1. EEE241: Fundamentals of Electromagnetics Introductory Concepts, Vector Fields and Coordinate Systems Instructor: Dragica Vasileska

2. Outline • Class Description • Introductory Concepts • Vector Fields • Coordinate Systems

3. Class Description Prerequisites by Topic: • University physics • Complex numbers • Partial differentiation • Multiple Integrals • Vector Analysis • Fourier Series

4. Class Description • Prerequisites: EEE 202; MAT 267, 274 (or 275), MAT 272; PHY 131, 132 • Computer Usage: Students are assumed to be versed in the use MathCAD or MATLAB to perform scientific computing such as numerical calculations, plotting of functions and performing integrations. Students will develop and visualize solutions to moderately complicated field problems using these tools. • Textbook: Cheng, Field and Wave Electromagnetics.

5. Class Description • Grading: Midterm #1 25% Midterm #2 25% Final 25% Homework 25%

6. Class Description

7. Why Study Electromagnetics?

8. Examples of Electromagnetic Applications

9. Examples of Electromagnetic Applications, Cont’d

10. Examples of Electromagnetic Applications, Cont’d

11. Examples of Electromagnetic Applications, Cont’d

12. Examples of Electromagnetic Applications, Cont’d

13. Research Areas of Electromagnetics • Antenas • Microwaves • Computational Electromagnetics • Electromagnetic Scattering • Electromagnetic Propagation • Radars • Optics • etc …

14. Why is Electromagnetics Difficult?

15. What is Electromagnetics?

16. What is a charge q?

17. Fundamental Laws of Electromagnetics

18. Steps in Studying Electromagnetics

19. SI (International System) of Units

20. Units Derived From the Fundamental Units

21. Fundamental Electromagnetic Field Quantities

22. Three Universal Constants

23. Fundamental Relationships

24. Scalar and Vector Fields • A scalar field is a function that gives us a single value of some variable for every point in space. • Examples: voltage, current, energy, temperature • A vector is a quantity which has both a magnitude and a direction in space. • Examples: velocity, momentum, acceleration and force

25. Example of a Scalar Field

26. Scalar Fields e.g. Temperature: Every location has associated value (number with units) 26

27. Scalar Fields - Contours • Colors represent surface temperature • Contour lines show constant temperatures 27

28. Fields are 3D • T = T(x,y,z) • Hard to visualize  Work in 2D 28

29. Vector Fields Vector (magnitude, direction) at every point in space Example: Velocity vector field - jet stream 29

30. Vector Fields Explained

31. Examples of Vector Fields

32. Examples of Vector Fields

33. Examples of Vector Fields

34. Choice is based on symmetry of problem VECTOR REPRESENTATION 3 PRIMARY COORDINATE SYSTEMS: • RECTANGULAR • CYLINDRICAL • SPHERICAL Examples: Sheets - RECTANGULAR Wires/Cables - CYLINDRICAL Spheres - SPHERICAL

35. Orthogonal Coordinate Systems: (coordinates mutually perpendicular) Cartesian Coordinates z P(x,y,z) y Rectangular Coordinates x P (x,y,z) z z P(r, θ, z) Cylindrical Coordinates P (r, Θ, z) y r x θ z Spherical Coordinates P(r, θ, Φ) θ r P (r, Θ, Φ) y x Φ Page 108

36. Parabolic Cylindrical Coordinates (u,v,z) • Paraboloidal Coordinates (u, v, Φ) • Elliptic Cylindrical Coordinates (u, v, z) • Prolate Spheroidal Coordinates (ξ, η, φ) • Oblate Spheroidal Coordinates (ξ, η, φ) • Bipolar Coordinates (u,v,z) • Toroidal Coordinates (u, v, Φ) • Conical Coordinates (λ, μ, ν) • Confocal Ellipsoidal Coordinate (λ, μ, ν) • Confocal Paraboloidal Coordinate (λ, μ, ν)

37. Parabolic Cylindrical Coordinates

38. Paraboloidal Coordinates

39. Elliptic Cylindrical Coordinates

40. Prolate Spheroidal Coordinates

41. Oblate Spheroidal Coordinates

42. Bipolar Coordinates

43. Toroidal Coordinates

44. Conical Coordinates

45. Confocal Ellipsoidal Coordinate

46. Confocal Paraboloidal Coordinate

47. z z Cartesian Coordinates P(x,y,z) P(x,y,z) P(r, θ, Φ) θ r y x y x Φ Cylindrical Coordinates P(r, θ, z) Spherical Coordinates P(r, θ, Φ) z z P(r, θ, z) y r x θ

48. Coordinate Transformation • Cartesian to Cylindrical (x, y, z) to (r,θ,Φ) (r,θ,Φ)to (x, y, z)

49. Coordinate Transformation • Cartesian to Cylindrical Vectoral Transformation

50. Coordinate Transformation • Cartesian to Spherical (x, y, z) to (r,θ,Φ) (r,θ,Φ)to (x, y, z)