Problem 104
A vibrating sonometer wire emits a note which gives a beat frequency of 6.0 Hz when sounded in unison with a standard tuning fork of frequency 256 Hz. When the fork is loaded, the beat frequency increases. What is the frequency of the note emitted by the sonometer?
Problem 105
A beam of microwaves of wavelength 3.1 cm is directed normally through a double slit in a metal screen and interference effects are detected in a plane parallel to the slit and at a distance of 40 cm from them. It is found that the distance between the centers of the first maximum in the interference pattern is 70 cm. Calculate an approximate value for the slit separation.
Problem 106
What is the wavelength of light which gives a first order maximum at an angle of 22° when incident normally on a grating with 600 lines/mm?
Problem 107
Light of wavelength 600 nm is incident normally on a diffraction grating of width 20.0 mm on which 10.04 lines have been ruled. Calculate the angular positions of the various orders.
Problem 108
A source emits spectral lines of wavelength 589 nm and 615 nm. This light is incident normally on a diffraction grating having 600 lines per mm. Calculate the angular separation between the first-order diffracted waves. Find the maximum order for each of the wavelengths.
Problem 109
When a certain grating is illuminated normally by monochromatic light of wavelength 600 nm, the first-order maximum is observed at an angle of 21.1°. If the same grating is now illuminated with light with wavelength from 500 nm to 700 nm, find the angular spread of the first-order spectrum.
Problem 110
When monochromatic light of wavelength 5.0 μm is incident normally on a plane diffraction grating, lines are formed at an angle of 30°. What is the number of lines per millimeter of the grating?
Problem 111
A spectral line of known wavelength (5.792 μm) emitted from the mercury vapor lamp is used to determine the spacing between the lines ruled on a plane diffraction grating. When the light is incident normally on the grating, the third-order spectrum, measured using a spectrometer, occurs at an angle of 60°19′ to normal. Calculate the grating spacing.
Problem 112
Light from a cadmium discharge lamp can be used to determine the spacing of the lines on a plane diffraction grating. This is done by measuring the angle θ between the diffracted beams either side of the normal in the first order spectrum for light incident normally on the grating.
(a) If the measured value of θ is 46°43′ and the red line used in the cadmium spectrum is of wavelength 644 nm, calculate the number of lines per meter on the grating.
(b) Make a suitable calculation to determine whether the second order spectrum of this line will be visible.
Problem 113
A light source emits two distinct wavelengths, one of which is 450 nm. When light from the source is incident normally on a diffraction grating, it is observed that the fourth order image is formed by the same angle of diffraction as the third order image for the other wavelength. If the angle of diffraction for each image is 46°, calculate:
- The second wavelength emitted by the source
- The number of lines per meter of the grating
Problem 114
A horizontal string is stretched between two points a distance 0.80 m apart. The tension in the string is 90 N and its mass is 4.5 g. Calculate:
- The speed of transverse waves along the string
- The wavelength and frequencies of the three lowest frequency modes of vibration of the string
Problem 115
The fundamental frequency of vibration of a stretched wire is 120 Hz. Calculate the new fundamental frequency if:
- The tension in the wire is doubled, the length remaining constant
- The length of the wire is doubled, the tension remaining constant
- The tension is doubled and the length of the wire is doubled
- The wavelength of waves with frequency 120 Hz
- The length of wire which when fixed at its ends gives a fundamental frequency of 120 Hz
Problem 116
The fundamental frequency of vibration of a stretched wire is 150 Hz. Calculate the new fundamental frequency if:
- The tension in the wire is tripled, the length remaining constant
- The length of wire is halved, the tension remaining constant
(c) The tension is tripled and the length of wire is halved.
Problem 117
A wire having a diameter of 0.80 mm is fixed in a sonometer and has a fundamental frequency of 256 Hz. Alongside it, a wire made of the same material but of diameter 0.60 mm is stretched over the same bridges on the sonometer but the thinner wire is subjected to only half the tension of the thicker wire. Calculate the fundamental frequency of vibration of the thinner wire.
Problem 118
A closed organ pipe is of length 0.60 m. Calculate the wavelengths and frequencies of the three lowest frequency modes of vibration. Take the speed of sound to be 345 m/s and neglect any end correction of the pipe.
Problem 119
Two open organ pipes are sounding together and produce a beat frequency of 12.0 Hz. If the longer pipe has length 0.400 m, calculate the length of the other pipe. Take the speed of sound as 340 m/s and ignore end corrections.
Problem 120
A piece of glass tubing is closed at one end by covering it with a sheet of metal. The fundamental frequency is found to be 280 Hz. If the metal sheet is now removed, calculate:
- The length of the tube
- The wavelength and frequencies of the fundamental and the first overtone (280 Hz). Ignore end corrections.
Problem 121
A tall vertical cylinder is filled with water and a tuning fork of frequency 512 Hz is held over its open end. The water is slowly run out and the first resonance of the air column is heard when the water level is 15.6 cm below the open end. Calculate:
- The end correction of the tube
- The position of the water level when the second resonance is heard
Problem 129
An open tube of length 30.0 cm has an end correction of 0.60 cm. Calculate its fundamental frequency.
Problem 130
Two open pipes of length 0.700 m and 0.750 m are sounded together and vibrate in their fundamental frequencies. Find the beat frequency, assuming that end corrections can be ignored.
Problem 131
Two identical closed pipes of length 0.322 m are each vibrating with their fundamental frequency. If one pipe is held at 0°C and the other at 17°C, calculate the beat frequency which is observed. Take the speed of sound at 0°C to be 331 m/s and ignore end corrections.
Problem 132
A closed pipe is of length 0.300 m. Calculate:
- Its fundamental frequency at 0°C, given that the speed of sound at 0°C is 331 m/s
- The temperature, in °C, at which it will be in unison with a tuning fork of frequency 288 Hz
Problem 133
Two open pipes of length 0.500 m and 0.550 m are sounded together and vibrate in their fundamental frequencies at 7°C. Calculate:
- The beat frequency, given that the speed of sound at 7°C is 335 m/s
- The value of the temperature of the air in the longer pipe at which the two pipes will be in unison
Problem 134
Stationary waves are set up in the space between a microwave transmitter and plane reflector. Successive minima are spaced 15 mm apart. What is the frequency of the microwave oscillator? Take the speed of electromagnetic waves as 3.0 × 108 m/s.
Problem 135
A system of stationary waves in which the nodes are 2 m apart are produced from progressive waves of frequency 200 Hz. Calculate the speed of the progressive waves.
Problem 136
A stretched wire of length 0.7 m vibrates in its fundamental mode with a frequency of 320 Hz. Calculate the velocity of waves along the wire. Why does such a vibration not continue indefinitely?
Problem 137
A wire of mass per unit length 5.0 g/m is stretched between two points 30 cm apart. The tension in the wire is 70 N. Calculate the frequency of the sound emitted by the wire when it oscillates in its fundamental mode.
Problem 138
- A string of unstretched length 2.0 m and mass 0.15 kg has a force constant of 25 N/m. For the experiment, the string is stretched to a total length of 3.0 m. Calculate the velocity of propagation of transverse waves along the string.
- The same string in (a) above is clamped between two rigid supports 3.0 m apart, and set in vibration. Calculate the wavelengths and frequencies of the five lowest frequency modes of vibration which can be excited on the string. When the vibrating string is held lightly at the centre, in which of these modes does the string continue to vibrate? Explain your reasoning.
Problem 139
A vertical steel wire is kept in tension by a piece of iron attached to one end. The wire is set in transverse vibration and emits a note of frequency 200 Hz. The iron is now completely immersed in water and the frequency of the note changes to 187 Hz. If the density of the water is 1000 kg/m³, calculate the density of the iron.
Problem 140
A resonance tube is held vertically in water and can be raised or lowered. A tuning fork of frequency 384 Hz is struck and held above the open end of the tube on a day when the speed of sound in air is 344 m/s. The shortest tube length at which resonance occurs is 21.6 cm and the corresponding length when the tube is filled with carbon dioxide is 16.7 cm. Calculate:
- The end correction for the tube
- The speed of sound in carbon dioxide
Problem 141
A tuning fork is sounded at the open end of a tube containing air, which is closed at the other end. Two successive positions of resonance are obtained when the length is 49.0 cm and 82.0 cm. Calculate:
- The wavelength of the sound waves in the tube
- The end correction of the tube
Problem 142
If the speed of sound in air is 340 m/s at a given temperature, calculate the length of an open pipe having a fundamental frequency of 192 Hz. If this pipe were sounded together with another open pipe of length 0.850 m at the same temperature, calculate the beat frequency. Ignore any end corrections.
Problem 143
An open-ended pipe of length 0.50 m is sounded at 20°C together with a tuning fork of slightly lower frequency. Five beats per second are heard. Calculate the change in temperature required to bring the pipe and fork back into unison. Neglect end corrections and assume that changes in temperature affect only the speed of sound in air, which is 340 m/s at 20°C.
Problem 144
A source of sound, when stationary, generates waves of frequency 500 Hz. The speed of sound is 340 m/s. Find from first principles the wavelength of the waves detected by the observer and the frequency observed when:
- The source is stationary and the observer moves towards it with speed 20.0 m/s
- The source moves away from a stationary observer with speed 30.0 m/s
- The source moves with speed 30.0 m/s in a direction away from the observer and the observer moves with speed 20.0 m/s
Problem 145
A train sounding its whistle (frequency 580 Hz) is traveling at 40.0 m/s along a long straight section of track, and passes an observer standing close to the track. Calculate the maximum change in frequency which the observer will hear. Take the speed of sound in air as 340 m/s.
Problem 146
A motorist approaches a road junction at a constant speed of 15 m/s. A policeman standing at the junction blows on a whistle with frequency 680 Hz.
- Find from first principles the frequency observed by the motorist. The motorist now reduces his speed at a rate of 5.0 m/s².
- Calculate the frequency he observes at subsequent 1-second intervals until he stops.
Problem 147
A train sounding its whistle moves at a constant speed of 20 m/s along a long straight section of track. The train passes under a low bridge on which stands an observer. If the observer records a maximum frequency of 638 Hz, calculate:
- The frequency of the whistle
- The minimum frequency the observer hears
Problem 148
A source of sound of frequency 400 Hz moves at steady speed of 15 m/s towards an observer. If the observer moves at a steady speed of 25 m/s towards the source, calculate the frequency he observes.
Problem 149
A loudspeaker which emits a note of frequency 250 Hz is attached to a wire and whirled in a vertical circle of radius 1.00 m at a steady rate of 20.0 revolutions per minute. Calculate:
- The speed of rotation of the loudspeaker in m/s
- The maximum and minimum frequency detected by a stationary observer
Problem 150
A train sounding its whistle travels at constant speed on a long straight section of track. An observer standing close to the track records a range of frequencies between 551 Hz and 658 Hz. Calculate:
- The speed of the train
- The frequency of its whistle
Problem 151
A hooter of frequency 360 Hz is sounded on a train approaching a tunnel in a cliff-face at 25 m/s, normal to the cliff. Calculate the observed frequency of the echo from the cliff-face, as heard by the train driver. Assume that the speed of sound in air is 330 m/s.
Problem 152
A car travels at a constant speed of 30 m/s towards a tunnel and sounds its horn, which has a frequency of 200 Hz. The sound is reflected from the tunnel entrance. Calculate the frequency of the echo observed by:
- The driver of the car
- A stationary observer standing close to the road
- The driver of the car travelling at 20 m/s which is following the first car
Problem 153
A train emerges from a tunnel at a speed of 20 m/s and sounds its whistle, which has a frequency of 450 Hz. Calculate the frequency of the echo from the tunnel entrance as observed by the train driver.
Problem 154
A source of sound which is stationary with respect to air emits a note of frequency 340 Hz. An observer receding at uniform speed from the source hears an apparent frequency of 300 Hz. Calculate the speed of the observer if the speed of sound in air is 340 m/s.
Problem 155
(a) Show from first principles that the frequency f of sound in still air, heard by a stationary observer as a source of sound of frequency f₀ approaches the observer with a velocity v, is given by:
Where C is the velocity of sound in still air.
(c) When f₀ = 1.0 × 10³ Hz and C = 300 m/s, what is the percentage change in the frequency heard by the stationary observer when the source velocity changes from 30 m/s to 35 m/s?
Problem 156
A model aircraft with an engine producing a note of constant frequency of value 400 Hz flies at constant speed in a horizontal circle of radius 12 m and completes one revolution in 3.0 s. An observer, situated in the plane of the circle and 30 m from its centre, monitors the frequency of the sound from the engine.
- Explain why the observed frequency shows periodic variations
- Derive a relation for the minimum observed frequency in terms of f, the true frequency of the engine, V, the speed of the aircraft and C, the speed of sound in air. Write down the corresponding relation for the maximum observed frequency.
- Taking C to be 340 m/s, calculate the maximum and minimum observed frequencies. Determine the time interval between the occurrence of a maximum frequency and the next minimum frequency.
Problem 157
A ship travelling at 3 m/s towards a cliff in still air and is sounding its siren at 1 kHz. Find from first principles the frequency of the echo as measured by an observer on the ship. Give sufficient detail for reasoning to be followed. The speed of sound in air is 330 m/s.
