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.emaN e1.sruIncrease oC eht fothe ezifont s tnosize f ehof t esthe aerCourse cnI.1 Name..egaP revoC e2.ht nuse i rethe daefollowing h a sa gniwasolaloheader f eht esin u the.2 Cover Page. TISREVINUALAGAPPA APPAGALAUNIVERSITY cyC drihT eht ni )46.3:APGC( CA[Accredited AN yb edarGwith ’+A’’A+’ htiwGrade detidby ercNAAC cA[ (CGPA:3.64) in the Third Cycle ]CGU-DRHM yb ytisrevinU I–yrogeand taC Graded sa dedarasG Category–I dna University by MHRD-UGC] 300 036 – IDUKIARA KARAIKUDI K – 630 003 NOITACUDE ECNATSIDDIRECTORATE FO ETAROTCEOF RIDDISTANCE EDUCATION M.Sc. [Physics] 345 23 ELECTROMAGNETIC THEORY II - Semester  ALAGAPPA UNIVERSITY [Accredited with ‘A+’ Grade by NAAC (CGPA:3.64) in the Third Cycle and Graded as Category–I University by MHRD-UGC] (A State University Established by the Government of Tamil Nadu) KARAIKUDI – 630 003 Directorate of Distance Education M.Sc. [Physics] II - Semester 345 23 ELECTROMAGNETIC THEORY Authors Dr. G Naveen Babu, Associate Professor, Shiv Nadar University, Gautam Budh Nagar, Greater Noida Units (1, 2.2-2.3, 3.3, 4-5, 6.2, 8.3) Dr. Partha Pratim Das, Assistant Professor, School of Applied Sciences, Haldia Institute of Technology, Haldia, (W. B.) Abhisek Chakraborty, Scientist, Space Application Centre (ISRO), Ahmedabad (Gujarat) Units (2.4, 3.2, 6.3) Rohit Khurana, Founder and CEO, ITL Education Solutions Ltd., New Delhi Units (10, 11.3) Vikas® Publishing House: Units (2.0-2.1, 2.5-2.9, 3.0-3.1, 3.4-3.8, 6.0-6.1, 6.4-6.8, 7, 8.0-8.2, 8.4-8.8, 9, 11.0-11.2, 11.4-11.8, 12-14) "The copyright shall be vested with Alagappa University" All rights reserved. No part of this publication which is material protected by this copyright notice may be reproduced or transmitted or utilized or stored in any form or by any means now known or hereinafter invented, electronic, digital or mechanical, including photocopying, scanning, recording or by any information storage or retrieval system, without prior written permission from the Alagappa University, Karaikudi, Tamil Nadu. Information contained in this book has been published by VIKAS® Publishing House Pvt. Ltd. and has been obtained by its Authors from sources believed to be reliable and are correct to the best of their knowledge. However, the Alagappa University, Publisher and its Authors shall in no event be liable for any errors, omissions or damages arising out of use of this information and specifically disclaim any implied warranties or merchantability or fitness for any particular use. Vikas® is the registered trademark of Vikas® Publishing House Pvt. Ltd. VIKAS® PUBLISHING HOUSE PVT. LTD. Phone: 0120-4078900 Fax: 0120-4078999 Website: www.vikaspublishing.com Email: [email protected] Work Order No. AU/DDE/DE1-291/Preparation and Printing of Course Materials/2018 Dated 19.11.2018 Copies - 500  SYLLABI-BOOK MAPPING TABLE Electromagnetic Theory Syllabi Mapping in Book BOLCK I: ELECTRO AND MAGNETOSTATICS MAXWELL’S EQUATIONS AND PROPAGATION OF EM WAVES UNIT I: Electro and Magnetostatics Basics-Electrostatics and Magnetostatics. - Wave equation in terms of Unit 1: Electro and Magnetostatics scalar and vector potential - Transverse nature of electromagnetic wave. (Pages 1-42); Unit 2: Field Equations and UNIT II: Field Equations and Conservation Laws Conservation Laws Maxwell's equations - Poynting theorem - Conservation of energy and (Pages 43-66); momentum, Continuity equation. Unit 3: Electromagnetic Waves and UNIT III: Electromagnetic Waves and Wave Propagation Wave Propagation Propagation of plane electromagnetic waves in (a) Free space, (b) Isotropic (Pages 67-94) and Anisotropic non-conducting medium and (c) Conducting medium- skin depth. BOLCK II: REFLECTION AND REFRACTION OF ELECTRO- MAGNETIC WAVES UNIT IV: Reflection and Refraction of Electromagnetic Waves Unit 4: Reflection and Refraction of Electromagnetic Waves Boundary conditions at the surface of discontinuity - Reflection and (Pages 95-110); refraction of electromagnetic waves at the interface of non-conducting Unit 5: Fresnel’s Equation media. (Pages 111-124); UNIT V: Fresnel's Equation Unit 6: Polarization Fresnel’s equations - Reflection and transmission coefficients at the (Pages 125-150) interface between two dielectric media. UNIT VI: Polarization Brewster's law and degree of polarization -Total internal reflection. BOLCK III: DISPERSION AND SCATTERING OF EM WAVES UNIT VII: Dispersion of Electromagnetic Waves Unit 7: Dispersion of Electromagnetic Normal and Anomalous dispersion - Dispersion in Gases - Experimental Waves demonstration of anomalous dispersion in gases, solids and liquids. (Pages 151-164); UNIT VIII: Clausius Mossotti Equation Unit 8: Clausius-Mossotti Equation Clausius-Mossotti relation - Lorentz formula. (Pages 165-180); UNIT IX: Scattering of Electromagnetic Waves Unit 9: Scattering of Electromagnetic Scattering and scattering parameters - Theory of scattering of EM waves Waves - Polarization of scattered Light - Coherence and incoherence of scattered (Pages 181-198) light. BOLCK IV: MICROWAVES, DYNAMICS OF CHARGED PARTICLES AND PLASMA PHYSICS UNIT X: Wave Guides Wave guides: Rectangular and Cylindrical waveguides. UNIT XI: Microwaves Unit 10: Wave Guides Generation of microwaves - Klystron - Magnetron - Gunn diodes - (Pages 199-228); Resonant cavities. Unit 11: Microwaves UNIT XII: Dynamics of Charged Particles (Pages 229-252); Lienard-Wiechert potential-EM fields from retarded potentials of Unit 12: Dynamics of Charged moving point charge-EM. fields of uniformly moving point charge- Particles Radiation from moving charges. (Pages 253-266); UNIT XIII: Plasma Physics Unit 13: Plasma Physics Introduction - Conditions for plasma existence - Occurrence of plasma (Pages 267-277); - Charged particles in uniform constant electric field, in homogeneous Unit 14: Magnetohydrodynamics magnetic fields, simultaneous homogeneous electric and magnetic (Pages 278-294) fields, in nonhomogeneous magnetic fields. UNIT XIV: Magnetohydrodynamics Magnetohydrodynamics - Magnetic Confinement-Pinch effect- Instabilities- Plasma waves.  CONTENTS INTRODUCTION BLOCK I: ELECTRO AND MAGNETOSTATICS MAXWELL’S EQUATIONS AND PROPAGATION OF EM WAVES UNIT 1 ELECTRO AND MAGNETOSTATICS 1-42 1.0 Introduction 1.1 Objectives 1.2 Basics-Electrostatics 1.2.1 Electric Flux Density 1.2.2 Gauss’s Law and Applications 1.2.3 Electric Potential (V) 1.2.4 Maxwell's Second Equation 1.2.5 Relation between and V 1.2.6 Electric Dipole 1.3 Magnetostatic 1.3.1 Ohm’s Law 1.3.2 Boundary Conditions of Current Density 1.3.3 Equation of Continuity and Kirchhoff’s Law 1.3.4 Postulates of Magnetostatics: Biot–Savart’s Law 1.3.5 Magnetic Potential 1.3.6 Forces Due to Magnetostatics 1.3.7 Ampere’s Circuit Law 1.4 Wave Equation in Terms of Scalar and Vector Potential 1.5 Answers to Check Your Progress Questions 1.6 Summary 1.7 Key Words 1.8 Self Assessment Questions and Exercises 1.9 Further Readings UNIT 2 FIELD EQUATIONS AND CONSERVATION LAWS 43-66 2.0 Introduction 2.1 Objectives 2.2 Maxwell’s Equations 2.3 Poynting Theorem 2.4 Conservation of Energy and Momentum, Continuity Equation 2.5 Answers to Check Your Progress Questions 2.6 Summary 2.7 Key Words 2.8 Self Assessment Questions and Exercises 2.9 Further Readings UNIT 3 ELECTROMAGNETIC WAVES AND WAVE PROPAGATION 67-94 3.0 Introduction 3.1 Objectives  3.2 Motion in Electromagnetic Waves 3.2.1 Propagation of a Wave 3.2.2 Progressive Wave and its Differential Form 3.2.3 Difference between Elastic (Mechanical) and Electromagnetic Waves 3.2.4 Standing Waves 3.3 Propagation of Plane Electromagnetic Waves 3.3.1 Wave Propagation in a Lossy Dielectric 3.3.2 Intrinsic Impedance 3.3.3 Wave Equation for Conducting Medium 3.3.4 Depth of Penetration or Skin Depth ( ) 3.4 Answers to Check Your Progress Questions 3.5 Summary 3.6 Key Words 3.7 Self Assessment Questions and Exercises 3.8 Further Readings BLOCK II: REFLECTION AND REFRACTION OF ELECTRO-MAGNETIC WAVES UNIT 4 REFLECTION AND REFRACTION OF ELECTROMAGNETIC WAVES 95-110 4.0 Introduction 4.1 Objectives 4.2 Boundary Conditions at the Surface of Discontinuity 4.3 Reflection and Refraction of Electromagnetic Waves at the Interface of Non-Conducting Media 4.4 Answers to Check Your Progress Questions 4.5 Summary 4.6 Key Words 4.7 Self Assessment Questions and Exercises 4.8 Further Readings UNIT 5 FRESNEL’S EQUATION 111-124 5.0 Introduction 5.1 Objectives 5.2 Reflection and Transmission Coefficients at the Interface between Two Dielectric Media 5.3 Fresnel’s Equations 5.4 Answers to Check Your Progress Questions 5.5 Summary 5.6 Key Words 5.7 Self Assessment Questions and Exercises 5.8 Further Readings UNIT 6 POLARIZATION 125-150 6.0 Introduction 6.1 Objectives 6.2 Polarization 6.2.1 Wave Polarization 6.2.2 Perpendicular Polarization  6.3 Brewster’s Law and Total Internal Reflection 6.3.1 Polarisation through Reflection and Brewster’s Law 6.3.2 Degree of Polarization 6.4 Answers to Check Your Progress Questions 6.5 Summary 6.6 Key Words 6.7 Self Assessment Questions and Exercises 6.8 Further Readings BLOCK III: DISPERSION AND SCATTERING OF EM WAVES UNIT 7 DISPERSION OF ELECTROMAGNETIC WAVES 151-164 7.0 Introduction 7.1 Objectives 7.2 Dispersion of Electromagnetic Waves 7.2.1 Normal Dispersion 7.2.2 Anomalous Dispersion 7.2.3 Dispersion in Gases 7.2.4 Experimental Demonstration of Anomalous Dispersion in Gases, Solids and Liquids 7.3 Answers to Check Your Progress Questions 7.4 Summary 7.5 Key Words 7.6 Self Assessment Questions and Exercises 7.7 Further Readings UNIT 8 CLAUSIUS-MOSSOTTI EQUATION 165-180 8.0 Introduction 8.1 Objectives 8.2 Clausius–Mossotti Relation 8.3 Lorentz Formula 8.4 Answers to Check Your Progress Questions 8.5 Summary 8.6 Key Words 8.7 Self Assessment Questions and Exercises 8.8 Further Readings UNIT 9 SCATTERING OF ELECTROMAGNETIC WAVES 181-198 9.0 Introduction 9.1 Objectives 9.2 Theory of Scattering of Electromagnetic or EM Waves 9.2.1 Scattering Parameters 9.2.2 Polarization of Scattered Light 9.3 Answers to Check Your Progress Questions 9.4 Summary 9.5 Key Words 9.6 Self Assessment Questions and Exercises 9.7 Further Readings BLOCK IV: MICROWAVES, DYNAMICS OF CHARGED PARTICLES AND PLASMA PHYSICS UNIT 10 WAVE GUIDES 199-228 10.0 Introduction 10.1 Objectives 10.2 Wave Guides 10.2.1 Rectangular Waveguides 10.2.2 Cylndrical or Circular Waveguides 10.3 Answers to Check Your Progress Questions 10.4 Summary 10.5 Key Words 10.6 Self Assessment Questions and Exercises 10.7 Further Readings UNIT 11 MICROWAVES 229-252 11.0 Introduction 11.1 Objectives 11.2 Generation of Microwaves - Klystron, Magnetron, Gunn Diodes 11.2.1 Generation of Microwaves Signals 11.3 Resonant Cavities 11.4 Answers to Check Your Progress Questions 11.5 Summary 11.6 Key Words 11.7 Self Assessment Questions and Exercises 11.8 Further Readings UNIT 12 DYNAMICS OF CHARGED PARTICLES 253-266 12.0 Introduction 12.1 Objectives 12.2 Charged Particles Dynamics 12.2.1 Lienard-Wiechert Potential 12.2.2 EM Fields from Retarded Potentials of Moving Point Charge 12.2.3 EM Fields of Uniformly Moving Point Charge 12.2.4 Radiation from Moving Charges 12.3 Answers to Check Your Progress Questions 12.4 Summary 12.5 Key Words 12.6 Self Assessment Questions and Exercises 12.7 Further Readings UNIT 13 PLASMA PHYSICS 267-277 13.0 Introduction 13.1 Objectives 13.2 Plasma Physics: Basics 13.2.1 Conditions for Plasma Existence 13.3 Answers to Check Your Progress Questions 13.4 Summary 13.5 Key Words  13.6 Self Assessment Questions and Exercises 13.7 Further Readings UNIT 14 MAGNETOHYDRODYNAMICS 278-294 14.0 Introduction 14.1 Objectives 14.2 Magnetohydrodynamics 14.2.1 Magneto-Convection 14.2.2 Pinch Effect 14.2.3 Instabilities and Plasma Waves 14.3 Answers to Check Your Progress Questions 14.4 Summary 14.5 Key Words 14.6 Self Assessment Questions and Exercises 14.7 Further Readings Introduction INTRODUCTION Electromagnetism is a branch of physics involving the study of the electromagnetic NOTES force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force is carried by electromagnetic fields composed of electric fields and magnetic fields, is responsible for electromagnetic radiation, such as light, and is one of the four fundamental interactions, commonly termed as the forces in nature. The other three fundamental interactions are the strong interaction, the weak interaction, and gravitation. At high energy the weak force and electromagnetic force are unified as a single electroweak force. Electromagnetic phenomena are defined in terms of the electromagnetic force, sometimes called the Lorentz force, which includes both electricity and magnetism as different manifestations of the same phenomenon. The electromagnetic force plays a major role in determining the internal properties of most objects encountered in daily life and are responsible for the chemical bonds between atoms which create molecules, and intermolecular forces. There are numerous mathematical descriptions of the electromagnetic field. In classical electrodynamics, electric fields are described as electric potential and electric current. In Faraday’s law, magnetic fields are associated with electromagnetic induction and magnetism, and Maxwell’s equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. This book, Electromagnetic Theory, is divided into four blocks, which are further subdivided into fourteen units. The concepts discussed include electrostatics and magnetostatics, wave equations in terms of scalar and vector potential, Maxwell’s equations, Poynting, continuity equation, electromagnetic waves and wave propagation, conducting medium, skin depth, reflection and refraction of electromagnetic waves, Fresnel’s equations, reflection and transmission coefficients at the interface between two dielectric media, Brewster’s law and degree of polarization, dispersion of electromagnetic waves, normal and anomalous dispersion, dispersion in gases, Clausius-Mossotti relation, Lorentz formula, scattering of electromagnetic waves, scattering and scattering parameters, scattering of EM waves, polarization of scattered light, wave guides - rectangular and cylindrical waveguides, generation of microwaves (klystron, magnetron and Gunn diodes), dynamics of charged particles, Lienard-Wiechert potential, plasma physics, magnetohydrodynamics and pinch effect. The book follows the self-instructional mode wherein each unit begins with an ‘Introduction’ to the topic. The ‘Objectives’ are then outlined before going on to the presentation of the detailed content in a simple and structured format. ‘Check Your Progress’ questions are provided at regular intervals to test the student’s understanding of the subject. ‘Answers to Check Your Progress Questions’, a ‘Summary’, a list of ‘Key Words’, and a set of ‘Self-Assessment Questions and Self-Instructional Exercises’ are provided at the end of each unit for effective recapitulation. 10 Material  Electro and BLOCK - I Magnetostatics ELECTRO AND MAGNETOSTATICS MAXWELL’S EQUATIONS AND PROPAGATION OF EM WAVES NOTES UNIT 1 ELECTRO AND MAGNETOSTATICS Structure 1.0 Introduction 1.1 Objectives 1.2 Basics-Electrostatics 1.2.1 Electric Flux Density 1.2.2 Gauss’s Law and Applications 1.2.3 Electric Potential (V) 1.2.4 Maxwell's Second Equation 1.2.5 Relation between 𝐸⃗ and V 1.2.6 Electric Dipole 1.3 Magnetostatic 1.3.1 Ohm’s Law 1.3.2 Boundary Conditions of Current Density 1.3.3 Equation of Continuity and Kirchhoff’s Law 1.3.4 Postulates of Magnetostatics: Biot – Savart’s Law 1.3.5 Magnetic Potential 1.3.6 Forces Due to Magnetostatics 1.3.7 Ampere’s Circuit Law 1.4 Wave Equation in Terms of Scalar and Vector Potential 1.5 Answers to Check Your Progress Questions 1.6 Summary 1.7 Key Words 1.8 Self Assessment Questions and Exercises 1.9 Further Readings 1.0 INTRODUCTION Electrostatics is a branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges. It has been observed that some material attract particles after rubbing. There are many examples of electrostatic phenomenon, such as attraction of plastic wrap to your hand after you remove it from a package, attraction of pieces of paper on a charged scale, etc. Electromagnetic theory exists due to a coupled presence of electric and magnetic field. The parameters of electric field include the potential or voltage and electric field intensity using Coulomb’s law, electric field due to a charge distribution. Self-Instructional Material 1 Electro and Magnetostatics is the study of magnetic fields in systems where the currents Magnetostatics are steady, i.e., not changing with time. It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur NOTES on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static — as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics, such as models of magnetic storage devices as in computer memory. Magnetostatics focusing can be achieved either by a permanent magnet or by passing current through a coil of wire whose axis coincides with the beam axis. In this unit, you will learn about the Coulomb’s law, Gauss law and static electricity principles that forms the basis of electromagnetic theory. Further, you will study about magnetic field and its components. 1.1 OBJECTIVES After going through this unit, you will be able to: Discuss the different postulates on electric field that includes Coulomb’s law, Gauss’s law and its applications Explain the various electric field components, such as electric potential, electric flux density, etc. Discuss the concept of magnetic field and its components, such as magnetic flux, magnetic flux density, magnetic vector potential, etc. Define Ohms law, Biot-Savart’s law and Ampere circuit law 1.2 BASICS-ELECTROSTATICS The term static means a situation where the field does not vary with time. Static electric field also referred as electrostatics is created by the fixed charges in space. There are many examples of electrostatic phenomena such as the attraction of the plastic wrap to your hand after you remove it from a package, and the attraction of paper to a charged scale, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturing, and photocopier and laser printer operation. Electrostatics involves the build-up of charge on the surface of objects due to contact with other surfaces. Although charge exchange happens whenever any two surfaces contact and separate, the effects of charge exchange are usually only noticed when at least one of the surfaces has a high resistance to electrical flow. This is because the charges that transfer are trapped there for a time long enough for their effects to be observed. These charges remains on the object until they either bleed off to ground or are quickly neutralized by a discharge: for example, the familiar phenomenon of a static ‘Shock’ Self-Instructional 2 Material is caused by the neutralization of charge built up in the body from contact with Electro and Magnetostatics insulated surfaces. Determination of the electrostatic field components, such as electric field, electric force, and electric flux density are explained by two important laws namely, NOTES Coulomb’s law and Gauss law. Coulomb’s Law Coulomb’s law provides the relation between forces experienced by the charges when they are separated by a distance. This theory was first proposed by Coulomb in 1785. This law states that, Force, F exerted between two point charges and as shown in Figure (1.1) is Directly proportional to the product of the two charges and Inversely proportional to the square of the distance between the two charges. The direction of the force will be in the same direction along the line joining the two charges. Mathematically, Coulomb’s law may be expressed as, Fig. 2.1 Coulomb’s Force Removing the proportionality, Where = Unit vector in the line of direction of force, F = Charges = Distance seprating the charges ( Where, = Permittivity in free space = Self-Instructional Material 3 Electro and Magnetostatics Now, assume two charges and at a distance of and , respectively,, from an observing point as shown in Figure (1.2). The force exerted by charge on is given by, NOTES Where 𝑅12⃗ 𝑎𝑅12⃗ = 𝑅12⃗ And 𝑅12⃗ = 𝑟2 − 𝑟1 Therefore, 𝑟2 − 𝑟1 𝑎𝑅12⃗ = ; |𝑟2 − 𝑟1 | 𝑄1 𝑄2 𝑅12⃗ 𝐹12⃗ = 2 3 4𝜋𝜖0 𝑅12 𝑅12 𝑄1 𝑄2 (𝑥2 − 𝑥1 )𝑎𝑥⃗ + (𝑦2 − 𝑦1 )𝑎𝑦⃗ Force, 𝐹⃗ = 3 4𝜋𝜖0 (𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 Fig. 1.2 Coulomb's Force on Charges at a Distance Similarly, for force exerted by charge Q2 on Q1 is given by, 𝐹21⃗ = −𝐹12⃗ ∵ 𝑎𝑅12⃗ = −𝑎𝑅21⃗ For Many Charges: Generalising, the above expression when many charges are present, 𝑁 𝑄 𝑟⃗ − 𝑟⃗𝑖 𝐹⃗ = 𝑄𝑖 3 4𝜋𝜖0 |𝑟⃗ − 𝑟⃗| 𝑖 𝑖=1 Self-Instructional 4 Material  Electro and Electric Field Intensity Magnetostatics Electric field intensity is defined as the strength of electric field at any point. It is equal to force per unit charge as experienced by test charge kept at that point. Therefore, it is expressed as, NOTES Also, Assuming that In general, 𝑁 1 𝑟⃗ − 𝑟⃗𝑖 𝐸⃗ = 𝑄𝑖 3 4𝜋𝜖0 |𝑟⃗ − 𝑟⃗| 𝑖 𝑖=1 Charge Distribution The presence of charge Q ensures the existence of electric field 𝐸⃗. The charges may be distributed on a line conductor, on a surface or inside a volume. Hence, based on the charge distribution, Along a line, charge, 𝑄 = 𝜌𝐿. 𝑑𝑙⃗ 𝐿 On a surface, charge, 𝑄 = 𝜌𝑆. 𝑑𝑠⃗ 𝑆 Inside a volume, charge, 𝑄 = 𝜌𝑉. 𝑑𝑣 𝑉 Where, 𝜌𝐿 = line charge density (𝐶/𝑚) 𝜌𝑆 = surface charge density (𝐶/𝑚2 ) 𝜌𝑉 = volume charge density (𝐶/𝑚3 ) Based on the above distribution of the charges on line, surface and volume, the electric field intensity, 𝐸⃗ can be given as, Self-Instructional Material 5 Electro and Magnetostatics 1 𝐸⃗ = 4𝜋𝜖 𝑅 2 ∫𝐿 𝜌𝐿. 𝑑𝑙⃗. 𝑎𝑅⃗ 0 1 𝐸⃗ = ∫𝑆 𝜌𝑆. 𝑑𝑠⃗. 𝑎𝑅⃗ NOTES 4𝜋𝜖 0 𝑅 2 1 𝐸⃗ = 4𝜋𝜖 𝑅 2 ∫𝑉 𝜌𝑉. 𝑑𝑣. 𝑎𝑅⃗ 0 Electric Field Intensity Due to a Line Charge In this section, let us derive the electric field intensity, 𝐸⃗ due to a line charge. Cartesian coordinate system is considered for the analysis. Consider a uniformly charged line of length 'L' with line charge density, 𝜌𝐿 (𝐶/𝑚). An incremental elemental length 'dl' is considred for the analysis from an observing point 'P' at a distance 'r'. The arrangement is depicted in Figure (1.3). The electric field along the line is given as, 1 𝐸⃗ = 𝜌𝐿. 𝑑𝑙⃗. 𝑎𝑟⃗ 4𝜋𝜖0 𝑟 2 𝐿 Fig. 1.3 Charge Distribution Due to Line Charge Electric field intensity due to a small elemental length 'dl' is given as, 1 𝑑𝐸⃗ = 𝜌. 𝑑𝑙⃗. 𝑎𝑟⃗ 4𝜋𝜖0 𝑟 2 𝐿 The electric field at point 'P'will be at an angle with respect to the normal axis. Hence, can be resolved in to x-component and y-component. Therefore, dEx = dE sin And, dEy = dE cos Self-Instructional 6 Material  Substituting the magnitude of dE in the above expressions, Electro and Magnetostatics 𝜌𝐿. 𝑑𝑙 𝑑𝐸𝑥 = sin 𝜃...(1.1) 4𝜋𝜖0 𝑟 2 NOTES To obtain Ex, then the above expression needs to be integrated over length 'L' and hence 'r' must be determined. From Figure (1.3), sin = h/r? ℎ Therefore, 𝑟= = ℎ 𝑐𝑜𝑠𝑒𝑐 𝜃 sin 𝜃 Also, from Figure (1.3), ℎ tan 𝜃 = 𝑥−𝑙 ℎ 𝑥−𝑙 = tan 𝜃 𝑥 − 𝑙 = ℎ cot 𝜃 −𝑑𝑙 = −ℎ 𝑐𝑜𝑠𝑒𝑐 2 𝜃 𝑑𝜃 Substituting dl and r in the Equation (1.1), −𝜌𝐿. sin 𝜃 𝑑𝐸𝑥 = (ℎ 𝑐𝑜𝑠𝑒𝑐 2 𝜃 𝑑𝜃) 4𝜋𝜖0 (ℎ2 𝑐𝑜𝑠𝑒𝑐 2 𝜃) −𝜌𝐿. sin 𝜃 𝑑𝐸𝑥 = 𝑑𝜃 4𝜋𝜖0 ℎ Integrating from 1 to – 2 for the entire length of the wire, 𝜋− 𝛼 2 −𝜌𝐿. sin 𝜃 𝐸𝑥 = 𝑑𝜃 𝛼1 4𝜋𝜖0 ℎ −𝜌𝐿 = [− cos 𝜃]𝜋− 𝛼1 𝛼2 4𝜋𝜖0 ℎ 𝜌𝐿 𝐸𝑥 = [cos 𝛼1 + cos 𝛼2 ] 4𝜋𝜖0 ℎ Self-Instructional Material 7 Electro and Similarly from dEy, Magnetostatics 𝜌𝐿. cos 𝜃 𝑑𝐸𝑦 = 𝑑𝜃 NOTES 4𝜋𝜖0 ℎ 𝜋− 𝛼 2 𝜌𝐿. cos 𝜃 ∴ 𝐸𝑦 = 𝑑𝜃 𝛼1 4𝜋𝜖0 ℎ 𝜋− 𝛼 2 𝜌𝐿 𝐸𝑦 = cos 𝜃 𝑑𝜃 4𝜋𝜖0 ℎ 𝛼1 𝜌𝐿 = [sin 𝜃]𝜋− 𝛼1 𝛼2 4𝜋𝜖0 ℎ 𝜌𝐿 𝐸𝑦 = [sin 𝛼2 − sin 𝛼1 ] 4𝜋𝜖0 ℎ There are two conditions associated with. They are, 𝜌𝐿 Case (i) If = 0, then Ey = 0 and 𝐸𝑥 = 𝐸 = 2𝜋𝜖0 ℎ 𝜌𝐿 Case (ii) If = = 1 2 , then Ey = 0 and 𝐸𝑥 = 𝐸 = cos 𝛼 2𝜋𝜖0 ℎ Electric Field Intensity Due to a Ring of Charge Consider a ring as shown in Figure (1.4), filled with charge Q. The x-axis is perpendicular to the ring and is at the center of the ring. The objective is to find the electric field at P due to the ring of radius 'R'. Fig. 1.4 Ring of Charge Self-Instructional 8 Material  Electro and Consider a small elemental charge, dQ on the ring. The electric field 𝑑𝐸⃗ at Magnetostatics point P is given as, 𝑑𝑄 𝑑𝑄 𝑑𝐸 = 𝑘 =𝑘 2 ℎ 2 (𝑅 + 𝑎2 ) NOTES The x-component of dE is dEx and is given as, 𝑑𝑄 𝑑𝐸𝑥 = 𝑑𝐸 cos 𝜃 = 𝑘 cos 𝜃 (𝑅 2+ 𝑎2 ) But, from Figure (1.4), 𝑎 cos 𝜃 = √𝑅 + 𝑎 2 2 𝑑𝑄 𝑎 𝑑𝐸𝑥 = 𝑘 1 (𝑅 2 2 + 𝑎 ) (𝑅 2 + 𝑎 2 )2 𝑑𝑄 (𝑎) 𝑑𝐸𝑥 = 𝑘 (𝑅 2 + 𝑎2 )3/2 Referring to the Figure (1.4), neither k,R or a changes. Hence, 𝑑𝑄 (𝑎) 𝐸𝑥 = ∫ 𝑑𝐸𝑥 = 𝑘 3 (𝑅 2 + 𝑎2 )2 (𝑎) 𝐸𝑥 = 𝑘 3 𝑑𝑄 (𝑅 2 + 𝑎2 )2 𝑄 (𝑎) 𝐸𝑥 = 𝑘 (𝑅 2 + 𝑎2 )3/2 When R 0, ring represents a point charge, therefore, 𝑘𝑄(𝑎) 𝑘𝑄 𝐸≈ ≈ 𝑎2 𝑎 1 Where 𝑘= 4𝜋𝜖0 Electric Field Intensity Due to a Circularly Charged Disc Unlike the previous structure of a ring, consider a disc of radius, R. The disc consists of a uniformly charged surface charge density of s C/m2. Consider an elemental ring of radius dr at a distance 'r' from the center. The electric field at a point P is given as, Self-Instructional Material 9 Electro and Magnetostatics 𝜌𝑆. 𝑑𝑠 𝑑𝐸 = 𝑘 ℎ2 The horizontal and vertical components of dE are dEx and dEy. The horizontal NOTES component dEx is zero and the vertical component is given as, dEy = dE cos 𝜌𝑆. 𝑑𝑠 𝑑𝐸𝑦 = 𝑘 cos 𝜃 ℎ2 We know that for the differential surface element ds, ds = 2 r dr 𝜌𝑆. (2𝜋𝑟 𝑑𝑟) 𝑑𝐸𝑦 = 𝑘 cos 𝜃 ℎ2 Fig. 1.5 Electric Field Due to a Circularly Charged Disc From Figure (1.5), 𝑟 tan 𝜃 = 𝑎 𝑟 = 𝑎 tan 𝜃 𝑑𝑟 = 𝑎 sec 2 𝜃 𝑑𝜃 𝑟 ℎ= sin 𝜃 Therefore, 1 𝜌𝑆. (2𝜋𝑟) (𝑎 sec 2 𝜃 𝑑𝜃) 𝑑𝐸𝑦 = cos 𝜃 4𝜋𝜖0 𝑟 2 sin 𝜃 𝜌𝑆 (2𝜋𝑟) (𝑎 𝑠𝑒𝑐 𝜃 sin2 𝜃 𝑑𝜃) 𝑑𝐸𝑦 = [∵ tan 𝜃 = 𝑟/𝑎] 2𝜖0 tan 𝜃 Self-Instructional 10 Material  𝜌𝑆 Electro and 𝑑𝐸𝑦 = sin 𝜃 𝑑𝜃 [tan 𝜃 = sec 𝜃 sin 𝜃] Magnetostatics 2𝜖0 Total electric field is given as, 𝛼 𝛼 NOTES 𝜌𝑆 𝐸= 𝑑𝐸𝑦 = sin 𝜃 𝑑𝜃 𝜃=0 2𝜖0 𝜃=0 𝜌𝑆 𝐸= (1 − cos 𝛼) 2𝜖 𝜌𝑆 𝑎 𝐸= 1− 2𝜖 √𝑎 + 𝑅 2 2 1.2.1 Electric Flux Density Electric flux density is an imaginary field lines that do not exist unlike magnetic field lines. Electric flux density do not exist practically and generally considered for theoretical reasoning only. Electric flux density is related to electric field by the following reason, 𝐷⃗ = 𝜖0 𝐸⃗ Electric flux density 𝐷⃗ is independent of the medium and may also be defined in terms of electric flux 𝜓 as, 𝜓 = ∫ 𝐷⃗. 𝑑𝑠⃗ All the electric field expressions derived earlier can be substituted in the electric flux density expressions. Therefore, electric flux density due to a long conductor of charges is given as, 𝐷⃗ = 𝜖0 𝐸⃗ Electric flux density due to a ring of charges is given by, 𝑎𝑄 𝐷⃗ = 4𝜋(𝑅 2 + 𝑎2 )3/2 Electric flux density due to a circularly charged disc is given by, 𝜌𝑆 𝑎 𝐷⃗ = 1− 2 √𝑎 + 𝑅 2 2 1.2.2 Gauss's Law and Applications Gauss' law is a powerful tool for the calculation of electric fields. The applications of Gauss law includes determination of electric field due to a point charge, sheet of charge, line charge on surface of conductor and sphere of charges. Gauss law states that total flux through a closed surface is equal to the charge enclosed by that surface. Mathematically, it is given as, Electric flux, 𝜓 = 𝑄 (Charge enclosed). Self-Instructional Material 11 Electro and Maxwell's Equation - I Magnetostatics From Gauss law, we know that, = Q. Also, from the basic definition for electric flux and charge Q on a volume, NOTES 𝜓= 𝑑𝜓 = 𝐷⃗. 𝑑𝑠⃗...(1.2) and 𝑄= 𝜌𝑣. 𝑑𝑣 𝑣...(1.3) Therefore, equating Equation (1.2) and Equation (1.3), 𝐷⃗. 𝑑𝑠⃗ = 𝜌𝑣. 𝑑𝑣 𝑆 𝑣...(1.4) Applying divergence theorem on the LHS of the above expression in Equation (1.4), ∇⃗. 𝐷⃗ 𝑑𝑣 = 𝜌𝑣. 𝑑𝑣 𝑣 𝑣 Therefore, ∇⃗. 𝐷⃗ = 𝜌𝑣...(1.5) Relating the units of the above expression in Equation (1.5), 𝜌𝑣 (𝐶/𝑚3 ) = ∇⃗. 𝐷⃗(𝐶/𝑚2 ) = ∇⃗. 𝐷⃗ (𝐶/𝑚3 ) Equation (1.5) is called as Maxwell's first equation expressed in differential form and Equations (1.4) is called Maxwell's first equation expressed in integral form. Gaussian Surfaces - Gauss's Law Application A mathematically closed surface is called as a Gaussian surface. These surfaces are assumed to have a uniform symmetric charge distribution which are ideal for determining the electric field vector, 𝐸⃗ by applying Gauss law. Also, the electric flux density vector, 𝐷⃗ is assumed to act tangentially or normally on the Gaussian surface. Therefore, accordingly, when 𝐷⃗ is normal, then 𝐷⃗. 𝑑𝑆⃗ = 𝐷𝑑𝑆 and when 𝐷⃗ is acting tangential, 𝐷⃗. 𝑑𝑆⃗ = 0 Self-Instructional 12 Material  Electro and (a) Determining 𝐷⃗ Due to a Point Charge Magnetostatics Consider a point charge, Q located at point P as shown in Figure (1.6). NOTES Fig. 1.6 Electric Flux Density 𝐷⃗ Due to a Point Charge According to Gauss law, =Q And 𝑄= 𝐷⃗. 𝑑𝑆⃗ Assuming that 𝐷⃗ is normal to the Gaussian surface, 𝑄= 𝐷. 𝑑𝑆 = 𝐷 𝑑𝑆 2𝜋 𝜋 𝑄=𝐷 𝑟 2 sin 𝜃 𝑑𝜃 𝑑𝜙 𝜙=0 𝜃=0 𝑄 = 𝐷(4𝜋𝑟 2 ) 𝑄 𝐷= 4𝜋𝑟 2 𝑄 𝐷⃗ = 𝑎⃗ 4𝜋𝑟 2 𝑛 (b) Determining 𝐷⃗ Due to Infinite Line Charge The infinite line conductor is a cylindrical surface and hence, 𝐷⃗ needs to be operated in cylindrical coordinate system, and hence assuming 𝐷⃗ to be normal to the Gaussian surface as shown in Figure (1.7), 𝐷⃗ = 𝐷𝜌 𝑎𝜌⃗ Also, we know that, 𝑄 = 𝜌𝐿. 𝑑𝑙 Self-Instructional Material 13 Electro and Since the length of the conductor is assumed to be infinite with length 'l', Magnetostatics 𝑄 = 𝜌𝐿. 𝑙 = 𝐷⃗. 𝑑𝑆⃗ = 𝐷𝜌. 𝑎𝜌⃗ 𝑑𝑆⃗ NOTES 𝑄 = 𝜌𝐿. 𝑙 = 𝐷𝜌 (2𝜋𝜌). 𝑙 𝑎𝜌⃗ ∵ 𝑑𝑆⃗ = 2𝜋𝜌𝑙 Therefore, 𝜌𝐿 𝐷⃗ = 𝑎⃗ 2𝜋𝜌 𝜌 Or 𝜌𝐿 𝐷= 2𝜋𝜌 Fig. 1.7 Electric Flux Density 𝐷⃗ Due to Infinite Line Charge (c) Determining 𝐷⃗ due to Charged Sphere Fig. 1.8 Electric Flux Density 𝐷⃗ Due to Charged Sphere Consider a sphere of radius, r. Electric flux density, 𝐷⃗ may either be inside the sphere (R < r) or outside the sphere (R > r). Hence accordingly, we have two cases to analysis as follows: Self-Instructional 14 Material Case (i) When R < a Electro and Magnetostatics We know that, =Q RHS: NOTES 𝑅 𝜋 2𝜋 𝑄= 𝜌𝑉. 𝑑𝑉 = 𝜌𝑉 𝑑𝑉 = 𝜌𝑉 𝜌2 sin 𝜃 𝑑𝜌 𝑑𝜃 𝑑𝜙 𝑉 𝑉 𝜌=0 𝜃=0 𝜙 =0 4 3 𝑄 = 𝜌𝑉 𝜋𝑅...(1.6) 3 LHS: 𝜋 2𝜋 𝜓= 𝐷⃗. 𝑑𝑆⃗ = 𝐷𝜌 𝑑𝑆 = 𝐷𝜌 𝜌2 sin 𝜃 𝑑𝜃 𝑑𝜙 𝑆 𝑆 𝜃=0 𝜙 =0 𝜓 = 𝐷𝜌 [4𝜋𝑅 2 ]...(1.7) Equating Equation (1.6) and Equation (1.7), 4 𝜌𝑉 3 𝜋𝑅 3 𝐷𝜌 = 4𝜋𝑅 2 𝜌𝑉 (𝑅) 𝐷𝜌 = 3 𝑅 𝐷⃗ = 𝜌𝑉. 𝑎𝜌⃗ 3 Case (ii) When R > a RHS: 4 3 𝑄= 𝜌𝑉. 𝑑𝑉 = = 𝜌𝑉 𝜋𝑅 𝑉 3 LHS: 𝜓 = 𝐷𝜌 [4𝜋𝑅 2 ] 𝑟3 𝑟3 𝐷𝜌 = 𝜌 𝑜𝑟 𝐷⃗ = 𝜌 𝑎⃗ 3𝑅 2 𝑉 3𝑅 2 𝑉 𝜌 𝑅 𝜌 𝑎⃗ 0