APPLIED PHYSICS - I

Manav Rachna Publishing House Pvt. Ltd.D E L H I

by

Dr. MEENAKSHI S KHURANAMSc., Ph. D., MISTEProfessor and Head

Department of Applied Sciences and HumanitiesManav Rachna College of Engineering, Faridabad

Published by :Manav Rachna Publishing House Pvt. Ltd.1172-F, Chitranjan Park,Delhi-110019.

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First Edition: Aug, 2009

Price : Rs. 225/-

Laser Type SettingAkriti Print & Graphics, DelhiPrinted BySagar Printer, Delhi

ToAnanya, Ayushmaan

& Upendra(v)

Preface to the First Edition

(vii)

CONTENTS

1. QUANTUM PHYSICS 1 — 44

1.1 Introduction 21.2 Historical Introduction 21.3 Failure of Classical Mechanics 31.4 Development of Old Quantum Theory 4

1.4.1. Black Body Radiation 41.4.2. Photoelectric Effect 71.4.3. The Compton Effect 10

1.5 Limitations of Old Quantum Theory 131.6 Wave Particle Duality : Dual Nature of Light 13

1.6.1. The Nature of Photons 141.7 de Broglie Waves : Dual Nature of Matter 15

1.7.1. Davisson and Germer’s Experiment : Detection of Matter Waves 161.7.2. de Broglie Waves 171.7.3. Phase and Group Velocities : Concept of a Wave Packet 19

1.8 Uncertainty Principle 251.8.1. Heisenberg’s Microscope : Thought Experiment 271.8.2. Applications of Uncertainty Principle 28

2. SCHRONDINGER EQUATION AND QUANTUM STATISTICS 45 — 84

2.1 Introduction 462.2 The Schrodinger Wave Equation : Time Dependent 462.3 The Time - Independent Schrodinger Equation 502.4 Physical Significance of Wave Functions 512.5 Physical Significance of Eigen Values 522.6 Statistical Physics 54

2.6.1. Phase Space 552.6.2. Basic Approach in Three Statistics 562.6.3. Three Kinds of Statistics 57

2.7 Bose-Einstein Statistics 582.7.1. Applications of Bose Einstein Statistics : Blackbody Radiation 62

2.8 Fermi-Dirac Statistics 672.8.1. Fermi-Dirac Distribution Function 702.8.2. Fermi-State and Fermi Energy 712.8.3. Fermi Function 732.8.4. Average Energy of Electrons at Absolute Zero 74

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3. BARRIERS AND WELLS 85 — 120

3.1 Introduction 863.2 Particle in a Box 863.3 Single step Barriers 923.4 Potential Barrier 993.5 Tunneling through the Potential Barrier 1043.6 Application of Tunneling 106

3.6.1 Electronics 1073.6.2 Behaviour of Molecules 1093.6.3 Nuclear Physics : Alpha Decay 110

4. L A S E R 121 — 140

4.1 Introduction 1224.2 Laser and Its Principle 122

4.2.1 Population Inversion 1244.2.2 Creating a Population Inversion 124

4.3 Einsteins’ coefficeints : Relation Between Spontaneousand Stimulated Emission Probabilities 126

4.4 Unique Properties of Laser Light 1284.5 Components of a Laser 1304.6 The Ruby Laser 1314.7 Helium - Neon Laser 1324.8 Semi-conductor Laser 1334.9 Applications of Lasers. 135

5. OPTICAL FIBRES 141 — 154

5.1 Introduction 1425.2 The Optical Fibre 1425.3 Principle of Light Transmission in an Optical Fibre 1435.4 The Numerical Aperture and Acceptance Cone 1445.5 Different types of Optical Fibres 145

5.5.1 Classification on the basis of refractive indices 1465.5.2 Classification on the basis of modes of propagation 146

5.6 Optial Fibre Communication System 1475.7 Advantages of Optical Fibre communication 1485.8 Disadvantages of an Optical Fibre 1495.9 Applications of Fibre Optics 149

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6. E L E C T R O S TAT I C S 155 — 184

6.1 Introduction 1566.2 The Coulomb’s Law 1566.3 Electric Field Strength or Field Intensity 157

6.3.1. Electric Field Lines 1586.3.2. Electric Field due to Distribution of Charges 158

6.4 Electric Potential 1596.4.1. Electric Potential due to a Distribution of Charges 1616.4.2. Equipotential Surface 1626.4.3. Field as a Potential Gradient 162

6.5 Divergence and Curl of Electrostatic Field 1636.6 Poisson’s Equation and Laplace’s Equation 1656.7 Laplace’s Equation in One Dimension 1666.8 Work done in Electrostatics 1676.9 Energy in Electrostatics 168

7. INTERACTION OF RADIATION WITH MATTER 185 — 204

7.1 Introduction 1867.2 Interaction of Heavy Charged Particle with Matter 1867.3 Interaction of Light Charged Particle with Matter 1877.4 Interaction of Gamma Radiation with Matter 189

7.4.1. Photoelectric Effect 1907.4.2. Compton Effect 1907.4.3. Pair Production 191

7.5 Detection of Nuclear Radiation 1917.5.1. Gas Filled Detectors 1917.5.2. Scintillation Counter 1967.5.3. Wilson’s Cloud Chamber 1977.5.4. Bubble Chamber 1987.5.5. Solid State Detectors 199

8. INTRODUCTION TO ASTROPHYSICS 185 — 204

8.1 Introduction 2068.2 Doppler Effect : Doppler Shift 2068.3 Effect of Gravity on Light 209

8.3.1. Bending of light 2098.3.2. Gravitional red shift 211

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8.4 An Introduction to Stars 2128.4.1 Stellar observational history 2138.4.2. Stellar composition 2148.4.3. Formation, evolution and aging of a star 2148.4.4. Red Giant Stars 2168.4.5. Stellar Structure : The Interior of a Star 2178.4.6. Nuclear fusion in a star 2198.4.7. The Hertzsprung-Russell Diagram 2208.4.8. X-ray Astronomy 221

8.5 White Dwarf 2228.5.1. Formation of White Dwarf 2238.5.2. Degenerate Stellar Matter 2248.5.3. Observations of White Dwarfs 226

8.6 Black Dwarf 2278.7 Neutron Stars 227

8.7.1. The Size of a degenerate star 2298.8 Black Holes 230

8.8.1. Existence of Black Holes? 2308.8.2. Seeing a black hole 231

8.9 Dark Matter 2328.9.1. Dark Matter Detection 2338.9.2. X-rays and Light-bending by dark matter 2348.9.3. Dark Matter and the Fate of the Universe 234

8.10 Anti Matter 2348.11 Introduction to Big Bang 235

8.11.1. What is Big Bang Theory? 2368.11.2. Development of the Theory 2368.11.3. Timeline of the Big Bang 2378.11.4. Evidence in support of Big Bang Theory 239

8.12 The Future Universe : Big Crunch or Big Freeze 241

1.1 Introduction

1.2 Historical Introduction

1.3 Failure of Classical Mechanics

1.4 Development of Old Quantum Theory

1.4.1. Black Body Radiation

1.4.2. Photoelectric Effect

1.4.3. The Compton Effect

1.5 Limitations of Old Quantum Theory

1.6 Wave Particle Duality : Dual Nature of Light

1.6.1. The Nature of Photons

1.7 de Broglie Waves : Dual Nature of Matter

1.7.1. Davisson and Germer’s Experiment : Detection of Matter Waves

1.7.2. de Broglie Waves

1.7.3. Phase and Group Velocities : Concept of a Wave Packet

1.8 Uncertainty Principle

1.8.1. Heisenberg ’s Microscope : Thought Experiment

1.8.2. Applications of Uncertainty Principle

ChapterChapterChapterChapterChapter

11111Quantum PhQuantum PhQuantum PhQuantum PhQuantum Physicsysicsysicsysicsysics

The more you see how strangely Nature behaves, the harder it is to make a modelthat explains how even the simplest phenomena actually work. So theoreticalphysics has given up on that ... What I am going to tell you about is what we teachour physics students in the third or fourth year of graduate school... It is my taskto convince you not to turn away because you don’t understand it. You see myphysics students don’t understand it. ... That is because I don’t understand it.Nobody does.

~Richard P. Feynman, The Strange Theory of Light and Matter.~Richard P. Feynman, The Strange Theory of Light and Matter.~Richard P. Feynman, The Strange Theory of Light and Matter.~Richard P. Feynman, The Strange Theory of Light and Matter.~Richard P. Feynman, The Strange Theory of Light and Matter.

2 Applied Physics - I

1.11.11.11.11.1 INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTIONClassical physics deals with objects at macroscopic level. In classical physics most of the

phenomenon are directly observable by naked eyes or by simple instruments like a microscope.Newton’s laws of motion easily explain these phenomenon. However, from the mid-19th throughthe early 20th century, scientists studied a set of new and puzzling phenomena concerning thenature of matter and indeed, of energy in all its forms. With the discovery of X-rays, radioactivity,photoelectric effect etc., it became evident that the description of the phenomenon on atomic andsubatomic scales do not fit well within the domains of classical laws. In microscopic world thevariables like energy and momentum were found to have discrete values and did not change ina continuous manner from one state to the other as in classical physics. The new concepts ledto the formulation of quantum mechanics.

The development of quantum mechanics took place in two stages. The first stage began withMax Planck’s hypothesis that radiation is emitted or absorbed by matter in discrete packets orquanta of energy, hν. The second stage began with the development of wave mechanics by ErwinSchrodinger in 1926. In the present chapter, we discuss the failure of classical mechanics and thephenomena of black body radiation which led to the development of quantum mechanics. Thelater part of the chapter deals with the dual nature of particles making the quantum system. Thevarious features of waves associated with these particles have been enumerated. The Heisenberg’suncertainity principle, an inherent property governing the behavior of quantum particles, isdiscussed at the end of the chapter.

Before moving on to the above mentioned topics, a brief historical introduction (1850s to1920s) of the field of quantum mechanics is given. Many different paths were followed by thescientists. There were many false starts and hence a person following the developments of the fieldcan easily get lost in the intricate maze.

1.21.21.21.21.2 HISTORICAL INTRODUCTIONHISTORICAL INTRODUCTIONHISTORICAL INTRODUCTIONHISTORICAL INTRODUCTIONHISTORICAL INTRODUCTIONThe path to quantum mechanics begins in 1859 with the work of the German physicist Gustav

Kirchhoff, who was studying the absorption of dark D-lines, emitted by the sun, by the sodiumflame produced in a bunsen burner. He observed selective absorption but could not explain it. Hebegan to consider the emission and absorption of radiation by heated materials in general.

Kirchoff imagined a cavity whose walls, when heated, emitted radiations that were trappedin the container. He assumed the system to be a perfect absorber. This perfectly absorbing bodybecame known as a blackbody blackbody blackbody blackbody blackbody and the radiation inside it as blackbody radiation. Kirchoffchallenged people to determine the intensity distribution of radiations inside the blackbody. It wasnot until the mid-1920s that the first theory that really accounted for the form of intensitydistribution function was created.

The next step in development was taken by the Austrian theoretical physicist LudwigBoltzmann. Boltzmann used the laws of thermodynamics to give total energy density as proportionalto fourth power of temperature. (Energy density = σT4, where σ is a universal constant known asthe Stefan-Boltzmann constant).

The next important step forward were taken a decade later by Wilhelm Wien. On the basisof thermodynamics alone, Wien observed that energy density function is of the form λ–5φ(λT). Hecould not determine the function φ. At this point entered Max Planck. He studied under Kirchoffat the University of Berlin and after Krichoffs death in 1887, succeded him as professor of physicsthere. He became fascinated by the subject of blackbody radiation and started working seriouslyon it in the mid 1890s. His initial goal was to derive Wien’s law from Maxwell’s electromagnetictheory and thermodynamics. The derivation kept eluding him.

In early 1900, Planck’s colleagues Otto Lummer and Ernst Pringsheim at Berlin didmeasurements of the long wavelength region of the blackbody spectrum. These observationsrevealed that Wien’s law broke down in the infrared region. Planck came up with a function thatreduced to Wien’s law at lower wavelength and fitted the experimental data at higher wavelengths.

Quantum Physics 3

By applying methods of statistical physics introduced by Boltzmann in the 1870s, Planckobtained the Kirchoff formula. He made a drastic assumption that a blackbody contained“resonators”“resonators”“resonators”“resonators”“resonators” or charges that could only emit and absorb energy of frequency ν in units of hν ,where h is a new universal constant with dimensions of energy multiplied by time. He called theseenergy units quanta.quanta.quanta.quanta.quanta.

During the same time, some months before publication of Planck’s paper, Lord Rayleigh alsopublished a brief paper on the subject. Einstein in 1905 decided to explore the consequences ofthe Wien end of the Planck formula. He proposed that if light beams really consisted of quantaof energy satisfying the relation E = hν, then such a beam falling on a metallic surface should beable to liberate electrons with an energy that depends only on the frequency of the light. Thisphenomenon is referred to as the photoelectric effectphotoelectric effectphotoelectric effectphotoelectric effectphotoelectric effect . It was not until about 1915 that theAmerican experimental physicist Robert Millikan demonstrated with finality that Einstein’s ideawas right. The photoelectric effect involves only the conservation of energy. But the quanta also

carry a momentum having magnitude p = νhc

and this must be conserved. It was not until 1922

that Arthur Compton considered the implications of both the conservation of energy and theconservation of momentum in an experiment on scattering of X-rays. He treated the X-rays asparticles with a definite momentum and energy. He introduced the photon, the quantum of light,as a full fledged elementary particle.

11111.3.3.3.3.3 FFFFFAILAILAILAILAILURE OF CLASSICURE OF CLASSICURE OF CLASSICURE OF CLASSICURE OF CLASSICAL MECHANICSAL MECHANICSAL MECHANICSAL MECHANICSAL MECHANICS1. The earliest evidence of the need for revision of classical concepts came from the field of

chemistry. It was long before realized that atoms or molecules of a substance, say oxygen,are composed of a number of positively and negatively charged particles held together byelectrostatic forces. However, a system of charged particles cannot remain at rest in stableequilibrium under the influence of purely electrostatic forces. Thus, the charges should bein a relative motion. If the particles are confined to a restricted region of space they mustfrequently or continuously change their direction of motion i.e., they should be accelerated.It is, however, a well known fact that accelerated charged particles radiate energy in theform of electromagnetic waves. Hence, the particles in a molecule should progressivelychange their state of motion in accordance with the loss of energy mentioned before. But,this is not in line with observations. This fact alone shows that stability of molecules cannot be understood on the basis of a classical model.

2. Light is emitted by substances which are raised to a high temperature or subjected to anelectric discharge. This light can be separated into its various wavelengths by using aprism or a diffraction grating. If the light source is a hot solid (e.g. light bulb filament), orliquid, the spectrum is continuous; i.e., light of all wavelength is present. But if the sourceis a gas carrying an electric discharge (as in a neon sign), or a salt heated in flame, onlyfew colours appear in the form of isolated sharp parallel lines, (each corresponding to adefinite wavelength and frequency) i.e, a line spectrum is obtained. The characteristicspectrum of an atom was presumably related to its internal structure. Attempts by scientiststo understand these solely on the basis of existing laws of classical mechanics andelectromagnetism were not successful.

3. In the year 1887, while working on electromagnetic-wave experiments, Hertz discoveredthe photoelectric effect. When light strikes a metal surface, some electrons near the metalsurface absorb enough energy to overcome the attraction of the positive ions in the metaland escape into the surrounding space. Classical optics failed to explain this effect.

4 Applied Physics - I

4. Scientists were unable to explain the production and scattering of X-rays discovered in1895. The X-rays are produced in high voltage electric discharge tubes and havewavelengths much shorter than those of visible light. On scattering by matter, the scatteredX-ray had larger wavelength than the incident X-ray.

5. The enormous range of electrical conductivities of solid materials can not be reasonablyexplained in terms of classical ideas. The conductivity of silver is more than 1024 timeslarger than that of fused quartz. A similar situation is found in magnetism: the magneticsusceptibility of iron is observed to be nearly 109 times larger than that of other metals.

All these phenomenon, and several others, pointed forcefully to the conclusion that classicalmechanics had its limitations.

1.41.41.41.41.4 DEVELOPMENT OF OLD QUANTUM THEORYDEVELOPMENT OF OLD QUANTUM THEORYDEVELOPMENT OF OLD QUANTUM THEORYDEVELOPMENT OF OLD QUANTUM THEORYDEVELOPMENT OF OLD QUANTUM THEORYFailure of classical mechanics to explain the phenomenon like continuous and line spectra,

photoelectric effect etc. led to development of new concepts which laid the foundations of quantummechanics. The path to quantum mechanics was laid by Gustav Kirchoff who posed the problem ofblack body radiations which eventually led to the quantum theory.

1.4.1 Blackbody Radiation

A hot piece of metal gives off visible light. As the temperature of the metal is increased, thecolour of the metal goes from red to yellow to white. This can easily be observed when a blacksmith heats his metal in a furnace. In fact, the heated metal gives off other wavelength also whichdo not lie in visible range. The objects do not need to be heated to emit energy. Whatever be thetemperature, all object radiate energy continuously, though the wavelength and the frequency ofradiation emitted depends upon the temperature of the body. At room temperature, most of theradiation emitted lies in the infrared part of the spectrum and hence is invisible.

The ability of a body to radiate is closely related to its ability to absorb radiation. An idealsurface that absorbs all wavelengths of electromagnetic radiation incident upon it is also thebest emitter of electromagnetic radiation at any wavelength. Such an ideal surface is called ablackbody,blackbody,blackbody,blackbody,blackbody, and the continuous spectrum radiation that it emits is called blackbody radiationblackbody radiationblackbody radiationblackbody radiationblackbody radiation. Inthe laboratory, a black body can be approximated by a hollow object with a very small holeleading to its interior. Any radiation entering the body is trapped by reflections back and forthand is completely absorbed. The black body radiates out the absorbed radiation and the radiationsare more when it is heated. By the year 1900, the radiations had been studied extensively, andseveral characteristics had been established. Fig. 1.1 shows the spectrum of a black body at threedifferent temperatures. Investigators discovered different properties of thermal radiations. Thesecan be summarized as :

1. The variation of energy density with wavelength is not uniform in the spectrum. At agiven temperature, the energy emitted increases with wavelength, reaches a maximumand then falls down.

2. The wavelength at which the energy emitted increases to its maximum value, is representedby λm. With increase in temperature, the peak of the curve shifts towards lower wavelengthside i.e., as temperature increases, λm decreases. This is given by the equation

λm T = constant ... (1.1)Equation (1.1) was deduced by Wilhelm Wien and is called Wien’s displacement law.Wien’s displacement law.Wien’s displacement law.Wien’s displacement law.Wien’s displacement law. Thelaw is clearly evident in Fig. 1.1 where the dashed line is showing the shift in λm towardsthe lower wavelength side as the temperature increases. The experimental value of theconstant in equation (1.1) is 2.90 � 10–3 mK.

Quantum Physics 5

3. As the temperature of a body is increased, the intensity of the emitted thermal radiationrises rapidly. The area under each curve represents the total energy emitted by the body ata particular temperature for the range of the wavelength considered. This area increaseswith temperature. This was summarized quantitatively as the Stefan-Boltzmann lawStefan-Boltzmann lawStefan-Boltzmann lawStefan-Boltzmann lawStefan-Boltzmann law(equation (1.2)). According to this law, the total radiated power, summed or integratedover all wavelengths, is proportional to fourth power of temperature, i.e.,

I ∝ T4

or I = σ T4 ... (1.2)where σ is a fundamental physical constant called the Stefan-Boltzmann constant. In SI units

σ = 5.670 � 10–8 W/m2 K4

Many attempts were made by the physicists duringthe last decade of the nineteenth century to derive theemperical results of the black body spectrum from theexisting basic principles. What was missing was anexpression for the intensity of the cavity radiation asa function of wavelength i.e. an equation thatcompletely fits the experimental results of the formplotted in Fig. 1.1.

Lord Rayleigh and James Jeans examined thisproblem by considering the radiation inside a cavityof absolute temperature T. The radiations wereconsidered to form a series of standing electromagneticwaves due to reflections from the walls of the cavity.The condition for formation of standing waves is thatthe path length should be an integral multiple of half

wavelengths i.e.2

nλ⎛ ⎞⎜ ⎟⎝ ⎠ so that nodes are formed at

each reflecting surface. The number of independent modes of vibration in the wavelength intervalλ and λ + dλ is

n(λ) dλ = 48πλ

dλ

According to classical physics, the theorem of equipartition of energy, the average energy per

degree of freedom is 12

kT. For the electromagnetic radiation in the cavity there are two degrees of

freedom associated with the wave and hence the classical energy per standing wave is ε = kTThe total energy density, therefore, is

I(λ) dλ = kT . n(λ)dλ

= 48πλ

kT dλ ... (1.3)

Equation (1.3) is called the Rayleigh-Jeans formula.Rayleigh-Jeans formula.Rayleigh-Jeans formula.Rayleigh-Jeans formula.Rayleigh-Jeans formula. It satisfies the experimental results wellat large wavelengths. However, the above result fails to satisfy the experimental curve at shorterwavelengths or in the ultra violet region. As λ → 0, the energy radiated should go to infinity. Inreality of course the energy density falls to 0 as λ → 0 (or ν → ∞). This discrepancy became knownas the ultraviolet catastropheultraviolet catastropheultraviolet catastropheultraviolet catastropheultraviolet catastrophe of classical physics.

T > T > T3 2 1

T3

T2

T1

Fig. 1.1.Fig. 1.1.Fig. 1.1.Fig. 1.1.Fig. 1.1. Spectral energy density for radiationfrom a black body at three different temperatures

6 Applied Physics - I

Wien also proposed an expression for the energy density. According to Wien’s lawWien’s lawWien’s lawWien’s lawWien’s law

I(λ) dλ = 58 hcπ

λ e–hc/λkT dλ ... (1.44444)

However, here I(λ) is still finite even at T → ∞ which is again in contradiction with the experimentalcurve for high frequency regions but not for low frequency regions.

Finally, in 1900 the German physicist Max Planck succeeded in developing a formula for theenergy density of black body radiation. The formula or function, now called the Planck’s radiationPlanck’s radiationPlanck’s radiationPlanck’s radiationPlanck’s radiationlaw,law,law,law,law, agreed very well with experimental intensity distribution curves.

For deriving the function, Planck made the assumption that the oscillators in the cavity walls cannot have a continuous distribution of all possible energies. Instead, they can radiate or absorb energyin units of magnitude hν. This unit is called the quantum of radiation and is named as photon photon photon photon photon.

i.e., oscillator energies εn = n hν ... (1.5)where n = 0, 1, 2, ...h now is called the Planck’s constantPlanck’s constantPlanck’s constantPlanck’s constantPlanck’s constant and has value

h = 6.626 � 10–34 JsThe Planck’s radiation lawPlanck’s radiation lawPlanck’s radiation lawPlanck’s radiation lawPlanck’s radiation law is

I(λ) dλ = 58 hcπ

λ ( )/ 1λ

λ

−hc kTd

e... (1.66666)

where, h is Planck’s constant, c is the speed of light, k is Boltzmann’s constant, T is the absolutetemperature and λ is the wavelength.

At low wavelengths (or high frequencies) ehc/λkT → ∞ and I(λ) → 0. Thus, the Planck’s law does notleads to ultraviolet catastrophe. At high wavelengths (or low frequencies ) we can approximate

ehc/λkT = 1 + ⎛ ⎞⎜ ⎟⎝ ⎠λ

hckT

+ 2⎛ ⎞

⎜ ⎟⎝ ⎠λhckT

+ ...

Taking only the first two terms, we have

ehc/λkT = 1 + λhckT

Thus, equation (1.6) at large wavelengths becomes

I(λ) dλ = 58 hcπ

λ

λ⎛ ⎞⎜ ⎟⎝ ⎠λ

dhckT

or I(λ) dλ = 48πλkT

dλ

which is the Rayleigh-Jeans lawRayleigh-Jeans lawRayleigh-Jeans lawRayleigh-Jeans lawRayleigh-Jeans law (see equation (1.3)).Planck’s radiation law also contains the Wien displacement law and the Stefan Boltzmann law.The Plancks hypothesis of emission and absorption of radiation in discrete units called photons,

was later on applied to other phenomenon which could not be explained by classical theory. Thesephenomenon, for example, are photoelectric effect, Compton effect etc. We will discus them briefly inthe next section.

Applied Physics - I

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