Fundamentals of Electromagnetics for Teaching and Learning: A Two-Week Intensive Course for Faculty in Electrical-, Electronics-, Communication-, and Computer- Related Engineering Departments in Engineering Colleges in India by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, India

Program for Hyderabad Area and Andhra Pradesh Faculty Sponsored by IEEE Hyderabad Section, IETE Hyderabad Center, and Vasavi College of Engineering IETE Conference Hall, Osmania University Campus Hyderabad, Andhra Pradesh June 3 – June 11, 2009 Workshop for Master Trainer Faculty Sponsored by IUCEE (Indo-US Coalition for Engineering Education) Infosys Campus, Mysore, Karnataka June 22 – July 3, 2009

1-2 Maxwell’s Equations Electric field intensity Magnetic flux density Charge density Magnetic field intensity Current density Displacement flux density E  dl  – d dt B  dS S  C  D  dS  dv V  S  H  dl  J  dS  d dt S  C  D  dS S  B  dS  0 S 

1-3 Module 1 Vectors and Fields 1.1 Vector algebra 1.2 Cartesian coordinate system 1.3 Cylindrical and spherical coordinate systems 1.4 Scalar and vector fields 1.5 Sinusoidally time-varying fields 1.6 The electric field 1.7 The magnetic field 1.8 Lorentz force equation

1-4 Instructional Objectives 1.Perform vector algebraic operations in Cartesian, cylindrical, and spherical coordinate systems 2.Find the unit normal vector and the differential surface at a point on the surface 3.Find the equation for the direction lines associated with a vector field 4.Identify the polarization of a sinusoidally time-varying vector field 5.Calculate the electric field due to a charge distribution by applying superposition in conjunction with the electric field due to a point charge 6.Calculate the magnetic field due to a current distribution by applying superposition in conjunction with the magnetic field due to a current element

1-5 Instructional Objectives (Continued) 7.Apply Lorentz force equation to find the electric and magnetic fields, for a specified set of forces on a charged particle moving in the field region

Vector Algebra (EEE, Sec. 1.1; FEME, Sec. 1.1) In this series of PowerPoint presentations, EEE refers to “Elements of Engineering Electromagnetics, 6th Edition,” Indian Edition (2006), and FEME refers to “Fundamentals of Electromagnetics for Engineering,” Indian Edition (2009). Also, all “D” Problems and “P” Problems are from EEE.

1-7 (1)Vectors (A)vs.Scalars (A) Magnitude and directionMagnitude only Ex: Velocity, Force Ex: Mass, Charge

1-8 (2)Unit Vectors have magnitude unity, denoted by symbol a with subscript. We shall use the right-handed system throughout. Useful for expressing vectors in terms of their components.

1-9 (3)Dot Product is a scalar A A B = AB cos   B Useful for finding angle between two vectors.

1-10 (4)Cross Product is a vector A A B = AB sin   B is perpendicular to both A and B. Useful for finding unit vector perpendicular to two vectors. anan

1-11 where (5)Triple Cross Product in general.

1-12 (6)Scalar Triple Product is a scalar.

1-13 Volume of the parallelepiped

1-14 D1.2 (EEE)A= 3a 1 + 2a 2 + a 3 B= a 1 + a 2 – a 3 C= a 1 + 2a 2 + 3a 3 (a) A+B – 4C =(3 + 1 – 4)a 1 + (2 + 1 – 8)a 2 + (1 – 1 – 12)a 3 =– 5a 2 – 12a 3

1-15 (b)A+2B – C =(3 + 2 – 1)a 1 + (2 + 2 – 2)a 2 + (1 – 2 – 3)a 3 =4a 1 + 2a 2 – 4a 3 Unit Vector =

1-16 (c)AC= =10 (d) = =5a 1 – 4a 2 + a 3

1-17 (e) =15 – = 8 Same as A (B C)=(3a 1 + 2a 2 + a 3 ) (5a 1 – 4a 2 + a 3 ) = (–4) =15 – =8

1-18 P1.5 (EEE) D=B – A( A + D = B) E=C – B( B + E = C) D and E lie along a straight line.

1-19 What is the geometric interpretation of this result?

1-20 E1.1 Another Example Given Find A.

1-21 To find C, use (1) or (2).

1-22 Review Questions 1.1. Give some examples of scalars Give some examples of vectors Is it necessary for the reference vectors a 1, a 2, and a 3 to be an orthogonal set? 1.4. State whether a 1, a 2, and a 3 directed westward, northward, and downward, respectively, is a right- handed or a left-handed set State all conditions for which A B is zero State all conditions for which A × B is zero What is the significance of A B × C = 0? 1.8. What is the significance of A × (B × C) = 0?

1-23 Problem S1.1. Performing several vector algebraic manipulations

1-24 Problem S1.1. Performing several vector algebraic manipulations (continued)

Cartesian Coordinate System (EEE, Sec. 1.2; FEME, Sec. 1.2)

1-26 Cartesian Coordinate System

1-27 Cartesian Coordinate System

1-28 Right-handed system xyz xy… a x, a y, a z are uniform unit vectors, that is, the direction of each unit vector is same everywhere in space.

1-29 Vector drawn from one point to another: From P 1 (x 1, y 1, z 1 ) to P 2 (x 2, y 2, z 2 )

1-30

1-31 P1.8 A(12, 0, 0), B(0, 15, 0), C(0, 0, –20). (a)Distance from B to C = (b)Component of vector from A to C along vector from B to C = Vector from A to C Unit vector along vector from B to C

1-32 (c)Perpendicular distance from A to the line through B and C =

1-33 (2)Differential Length Vector (dl)

1-34 dl=dx a x + dy a y =dx a x + f (x) dx a y Unit vector normal to a surface

1-35 D1.5Find dl along the line and having the projection dz on the z-axis. (a) (b)

1-36 (c)Line passing through (0, 2, 0) and (0, 0, 1).

1-37 (3)Differential Surface Vector (dS) Orientation of the surface is defined uniquely by the normal ± a n to the surface. For example, in Cartesian coordinates, dS in any plane parallel to the xy plane is

1-38 (4)Differential Volume (dv) In Cartesian coordinates,

1-39 Review Questions 1.9. What is the particular advantageous characteristic associated with unit vectors in the Cartesian coordinate system? What is the position vector? What is the total distance around the circumference of a circle of radius 1 m? What is the total vector distance around the circle? Discuss the application of differential length vectors to find a unit vector normal to a surface at a point on the surface Discuss the concept of a differential surface vector What is the total surface area of a cube of sides 1 m? Assuming the normals to the surfaces to be directed outward of the cubical volume, what is the total vector surface area of the cube?

1-40 Problem S1.2. Finding the unit vector normal to a surface and the differential surface vector, at a point on it

Cylindrical and Spherical Coordinate Systems (EEE, Sec. 1.3; FEME, Appendix A)

1-42 Cylindrical Coordinate System

1-43 Spherical Coordinate System

1-44 Cylindrical (r, , z)Spherical (r, ,  ) Only a z is uniform.All three unit a r and a  arevectors arenonuniform. Cylindrical and Spherical Coordinate Systems

1-45 x=r cos  x=r sin  cos  y=r sin  y=r sin  sin  z=zz=r cos  D1.7(a)(2, 5  /6, 3) in cylindrical coordinates

1-46 (b)

1-47 (c)

1-48 (d)

1-49 Conversion of vectors between coordinate systems

1-50 P1.18A=a r at(2,  /6,  2) B=a  at(1,  /3, 0) C=a  at(3,  /4, 3  /2)

1-51

1-52 (a) (b)

1-53 (c) (d)

1-54 Differential length vectors: Cylindrical Coordinates: dl = dr a r + r d  a  + dz a z Spherical Coordinates: dl = dr a r + r d  a  + r sin  d  a 

1-55 Review Questions Describe the three orthogonal surfaces that define the cylindrical coordinates of a point Which of the unit vectors in the cylindrical coordinate system are not uniform? Explain Discuss the conversion from the cylindrical coordinates of a point to its Cartesian coordinates Describe the three orthogonal surfaces that define the spherical coordinates of a point Discuss the nonuniformity of the unit vectors in the spherical coordinate system Discuss the conversion from the cylindrical coordinates of a point to its Cartesian coordinates.

1-56 Problem S1.3. Determination of the equality of vectors specified in cylindrical and spherical coordinates

1-57 Problem S1.4. Finding the unit vector tangential to a curve, at a point on it, in spherical coordinates

Scalar and Vector Fields (EEE, Sec. 1.4; FEME, Sec. 1.3)

1-59 FIELD is a description of how a physical quantity varies from one point to another in the region of the field (and with time). (a)Scalar fields Ex:Depth of a lake, d(x, y) Temperature in a room, T(x, y, z) Depicted graphically by constant magnitude contours or surfaces.

1-60 (b)Vector Fields Ex:Velocity of points on a rotating disk v(x, y) = v x (x, y)a x + v y (x, y)a y Force field in three dimensions F(x, y, z)= F x (x, y, z)a x + F y (x, y, z)a y + F z (x, y, z)a z Depicted graphically by constant magnitude contours or surfaces, and direction lines (or stream lines).

1-61 Example: Linear velocity vector field of points on a rotating disk

1-62 (c)Static Fields Fields not varying with time. (d)Dynamic Fields Fields varying with time. Ex:Temperature in a room, T(x, y, z; t)

1-63 D1.10 T(x, y, z, t) Constant temperature surfaces are elliptic cylinders, (a)

1-64 (b) Constant temperature surfaces are spheres (c) Constant temperature surfaces are ellipsoids,

1-65 Procedure for finding the equation for the direction lines of a vector field The field F is tangential to the direction line at all points on a direction line.

1-66 Similarly cylindrical spherical

1-67 P1.26(b) (Position vector)

1-68  Direction lines are straight lines emanating radially from the origin. For the line passing through (1, 2, 3),

1-69 Review Questions Discuss briefly your concept of a scalar field and illustrate with an example Discuss briefly your concept of a vector field and illustrate with an example How do you depict pictorially the gravitational field of the earth? Discuss the procedure for obtaining the equations for the direction lines of a vector field.

1-70 Problem S1.5. Finding the equation for direction line of a vector field, specified in spherical coordinates

Sinusoidally Time-Varying Fields (EEE, Sec. 3.6; FEME, Sec. 1.4)

1-72 Sinusoidal function of time

1-73 Polarization is the characteristic which describes how the position of the tip of the vector varies with time. Linear Polarization: Tip of the vector describes a line. Circular Polarization: Tip of the vector describes a circle.

1-74 Elliptical Polarization: Tip of the vector describes an ellipse. (i)Linear Polarization Linearly polarized in the x direction. Direction remains along the x axis Magnitude varies sinusoidally with time

1-75 Linear polarization

1-76 Direction remains along the y axis Magnitude varies sinusoidally with time Linearly polarized in the y direction. If two (or more) component linearly polarized vectors are in phase, (or in phase opposition), then their sum vector is also linearly polarized. Ex:

1-77 Sum of two linearly polarized vectors in phase (or in phase opposition) is a linearly polarized vector

1-78 (ii) Circular Polarization If two component linearly polarized vectors are (a) equal in amplitude (b) differ in direction by 90˚ (c) differ in phase by 90˚, then their sum vector is circularly polarized.

1-79 Circular Polarization

1-80 Example:

1-81 (iii) Elliptical Polarization In the general case in which either (i) or (ii) is not satisfied, then the sum of the two component linearly polarized vectors is an elliptically polarized vector. Example:

1-82 Example:

1-83 D3.17 F 1 and F 2 are equal in amplitude (= F 0 ) and differ in direction by 90˚. The phase difference (say  ) depends on z in the manner –2  z – (–3  z) =  z. (a)At (3, 4, 0),  =  (0) = 0. (b)At (3, –2, 0.5),  =  (0.5) = 0.5 .

1-84 (c) At (–2, 1, 1),  =  (1) = . (d)At (–1, –3, 0.2) =  =  (0.2) = 0.2 .

1-85 Review Questions A sinusoidally time-varying vector is expressed in terms of its components along the x-, y-, and z- axes. What is the polarization of each of the components? What are the conditions for the sum of two linearly polarized sinusoidally time-varying vectors to be circularly polarized? What is the polarization for the general case of the sum of two sinusoidally time-varying linearly polarized vectors having arbitrary amplitudes, phase angles, and directions? Considering the seconds hand on your analog watch to be a vector, state its polarization. What is the frequency?

1-86 Problem S1.6. Finding the polarization of the sum of two sinusoidally time-varying vector fields

The Electric Field (EEE, Sec. 1.5; FEME, Sec. 1.5)

1-88 The Electric Field is a force field acting on charges by virtue of the property of charge. Coulomb’s Law

1-89 D1.13(b) From the construction, it is evident that the resultant force is directed away from the center of the square. The magnitude of this resultant force is given by Q 2 /4  0 (2a 2 ) Q 2 /4  0 (4a 2 ) Q 2 /4  0 (2a 2 )

1-90

1-91 Electric Field Intensity, E is defined as the force per unit charge experienced by a small test charge when placed in the region of the field. Thus Units: Sources:Charges; Time-varying magnetic field

1-92 Electric Field of a Point Charge (Coulomb’s Law)

1-93 Constant magnitude surfaces are spheres centered at Q. Direction lines are radial lines emanating from Q. E due to charge distributions (a) Collection of point charges

1-94 E1.2  Electron (charge e and mass m) is displaced from the origin by  (<< d) in the +x-direction and released from rest at t = 0. We wish to obtain differential equation for the motion of the electron and its solution.

1-95 For any displacement x, is directed toward the origin, and

1-96 The differential equation for the motion of the electron is Solution is given by

1-97 Using initial conditions and at t = 0, we obtain which represents simple harmonic motion about the origin with period

1-98 (b)Line Charges Line charge density,  L (C/m) (c)Surface Charges Surface charge density,  S (C/m 2 ) (d)Volume Charges Volume charge density,  (C/m 3 )

1-99 E1.3 Finitely-Long Line Charge  

1-100

1-101 Infinite Plane Sheet of Charge of Uniform Surface Charge Density 

1-102

1-103

1-104 D1.16 Given

1-105 Solving, we obtain (d) (a) (c) (b)

1-106 Review Questions State Coulomb’s law. To what law in mechanics is Coulomb’s law analogous? What is the value of the permittivity of free space? What are its units? What is the definition of electric field intensity? What are its units? Describe the electric field due to a point charge Discuss the different types of charge distributions. How do you determine the electric field due to a charge distribution? Describe the electric field due to an infinitely long line charge of uniform density Describe the electric field due to an infinite plane sheet of uniform surface charge density.

1-107 Problem S1.7. Determination of conditions for three point charges on a circle to be in equilibrium

1-108 Problem S1.8. Finding the electric field due to an infinite plane slab charge of specified charge density

The Magnetic Field (EEE, Sec. 1.6; FEME, Sec. 1.6)

1-110 The Magnetic Field acts to exert force on charge when it is in motion. B = Magnetic flux density vector Alternatively, since charge in motion constitutes current, magnetic field exerts forces on current elements.

1-111 Units of B: Sources:Currents; Time-varying electric field

1-112 Ampère’s Law of Force

1-113 Magnetic field due to a current element (Biot-Savart Law) B right-circular to the axis of the current element  Note

1-114 E1.4

1-115

1-116 Current Distributions (a) Filamentary Current I (A) (b)Surface Current Surface current density, J S (A/m)

1-117 (c) Volume Current Density, J (A/m 2 )

1-118 P1.44 11  22 P(r, , z)

1-119

1-120 For infinitely long wire,

1-121 Magnetic Field Due to an Infinite Plane Sheet of Uniform Surface Current Density This can be found by dividing the sheet into infinitely long strips parallel to the current density and using superposition, as in the case of finding the electric field due to an infinite plane sheet of uniform surface charge density. Instead of going through this procedure, let us use analogy. To do this, we first note the following:

1-122 Point Charge Current Element (a)

1-123 (b)Line Charge Line Current

1-124 Then, (c) Sheet Charge Sheet Current

1-125 Review Questions How is magnetic flux density defined in terms of force on a moving charge? Compare the magnetic force on a moving charge with electric force on a charge How is magnetic flux density defined in terms of force on a current element? What are the units of magnetic flux density? State Ampere’s force law as applied to current elements. Why is it not necessary for Newton’s third law to hold for current elements? Describe the magnetic field due to a current element What is the value of the permeability of free space? What are its units?

1-126 Review Questions (continued) Discuss the different types of current distributions. How do you determine the magnetic flux density due to a current distribution? Describe the magnetic field due to an infinitely long, straight, wire of current Discuss the analogies between the electric field due to charge distributions and the magnetic field due to current distributions.

1-127 Problem S1.9. Finding parameters of an infinitesimal current element that produces a specified magnetic field

1-128 Problem S1.10. Finding the magnetic field due to a specified current distribution within an infinite plane slab

Lorentz Force Equation (EEE, Sec. 1.7; FEME, Sec. 1.6)

1-130 Lorentz Force Equation For a given B, to find E,

1-131 D1.21 Find E for which acceleration experienced by q is zero, for a given v. (a)

1-132 (b) (c)

1-133 For a given E, to find B, One force not sufficient. Two forces are needed.

1-134 provided, which means v 2 and v 1 should not be collinear.

1-135 P1.54 For v = v 1 or v = v 2, test charge moves with constant velocity equal to the initial value. It is to be shown that for the same holds. (1) (2) (3)

1-136 Alternatively,

1-137

1-138 Review Questions State Lorentz force equation If it is assumed that there is no electric field, the magnetic field at a point can be found from the knowledge of forces exerted on a moving test charge for two noncollinear velocities. Explain Discuss the determination of E and B at a point from the knowledge of forces experienced by a test charge at that point for several velocities. What is the minimum number of required forces? Explain.

1-139 Problem S1.11. Finding the electric and magnetic fields from three forces experienced by a test charge

1-140 The End