by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus

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. Module 6 • Statics, Quasistatics, and Transmission Lines • 6.1 Gradient and electric potential • 6.2 Poisson’s and Laplace’s equations • 6.3 Static fields and circuit elements • 6.4 Low-frequency behavior via quasistatics • 6.5 Condition for the validity of the quasistatic approximation • 6.6 The distributed circuit concept and the transmission-line

4. Instructional Objectives • 42. Understand the geometrical significance of the gradient • operation • 43. Find the static electric potential due to a specified charge • distribution by applying superposition in conjunction • with the potential due to a point charge, and further find • the electric field from the potential • 44. Obtain the solution for the potential between two • conductors held at specified potentials, for one- • dimensional cases (and the region between which is filled • with a dielectric of uniform or nonuniform permittivity, • or with multiple dielectrics) by using the Laplace’s • equation in one dimension, and further find the • capacitance per unit area (Cartesian) or per unit length • (cylindrical) or capacitance (spherical) of the • arrangement

5. Instructional Objectives (Continued) • 45. Perform static field analysis of arrangements consisting • of two parallel plane conductors for electrostatic, • magnetostatic, and electromagnetostatic fields • 46. Perform quasistatic field analysis of arrangements • consisting of two parallel plane conductors for • electroquastatic and magnetoquasistatic fields • 47. Understand the condition for the validity of the quasistatic • approximation and the input behavior of a physical • structure for frequencies beyond the quasistatic • approximation • 48. Understand the development of the transmission-line • (distributed equivalent circuit) from the field solutions • for a given physical structure and obtain the transmission- • line parameters for a line of arbitrary cross section by • using the field mapping technique

6. 6.1 Gradient and Electric Potential(EEE, Secs. 5.1, 5.2; FEME, Sec. 6.1)

7. Gradient and the Potential Functions

8. 6-7 B can be expressed as the curl of a vector. Thus A is known as the magnetic vector potential. Then

9. F is known as the electric scalar potential. is the gradient of F.

11. 6-10 Basic definition of : For a constant  surface, d = 0. Therefore is normal to the surface.

12. 6-11 Thus, the magnitude of at any point P is the rate of increase of  normal to the surface, which is the maximum rate of increase  at that point. Thus Useful for finding unit normal vector to the surface.

13. 6-12 D5.1 Finding unit normal vectors to the surface at several points:

14. 6-13 6-13

15. 6-14 (1) (2) (3) (4) (4) (1) (3)

16. 6-15 (2) Potential function equations

17. Laplacian of scalar Laplacian of vector In Cartesian coordinates,

18. 6-17 For static fields, But, also known as the potential difference between A and B, for the static case.

19. Given the charge distribution, find V using superposition. Then find E using the above. For a point charge at the origin, since agrees with the previously known result.

20. Thus for a point charge at an arbitrary location P P R Q P5.9

21. Considering the element of length dz at (0, 0, z), we have Using

22. 6-21

23. Magnetic vector potential due to a current element P R Analogous to

24. Review Questions • 6.1. What is the divergence of the curl of a vector? • 6.2. What is the expansion for the gradient of a scalar in • Cartesian coordinates? When can a vector be expressed • as the gradient of a scalar? • 6.3. Discuss the basic definition of the gradient of a scalar. • 6.4. Discuss the application of the gradient concept for the • determination of unit vector normal to a surface. • 6.5. Define electric potential. What is its relationship to the • electric field intensity? • 6.6. Distinguish between voltage as applied to time-varying • fields and potential difference. • 6.7. What is the electric potential due to a point charge? • Discuss the determination of electric potential due to a • charge distribution.

25. Review Questions (Continued) • 6.8. What is the Laplacian of a scalar? What is the expansion • for the Laplacian of a scalar in Cartesian coordinates? • 6.9. What is the magnetic vector potential? How is it related • to the magnetic flux density?

26. Problem S6.1. Finding the gradient of a two-dimensional function and associated discussion

27. Problem S6.2. Finding the angle between two plane surfaces, by using the gradient concept

28. Problem S6.3. Finding the image charge(s) for a point charge in the presence of a conductor

29. Problem S6.3. Finding the image charge(s) for a point charge in the presence of a conductor (Continued)

30. 6.2 Poisson’s and Laplace’s Equations(EEE, Sec. 5.3; FEME, Sec. 6.2)

31. Poisson’s Equation For static electric field, Then from If e is uniform, Poisson’s equation

32. If eis nonuniform, then using Thus Assuming uniform e, we have For the one-dimensional case of V(x),

33. D5.7 Anode, x = d V = V0 Vacuum Diode Cathode, x = 0 V = 0 (a)

34. (b)

35. (c)

36. 6-35 Laplace’s Equation If r = 0, Poisson’s equation becomes Let us consider uniform efirst. E6.1. Parallel-plate capacitor x = d, V= V0 x = 0, V = 0

37. Neglecting fringing of field at edges, General solution

38. Boundary conditions Particular solution

40. area of plates For nonuniforme, For

41. E6.2 x = d, V = V0 x = 0, V = 0

44. Review Questions • 6.10. State Poisson’s equation for the electric potential. How • is it derived? • 6.11. Outline the solution of the Poisson’s equation for the • potential in a region of known charge density varying in • one dimension. • 6.12. State Laplace’s equation for the electric potential. In • what regions is it valid? • 6.13. Outline the solution of Laplace’s equation in one • dimension by considering a parallel-plate arrangement. • 6.14. Outline the steps in the determination of the capacitance • of a parallel-plate capacitor.

45. Problem S6.4. Solution of Poisson’s equation for a space charge distribution in Cartesian coordinates

46. Problem S6.5. Finding the capacitance of a spherical capacitor with a dielectric of nonuniform permittivity

47. 6.3 Static Fields and Circuit Elements(EEE, Sec. 5.4; FEME, Sec. 6.3)

48. Classification of Fields 6-47 Static Fields ( No time variation; ) Static electric, or electrostatic fields Static magnetic, or magnetostatic fields Electromagnetostatic fields Dynamic Fields (Time-varying) Quasistatic Fields (Dynamic fields that can be analyzed as though the fields are static) Electroquasistatic fields Magnetoquasistatic fields

49. Static Fields 6-48 6-48 For static fields, , and the equations reduce to × ò E d l = 0 C × × ò H d l = ò J d S C S × ò D d S = ò r dv S V × ò B d S = 0 S × ò J d S = 0 S

50. 6-49 Solution for Potential and Field Solution for charge distribution Solution for point charge Electric field due to point charge