Nature and Propagation of Light - Class 12 Physics

Topics to be covered on Nature and Propagation of Light:

1. Theories of Light
2. The Electromagnetic Spectrum
3. Wave fronts and Rays
4. Huygen’s Principle

Nature and Sources of Light

Optics is a branch of physics which deals with study of phenomena like reflection, refraction, dispersion, interference, diffraction etc. The subject of optics has been divided into ray optics and wave optics. The branch of optics that deals with the production, emission and propagation of light, its nature and the study of the phenomenon of interference, diffraction and polarization is called wave optics.

Nature and Sources of Light

Scientists have made discussion on true nature of light and have postulated various theories. These are discussed below:

1. Newton’s Corpuscular Theory
   Newton put forward this theory that consist of tiny particles, called corpuscles, which are shot out by luminous objects such as an electric lamp, candle, etc. According to Newton, these particles travel in straight lines and bounce off or pass into an object striking off or pass into an object on striking it. According to Newton, these particles travel in straight lines and bounce off or pass into an object striking on it. The speed of light is increased in denser medium and this resulted in a change in direction as corpuscles are attracted by the material of denser medium. However, speed of light decreases in denser medium. This theory cannot explain interference, diffraction and polarization of light. So, this theory is incomplete and inaccurate.
2. Huygens’ Wave Theory
   According to this theory, light propagates from the source in the form of a wave. For the propagation of wave a medium is necessary. So, it was assumed that space is filled with medium called ether, which has the property of both elasticity and inertia. This theory says that light wave is longitudinal but actually light wave is transverse.
3. Electromagnetic Theory
   Theoretically light gets propagated in the form of electromagnetic waves, consisting of electric and magnetic fields mutually perpendicular as well as transverse to the direction of propagation of light. The electromagnetic waves propagate in free space with the velocity of light. Hertz demonstrated experimentally that electromagnetic waves propagate with velocity equal to that of light.
4. Quantum Theory
   in 1905 AD Einstein proposed a new theory of light called a quantum theory, in order to explain the photoelectric effect. According to this theory, light is transmitted as tiny packets of energy called photons and energy of each photon is given by
   $$ E = hf $$
   where f is the frequency of light and h is Planck’s constant.

5. Duel Nature of Light
   Light can exist in both particle and wave form. Using quantum theory, we can explain photoelectric effect. But this theory could not explain interference, diffraction and polarization. These phenomena can be explained using wave theory but wave theory could not explain photoelectric effect.

Wave front

According to wave theory, a source of light sends out disturbance or waves in all directions. A wave front as the locus of all adjacent points at which the phase of vibrations of a physical quantity associated with the wave is the same.

Types of Wave front

Depending on the nature of the source of light, there are three types of wave fronts:

1. Spherical Wave front
   It is produced by the point source of light. This is all because such points which are equidistant from the point source will lie on a sphere. A spherical wave front is shown in figure.
2. Cylindrical Wave front
   When the source of light is linear in shape (e.g a slit), all points equidistant from the linear source lie on the surface of a cylinder. Such a wave front is called a cylindrical wave front as shown in the figure.
3. Plane Wave front
   A small portion of cylindrical wave front or spherical wave front originating from a distant source will appear a plane and hence termed as plane wave front.
   
   

Electromagnetic Spectrum

Electromagnetic Spectrum

The orderly distribution of electromagnetic radiations according to their wavelength or frequency as called the electromagnetic spectrum. Electromagnetic spectrum has a very wide range with wavelength variation from 10^-13m to 6×10⁶m.

The usual classification of the electromagnetic spectrum is explained below

1. Radio waves

   These are electromagnetic waves of the frequency range from few Hz to 10⁶ Hz.These waves, which are used in television and radio broadcasting systems, are generated by electronic devices, mainly oscillating circuits having an inductor and capacitor.
   
   

2. Microwaves

   The wavelength of microwaves is greater than 1.0 mm and less than 30 cm. The frequency range of microwaves is 10⁹Hz to 3.0×10¹¹Hz.They are produced by oscillating electrons in a cavity. The commonly used oscillators to produce microwaves are Klystron and Magnetron.
   
   Uses of Microwaves

• Microwaves are used in a radar communication.
• These are used for atomic and molecular research.
• These are used for aircraft navigation.
• These are used in microwave ovens for cooking and warming food.

3. Infra-red (IR) rays

   The wavelength range of infra-red rays 1 nm to 700 nm and the frequency range is 3.0 × 20¹¹Hz to 4.3 × 10¹⁴Hz Infrared rays are produced by the excitation of atoms and molecules. Hot bodies also radiate infra-red rays.
   
   Uses of Infra-red Rays

• These rays can pass through the haze, fog, and mist, so these are used in night vision devices during warfare and for taking photographs of the earth under foggy conditions from a great height.
• These rays are used to keep the greenhouse warm.
• They are used in revealing the secret writings on the ancient walls.
• They are used to treat muscular strains.
• The infra-red rays from the sun keep the earth warm.
  
  

4. Visible Light

   The range of visible light is 400 nm to 60 nm and their frequency range is 4.3×10¹⁴Hz to 7.5×10¹⁴Hz.Visible light is emitted when an electron jumps from higher energy level to lower energy level of an atom.
   
   Uses of Visible Light

• Visible light stimulates the sense of sight so we can see the beautiful world in the presence of visible light.
• Visible light is useful in photography.
• It is useful in optical microscopy and
• It is useful astronomy.
  
  

5. Ultra-violet (UV) rays

   The range of ultra-violet rays is 400 nm to 60 nm and their frequency range is 7.5×10¹⁴Hz to 5×10¹⁵Hz.The sun is the most important source of ultra-violet rays. Ultra-violet rays are produced by the spark of welding. These rays are harmful to the living tissues.
   
   Uses of Ultra Violet Rays

• They are used to preserve food stuff and make drinking water free from bacteria as these rays can kill bacteria, germs etc.
• They are used for sterilizing the surgical instruments.
• They are used in detecting the invisible writings, forged documents, and finger prints.
• They are used to study the structure of molecules.

6. X-rays

   X-rays can be produced by bombarding a target of high atomic number (Z) with a beam of fast-moving electrons. The range of the wavelength of X-rays varies from 60 nm to 10^-8.The frequency of these waves varies from 5.0×0¹⁵Hz to 3.0×10¹⁸Hz. X-rays can penetrate through the human flesh, but bones or metallic materials block these days.
   
   Uses of X-rays

• These are used in medical diagnosis like locating the fracture in the bone, foreign materials, like coin or bullet in the body.
• These are used in radiotherapy to cure skin diseases, cancer, and tumors.
• These are used in engineering for locating the faults, cracks, and flaws in the finished metallic materials.
• X-rays are used by detective agencies to detect gold, silver, diamonds and other contraband goods etc. concealed in bags or the body of a person.

7. Gamma rays

   γ-rays are produced during radioactive decay of nuclei and nuclear reactions. The wavelength of γ-rays is the shortest of all the electromagnetic waves. The range of the wavelength of these rays varies from 1.0 nm to 10^-5On the other hand, the frequency of γ-rays in the highest of all the electromagnetic waves. The range of the frequency of γ-rays is between 3.0¹⁸Hz to 3.0×10²² Hz.
   
   Uses of γ-ray

• In the treatment of cancer and tumors.
• To pressure the foodstuffs for a long time as the soft γ-rays can kill microorganisms easily.
• To produce nuclear reactions.
• To provide valuable information about the structure of the atomic nucleus.

Huygens‘ Principle

Huygens ’ Principle enables us to determine what its position will be at some later time. In other words, the principle gives a method to know as to how light spread out in the medium. A source of light sends out wave front is propagated forwards through a homogeneous isotropic medium, Christian Huygens made the following assumptions.

1. Each point on a wave front acts as a new source of the disturbance. The disturbances from these points are secondary wave lets. These wavelets spread out in all directions in the medium with the velocity of light.
2. The new wave front is then obtained by constructing a tangential plane to all the secondary wavelets. The new wave front is the envelope of to secondary wavelets at that instant.

Laws of Reflection on the Basis of Wave Theory

Conclusion of laws of reflection on the basis of wave theory:

1. The angle of reflection r is equal to the angle of incidence i for all wavelengths and for any pairs of materials.
   $$ r = i$$
2. The incident ray, reflected ray and normal to the reflecting surface all lie on the same plane.

Consider a plane wave front AB incident on the reflecting surface XY at an angle of incidence I as shown in the figure. The lines 1, 2 and 3 which are perpendicular to the wave font AB represent incident rays. AN is normal tot the reflecting surface. Point A of the wave front reaches the reflecting surface at times t = 0. By the time, point B, of the wave front reaches point A’ (t=t), the secondary wavelets from A spread out in the form of a sphere. There A’B’ = BA’ = ct, where c is the velocity of light. From A’ draw a tangent to the sphere. Then A’B’ represents reflected wave front. Similarly, the wavelets from C reach point D and from D reach E of the reflected wave from in time t. reflected rays must be at right angles to the wave A’B’. In the figure, reflected rays are represented by 1’ 2’ and 3’.

Draw A’N’ normal to the reflecting surface. Now, ÐB’A’A = r, is the angle of reflection. From the right angle triangles ABA’ and AB”A”, we have,

1. $$ AB’ = BA’ $$$$\angle AB’A’ = \angle ABA’ = 90^0 $$
2. $$ AA’ \: \text {is common. Therefore the two triangles are congruent.} $$

\begin{align*} \text {Hence,} \angle BAA’ = \angle B’A’A \\ \text {or,} \: i &= r \\\end{align*}

i.e the angle of incidence is equal to the angle of reflection which is the first law of reflection.

The incident wave front, AB, the reflected wave front A’B’ and the reflecting surface AA’ are all perpendicular to the plane of the paper. So the incident ray, normal to the reflecting surface and the reflected ray all lie in the same plane. This proves the second law of reflection.

Laws of Refraction on the Basis of Wave Theory

The refraction of light when light enters from a rarer medium to a denser medium. Let v and c be the velocity of the light in the denser medium and in air respectively. The laws of refraction are:

1. The ratio of the sine of angle incidence to the sine of the angle of refraction is constant for any two given media. Therefore,
   $$ \frac {\sin i}{\sin r} = \frac cv = \mu $$
   where µ is constant called the refractive index of the medium with respect to air.
2. The incident ray, the refracted rays and the normal at the point of an incident on the refracting surface lie on the same plane.

Consider a plane wave front AB incident on a refracting plane surface XY separating two different media. The lines 1, 2, and 3 which are perpendicular to the wavefront AB represent incident rays. AN is normal to the surface XY and I is the angle on incidence which is equal to the angle made by incident wavefront AB with the surface XY as shown in the figure. Let v and c be the velocity of light in the denser and rarer medium respectively, such that v<c.

According to Huygens principle, every point on the wave front AB is a source of secondary wavelets. At an instant, the wavelet to reach A’ from B, then BA’ = ct. During this time, the secondary if t is the time taken by the secondary wavelet originating at A has traveled a distance vt = AB’ in the denser medium. Similarly, the wavelet from point C of the wave front reaches D and the wavelet from D reach point E in the same time. If we draw a sphere of radius -= AB’ = vt, and draw a tangent A’B’ to this sphere, wavelets form points between A and A’ reach to the surface of spheres to which A’B’ is tangent. Then, A’B’ is the new wave front in the denser medium. The rays 1’, 2’ and 3’, normal to the wavefront AA’, are refracted rays.

Draw N’A’ normal to the refracting surface at A’. The angle, ∠AA’B’ = r is the angle of refraction.

$$ \begin{align*} \text {In right angled triangle} \Delta \text {ABA’} \\ \sin i &= \frac {A’B}{AA’} = \frac {ct}{AA’} \dots (i) \\ \sin r &= \frac {AB’}{AA’} = \frac {vt}{AA’} \dots (ii) \\ \text {Dividing} \: (i) \: \text {by} \: (ii) \: \text {we get,} \\ \frac {\sin i}{\sin r} = \frac cv \dots {iii} \\ \text {where} \frac cv = \mu,\\ \end{align*} $$

is the refractive index of denser medium with respect to rarer medium.}

$$ \begin{align*} \therefore \frac {\sin i}{\sin r} &= \mu \\ \end{align*} $$

The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for the given pair of media which proves Snell’s law.

Further, the incident ray refracted ray and the normal to the surface of separation at the point incidence all lie in the plane of the paper. This verifies the second law of refraction.

Measurement of Velocity of Light

Velocity of Light

Many experiments were designed to measure the velocity of light. The first attempt was made by Galileo in 1600. Galileo attempted to measure the velocity of light by covering and uncovering a lantern at night and timing how long the light took to reach an observer a few miles away. As the velocity of light is very large, the time was taken to transverse the distance was too small to measured and thus this method failed to measure the velocity of light. The first successful attempt to measure the velocity of light was made by Romer, observing the eclipses on a satellite of Jupiter. However, the first measurement of the velocity of light was made by Fizeau in 1849.

Foucault ‘s Method for the Velocity of Light

The rays of light from a source S are allowed to fall on a convex lens L, after passing through the glass plate P. these rays are converged to the point I by the lens L. If a mirror M­₁ is placed at A as shown in figure, light after reflection from the mirror M₁, converges at the pole of the concave mirror C, whose distance from A is adjusted such equal to its radius of curvature. Light is reflected back from C along its original path and finally the image is formed on the source S. As there is a half silvered glass plate P inclined at an angle of 45⁰ C to the axis of the lens, the image if formed at B₁, which can be viewed with the help of micrometer eyepiece.

Suppose M₁ is rotated at the uniform angular speed about an axis passing through A and the rays after reflection from the concave mirror C, find the plane mirror displaced by an angle θ to a new position M₂. The image is now observed at B₂. The displacement B­₁B₂ of the image is measured and velocity of light is calculated

Theory

Consider a point E on a concave mirror from which light is reflected back to the rotating mirror. When the plane mirror is at the position M₁, the rays reflected from it to the lens L seem to come from I which is an image E in M₁. When it has rotated through an angle θ, the rays reflected by it to the lens L appears to come from I₁ which is the image of E in the new position M₂ of the mirror. The distance is set as AE = AI = a, where a is a radius of curvature of the concave mirror. Thus, we find that A acts as the centre of curvature of the concave mirror.

If the mirror M₁ is turned through an angle θ, the reflected ray is turned through an angle 2θ and hence I₁ AI = 2θ. So,

$$ \begin{align*} \therefore II_1 = a \times 2\theta = 2a\theta \\ \text {As} \:S \: \text { and } S_1 \: \text {are conjugate points with respect to} \: I\: \text {and } \: I_1, \text {for the lens L} \\ \frac {S_1S}{l} = \frac {II_1}{(a + b)} \\ \text {or,} \: S_1S &= \frac {2a\theta \times l}{(a + b)} \\ \end{align*} $$ where l is distance between the lens and source S and b is the distance between the lens and the mirror M.$$ \begin{align*} \text {Suppose,} S_1S = BB_2 = y. \\ \text {Then,} \\ y &= \frac {2a\theta \times l}{(a + b)} \\ \text {or,} \: \theta &= \frac {y(a + b)}{2al} \dots (i) \\ \end{align*} $$  If n be the number of revolutions made per second by the rotating mirror, then the time taken by the plane mirror to rotate through an angle\(\: \theta \: \text {is} \) $$ \begin{align*} t = \frac {theta }{2\pi n} \dots (ii) \\ \text { If the velocity of light is c,} \\ \text {the time taken by the light to covered distance 2a,} \\ \end{align*} $$ 

$$ \begin{align*}\text {i.e. from A to E and back to A is given by} \\ \therefore t = \frac {2a}{c} \dots (iii) \\ \text {Therefore from equations} \: (ii) \: \text {and} \: (iii) \: , \text {we have} \\ \frac {2a}{c} = \frac {theta }{2\pi n} \\ \text {or,} \: \theta = \frac {4\pi na}{c} \dots (iv) \\ \text {From equations} \: (ii) \: \text {and} \: (iii) \: , \text {we have}\\ \frac {y(a + b)}{2al} = \frac {4\pi na}{c} \\ \therefore c = \frac {8\pi na^2l}{y(a+b)} \\ \end{align*} $$

As n, a, l, y and b are measurable quantities, the speed of light c can be calculated. The value of light found by Foucault was 2.98×10⁸ m/s.

Advantage

1. It can be used to measure the velocity of light in a liquid by putting it in a tube that is placed between two mirrors.
2. Space required in this required in this experiment is quite small and hence it may be performed inside a laboratory.

Drawbacks

1. The shift in the image is very small.
2. The image obtained is not very bright due to reflection and refraction of light at various surfaces.

Michelson’s Method for the Velocity of Light

An octagonal mirror M₁ is mounted on the shaft of a variable speed motor. Light from a bright source S is focused at an angle of 45⁰ on one of the faces of mirror M₁ after passing through a slit S₁. The reflected light falls on a distant concave mirror M₂. In the figure, M₃ is a plane mirror and with the help of this mirror M₃ plane mirror and with the help of the mirror M₃ placed at the centre of curvature of mirror M₂ the beam of light is returned back and falls on face 3 of the octagonal mirror M₁ again at an angle of 45⁰. The light reflected by this face is then collected by a telescope T and the eye at the position. At the rest position of the mirror M+1+, an image of the light source can be observed in the telescope.

If the mirror M₁ is rotated, the light returning to it from the mirror M₂ will not be incident at an angle of 45⁰ , and hence will not enter the telescope. When the speed of rotation of mirror M₁ is so adjusted that the face 2 of mirror occupies exactly the same position as was occupied by face 3 earlier during the time travels from M₁ to M₂ and back to M₁, then the image of a source will reappear.

If d be the distance between the mirror M₁ and M₂ and c be the speed of light, then the time taken by the light to travel from M₁ to M₂ and back to M₁ is

$$ \begin{align*} t = \frac {2d}{c} \\\end{align*} $$ If f is the number of revolutions per second of mirror M₁and m is the number of faces of this mirror then the angle rotated by the mirror during the time t is $$ \begin{align*}\theta = \frac {2\pi }{m} \\ t = \frac {\theta }{2 \pi f} = \frac {2\pi }{2\pi f m} = \frac {1}{mf} \\ \text {or,} \: \frac {2d}{c} = \frac {1}{mf} \\ \therefore c &= 2mfd \\ \end{align*} $$

The value obtained is 2.99775×10⁸ m/s.

Advantage of Michelson’s Method

1. The distance d between the two stations is very large.
2. This null method and so, their measurement of the image.
3. The image is very bright so the result is also accurate.

Disadvantage of Michelson’s Method

1. It is very difficult to keep the high speed of rotation of the mirror constant for a long time.
2. At high speed, the rotating may break. But speed can be reduced by increasing the number of faces of the mirror.

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