by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus

1. Fundamentals of Electromagnetics:A Two-Week, 8-Day, Intensive Course for Training Faculty in Electrical-, Electronics-, Communication-, and Computer- Related Engineering Departments 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 Amrita Viswa Vidya Peetham, Coimbatore August 11, 12, 13, 14, 18, 19, 20, and 21, 2008

2. Module 6 • Statics, Quasistatics, and Transmission Lines • Gradient and electric potential • Poisson’s and Laplace’s equations • Static fields and circuit elements • Low-frequency behavior via quasistatics • The distributed circuit concept and the transmission line

3. Instructional Objectives • 25. 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 • 26. Obtain the solution for the potential between two conductors held at specified potentials, for one-dimensional cases in the Cartesian coordinate system (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 length (or capacitance in the case of spherical conductors) of the arrangement

4. Instructional Objectives (Continued) • 27. Perform static field analysis of arrangements consisting of two parallel plane conductors for electrostaic, magnetostatic, and electromagnetostatic fields • 28. Perform quasistatic static field analysis of arrangements consisting of two parallel plane conductors for electroquastatic and magnetoquasistatic fields • 29. Understand the development of the transmission-line (distributed equivalent circuit) from the field solutions for a given physical structure

5. Gradient and Electric Potential(FEME, Sec. 6.1; EEE6E, Secs. 5.1, 5.2)

6. Gradient and the Potential Functions

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

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

10. Basic definition of From this, we get

11. Potential function equations

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

13. For static fields,

14. But, also known as the potential difference between A and B, for the static case.

15. 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.

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

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

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

20. Poisson’s and Laplace’s Equations(FEME, Sec. 6.2; EEE6E, Sec. 5.3)

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

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

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

24. (b)

25. (c)

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

27. Neglecting fringing of field at edges, General solution

28. Boundary conditions Particular solution

30. area of plates For nonuniform e, For

31. Example x = d, V = V0 x = 0, V = 0

34. Static Fields and Circuit Elements(FEME, Sec. 6.3; EEE6E, Sec. 5.4)

35. Classification of Fields Static Fields ( No time variation; t = 0) 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

36. Ñ x E = 0 Ñ x H = J Static Fields For static fields,t = 0, 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

37. Solution for Potential and Field Solution for charge distribution Solution for point charge Electric field due to point charge

38. Laplace’s Equation and One-Dimensional Solution For Poission’s equation reduces to Laplace’s equation

39. r S r S Example of Parallel-Plate Arrangement;Capacitance 

40. Electrostatic Analysis of Parallel-Plate Arrangement Capacitance of the arrangement, F

41. Ñ x H = J Magnetostatic Fields × × ò ò H d l = J d S C S × B d S = 0 ò S Poisson’s equation for magnetic vector potential

42. m I d l ( r ¢ ) x ( r - r ¢ ) B ( r ) = 3 4 p r - r ¢ Solution for Vector Potential and Field Solution for current distribution Solution for current element Magnetic field due to current element 2A = 0 For current-free region

43. Example of Parallel-Plate Arrangement;Inductance 

44. Magnetostatic Analysis of Parallel-Plate Arrangement

45. Magnetostatic Analysis of Parallel-Plate Arrangement Inductance of the arrangement, H

46. Ñ x E = 0 Ñ x H = J = s E c Electromagnetostatic Fields × ò E d l = 0 C × × × ò H d l = ò J d S = s ò E d S c C S S × ò D d S = 0 S × ò B d S = 0 S

47. r S r S Example of Parallel-Plate Arrangement 

48. Electromagnetostatic Analysis of Parallel-Plate Arrangement

49. Electromagnetostatic Analysis of Parallel-Plate Arrangement Conductance, S Resistance, ohms

50. H = H ( z ) a y y [m H d(dz¢)] y Electromagnetostatic Analysis of Parallel-Plate Arrangement 0 1 – z ¢ æ ö = ò è ø I l – ¢ = z l c Internal Inductance