Faraday s Law & Maxwell s Equations (Griffiths Chapter 7: Sections 2-3) B t da = S

Size: px
Start display at page:

Download "Faraday s Law & Maxwell s Equations (Griffiths Chapter 7: Sections 2-3) B t da = S"

Transcription

1 Dr. Alain Brizard Electromagnetic Theory I PY 3 Faraday s Law & Maxwell s Equations Griffiths Chapter 7: Sections -3 Electromagnetic Induction The flux rule states that a changing magnetic flux Φ B = S B da through a surface S induces an emf E on the boundary S: E = dφ B = S B da = S E dl. 1 The flux rule is also known as Faraday s Law in integral form. The minus sign in Faraday s Law is associated with Lenz s Law: the induced current on the boundary S generated by the induced emf flows in a direction which opposes the change in magnetic flux. It is worth mentioning, here, that there is an infinite number of open surfaces S with boundary S but that the magnetic flux S B da through surface S with boundary S is equal to the magnetic flux S B da through surface S with boundary S if the boundaries are identical S = S. This statement is proved by considering the volume V enclosed by the two surfaces S and S such that V = S S : B da B da = B da = B dτ =, S S V V which follows from the Divergence Theorem and Gauss s Law for magnetic fields. Returning to Faraday s Law 1, we use Stoke s Theorem on the last integral to obtain B S S da = E dl = E da, S which becomes Faraday s Law in differential form: E = B. Hence, the presence of a time-dependent magnetic field induces an electric field whose curl is not zero. Consequently, this induced electric field cannot be expressed in terms of the gradient of a scalar field since E. The clue to the nature of this new electric field is provided by the representation of the magnetic field B = A in terms of the vector potential A: B = A = E 1 E + A =.

2 From the Helmholtz Decomposition Theorem for vector fields, the electric and magnetic fields are, therefore, expressed in terms of the scalar potential Φ and vector potential A as E = Φ A and B = A, 3 where the new term in the electric field is called the inductive electric field. Note that these definitions immediately imply Faraday s Law and Gauss s Law for magnetic fields: E = B/ and B =, respectively. For example, when a uniform magnetic field Bt is time dependent, it induces a timedependent nonuniform electric field Er,t. To show this, we invoke the formula Ar,t = r Bt for the vector potential of a uniform magnetic field; note, here, that A = Bt r + Bt The inductive electric field is, thus, given as Er,t = A and that this field is divergenceless E = 1 r = 1 Bt + 3 Bt = Bt. dbt = r dbt, r =. Inductance Suppose now we are interested in calculating the magnetic flux Φ B through a wire loop labeled due to the magnetic field B 1 produced by another wire loop labeled 1: Φ B = B 1 da = M 1 I 1, S where the magnetic flux Φ B must clearly be proportional to the current I 1 flowing in loop 1 and the constant of proportionality M 1, known as the mutual inductance, is a purely geometric quantity which depends on the relative orientation and positions of the loops as well as their shapes. A similar argument leads to the formula Φ B 1 = B da 1 = M 1 I. S 1 If we now replace the magnetic field B j = A j with the vector potential A j = µ 4π I j S j dl j r r j,

3 we find Φ Bk = A j dl k = µ S k 4π I dl j dl k j = M j k I j. S j S k r j r k Hence, the mutual inductance is symmetric, M 1 = M 1 = M 1, and is expressed as M 1 = µ 4π S dl 1 dl S r r = µ 4πI 1 I V dτ V dτ J 1 r J r r r For example, we consider the mutual inductance of two current loops of radii a and b separated by a distance Z; furthermore, the two loops are parallel and share the same axis see Figure below.. Here, the current densities are J 1 r = I 1 δr aδz Z θ 1 θ and J r = I δr bδz θ θ, where θ 1 θ = cosθ θ, so that M = µ π 4π ab = µ abq dθ π π dθ cosθ θ Z + a + b ab cosθ θ cos χdχ = µ abq d 1 q cos χ dq π 1 q cos χdχ where we used the substitution θ, θ θ, χ = θ θ, and integrated over θ, and introduced the dimensionless quantity q = ab/z +a +b. Note that the Taylor expansion of 1 q cos χ yields, 1 q cos χ = 1 q cos χ q cos χ q3 cos3 χ 5 q4 8 cos 4 χ, 3

4 so that d π 1 q cos χdχ = πq dq 8 q +, and the mutual inductance between the two circular loops is M = µ π abq q + µ πa πb, π Z 3 which is valid when Z a, b with q ab/z 1. A single wire loop also experiences a self-inductance, labeled L, which opposes a change in the current flowing through it: Φ B = LI E = dφ B = L di. Here, we note that both self-inductance L and mutual inductance M are measured in henries SI units, abbreviated H, and defined as H = V s. For example, the self-inductance of a rectangular toroidal coil height h, inner radius a, and outer radius b with N turns is L = N I B da = N I b a dr h dz µ NI πr = µ π N h lnb/a. The self-inductance, thus, plays the role of electromagnetic inertia, i.e., the large selfinductance of an electrical circuit makes it difficult to either initiate current flow in the circuit or vary the current flowing through the circuit. Magnetic Energy in a Current Distribution The rate of work done in allowing a current I to flow through a circuit with selfinductance L is dw = EI = LI di = d L I, and, thus, the work done in setting a current distribution is W = 1 Φ B I = I A dl = 1 A J dτ. If we now make use of Ampère s Law, J = µ 1 B, and integrate by parts, we find W = 1 A B dτ = 1 [ ] B dτ A B dτ µ V µ V V = 1 [ ] B dτ A B da. µ V V If the integration volume V is R 3, for which the magnetic field B must vanish on V, the magnetic energy stored in a current distribution is W = 1 µ 1 R 3 B dτ. 4 4

5 Maxwell s Equation Following up on Faraday s Law, which states that a time-dependent magnetic field generates a nonuniform electric field, we naturally ask the question: Will a time-dependent electric field generate a nonuniform magnetic field? In order to provide an answer, we begin with the following puzzle involving Ampère s Law: = B = µ J, where J does not vanish for time-dependent charge and currents distributions. Hence, Ampère s Law is incomplete for time-dependent fields. Fortunately, from the charge conservation law and Gauss s Law for the electric field, we find J = ρ = ɛ, and, therefore, the solution proposed by Maxwell is to modify Ampère s Law and write B = µ J + µ ɛ, 5 which could be called the Ampère-Maxwell Law. It states that, even in the absence of a current density J, a time-dependent electric field will produce a nonuniform magnetic field. In summary, Maxwell s equations are E = ɛ 1 ρ Gauss s Law for E B = Gauss s Law for B E = B B = µ J + µ ɛ Faraday s Law Ampère-Maxwell s Law An important note must be made concerning the charge conservation law used above. On the one hand, the charge density ρ is known to be the sum of the free-charge density ρ f and the bound-charge density ρ b = P expressed in terms of the polarization P, so that ρ = ρ f P. On the other hand, the current density must now be the sum of the free-current density J f, the bound-current density J b = M a divergenceless current expressed in terms of the 5

6 magnetization M, and a polarization current J p associated with the fact the polarization P is time dependent. Hence, J = J f + J p = ρ = ρ f ρ b, and, since the free-charge and free-current distributions must satisfy the charge conservation law separately i.e., J f = ρ f /, we find J p = ρ b = P J p = P. From these definitions, we may now express the Ampère-Maxwell Law solely in terms of the free-current density as H = J f + D, where D = ɛ E + P = ɛ E and H = µ 1 B M = µ 1 B. Energy-Momentum Transport When we combine the electric and magnetic energies found in a disbtribution of charges and currents, we find W = 1 ɛ E + µ 1 R 3 B dτ = 1 D E + H B dτ, R 3 where the last expression takes into account the permitivity ɛ and permeability µ of the medium. If we introduce the electromagnetic energy density E = 1 ɛ E + µ 1 B, we find, by using Maxwell s equations, that = ɛ E + µ 1 B B = E µ 1 B J µ 1 B E, which can be re-arranged as + S = E J, 6 where S = E µ 1 B represents the Poynting electromagnetic-energy flux. Hence, the total electromagnetic energy E = V Edτ is dissipated at a rate E J. Since the electromagnetic field carries energy, it also carries momentum, with momentum density defined as Π = ɛ E B. Hence, using Maxwell s equations once again, we find Π = ɛ B + ɛ E B = µ 1 B J B ɛ E E, 6

7 which can be re-arranged as Π + T = ρ E J B, 7 where the momentum-stress tensor for the electromagnetic field is defined as T = E I ɛ EE + µ 1 BB. Electromagnetic Waves in Free Space An important limit of Maxwell s equations is provided by the case of electromagnetic fields in free space defined by the absence of charge and current densities. In free space, Maxwell s equations are thus E = = B E = B and B = 1 c, where we note that µ ɛ =1/c c denotes the speed of light. By taking the curl of Faraday s Law and using the Ampère-Maxwell Law, we find E = B which becomes 1 c = 1 c E, E =, after using the vector identity E = E E = E, where the second equality follows from Gauss s Law for E in free space. Note that the magnetic field satisfies a similar equation 1 B =. c The plane-wave solutions for the electric and magnetic fields are Ex,t Ẽ Ẽ = e i k x ωt + e i k x ωt, Bx,t B B where denotes the complex conjugate and the electromagnetic waves satisfy the dispersion relation ω = k c, i.e., electromagnetic waves are light waves. Lastly, we find that the time-averaged energy density and Poynting flux are E = ɛ 1 + k c ω Ẽ = ɛ Ẽ and S = kc ω E = c Π. 7

Edmund Li. Where is defined as the mutual inductance between and and has the SI units of Henries (H).

Edmund Li. Where is defined as the mutual inductance between and and has the SI units of Henries (H). INDUCTANCE MUTUAL INDUCTANCE If we consider two neighbouring closed loops and with bounding surfaces respectively then a current through will create a magnetic field which will link with as the flux passes

More information

Lecture 22. Inductance. Magnetic Field Energy. Outline:

Lecture 22. Inductance. Magnetic Field Energy. Outline: Lecture 22. Inductance. Magnetic Field Energy. Outline: Self-induction and self-inductance. Inductance of a solenoid. The energy of a magnetic field. Alternative definition of inductance. Mutual Inductance.

More information

6 J - vector electric current density (A/m2 )

6 J - vector electric current density (A/m2 ) Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J - vector electric current density (A/m2 ) M - vector magnetic current density (V/m 2 ) Some problems

More information

Chapter 30 Inductance

Chapter 30 Inductance Chapter 30 Inductance - Mutual Inductance - Self-Inductance and Inductors - Magnetic-Field Energy - The R- Circuit - The -C Circuit - The -R-C Series Circuit . Mutual Inductance - A changing current in

More information

Eðlisfræði 2, vor 2007

Eðlisfræði 2, vor 2007 [ Assignment View ] [ Print ] Eðlisfræði 2, vor 2007 30. Inductance Assignment is due at 2:00am on Wednesday, March 14, 2007 Credit for problems submitted late will decrease to 0% after the deadline has

More information

Solution Derivations for Capa #11

Solution Derivations for Capa #11 Solution Derivations for Capa #11 Caution: The symbol E is used interchangeably for energy and EMF. 1) DATA: V b = 5.0 V, = 155 Ω, L = 8.400 10 2 H. In the diagram above, what is the voltage across the

More information

Electromagnetism Laws and Equations

Electromagnetism Laws and Equations Electromagnetism Laws and Equations Andrew McHutchon Michaelmas 203 Contents Electrostatics. Electric E- and D-fields............................................. Electrostatic Force............................................2

More information

Inductance and Magnetic Energy

Inductance and Magnetic Energy Chapter 11 Inductance and Magnetic Energy 11.1 Mutual Inductance... 11-3 Example 11.1 Mutual Inductance of Two Concentric Coplanar Loops... 11-5 11. Self-Inductance... 11-5 Example 11. Self-Inductance

More information

Electromagnetism - Lecture 2. Electric Fields

Electromagnetism - Lecture 2. Electric Fields Electromagnetism - Lecture 2 Electric Fields Review of Vector Calculus Differential form of Gauss s Law Poisson s and Laplace s Equations Solutions of Poisson s Equation Methods of Calculating Electric

More information

Fundamentals of Electromagnetic Fields and Waves: I

Fundamentals of Electromagnetic Fields and Waves: I Fundamentals of Electromagnetic Fields and Waves: I Fall 2007, EE 30348, Electrical Engineering, University of Notre Dame Mid Term II: Solutions Please show your steps clearly and sketch figures wherever

More information

Divergence and Curl of the Magnetic Field

Divergence and Curl of the Magnetic Field Divergence and Curl of the Magnetic Field The static electric field E(x,y,z such as the field of static charges obeys equations E = 1 ǫ ρ, (1 E =. (2 The static magnetic field B(x,y,z such as the field

More information

Problem 1 (25 points)

Problem 1 (25 points) MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2012 Exam Three Solutions Problem 1 (25 points) Question 1 (5 points) Consider two circular rings of radius R, each perpendicular

More information

Scott Hughes 7 April 2005. Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005. Lecture 15: Mutual and Self Inductance.

Scott Hughes 7 April 2005. Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005. Lecture 15: Mutual and Self Inductance. Scott Hughes 7 April 2005 151 Using induction Massachusetts nstitute of Technology Department of Physics 8022 Spring 2005 Lecture 15: Mutual and Self nductance nduction is a fantastic way to create EMF;

More information

Chapter 4. Electrostatic Fields in Matter

Chapter 4. Electrostatic Fields in Matter Chapter 4. Electrostatic Fields in Matter 4.1. Polarization A neutral atom, placed in an external electric field, will experience no net force. However, even though the atom as a whole is neutral, the

More information

Inductors & Inductance. Electronic Components

Inductors & Inductance. Electronic Components Electronic Components Induction In 1824, Oersted discovered that current passing though a coil created a magnetic field capable of shifting a compass needle. Seven years later, Faraday and Henry discovered

More information

Magnetic Field of a Circular Coil Lab 12

Magnetic Field of a Circular Coil Lab 12 HB 11-26-07 Magnetic Field of a Circular Coil Lab 12 1 Magnetic Field of a Circular Coil Lab 12 Equipment- coil apparatus, BK Precision 2120B oscilloscope, Fluke multimeter, Wavetek FG3C function generator,

More information

April 1. Physics 272. Spring 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html. Prof. Philip von Doetinchem philipvd@hawaii.

April 1. Physics 272. Spring 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html. Prof. Philip von Doetinchem philipvd@hawaii. Physics 272 April 1 Spring 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html Prof. Philip von Doetinchem philipvd@hawaii.edu Phys272 - Spring 14 - von Doetinchem - 164 Summary Gauss's

More information

Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation

Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of

More information

arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014

arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014 Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014 Abstract We discuss the theory of electromagnetic

More information

Magnetic Circuits. Outline. Ampere s Law Revisited Review of Last Time: Magnetic Materials Magnetic Circuits Examples

Magnetic Circuits. Outline. Ampere s Law Revisited Review of Last Time: Magnetic Materials Magnetic Circuits Examples Magnetic Circuits Outline Ampere s Law Revisited Review of Last Time: Magnetic Materials Magnetic Circuits Examples 1 Electric Fields Magnetic Fields S ɛ o E da = ρdv B V = Q enclosed S da =0 GAUSS GAUSS

More information

Elasticity Theory Basics

Elasticity Theory Basics G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold

More information

Eðlisfræði 2, vor 2007

Eðlisfræði 2, vor 2007 [ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline

More information

Chapter 30 - Magnetic Fields and Torque. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Chapter 30 - Magnetic Fields and Torque. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University Chapter 30 - Magnetic Fields and Torque A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University 2007 Objectives: After completing this module, you should

More information

12. The current in an inductor is changing at the rate of 100 A/s, and the inductor emf is 40 V. What is its self-inductance?

12. The current in an inductor is changing at the rate of 100 A/s, and the inductor emf is 40 V. What is its self-inductance? 12. The current in an inductor is changing at the rate of 100 A/s, and the inductor emf is 40 V. What is its self-inductance? From Equation 32-5, L = -E=(dI =dt) = 40 V=(100 A/s) = 0.4 H. 15. A cardboard

More information

Electrostatic Fields: Coulomb s Law & the Electric Field Intensity

Electrostatic Fields: Coulomb s Law & the Electric Field Intensity Electrostatic Fields: Coulomb s Law & the Electric Field Intensity EE 141 Lecture Notes Topic 1 Professor K. E. Oughstun School of Engineering College of Engineering & Mathematical Sciences University

More information

Module 22: Inductance and Magnetic Field Energy

Module 22: Inductance and Magnetic Field Energy Module 22: Inductance and Magnetic Field Energy 1 Module 22: Outline Self Inductance Energy in Inductors Circuits with Inductors: RL Circuit 2 Faraday s Law of Induction dφ = B dt Changing magnetic flux

More information

Inductance. Motors. Generators

Inductance. Motors. Generators Inductance Motors Generators Self-inductance Self-inductance occurs when the changing flux through a circuit arises from the circuit itself. As the current increases, the magnetic flux through a loop due

More information

EEE1001/PHY1002. Magnetic Circuits. The circuit is of length l=2πr. B andφ circulate

EEE1001/PHY1002. Magnetic Circuits. The circuit is of length l=2πr. B andφ circulate 1 Magnetic Circuits Just as we view electric circuits as related to the flow of charge, we can also view magnetic flux flowing around a magnetic circuit. The sum of fluxes entering a point must sum to

More information

CBE 6333, R. Levicky 1 Differential Balance Equations

CBE 6333, R. Levicky 1 Differential Balance Equations CBE 6333, R. Levicky 1 Differential Balance Equations We have previously derived integral balances for mass, momentum, and energy for a control volume. The control volume was assumed to be some large object,

More information

5. Measurement of a magnetic field

5. Measurement of a magnetic field H 5. Measurement of a magnetic field 5.1 Introduction Magnetic fields play an important role in physics and engineering. In this experiment, three different methods are examined for the measurement of

More information

Slide 1 / 26. Inductance. 2011 by Bryan Pflueger

Slide 1 / 26. Inductance. 2011 by Bryan Pflueger Slide 1 / 26 Inductance 2011 by Bryan Pflueger Slide 2 / 26 Mutual Inductance If two coils of wire are placed near each other and have a current passing through them, they will each induce an emf on one

More information

The purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law.

The purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law. 260 17-1 I. THEORY EXPERIMENT 17 QUALITATIVE STUDY OF INDUCED EMF Along the extended central axis of a bar magnet, the magnetic field vector B r, on the side nearer the North pole, points away from this

More information

EE301 Lesson 14 Reading: 10.1-10.4, 10.11-10.12, 11.1-11.4 and 11.11-11.13

EE301 Lesson 14 Reading: 10.1-10.4, 10.11-10.12, 11.1-11.4 and 11.11-11.13 CAPACITORS AND INDUCTORS Learning Objectives EE301 Lesson 14 a. Define capacitance and state its symbol and unit of measurement. b. Predict the capacitance of a parallel plate capacitor. c. Analyze how

More information

Chapter 11. Inductors ISU EE. C.Y. Lee

Chapter 11. Inductors ISU EE. C.Y. Lee Chapter 11 Inductors Objectives Describe the basic structure and characteristics of an inductor Discuss various types of inductors Analyze series inductors Analyze parallel inductors Analyze inductive

More information

Teaching Electromagnetic Field Theory Using Differential Forms

Teaching Electromagnetic Field Theory Using Differential Forms IEEE TRANSACTIONS ON EDUCATION, VOL. 40, NO. 1, FEBRUARY 1997 53 Teaching Electromagnetic Field Theory Using Differential Forms Karl F. Warnick, Richard H. Selfridge, Member, IEEE, and David V. Arnold

More information

1. A wire carries 15 A. You form the wire into a single-turn circular loop with magnetic field 80 µ T at the loop center. What is the loop radius?

1. A wire carries 15 A. You form the wire into a single-turn circular loop with magnetic field 80 µ T at the loop center. What is the loop radius? CHAPTER 3 SOURCES O THE MAGNETC ELD 1. A wire carries 15 A. You form the wire into a single-turn circular loop with magnetic field 8 µ T at the loop center. What is the loop radius? Equation 3-3, with

More information

Chapter 7: Polarization

Chapter 7: Polarization Chapter 7: Polarization Joaquín Bernal Méndez Group 4 1 Index Introduction Polarization Vector The Electric Displacement Vector Constitutive Laws: Linear Dielectrics Energy in Dielectric Systems Forces

More information

Last time : energy storage elements capacitor.

Last time : energy storage elements capacitor. Last time : energy storage elements capacitor. Charge on plates Energy stored in the form of electric field Passive sign convention Vlt Voltage drop across real capacitor can not change abruptly because

More information

Direction of Induced Current

Direction of Induced Current Direction of Induced Current Bar magnet moves through coil Current induced in coil A S N v Reverse pole Induced current changes sign B N S v v Coil moves past fixed bar magnet Current induced in coil as

More information

Physics 25 Exam 3 November 3, 2009

Physics 25 Exam 3 November 3, 2009 1. A long, straight wire carries a current I. If the magnetic field at a distance d from the wire has magnitude B, what would be the the magnitude of the magnetic field at a distance d/3 from the wire,

More information

Faraday s Law of Induction

Faraday s Law of Induction Chapter 10 Faraday s Law of Induction 10.1 Faraday s Law of Induction...10-10.1.1 Magnetic Flux...10-3 10.1. Lenz s Law...10-5 10. Motional EMF...10-7 10.3 Induced Electric Field...10-10 10.4 Generators...10-1

More information

Figure 1.1 Vector A and Vector F

Figure 1.1 Vector A and Vector F CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

More information

Lecture 5. Electric Flux and Flux Density, Gauss Law in Integral Form

Lecture 5. Electric Flux and Flux Density, Gauss Law in Integral Form Lecture 5 Electric Flux and Flux ensity, Gauss Law in Integral Form ections: 3.1, 3., 3.3 Homework: ee homework file LECTURE 5 slide 1 Faraday s Experiment (1837), Flux charge transfer from inner to outer

More information

Chapter 28 Fluid Dynamics

Chapter 28 Fluid Dynamics Chapter 28 Fluid Dynamics 28.1 Ideal Fluids... 1 28.2 Velocity Vector Field... 1 28.3 Mass Continuity Equation... 3 28.4 Bernoulli s Principle... 4 28.5 Worked Examples: Bernoulli s Equation... 7 Example

More information

Physics of the Atmosphere I

Physics of the Atmosphere I Physics of the Atmosphere I WS 2008/09 Ulrich Platt Institut f. Umweltphysik R. 424 Ulrich.Platt@iup.uni-heidelberg.de heidelberg.de Last week The conservation of mass implies the continuity equation:

More information

Chapter 22: The Electric Field. Read Chapter 22 Do Ch. 22 Questions 3, 5, 7, 9 Do Ch. 22 Problems 5, 19, 24

Chapter 22: The Electric Field. Read Chapter 22 Do Ch. 22 Questions 3, 5, 7, 9 Do Ch. 22 Problems 5, 19, 24 Chapter : The Electric Field Read Chapter Do Ch. Questions 3, 5, 7, 9 Do Ch. Problems 5, 19, 4 The Electric Field Replaces action-at-a-distance Instead of Q 1 exerting a force directly on Q at a distance,

More information

Chapter 27 Magnetic Field and Magnetic Forces

Chapter 27 Magnetic Field and Magnetic Forces Chapter 27 Magnetic Field and Magnetic Forces - Magnetism - Magnetic Field - Magnetic Field Lines and Magnetic Flux - Motion of Charged Particles in a Magnetic Field - Applications of Motion of Charged

More information

Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 25 Notes

Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 25 Notes Jim Lambers MAT 169 Fall Semester 009-10 Lecture 5 Notes These notes correspond to Section 10.5 in the text. Equations of Lines A line can be viewed, conceptually, as the set of all points in space that

More information

Exercises on Voltage, Capacitance and Circuits. A d = (8.85 10 12 ) π(0.05)2 = 6.95 10 11 F

Exercises on Voltage, Capacitance and Circuits. A d = (8.85 10 12 ) π(0.05)2 = 6.95 10 11 F Exercises on Voltage, Capacitance and Circuits Exercise 1.1 Instead of buying a capacitor, you decide to make one. Your capacitor consists of two circular metal plates, each with a radius of 5 cm. The

More information

CHAPTER 24 GAUSS S LAW

CHAPTER 24 GAUSS S LAW CHAPTER 4 GAUSS S LAW 4. The net charge shown in Fig. 4-40 is Q. Identify each of the charges A, B, C shown. A B C FIGURE 4-40 4. From the direction of the lines of force (away from positive and toward

More information

A METHOD OF CALIBRATING HELMHOLTZ COILS FOR THE MEASUREMENT OF PERMANENT MAGNETS

A METHOD OF CALIBRATING HELMHOLTZ COILS FOR THE MEASUREMENT OF PERMANENT MAGNETS A METHOD OF CALIBRATING HELMHOLTZ COILS FOR THE MEASUREMENT OF PERMANENT MAGNETS Joseph J. Stupak Jr, Oersted Technology Tualatin, Oregon (reprinted from IMCSD 24th Annual Proceedings 1995) ABSTRACT The

More information

Phys222 Winter 2012 Quiz 4 Chapters 29-31. Name

Phys222 Winter 2012 Quiz 4 Chapters 29-31. Name Name If you think that no correct answer is provided, give your answer, state your reasoning briefly; append additional sheet of paper if necessary. 1. A particle (q = 5.0 nc, m = 3.0 µg) moves in a region

More information

ElectroMagnetic Induction. AP Physics B

ElectroMagnetic Induction. AP Physics B ElectroMagnetic Induction AP Physics B What is E/M Induction? Electromagnetic Induction is the process of using magnetic fields to produce voltage, and in a complete circuit, a current. Michael Faraday

More information

Homework #11 203-1-1721 Physics 2 for Students of Mechanical Engineering

Homework #11 203-1-1721 Physics 2 for Students of Mechanical Engineering Homework #11 203-1-1721 Physics 2 for Students of Mechanical Engineering 2. A circular coil has a 10.3 cm radius and consists of 34 closely wound turns of wire. An externally produced magnetic field of

More information

Magnetostatics (Free Space With Currents & Conductors)

Magnetostatics (Free Space With Currents & Conductors) Magnetostatics (Free Space With Currents & Conductors) Suggested Reading - Shen and Kong Ch. 13 Outline Review of Last Time: Gauss s Law Ampere s Law Applications of Ampere s Law Magnetostatic Boundary

More information

Physics 210 Q1 2012 ( PHYSICS210BRIDGE ) My Courses Course Settings

Physics 210 Q1 2012 ( PHYSICS210BRIDGE ) My Courses Course Settings 1 of 11 9/7/2012 1:06 PM Logged in as Julie Alexander, Instructor Help Log Out Physics 210 Q1 2012 ( PHYSICS210BRIDGE ) My Courses Course Settings Course Home Assignments Roster Gradebook Item Library

More information

Physics 2102 Lecture 19. Physics 2102

Physics 2102 Lecture 19. Physics 2102 Physics 2102 Jonathan Dowling Physics 2102 Lecture 19 Ch 30: Inductors and RL Circuits Nikolai Tesla What are we going to learn? A road map Electric charge Electric force on other electric charges Electric

More information

Ampere's Law. Introduction. times the current enclosed in that loop: Ampere's Law states that the line integral of B and dl over a closed path is 0

Ampere's Law. Introduction. times the current enclosed in that loop: Ampere's Law states that the line integral of B and dl over a closed path is 0 1 Ampere's Law Purpose: To investigate Ampere's Law by measuring how magnetic field varies over a closed path; to examine how magnetic field depends upon current. Apparatus: Solenoid and path integral

More information

DEGREE: Bachelor's Degree in Industrial Electronics and Automation COURSE: 1º TERM: 2º WEEKLY PLANNING

DEGREE: Bachelor's Degree in Industrial Electronics and Automation COURSE: 1º TERM: 2º WEEKLY PLANNING SESSION WEEK COURSE: Physics II DEGREE: Bachelor's Degree in Industrial Electronics and Automation COURSE: 1º TERM: 2º WEEKLY PLANNING DESCRIPTION GROUPS (mark ) Indicate YES/NO If the session needs 2

More information

Force on a square loop of current in a uniform B-field.

Force on a square loop of current in a uniform B-field. Force on a square loop of current in a uniform B-field. F top = 0 θ = 0; sinθ = 0; so F B = 0 F bottom = 0 F left = I a B (out of page) F right = I a B (into page) Assume loop is on a frictionless axis

More information

Chapter 7. Magnetism and Electromagnetism ISU EE. C.Y. Lee

Chapter 7. Magnetism and Electromagnetism ISU EE. C.Y. Lee Chapter 7 Magnetism and Electromagnetism Objectives Explain the principles of the magnetic field Explain the principles of electromagnetism Describe the principle of operation for several types of electromagnetic

More information

CHAPTER - 1. Chapter ONE: WAVES CHAPTER - 2. Chapter TWO: RAY OPTICS AND OPTICAL INSTRUMENTS. CHAPTER - 3 Chapter THREE: WAVE OPTICS PERIODS PERIODS

CHAPTER - 1. Chapter ONE: WAVES CHAPTER - 2. Chapter TWO: RAY OPTICS AND OPTICAL INSTRUMENTS. CHAPTER - 3 Chapter THREE: WAVE OPTICS PERIODS PERIODS BOARD OF INTERMEDIATE EDUCATION, A.P., HYDERABAD REVISION OF SYLLABUS Subject PHYSICS-II (w.e.f 2013-14) Chapter ONE: WAVES CHAPTER - 1 1.1 INTRODUCTION 1.2 Transverse and longitudinal waves 1.3 Displacement

More information

1 Introduction. 2 Electric Circuits and Kirchoff s Laws. J.L. Kirtley Jr. 2.1 Conservation of Charge and KCL

1 Introduction. 2 Electric Circuits and Kirchoff s Laws. J.L. Kirtley Jr. 2.1 Conservation of Charge and KCL Massachusetts Institute of Technoloy Department of Electrical Enineerin and Computer Science 6.061 Introduction to Power Systems Class Notes Chapter 6 Manetic Circuit Analo to Electric Circuits J.L. Kirtley

More information

ES250: Electrical Science. HW7: Energy Storage Elements

ES250: Electrical Science. HW7: Energy Storage Elements ES250: Electrical Science HW7: Energy Storage Elements Introduction This chapter introduces two more circuit elements, the capacitor and the inductor whose elements laws involve integration or differentiation;

More information

Mechanics 1: Conservation of Energy and Momentum

Mechanics 1: Conservation of Energy and Momentum Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation

More information

Electromagnetic waves

Electromagnetic waves Chapter 8 lectromagnetic waves David Morin, morin@physics.harvard.edu The waves we ve dealt with so far in this book have been fairly easy to visualize. Waves involving springs/masses, strings, and air

More information

Force on Moving Charges in a Magnetic Field

Force on Moving Charges in a Magnetic Field [ Assignment View ] [ Eðlisfræði 2, vor 2007 27. Magnetic Field and Magnetic Forces Assignment is due at 2:00am on Wednesday, February 28, 2007 Credit for problems submitted late will decrease to 0% after

More information

Gauss Formulation of the gravitational forces

Gauss Formulation of the gravitational forces Chapter 1 Gauss Formulation of the gravitational forces 1.1 ome theoretical background We have seen in class the Newton s formulation of the gravitational law. Often it is interesting to describe a conservative

More information

Coupled Inductors. Introducing Coupled Inductors

Coupled Inductors. Introducing Coupled Inductors Coupled Inductors From power distribution across large distances to radio transmissions, coupled inductors are used extensively in electrical applications. Their properties allow for increasing or decreasing

More information

13 ELECTRIC MOTORS. 13.1 Basic Relations

13 ELECTRIC MOTORS. 13.1 Basic Relations 13 ELECTRIC MOTORS Modern underwater vehicles and surface vessels are making increased use of electrical actuators, for all range of tasks including weaponry, control surfaces, and main propulsion. This

More information

Examples of Uniform EM Plane Waves

Examples of Uniform EM Plane Waves Examples of Uniform EM Plane Waves Outline Reminder of Wave Equation Reminder of Relation Between E & H Energy Transported by EM Waves (Poynting Vector) Examples of Energy Transport by EM Waves 1 Coupling

More information

RUPHYS2272015 ( RUPHY227F2015 ) My Courses Course Settings University Physics with Modern Physics, 14e Young/Freedman

RUPHYS2272015 ( RUPHY227F2015 ) My Courses Course Settings University Physics with Modern Physics, 14e Young/Freedman Signed in as Jolie Cizewski, Instructor Help Sign Out RUPHYS2272015 ( RUPHY227F2015 ) My Courses Course Settings University Physics with Modern Physics, 14e Young/Freedman Course Home Assignments Roster

More information

Objectives. Capacitors 262 CHAPTER 5 ENERGY

Objectives. Capacitors 262 CHAPTER 5 ENERGY Objectives Describe a capacitor. Explain how a capacitor stores energy. Define capacitance. Calculate the electrical energy stored in a capacitor. Describe an inductor. Explain how an inductor stores energy.

More information

Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows

Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows 3.- 1 Basics: equations of continuum mechanics - balance equations for mass and momentum - balance equations for the energy and the chemical

More information

F = 0. x ψ = y + z (1) y ψ = x + z (2) z ψ = x + y (3)

F = 0. x ψ = y + z (1) y ψ = x + z (2) z ψ = x + y (3) MATH 255 FINAL NAME: Instructions: You must include all the steps in your derivations/answers. Reduce answers as much as possible, but use exact arithmetic. Write neatly, please, and show all steps. Scientists

More information

Review Questions PHYS 2426 Exam 2

Review Questions PHYS 2426 Exam 2 Review Questions PHYS 2426 Exam 2 1. If 4.7 x 10 16 electrons pass a particular point in a wire every second, what is the current in the wire? A) 4.7 ma B) 7.5 A C) 2.9 A D) 7.5 ma E) 0.29 A Ans: D 2.

More information

Solutions to Homework 10

Solutions to Homework 10 Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

More information

Mutual Inductance and Transformers F3 3. r L = ω o

Mutual Inductance and Transformers F3 3. r L = ω o utual Inductance and Transformers F3 1 utual Inductance & Transformers If a current, i 1, flows in a coil or circuit then it produces a magnetic field. Some of the magnetic flux may link a second coil

More information

This makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5

This makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5 1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,

More information

potential in the centre of the sphere with respect to infinity.

potential in the centre of the sphere with respect to infinity. Umeå Universitet, Fysik 1 Vitaly Bychkov Prov i fysik, Electricity and Waves, 2006-09-27, kl 16.00-22.00 Hjälpmedel: Students can use any book. Define the notations you are using properly. Present your

More information

Chapter 22: Electric Flux and Gauss s Law

Chapter 22: Electric Flux and Gauss s Law 22.1 ntroduction We have seen in chapter 21 that determining the electric field of a continuous charge distribution can become very complicated for some charge distributions. t would be desirable if we

More information

m i: is the mass of each particle

m i: is the mass of each particle Center of Mass (CM): The center of mass is a point which locates the resultant mass of a system of particles or body. It can be within the object (like a human standing straight) or outside the object

More information

A RIGOROUS AND COMPLETED STATEMENT ON HELMHOLTZ THEOREM

A RIGOROUS AND COMPLETED STATEMENT ON HELMHOLTZ THEOREM Progress In Electromagnetics Research, PIER 69, 287 304, 2007 A RIGOROU AND COMPLETED TATEMENT ON HELMHOLTZ THEOREM Y. F. Gui and W. B. Dou tate Key Lab of Millimeter Waves outheast University Nanjing,

More information

Candidate Number. General Certificate of Education Advanced Level Examination June 2014

Candidate Number. General Certificate of Education Advanced Level Examination June 2014 entre Number andidate Number Surname Other Names andidate Signature General ertificate of Education dvanced Level Examination June 214 Physics PHY4/1 Unit 4 Fields and Further Mechanics Section Wednesday

More information

Practice Final Math 122 Spring 12 Instructor: Jeff Lang

Practice Final Math 122 Spring 12 Instructor: Jeff Lang Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6

More information

CONSERVATION LAWS. See Figures 2 and 1.

CONSERVATION LAWS. See Figures 2 and 1. CONSERVATION LAWS 1. Multivariable calculus 1.1. Divergence theorem (of Gauss). This states that the volume integral in of the divergence of the vector-valued function F is equal to the total flux of F

More information

Physics 221 Experiment 5: Magnetic Fields

Physics 221 Experiment 5: Magnetic Fields Physics 221 Experiment 5: Magnetic Fields August 25, 2007 ntroduction This experiment will examine the properties of magnetic fields. Magnetic fields can be created in a variety of ways, and are also found

More information

Chapter 12 Driven RLC Circuits

Chapter 12 Driven RLC Circuits hapter Driven ircuits. A Sources... -. A ircuits with a Source and One ircuit Element... -3.. Purely esistive oad... -3.. Purely Inductive oad... -6..3 Purely apacitive oad... -8.3 The Series ircuit...

More information

Basic Equations, Boundary Conditions and Dimensionless Parameters

Basic Equations, Boundary Conditions and Dimensionless Parameters Chapter 2 Basic Equations, Boundary Conditions and Dimensionless Parameters In the foregoing chapter, many basic concepts related to the present investigation and the associated literature survey were

More information

Exam 1 Practice Problems Solutions

Exam 1 Practice Problems Solutions MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8 Spring 13 Exam 1 Practice Problems Solutions Part I: Short Questions and Concept Questions Problem 1: Spark Plug Pictured at right is a typical

More information

SIO 229 Gravity and Geomagnetism: Class Description and Goals

SIO 229 Gravity and Geomagnetism: Class Description and Goals SIO 229 Gravity and Geomagnetism: Class Description and Goals This graduate class provides an introduction to gravity and geomagnetism at a level suitable for advanced non-specialists in geophysics. Topics

More information

State of Stress at Point

State of Stress at Point State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

More information

Magnetism Basics. Magnetic Domains: atomic regions of aligned magnetic poles Random Alignment Ferromagnetic Alignment. Net Effect = Zero!

Magnetism Basics. Magnetic Domains: atomic regions of aligned magnetic poles Random Alignment Ferromagnetic Alignment. Net Effect = Zero! Magnetism Basics Source: electric currents Magnetic Domains: atomic regions of aligned magnetic poles Random Alignment Ferromagnetic Alignment Net Effect = Zero! Net Effect = Additive! Bipolar: all magnets

More information

PHYS 222 Spring 2012 Final Exam. Closed books, notes, etc. No electronic device except a calculator.

PHYS 222 Spring 2012 Final Exam. Closed books, notes, etc. No electronic device except a calculator. PHYS 222 Spring 2012 Final Exam Closed books, notes, etc. No electronic device except a calculator. NAME: (all questions with equal weight) 1. If the distance between two point charges is tripled, the

More information

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

More information

An equivalent circuit of a loop antenna.

An equivalent circuit of a loop antenna. 3.2.1. Circuit Modeling: Loop Impedance A loop antenna can be represented by a lumped circuit when its dimension is small with respect to a wavelength. In this representation, the circuit parameters (generally

More information

Chapter 10. Faraday s Law of Induction

Chapter 10. Faraday s Law of Induction 10 10 10-0 Chapter 10 Faraday s Law of Induction 10.1 Faraday s Law of Induction... 10-3 10.1.1 Magnetic Flux... 10-5 10.2 Motional EMF... 10-5 10.3 Faraday s Law (see also Faraday s Law Simulation in

More information

Electromagnetic Induction: Faraday's Law

Electromagnetic Induction: Faraday's Law 1 Electromagnetic Induction: Faraday's Law OBJECTIVE: To understand how changing magnetic fields can produce electric currents. To examine Lenz's Law and the derivative form of Faraday's Law. EQUIPMENT:

More information

Scalars, Vectors and Tensors

Scalars, Vectors and Tensors Scalars, Vectors and Tensors A scalar is a physical quantity that it represented by a dimensional number at a particular point in space and time. Examples are hydrostatic pressure and temperature. A vector

More information