Physics Galaxy Magnetism, EMI & AC PDF

Study Physics Galaxy with www.puucho.com (b GKP Volu111e 111B Magnetism, EMI 8 Alternating Current 'Physics Galaxy' iOS and Android Application , !......

Study Physics Galaxy with www.puucho.com (b GKP Volu111e 111B Magnetism, EMI 8 Alternating Current 'Physics Galaxy' iOS and Android Application , !... 'Physics Galaxy' Mobile Application is a quick way to access World's Largest Video Encyclopedia of Online Video AppStore Lectures by Ashish Arora Sir on High School Physics to prepare for JEE (Mains), NEET and BITSAT through 3500+ video lectures. For JEE (Advance) and AIIMS, 700+ additional advance concept video illustrations will help students to excel in ,."- t ·'"',:I, ' their concept applications. Download the application on your device to access these Video Lectures of Physics Galaxy for FREE. www.puucho.com  Study Physics Galaxy with www.puucho.com Q r.t. '- -4 PG Educate by Phys,csGala v com ftee Access to QUEGRID Rank Boosters Click Quegrid on contents tab on www.physicsgalaxy.com Students of IX to XI I for f/arious competitifle exams includin9 JEE-MAIN I JEE-ADVANCE BITSAT I NEET I A/IMS I NTSE & KVPY ,\'ou: ai'aifabfe on u:eb and mobile apps U,'orld's far9est poof of FREE Onfine Video Lectures on Hi9h Schoof and Competiticle PhtJsics _II_ o~ a o~ PG website : www.physicsgalaxy.com T ® Ph9sics Galax9 is a cross platform app to watch FREE Online PG Video Lectures. LIVE STREAM Watch 7000+ Video Lectures by Ashish Arora Sir on Physics Galaxy youtube Channel or on website www.physicsgalaxy.com and you can also attempt 8000+ online MCOs and access to lots of academic downloads fromthe website : : ~~ -,, Physics Galaxy on YouTube PG Courses Largest Channel on high school physics - Onli ne and Offline PG Classes with Synchronised Practice Exercises 10000+ Online MCQs PG Streaming Classes Live Streaming on PG LIVE app D Avai lable for on line atte mpt on video lecture dashboard or direct ly on www.quegrid.com For any further Information you can mail at [email protected] www.puucho.com Study Physics Galaxy with www.puucho.com J ~· Physics Galaxv Volume 111B Magnetism, EM_I & _Alternating Current Ashish Arora Mentor & Founder PHYSICSGALAXY.COM Worlds largest mcyclopedia ofonlinc video lectures 011 Hi§I School Phy,ics {I) G K Publications (P) Ltd www.puucho.com  Study Physics Galaxy with www.puucho.com CL MEDIA (P) LTD. First Edition 2000 Edition 2019-20 ©AUTHOR Administrative and Production Offices No part of this book may be reproduced in a retrieval syst.em Published by : CL Media (P) Ltd. or transmitted, in any form or by any means, electronics, A-41, Lower Ground Floor, mechanical, photocopying, recording, scanning and or Espire Building, without the written permission of the author/publisher. Mohan Cooperative Industrial Area, Main Mathura Road, New Delhi - 110044 Marketed by : G.K. Publications (P) Ltd. A-41, Lawer Ground Floor, Espire Building, Mohan Cooperative Industrial Area, · Main Mathura Road, Typeset by : CL Media DTP Unit New Delhi - 110044 For product information : Visit www.gkpublications.com or email to [email protected] www.puucho.com Study Physics Galaxy with www.puucho.com PlJedt,'oated to, G4ty fJ?la1l!physicsgalaxy.com also has teaching sessions dubbed in American accent and subtitles in 87 languages. www.puucho.com  Study Physics Galaxy with www.puucho.com FOREWORD It has been pleasure for me to follow the progress Er. Ashish Arora has made in teaching and professional career. In the last about two decades he has actively contributed in developing several new techniques for teaching & learning of Physics and driven important contribution to Science domain through nurturing young students and budding scientists. Physics Galaxy is one such example ofnumerous effurts he has undertaken. The Physics Galaxy series_provides a good coverage of various topics of Mechanics, Thermodynamics and Waves, Optics & Modern Physics and Electricity & Magnetism through dedicated volumes. It would be an important resource for students appearing in competitive examination for seeking admission in engineering and medical streams. "E-version" ofthe book is also being launched to allow easy access to all. After release ofphysics galaxymobile app on both iOS and Android platforms it has now become very easy and on the go access to the online video lectures by Ashish Arora to all students and the most creditable and appreciable thing about mobile app is that it is free for everyone so that anytime anyone can refer to the high quality content ofphysics for routine school curriculum as well as competitive preparation along with this book. The structure of book is logical and the presentation is innovative. Importantly the book covers some ofthe concepts on the basis ofrealistic experiments and examples. The book has been written in an informal style to help students learn faster and more interactively with better diagrams and visual appeal of the content. Each chapter has variety of theoretical and numerical problems to test the knowledge acquired by students. The book also includes solution to all practice exercises with several new illustrations and problems for deeper learning. I am sure the book will widen the horizons ofknowledge in Physics and will be found very useful by the students for developing in-depth understanding of the subject. Prof. Sandeep Sancheti Ph. D. (U.K), B.Tech. FIETE, MJEEE www.puucho.com  Study Physics Galaxy with www.puucho.com PREFACE For a science student, Physics is the most important subject, unlike to other subjects it requires logical reasoning and high imagination of brain. Without improving the level _ofphysics it is very difficult to achieve a goal in the present age of competitions. To score better, one does not require hard working at least in physics. It just requires a simple understanding and approach to think a physical situation. Actuallyphysics is the surrounding of our everyday life. All the six parts of general physics-Mechanics, Heat, Sound, Light, Electromagnetism and Modern Physics are the constituents of our surroundings. If you wish to make the concepts of physics strong, you should try to understand core concepts of physics in practical approach rather than theoretical. Whenever you try to solve a physics problem, first create a hypothetical approach rather than theoretical. Whenever you try to solve a physics problem, first create a hypothetical world in your imagination about the problem and try to think psychologically, what the next step should be, the best answer would be given by your brain psychology. For making physics strong in all respects and you should try to merge and understand all the concepts with the brain psychologically. The book PHYSICS GALAXY is designed in a totally different and friendly approach to develop the physics concepts psychologically. The book is presented in four volumes, which covers almost all the core branches of general physics. First volume covers Mechanics. It is the most important part of physics. The things you will learn in this book will form a major foundation for understanding of other sections of physics as mechanics is used in all other branches of physics as a core fundamental. In this book every part ofmechanics is explained in a simple and interactive experimental way. The book is divided in seven major chapters, covering the complete kinematics and dynamics of bodies with both translational and rotational motion then gravitation and complete fluid statics and dynamics is covered with several applications. The best way of understanding.physics is the experiments and this methodology I am using in my lectures and I found that it helps students a lot in concept visualization. In this book I have tried to translate the things as I.used in lectures. After every important section there are several solved examples included with simple and interactive explanations. It might help a student in a way that the student does not require to consult any thing with the teacher. Everything is self explanatory and in simple language. One important factor in preparation of physics I wish to highlight that most of the student after reading the theory of a concept start working out the numerical problems. This is not the efficient way of developing concepts in brain. To get the maximum benefit of the book students should read carefully the whole chapter at least three or four times with all the illustrative examples and with more stress on some illustrative examples included in the chapter. Practice exercises included after every theory section in each chapter is for the purpose of in-depth understanding of the applications of concepts covered. Illustrative examples are explaining some theoretical concept in the form of an example. After a thorough reading of the chapter students can start thinking on discussion questions and start working on numerical problems. Exercises given at the end of each chapter are for circulation of all the concepts in mind. There are two sections, first is the discussion questions, which are theoretical and help in understanding the concepts at root level. Second section is of conceptual MCQs which helps in enhancing the theoretical thinking of students and building logical skills in the chapter. Third section of numerical MCQs helps in the developing scientific and analytical application of concepts. Fourth section of advance MCQs with one or more options correct type questions is for developing advance and comprehensive thoughts. Last section is the Unsolved Numerical Problems which includes some simple problems and some tough problems which require the building fundamentals of physics from basics to advance level problems which are useful in preparation ofNSEP, INPhO or !Ph0. In this second edition of the book I have included the solutions to all practice exercises, conceptual, numerical and advance MCQs to support students who are dependent on their self study and not getting access to teachers for their preparation. This book has taken a shape jnst because of motivational inspiration by my mother 20 years ago when I just thought to write something for my students. She always moiivated and was on my side whenever I thought to develop some new learning methodology for my students. www.puucho.com  Study Physics Galaxy with www.puucho.com I don't have words for my best friend mywifeAnuja for always being together with me to complete this book in the unique style and format. I would like to pay my gratitude to Sh. Dayasbankar Prajapati in assisting me to complete the task in Design Labs of PHYSICSGALAXY.COM and presenting the book in totallynewformat of second edition. At last bot the most important person, my father who has devoted bis valuable time to finally present the book in such a format and a simple language, thanks is a very small word for his dedication in this book. In this second edition I have tried my best to make this book error free but owing to the nature of work, inadvertently, there is possibility of errors left untouched. I shall be grateful to the readers, ifthey point out me regarding errors and oblige me by giving their valuable and constructive suggestions via emails for further improvement of the book. Ashish Arora PHYSICSGALAXY.COM B-80, Model Town, Ma/viya Nagar, Jaipur-302017 e-mails: [email protected] [email protected] www.puucho.com  Study Physics Galaxy with www.puucho.com CONTENTS =-----~----- ------ Chapter4 Mag,1etic_:Efft;C!S. of,C11rrent &,,!{ag,i~m.' 1-ui! 4.1 Biot Savart's Law 2 4.1.I Magnetic Induction Direction by Right Hand P~lm Rule 3 4.1.2 MagneticJnduction due to Moving Charges 4 4.1.3 Units used for magnetic induch"on 4 4.2 Magnetic Induction due to di.fferent Current Carrying Conductors Configurations 4 4.2.1 Magnetic Induction due to a Long Straight Cu"ent Carrying Conductor 5 4.2.2 Magnetic Induction due to a Finite Length Current Carrying Wire 6 4.2.3 Magnetic Induction due to a Semi-Infinite Straight Wire 6 4.2.4 Magnetic Induction at the Center of a Circular Coil 7 4.2.5 Magnetic Induction at Axial Point of a Circular Coil 8 4.2.6 Helmholtz Coils 9 4.2.7 Magnetic Induction at Center of a Circular Arc 10 4.3 Magnetic lnducdon due to Extended Current Configurations 17 4.3.l Magnetic Induction Inside a Long Solenoid 17 4.3.2 Direction of Magnetic Induction Inside a Long Solenoid 18 4.3.3 Magnetic Induction due to a Semi-Infinite Solenoid 18 4.3.4 Magnetic Induction due to a finite length Solenoid 18 4.3.5 Magnetic Induction due to a Large Toroid 19 4.3.6 Magnetic Induction due to a Large Cu"ent Carrying Sheet 20 4.3. 7 Magnetic Induction due to a Revolving Charge Particle 21 4.3.8 Magnetic Induction due to a Surface Charged Rotating Disc 22 4.4 Ampere's Law 24 4.4.1 Applications of Ampere"s Law 25 4.4.2 Magnetic Induction due to a Long Straight Wire 25 4.4.3 Magnetic Induction due to a Long Solenoid 26 4.4.4 Magnetic Induction due to a Tightly Wound Toroid 27 4.4.5 Magnetic Induction due to a Cylindrical Wire 28 4.4.6 Magnetic Induction due to a Hollow Cylindrical W,re 29 4.4.7 Magnetic Induction due to a Large Cu"ent Carrying Sheet 30 4.4.8 Magnetic Induction due to a Large and Thick Current Carrying Sheet 30 4.4.9 Unidirectional Magnetic Field in a Region 31 4. S Electromagnetic Interactions 36 4.5.l Electromagnetic Force on a Moving Charge 36 4.5.2 Direction of Magnetic Force on Moving Charges 37 4.5.3 Work Done by Magnetic Force on Moving Charges 37 4.5.4 Projection of a Charge Particle in Uniform Magnetic Field in Perpendicular Direction 37 4.5.5 Projection of a Charge Particle_from Outside into a Unifonn Magnetic Field in Perpendicular Direction 3 8 4.5.6 Projection of a Charge Particle in Direction PQrallel to the Magnetic Field 39 4.5.7 Projection of a Charge Particle at Some Angle to the Direction of Magnetic Field 39 4.5.8 Deflection of a Moving Charge by a Sector of Magnetic Field 40 4.6 Magnetic Force on Cu"1mt Carrying Conductors 45 4.6.l Force on a Wire in Uniform Magnetic Field 45 4.6.2 Direction of Magnetic Force on Currents in Magnetic Field 46 4.6.3 Force Between 1wo Parallel Current Carrying Wires 47 4.6.4 Magnetic Force on a Random Shaped Cu"ent Carrying Wire in Uniform Magn"etic Field 47 4. 7 Motion of a Charged Particle in Electromagnetic Field 53 4.7.1 Undeflected motion of a charge particle in Electric and Magnetic Fields 53 4.7.2 Cycloidal Motion of a Charge Particle in Electromagnetic Field 53 4.8 A Closed Cu"ent Carrying Coll placed in Magnetic Fteld 60 4.8.1 Torque due to Magnetic Forces on a Current Carrying Loop in Magnetic Field 61 4.8.2 Interaction Energy of a Cu"ent Carrying Loop in Magnetic Field 61 4.8.3 Work Done In Changing Orientation of a Cu"ent Carrying Coil in Magnetic Field 62 4.8.4 Magnetic Flur through a surface in Magnetic Field 62 www.puucho.com  Study Physics Galaxy with www.puucho.com 4.8.5 Stable and Unstable Equilibrium of a Current Can-ying Loop in Magnetic Field 63 4.8.6 Moving Coil Galvanometer 64 4.9 Relation in Magnetic Moment and Angular Momentum of uniformly charged and r,niform dense rotating bodies 69 4.10 Magnetic Pressure and Field· Energy of Magnetic Field 71 4.11 Classical Magnetism 75 4.11.1 Pole Strength ofa Magnetic Pole 15 4.11.2 Coulomb's Law for Magnetic Forces 76 4.11.3 Magnetic Induction in terms of Force on Poles 16 4.11.4 MagneJic Induction due to a Magnetic Pole 77 4.11.5 Bar Magnet 77 4.11.6 Magnetic Induction due to a Bar Magnet on its Axis 78 4.11.7 Magnetic Induction due to a Bar Magnet on its Equatorial Line 79 4.11.8 Analogy between Electric and Magnetic Dipoles 79 4.11.9 A Small Current Carrying Coil as a Magnetic Dipole 80 4.11.10 Force on a Magnetic Dipole in Magnetic Field 8I 4.12 Terrestrial Magnetism 84 4.12.1 Direction of Earth's Magnetic Field on Surface 84 4.12.2 Some Definitions and Understanding Earth's Magnetic Field 85 4.12.3 Tangent Galvanometer 86 4.12.4 Deflection Magnetometer 87 4.12.5 Earth's Magnetic Field in Other Units 88 4.12.6 Apparent angle of dip at a point on Earth S Surface 88 DISCUSSION QUESTION 91 CONCEPTUAL MCQS SINGLE OPTION CORRECT 92 NUMERICAL MCQS SINGLE OPTIONS CORRECT 96 ADVANCE MCQS WITH ONE OR MORE OPTIONS CORRECT 105 UNSOLVED NUMERICAL PROBLEMS FOR PREPARATION OF NSEP. INPHO & !PHO l l0 - - - - - ~"""",-"" ___ " _ _ _ _ ~"-·""'-- · - - · - - - -. l ' Chapter 5 Eleiitr;,111agnetic. -~~·---o" Induiiti~n a11d AlternatilJ;f: ,--·------,a~~-----_.M,_ Current ~-.- S "-~ ·~·,~ ~--·----. '+~-w~ 119':.2601 ·- =·--- __,,_,., o; 5.1 Faraday ·s Law of Electromagnetic Induction 120 5.1.1 Lenz's Law 121 5.1.2 Motion EMF in a Straight Conductor Moving in Unifonn Magnetic Field 123 5.1.3 Moiional EMF by Faraday's Law 124 5.1.4 Motional EMF as an Equivalent Battery 124 5.1.5 EMF Induced in a Rotating Conductor in Unifonn Magnetic Field 125 5.1.6 EMF induced in a Conductor in Magnetic Field which is Moving in Different Directions 126 5.1. 7 Motional EMF in a Random Shaped Wire Moving in Magnetic Field 126 5.1.8 EMF induced in a Rotating Coil i,i Unifonn Magnetic Field 127 5.1.9 Motional EMF under External Force and Power Transfer 128 5.2 Time Varying Magnetic Fields (TVMFJ 140 5.2.1 Charge Flown through a Coil due to Change in Magnetic Flux 140 5.2.2 Induced EMF in a Coil in Time Varying Magnetic Field 140 5.2.3 Induced Electric Field in Cylindrical Region nme Varying Magnetic Field 141 5.2.4 Induced Electric Field outside a Cylindrical Region Time Parying Magnetic Field 142 5.2.5 Electric Potential in Region of Time Mirying Magnetic Field 142 5.2.6 Eddy Currents 143 5.3 Self Indr,ction 150 5.3.1 Understanding Self Induction 15 l 5.3.2 Self Inductance of a Solenoid 157 5.3.3 Behaviour of an Inductor in Electrical Circuits 152 5.4 Growth ofCu"ent in an Inductor 155 5.4.1 Growth of Current in an Inductor Connected Across a Battery 155 5.4.2 Growth of Current in RL Circuit across a battery 155 5.4.3 Time Constant of RL Circuit 156 5.4.4 Instantaneous Behaviour of an Inductor 156 5.4.5 Transient Analysis of Advance RL Circuits 157 www.puucho.com Study Physics Galaxy with www.puucho.com 5.5 Energy Stored in an Inductor 160 5.5.1 Magnetic Field Energy Density 160 5.5.2 Field Energy Stored in a Current Carrying Inductor 161 5.6 Decay ofCu"ent in RL Circuit 162 5. 7 Mutual Induction 168 5.7.1 Total Flux linked to a Coil with Mutual Induction 169 5.7.2 Relation between Coefficients of Self and Mutual Induction for a given Pair of Coils 169 5.7.3 Inductors in Series Combination 170 5.7.4 Inductors in Parallel Combination 171 5.8 LC Oscillations 174 5.8.1 Oscillation Period of LC Oscillations 175 5. 9 Magnetic Properties of Matter 181 5.9.l Magnetic Moment of Hydrogen Atom 181 5.9.2 Natural Magnetic Moment in Different Materials 18\ 5.9.3 Diamagnetism 182 5.9.4 Response of Diamagnetic Materials to External Magnetic Fields 182 5.9.5 Magnetic Levitation in Diamagnetic Materials 183 5.9.6 Paramagnetism 183 5.9.7 Response of Paramagnetic Materials to External Magnetic Fields 183 5.9.8 Magnetic Induction inside a Paramagnetic and Diamagnetic Materials 184 5.9.9 Intensity of Magnetization 184 5.9.10 Magnetic Field ~ctors B and H 1S5 5.9. l I Relation in B H and I in Magnetization of Materials 185 5.9.12 Magnetic Susceptibility 185 5.9.13 Curie's Law 185 5.9.14 Ferromagnetic Materials 186 5.9.15 Magnetiza.tion of Ferromagnetic Materials 186 5.9.16 Superconductor as a Perfectly Diamagnetic Material 186 5.9.17 Magnetic Hysteresis 187 5.10 Alternating Cu"ent 189 5.10.1 Mean or Average J'izlue of an Alternating Current 190 5.10.2 Mean Value ofa Sinusoi/dal Alternating Current 190 5.10.3 Root Mean Square (RMS) value of an Alternating Current 191 5.10.4 RMS value of a Sinusoidal Alternating Current 191 5.11 AC Circuit Components 193 5.11.1 A Resistor Across an AC EMF 193., 5.ll.'2 A Capacitor Across an AC EMF 194 5.11.3 An Inductor Across an AC EMF 194 5.Jl.4 Comparison of AC Circuit Elements 195 5.11.5 Behaviour of L and Cat High and Low Frequency AC · 196 5.11.6 LR Series Circuit Across an AC EMF 196 5.11.7 RC Series Circuit Across an AC EMF 198 5.12 Phasor Analysis 202 5.12.l Standard Phasorsfor Resistances and Reactances in AC Circuits 203 5.12.2 Phasor Algebra 203 5.12.3 Phasor Analysis of a Resistance Across an AC EMF 205 5.12.4 Phasor Analysis of a Capacitor Across an AC EMF 205 5.12.5 Phasor Analysis of an Inductor Across an AC EMF 205 5.12.6 Impedance of an AC Circuit 206 5.12.7 Phasor Analysis of S~ries RL Circuit Across AC EMF 207 5.12.8 Phasor Analysis of Series RC Circuit Across AC EMF 208 5.12.9 Phasor Analysis of Series LC Circuit Across AC EMF 208 5.12.10 Series RLC Circuit Across AC EMF 209 5.12.11 Phase Relations in Series RLC Circuit 209 5.13 Power in AC Circuits 213 5.13.1 Average Power in AC Circuits 213 5.13.2 Wattless Current in AC Circuits 213 5.13.3 Maximum Power in an AC Circuit 214 www.puucho.com  Study Physics Galaxy with www.puucho.com 5.13.4 Resonance in Series RLC Circuit 214 5.13.5 Variation of Current with Frequency itz Series RLC Circuit 214 5.13.6 Half Power Frequencies in Series RLC Circuit 215 5.13. 7 Bandwidth for a Series RLC Circuit 215 5.13.8 Quality Factor of Series RLC Circuit 216 5.13.9 Selectivity of a Resonance Circuit 216 5.13.10 Parallel RLC Circuit 216 S.1./ Tram.former 220 5.14. / Poirer Relations in a TransfOrmer 221 5.14.2 Types of Transformers 221 5,14.3 Losses in Transformers 221 5./4.4 Efficiency ofa Tra11sjOrmer 222 DISCUSSION QUESTION 225 CONCEPTUAL MCQS SINGLE OPTION CORRECT 227 NUMERICAL MCQS SINGLE OPTIONS CORRECT 237 ADVANCE MCQS WITH ONE OR MORE OPTIONS CORRECT 245 UNSOLVED NUMERICAL PROBLEMS FOR PREPARATION OF NSEP, INPHO & /PHO 250 ,:., , -~NSWERS&SOLUTIONS. ' - ' Chapter4 Magnetic Effects of Current & Magnetism 261 - 294 Chapters Electromagnetic Induction and Alternating Current 295 - 336 www.puucho.com  Study Physics Galaxy with www.puucho.com Magnetic Effects of Current & Magnetism 4 FEW WORDS FOR STUDENTS First time playing with magnets is always a fim for every child. Wit!, age every011e gets aware t!,at magnets !,ave two poles, designated as north and sollt/, poles. All ofyou might have seen a compass used by travellers for tl,e pmpose of navigation. 111 ancient times compass was tl,e only tool available for determini11g the directions. TJ,e 11eedle ofsuc/1 a compass is a small, tl,i11 magnet. Magnets and magnetic effects are very i111po11a11t in 1liffere11t industrial applications as well as t/,ese are very important in 1111derslm11li11g 1111111y 1111t11ral a111I lab phenome11011 relt,ted to magnetism. Even many birds use tl,e magnetic field ofE11rth for 1wvig11tio11 along with 1/irectiol!s from the location of Sun and ~·tars d111·i11g long tlist1111ce migmtions. This wl,ole cl,apter is co1 ering effects and pl,e11omenon of magnetism to strengthen 1111tlerst11mling of this topic to a decent level. CHAPTER CONTENTS 4.8 A Closed Current Carryillg Coil placed in 4.1 Biot Savart's Law Magnetic Field 4.2 Magnetic Induction due to different Current 4.9 Relation in Magnetic JI-Ioment and Angular Carrying Conductors Configurations Momentum of1111iformly ch11rged and uniform dense rotatillg bodies 4.3 Magnetic Induction due to Extended Current Configurations 4.10 M11g11etic Pressure and Field Energy of Magnetic Field 4.4 Ampere's Law 4.11 Classical Magnetism 4.5 Electromagnetic Illteractions 4.12 Terrestrial Magnetism 4.6 Magnetic Force 011 Current Carrying Conductors 4.7 Motion of a Charged Particle in Electromagnetic Field COVER APPLICATION Mirror Translucent sc,k Fixed core Moving coil Figure-{a) Figure-{b) A ballistic galvanometer is used to measure charge which passes through the device. When charge is suddenly passed through a coil in magnetic field, it imparts an angular impulse to the coil which deflects the coil once and a small mirror attached to the coil axis also rotates along with the coil due to this. Torsional springs restores the position of coil back to normal but the deflection ofa light beam falling on the mirror is measured on a scale and amount of charge can be calculated. Figure-{a) shows the setup of ballistic galvanometer experiment and figure-(b) shows the industrial orfab ballistic galvanometer at the center of which there is a small galss window through which a narrow light beam falls on the mirror and on the path of reflected beam a scale is placed. www.puucho.com  Study Physics Galaxy with www.puucho.com Magnetic. Effects"ot Currents and Classical Magn~tif!!'] 'Magnetic forces of attraction and repulsion between magnetic (3) A current carrying wire produces only magnetic field in poles are similar to the interaction between charges but its surroundings and only magnetic field can exert a force on magnetic poles and charges are not the same thing. However current carrying wires. the way electric charges produce electric field in their surrounding is similar to that magnetic poles also have an 4.1 Biot Savart's Law associated magnetic field in their surrounding. Magnetic lines of forces are also similar to electric lines of forces used to Biot Savart's Law is a basic law concerning electricity and trace the pattern of electric field in surrounding of charges. magnetism which describes the magnetic field generated by a Tang~nt to a magnetic line of force in space gives the direction current carrying wire in its surrounding. The equation ofBiot of magnetic field strength vector at that point and density of Savart's Law gives the strength of magnetic field at a specific magnetic field lines in a region is a measures of the magnitude point in surrounding of a current carrying wire. The magnetic of magnetic field in that region. Similar to charges as we field which is associated with electric field at any point for approach closer to a magnetic pole, density of magnetic lines which we were discussing before this article is often referred increase. as 'Magnetic Induction' instead ofmagnetic field in technical It is also observed that in a region variation of electric field terms. This is denoted by 'B ' and also referred as magnetic gives rise to induction of magnetic field and vice versa. A flux density like electric field intensity ' E '. danish scientist Oersted observes magnetic field in the surrounding of current carrying conductor. Figure-4.1 shows This law analyzes that the magnetic induction produced due the setup of Oersted's Experiment in which when a current is to an elemental length ofa current carrying wire depends upon switched on, deflection in magnetic needle is observed which four factors. Consider a wire XY carrying a current I. To find verified the presence ofmagnetic field in surrounding ofcurrent the magnetic induction at a point Pin its neighbourhood, we carrying wires. This is called 'Magnetic Effects of Electric consider an elemental segment AB on this wire oflength di as Current'. shown. y I Figure 4.1 X In this experiment it is also seen that the direction of deflection in magnetic needle is opposite when the needle is placed below Figure 4.2 or above the current carrying wire which indicates that magnetic field direction is opposite on the two sides of a current According to Biot Savart's Law the magnetic induction dB at carrying wire. point P due to the elemental wire segment AB depends upon four factors which are given as In previous chapters we've already studied about electric field and electric forces. There are some facts listed below (i) dB is directly proportional to the current in the element. concerning to electric and magnetic field and forces. Students are advised to keep these below points always in mind as these dBocl... (4.1) points form the basis of understanding magnetic effects. (ii) dB is directly proportional to the length of the element (I) A static charge produces only electric field in its dB oc di... (4.2) surrounding and only electric field can exert a force on static charges. (iii) dB is inversely proportional to the square of the distance r of the point P from the element (2) A moving charge produces both electric and magnetic fields in its surrounding and both electric and magnetic fields dBoc _.!:.... (4.3) can exert force on moving charges. r' www.puucho.com  Study Physics Galaxy with www.puucho.com IMagnetic-Effects of Currents an·iclassical Magnetism '-~---------- - - 3j (iv) dB is directly proportional to the sine of the angle 0 current, then the direction in which the fingers curl, gives the between the direction of the current flow and the line joining direction ofmagnetic filed lines". This is shown in figure-4.3. the element to the point P dB oc sin 0... (4.4) Combining above factors, we have 0 (8) dB oc Id/ s:n 0 r \ ,I --..------- ' [ , ------~ "fl................-_.;.::: :::---...:~...... , -... ', '1....... --..,....... l,,,JII' I i...... _----......:-_-_-_ :_:..."!°... ,..-'__.,.. ,... dB=Kldlsin0... (4.5) I / -------- -------~ r' Where K is a proportionality constant and its value depends upon the nature of the medium surrounding the current carrying wire. Ifthe current carrying wire is placed in vacuum, then in SI Units its value is given as Figure4.3 K= &_ = 10-7 T-m/A 41t In the figure-4.2 using above rule we can see that direction of Here µ 0 is called permeability of vacuum or free space. It is a vector dB is into the plane of paper which is represented by physical quantity for a given medium which is a measure ofa ® similarly on the points located on the left side of wire medium's ability on the extent to which it allows external direction of dB is in outward direction from the plane ofpaper magnetic field to polarize the material inside the volume of and it is represented by 0. We can accommodate this direction the body. Therefore Biot Savert's Law is written as in equation-(4.6) which can be rewritten as dB= µ,Id/sine 41t r 2... (4.6) - (µ dB= 41t Jd/x; -0) - r3 -... (4.8) This constant µ 0 is similar in nature to the constant e 0 we In above equation you can see that the direction of cross product discussed in the topic of electrostatics. For a given medium its of the vector of elemental length and position vector of point magnetic permeability is defined as P with respect to the element is giving the same direction as stated by right hand thumb rule. µ= µoµ, At point Pin figure-4.2 magnetic induction due to whole wire Where µ, is called 'Relative Permeability of Medium' which XY can be obtained by integrating the above expression for is casually also referred as diamagnetic constant ofthe medium. the entire length of the wire within proper limits according to Unlike to the case of electrostatics where dielectric constant the shape of the wire and it is given as of any medium is always greater than unity, in case of magnetism, depending upon the types of mediums value ofµ,... (4.9) can be greater or smaller ihan unity. In the topic of magnetic properties of material in next chapter we will discuss about In above equations many times the term 'Id/' which is the the magnetic permeability in detail. As of now students can product of current and length of element is also referred as a consider that if the current carrying wire is placed in a material separate physical quantity called 'Current Element' which is medium then equation-(4.6) can be rewritten as measured in units of 'Ampere-meter' or 'A-m', In later articles of the chapter we will discuss its physical significance. dB= µ /d/sin0 µ 0 µ, /dlsin0... (4.7) 41t r 2 r' 4.1.1 Magnetic Induction Direction by Right Hand Palm Rule Above equation-(4.7) gives the magnitude of magnetic induction produced due to a small current element. The In previous article we studied that the direction of magnetic direction of this magnetic induction is given by right hand induction in surrounding of a current carrying wire can be thumb rule stated as "Grasp the conductor in the palm ofright obtained by right hand thumb rule. Same can also be obtained hand so that the thumb points in the direction of the flow of by 'Right Hand Palm Rule' stated as "Stretch your palm of www.puucho.com Study Physics Galaxy with www.puucho.com right hand keeping fingers straight and put the thumb along (iii) Bis inversely proportional to the square of the distance r the direction ofcurrent and point yourfingers toward the point of the point P from the element of application. 11,e area vector of your palm face gives you I the direction ofmagnetic induction vector at that point". This Boc -... (4.12) r' is shown in figure-4.4. (iv) Bis directlyproportional to the sine ofthe angle 0 between the direction of the motion of positive charge and the line joining the charge to the point P I Bocsin0... (4.13) Combining above factors we have, qvsin0 Boc..,___ r' B=Kqvsin0 ·... (4.14) r' µ,. For free space we can use K = so we have 411 Figure 4.4 B= &_ qvsin0... (4.15) 4.1.2 Magnetic Induction doe to Moving Charges 41t r 2 Vectorially above resnlt can be rewritten including the direction As already discussed that moving charges produce magnetic of direction vector of magnetic induction is given as field in their surrounding. Biol Savert's law is also defined for moving charges like current carrying wires we studied in - µ, q(iixr) article-4.1. To analyze the magnetic induction in surrounding B--... (4.16) - 411 r3 of a moving charge we consider a charge q moving at velocity v as shown in figure-4.5. In this situation we will determine In above expression of magnetic induction at point P due to the magnetic induction at a point P located at a position vector the moving charge the cross product of velocity and position r from the instantaneous position of charge as shown in figure. vector is written in such a way that the directioµ of magnetic field obtained by right hand thumb rule is same as that given by this cross product. In figure--4.5 direction of magnetic p induction at point P is into the plane of paper (inward) as ®it shown in this figure. +q 4.1.3 Units used for magnetic induction ' '' In SI units magnetic induction is measured in 'Tesla' denoted '' '' as 'T and magnetic flux in space through any area is measured '' I' in units of'weber' and denoted as 'Wb'. As magnetic induction I is also referred as magnetic flux density or _flux passing through a unit normal area in a region like electric field was defined. Figure4.5 The units tesla(T) and weber(Wb) are related as The magnetic induction B due to moving charges in 1 T= I Wb/m2... (4.17) surrounding depends upon four factors which are given as (i) B is directly proportional to the charge 4.2 Magnetic Induction due to different Current Carrying Conductors Configurations Boc q... (4.10) Equation of Biol Savarts Law gives the magnetic induction (ii) B is directly proportional to the speed ofcharge due to a current element in its surrounding. For different shapes Bocv... (4.11) of current carrying conductors this needs to be integrated www.puucho.com  Study Physics Galaxy with www.puucho.com !Magnetici- Etf&i~;of Currents~~~]~~~!~~! M8g~etism differently to obtain the net magnetic induction due to the To integrate the above expression we substitute conductors in their surrounding at specific points. Next we y= rtan0 are going to discuss some standard configurations ofthe current carrying conductors. dy= r sec:20 d0 4.2.1 Magnetic Induction due to a Long Straight Current With the above substitution limits of integration also changes Carrying Conductor as 2" Figure-4.6 shows a long straight wire carrying a current/. To at y=-oo ~ 0= - determine the magnetic induction due to the wire at point P situated at a normal distance r from the wire we consider an element oflength di in the wire at a distance y from the point and at y=+oo ~ 0= " +- 2 0 as shown. By right hand palm rule we can see that the direction ofmagnetic field at pointP is into the plane of paper tt (inward).. µ,Ir f +- 2 rsec 0d0 2 B = 4n -~ (r' + r tan' 0)312 2 I B = µo l r f rs3ec +tt/2· 0d0 2 3 4n -n/2 r sec e 01+----'-'---~®:B I +nil B= &_ 4nr Jcos0d0 -nil y. 0].. B = -µ,I [sm " _7112 e · 4nr dy B= µ,I[!-(-!)] 4nr µ,I B= 2nr... (4. 19) Figure 4.6 The magnetic induction dB at point P is into the plane of As already discussed that the direction of magnetic induction paper and given by Biol Savart's Law using equation-(4.6) as due to a long straight wire in its surrounding is along the tangent to the concentric magnetic lines as shown in figure-4. 7. µ, Idysin0 This is given by either of right hand thumb rule or right hand dB= - X - -2 " - ~ 2 4n (r +y ) palm rule. In above equation we can substitute sin0 = F'+J r2 + y2 gives µ0 ldyr dB= 4 ,, x (r' + y')'"... (4.18) As due to all the elements in the wire the magnetic induction is in inward direction at point P, the net magnetic induction at point Pis given by integrating above equation-(4.18) for the whole length of wire.within limits of y from --oo to -too as y-+oo B= fdB= -J µ, x I dyr 2 ,·- 41t (r + y' )'" µ 0 /r -J dy B = 41t _ (r 2 + y')'" Figure4.7 www.puucho.com  Study Physics Galaxy with www.puucho.com 4.2.2 Magnetic Induction due to a Finite Length Current Carrying Wire µ 0 /r s'' r csc' 0d0 B = - 471 -o, (r' + r 2 cot' 0) 312 Figure-4.8 shows a wire AB of length L carrying a· current/. In the surrounding of wire consider a point Pas shown which is located at a perpendicular distance r from the wire such that the point is subtending angles 0 1 and 02 at the end points of the wire as shown in figure. / +81. A , B=-.&_ f sin0d0 471r -o, 81 --..................... I ---- ---,,, dB r - p I / / /® µ/ ° 1 0 ~,, +;,'-// B= - 4 71r [cos0 1 -(-cos02 )] B= µ,I [cos0 1 +cos0,J t ///// 471r... (4.20) The direction of magnetic induction in surrounding ofa finite B length current carrying wire can also be given by right hand Figure 4.8 thumb rule as shown in figure-4.7. Equation-(4.20) can also be used to calculate the magnetic induction due to an infinite To calculate the magnetic induction at point P we consider an current carrying straight wire in which both the side angles 0 1 element oflength di at a distance y from the point O as shown and 02 will tend to zero. In this equation if we substitute both in figure-4.8. By using Biot Savart's law the magnetic induction angles zero then it gives the expression given in dB at point P due to this current element is given as equatiori-(4.19). 0 µ Id/sine dB=-· 4.2.3 -Magnetic Induction·due to a Semi-Infinite Straight 471 (r 2 + y 2 ) Wire As due to all the elements in the wire the magnetic induction is in inward direction at point P, the net magnetic induction at Figure-4.9 shows a very long wire AB carrying a current/ and point P due to wire AB is given by integrating above Pis a point.at a perpendicular distance r from the end A of the equation-(4.18) for the whole length of wire AB within limits wire. Using right hand thumb rule here we can state thai the ofy from-r cot02 to +r cot0 1·as magnetic induction at point P is in outward direction. The magnitude ofthe magnetic Induction at P can be calculated by using equation-(4.20). Ifwe look at figure-4.9(b), we can see that at point A point P is making an angle 90° with the µ,,r J: +rcot81 f d y length ofwire and at point B which is located fur away distance, => B= '471 -rcot8 2 (r'+y')'" the angle between the line PB and wire can be considered as To integr·ate the above expression , , I, substitute ~ 0°. Thus magn~ic induction at point P due to this semi infinite wire is given as y=rcot0 dy = - r cosec20 d0 B= µ,I[cos(90°)+cos(0°)] 471r With the above substitution limits of integration also changes as => ·B= µ,I [0+1] 471r at y=-rcot02 -> 0=-02 µ,/ and at y=+rcot0 1 -> 0=+0 1 B=-... (4.21). 471r www.puucho.com  Study Physics Galaxy with www.puucho.com lf§i,etlc EffecfofCurients and Classliai Magnetism µI 0B, B= - ° (cos 0 + cos {0°)] pt 4 nr i ' µI ri' B= - °[I +cos0]... (4.22) '' f 90° 4 nr A n In both of above cases the direction of magnetic induction is. I given by right hand thumb rule as shown in these fignres. ·(a) 4.2.4 Magnetic Induction at the Center of a Circular Coil Figure-4.ll(a) shows a circular coil of radius R carrying a current I. If we consider an element of length di along its circumference as shown then due to this element by right hand thumb rule or palm rule we can see that the direction of (b) magnetic induction dB at the center of coil is in outward Flgure4.9 direction as shown in figure-4. I l{b). By Bio! Savart's Law the magnetic induction at the center of this coil due to this current , The magnetic induction given in above expression given in element is given as equation-(4.21) is half of that obtained dueto an infinitely _ µ Id/ sin(9O°) long wire as given in equation-(4.19) which can be directly dB- 4n0 R' stated as both the halves of an infinite wires are identical with respect to point Pin jigure-4.Q on the two sides of it and if one µ,Id/ \o half is removed then' due the remaining half the magnetic => dB= 4nR'... (4.23) induction is also reduced.to half.. · I dB ® ,, e· A"--'------+''-------------------B I.. ' (a) (a) -- - - - - - - - - - - ~ - - ~ - - ' - - - - - - - - - - - - B A I (b) Figure 4;t0· Fig~re-4.IO(aj and (b) shows another semi infinite wire but in this case we ~11 ca[culate the magnetic induction at points P 1 andP2 which are located at same distance from end p~intA of _wire at same perpendicular distance r from the line of wire. (b) Again using equation-(4.20) we can calculate the magnetic Figure 4.11 induction at these points which is given as www.puucho.com  Study Physics Galaxy with www.puucho.com '----~---- 8. As shown in figure-4.11 (b) the direction of magnetic induction ® ® due to all the elements ofthe coil are in same outward direction ® at every interior point of the coil and the magnetic lines are making closed loops from outside of the coil as shown. Thus 0 0 net magnetic induction at center of coil can be given by 0 ® 0 integrating equation-(4.20) for the whole length of the coil 0 which is given as ® ® µ 0 / '"" (b) J B= dB= - - 2 di J 41tR 0 ----------.. Magnetic Lines B = µ,I [21tR - OJ of Force 41tR' µ,I B=-... (4.24) 2R If there areNturns of wire in the coil then total length of wire will be 21tRN so for N turns in coil, the magnetic induction at Current Flow the center of coil is given as (c) Figure 4.12 µ,IN B=--... (4.25) 2R Figure-4.12(c) shows the configuration of magnetic lines in a diametrical plane of a circular current carrying coil which is Like a circular coil for any closed current carrying coil of any explained in figure-4.11. shape in a plane the direction of magnetic induction can also be calculated by using right hand thumb rule in a different 4.2.5 Magnetic Induction at Axial,Point of a Circular Coil way. For this we circulate our right hand fingers along the direction of current in the loop as shown in figure-4.12(a) Figure-4.13 shows a current carrying circular coil ofradius R then the direction of our right hand thumb gives the direction mounted in 17 plane carrying a current I. We will calculate. the magnetic induction due to this coil at a point P located on of magnetic induction at interior points in the plane of the the axis of coil at a distance x from its center as shown. By loop and at all the points outside the loop in its plane the using right hand thumb rule as described in previous article, direction of magnetic induction is opposite as shown in we can see that the direction of magnetic induction at point P figure-4.12(b). due to the current in this coil is in rightward direction. y / - (----::,--~7-1---.--rt-,-,--7 -,_........,,\ z '-·... __:;_,;,------1-·--11c_______)...,.._.,,/,, Figure 4.13 ,_,. '' '' To calculate the magnitude of magnetic induction at P due to the coil we consider a small element oflength di at the top of (a) the coil as shown in figure-4.14. Due to the current flowing in www.puucho.com  Study Physics Galaxy with www.puucho.com ~tlcj:ff~~ of Currents andCiassical_ Magnetism this element ifwe find the magnetic induction dB at point P Above expression given in equation-(4.27) will be used as a then its direction can be given by right hand thumb rule as standard result for many advance cases of determining shown in figure. To understand the direction of dB at point P magnetic induction. This result can be modified if coil has N students can imagine their right hand palm placed in the figure turns and it can be given as with thumb along the direction of current in di and fingers. µ 0 NJR' pointing toward the point Pthen you can feel that the direction B= Z(R' +x')'"... (4.28) of dB is perpendicular to the palm of your right hand. y'i,;~/' 4.2.6 Helmholtz Coils di,,/~ A setup of two identical circular coaxial coils separated by a ----............................ /;:;:;- dB cos St dB distance equal to their radius when carries current in same ,,./ -----t:............... le I direction then in the region between the two coils near their ,, 8 -......... I dB sin8 common axis magnetic induction is found to be uniform. Such p X a system oftwo coils is called 'Helmholtz Coils'. Figure-4. lS(a) shows the setup ofHehnholtz coils and the variation graph of magnetic field with distance of the two coils on their common z axis. B Flgure 4.14 If we resolve the magnetic induction dB in mutually perpendicular directions along the axis and perpendicular to the axis as shown in figure then the component dB cos0 which is normal to axis gets cancelled out due to the element on coil which is diametrically opposite to the element considered and the other component dB sin0 which is along the axis will all get added up as due to all the elements on the coil these components are in same direction and the resultant magnetic induction at point Pis in the direction as stated in figure-4.13. The magnetic induction at point P due to the element considered can be given by Biot Savart's law as µ 0 Id/ sin ( 90°) dB= 41t (R' + x') µ0 Id/ (a)... (4.26) dB= 41t (R' +x') The magnetic induction at P due to the coil is given by integrating the above expression for the whole length of the circumference of the coil. Thus it is given as B= JdBsin0 's"' µ 0 Id/ R B= o 41t.(R'+x') ,/R'+x' => - µ,JR 's"' di B - 41t(R' + x2 ) 312 0 - µ,JR [21tR-O] B - 41t(R' + x' )'" (b) Figure4.15 µ,JR' B = 2(R 2 + x' )312.. · (4.27) In above graphs it is observed that the drop in magnetic www.puucho.com  Study Physics Galaxy with www.puucho.com '.. [ 10 M,ag9~ti~'§!fecls -ofCOrr~~ts ah_d Qlassical·Mag'ri~Ysffi'.J induction due to one coil on either side from ± R/2 position Using above expression given in equation-(4.30) we can (mid point) is equal to the increase in magnetic induction _due calculate the.magnetic induction due to a semicircular current to other coil because of which the overall magnetic induction carrying wire which is given by substituting 0 = _211 as between the two coils remains almost uniform. The magnitude µ,I of this magnetic induction can be ca_lculated by using B= 4R.. (4.31) equation-(4.28) at x = R/2 due to both the coils. Above expression- in equation,(4.31) is halfof the magnetic BHelmholtz = 2 Bx=R/2 induction que to a circular coil.as __given in equation-(4.24) because due to all elements of a circular coil magnetic induction B _ 2[ µ0 N/R 2 J. is in same direction so making the coil half reduced the Helmholtz- 2(R 2 +x 2 )"' x=R/2 magnetic induction also to lialt: Similarly we can state 'that due to a quarter circular arc carrying a current i magnetic induction at its center will be one fourth of that of a circular · · · B µ0 NIR =-~~--~ 2. coil given as Helmholtz (R' +(R/2)')'" µ,I B=-... (4.32) BHelmholtz = ( )µ s..[5 R 8 0 N/... (4.29) _8R # lllustrative Example 4.1 Above magnitude of magnetic induction as given in 0 equation-(4.29) is approximately constant, in the region A current i = IA circulates in a round thin wire loop ofradius I between the two coils as shown in fignre-4.15(b). This setup r "" l 00riim. Find the magnetic induction , of Helmholtz coils is used to setup uniform magnetic field;in (a) At the centre of the loop lab frame for various experiments and experi~enial demonstrations. (b) ·At a pdint lying on 'the axis of the loop at a distance , x = 100mm from.its centre., 4.2. 7 Magnetic Induction at Center of a Circular Arc,.Solution. Figure-4.16 shows a circular arc of radius R carrying a curr.;;.'t · I. Due to all the current elements on this arc magnetic induction- ·(a) At the center of a circular loop magnetic induction is at its center O are in same inward direction which is given _by. given as right hand thumb rule as shown. For the angle 0 = 211 it wiil' be a circular coil and for angle 0 subtended by arc at its center· the magnetic induction can be directly given by using the·. expression of magnetic induction due to a circular coil 1g}ven~ as · 411;;10-1 B=----- xi.a 2x0,1 µ,J 0 B=-X- 2R 211 µ,10 B= 41tR... (4.30) (b) As studied at a point on the axis of the loop magnetic induction is given as I µ ir 2 B =0 - X - - - - 411 (x2 +r2)312 R.......... 411x!0-7 xl.0x(0.1) 2............ B = - - - - ~312 ~- ---............. 2 X (0.02) ,, --,,' ' B ~ B=2.22x 10-,;T Figure 4.16 B= 2.22 µT www.puucho.com  Study Physics Galaxy with www.puucho.com [Magnetic Eiiec(i,icurren~in5!.El':"sical Magnetism _ -- -- ---- __ --- - - - - ~ 1 1 j # lllustrative Example 4.Z (b) Field at O due to arc KL is given as µ 0 i 31t Find the magnetic induction at the point O due to the loop B=- X- 1 41ta 2 current i in the two cases given.below. The shape of the loops are illustrated as Field at O due to parts LM and PK is zero because the point 0 is located on the line of current. (a) In figure-4.17(a), the radii a and b, as well as the angle are known. Field at O due to part MN is given as B = µoi xsin45o= µoi x-1- 2 41th 41th.J2 Field at O due to part NP is given as µoi. µoi I B = - X S l i l 45 0 =- x-