by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus of Electrical and Computer Engineering University of Ill

1. Fundamentals of Electromagneticsfor Teaching and Learning:A Two-Week Intensive Course for Faculty inElectrical-, 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

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

3. × ò ò D d S = r dv S V Maxwell’s Equations d × × ò E d l = – ò B d S dt C S Charge density Magnetic flux density Electric field intensity d × × × × ò B d S = 0 ò ò ò H d l = J d S + D d S dt S C S S Current density Magnetic field intensity Displacement flux density

4. 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

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

6. 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

7. 1.1 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.

8. (1)Vectors (A) vs. Scalars (A) Magnitude and direction Magnitude only Ex: Velocity, Force Ex: Mass, Charge

9. 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.

10. (3)Dot Product is a scalar A A • B = AB cos a B Useful for finding angle between two vectors.

11. (4)Cross Product is a vector A A B = AB sin a B is perpendicular to both A and B. Useful for finding unit vector perpendicular to two vectors. an

12. where (5)Triple Cross Product in general.

13. (6)Scalar Triple Product is a scalar.

14. Volume of the parallelepiped

15. D1.2 (EEE)A = 3a1 + 2a2 + a3 B = a1 + a2 – a3 C = a1 + 2a2 + 3a3 (a)A + B – 4C = (3 + 1 – 4)a1 + (2 +1 – 8)a2 + (1 – 1 – 12)a3 = – 5a2 – 12a3

16. (b)A + 2B – C = (3 + 2 – 1)a1 + (2 + 2 – 2)a2 + (1 – 2 – 3)a3 = 4a1 + 2a2 – 4a3 Unit Vector = =

17. (c)A • C = 3 1 + 2 2 + 1 3 = 10 (d) = = 5a1 – 4a2 + a3

18. (e) = 15 – 8 + 1 = 8 Same as A • (B C) = (3a1 + 2a2 + a3) • (5a1 – 4a2 + a3) = 3 5 + 2 (–4) + 1 1 = 15 – 8 + 1 = 8

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

20. What is the geometric interpretation of this result?

21. E1.1 Another Example Given Find A.

22. To find C, use (1) or (2).

23. Review Questions 1.1. Give some examples of scalars. 1.2. Give some examples of vectors. 1.3. Is it necessary for the reference vectors a1, a2, and a3 to be an orthogonal set? 1.4. State whether a1, a2, and a3 directed westward, northward, and downward, respectively, is a right- handed or a left-handed set. 1.5. State all conditions for which A•B is zero. 1.6. State all conditions for which A×B is zero. 1.7. What is the significance of A•B×C =0? 1.8. What is the significance of A× (B×C)=0?

24. Problem S1.1. Performing several vector algebraic manipulations

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

26. 1.2 Cartesian Coordinate System (EEE, Sec. 1.2; FEME, Sec. 1.2)

27. Cartesian Coordinate System

28. Cartesian Coordinate System

29. Right-handed system xyz xy… ax, ay, az are uniform unit vectors, that is, the direction of each unit vector is same everywhere in space.

30. Vector drawn from one point to another:FromP1(x1, y1, z1) toP2(x2, y2, z2)

32. P1.8A(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

33. (c) Perpendicular distance from A to the line through B and C =

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

35. dl = dx ax + dy ay = dx ax + f¢(x) dx ay Unit vector normal to a surface

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

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

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

39. (4)Differential Volume (dv) In Cartesian coordinates,

40. Review Questions 1.9. What is the particular advantageous characteristic associated with unit vectors in the Cartesian coordinate system? 1.10. What is the position vector? 1.11. What is the total distance around the circumference of a circle of radius 1 m? What is the total vector distance around the circle? 1.12. Discuss the application of differential length vectors to find a unit vector normal to a surface at a point on the surface. 1.13. Discuss the concept of a differential surface vector. 1.14. 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?

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

42. 1.3 Cylindrical and Spherical Coordinate Systems (EEE, Sec. 1.3; FEME, Appendix A)

43. Cylindrical Coordinate System

44. Spherical Coordinate System

45. 1-44 Cylindrical and Spherical Coordinate Systems Cylindrical (r, f, z) Spherical (r, q, f) Only az is uniform. All three unit ar and af are vectors are nonuniform. nonuniform.

46. x = r cos fx = r sin q cos f y = r sin fy = r sin q sin f z = zz = r cos q D1.7 (a) (2, 5p/6, 3) in cylindrical coordinates 1-45

47. (b) 1-46

48. (c) 1-47

49. 1-48 (d)

50. Conversion of vectors between coordinate systems