1. Electromagnetic Fields ELC205B-Spring 2012 Department of Electronics and Electrical Communications Engineering Faculty of Engineering – Cairo University

2. Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

3. Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

4. Mathematical Operators  © Mamdouh Hassan Abbas,2012 Positive Source Negative Sink Zero No charge S S S n n n

5. Mathematical Operators  n Always Zero Circulating quantity S © Mamdouh Hassan Abbas,2012

6. Mathematical Operators  Always Zero Flowing quantity Non-zero Circulating quantity E B © Mamdouh Hassan Abbas,2012

7. Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

8. Fundamental Laws of Electro and Magneto Statics  © Mamdouh Hassan Abbas,2012

9. Fundamental Laws of Electro and Magneto Statics  © Mamdouh Hassan Abbas,2012

10. Fundamental Laws of Electro and Magneto Statics  © Mamdouh Hassan Abbas,2012

11. Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

12. Faraday’s Experiment battery switch toroidal iron core compass primary coil secondary coil © Mamdouh Hassan Abbas,2012

13. Faraday’s Experiment battery switch toroidal iron core compass primary coil secondary coil © Mamdouh Hassan Abbas,2012

14. Faraday’s Experiment battery switch toroidal iron core compass primary coil secondary coil © Mamdouh Hassan Abbas,2012

15. Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

16. Lenz’s Law  © Mamdouh Hassan Abbas,2012

17. Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

18. Faraday’s Law  © Mamdouh Hassan Abbas,2012

19. Faraday’s Law  Illustrative examples © Mamdouh Hassan Abbas,2012

20. Faraday’s Law  © Mamdouh Hassan Abbas,2012

21. Faraday’s Law  © Mamdouh Hassan Abbas,2012

22. Faraday’s Law  B(Outward flux) Conducting path Conducting rod © Mamdouh Hassan Abbas,2012

23. Faraday’s Law  - - B(Outward flux) U F Induced (inward flux) © Mamdouh Hassan Abbas,2012

24. Faraday’s Law  © Mamdouh Hassan Abbas,2012

25. Faraday’s Law  © Mamdouh Hassan Abbas,2012

26. Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

27. Induced Electric Field  © Mamdouh Hassan Abbas,2012

28. Induced Electric Field  Faraday’s law states that a changing magnetic field induces an electric field.  The induced electric field is non- conservative. © Mamdouh Hassan Abbas,2012

29. Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

30. Electric Field in terms of Potential Functions  Electric scalar potential © Mamdouh Hassan Abbas,2012

31. Electric Field in terms of Potential Functions  © Mamdouh Hassan Abbas,2012

32. Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

33. Ampere’s Law and the Continuity Equation  Q S n J © Mamdouh Hassan Abbas,2012

34. Ampere’s Law and the Continuity Equation  © Mamdouh Hassan Abbas,2012

35. Ampere’s Law and the Continuity Equation  Conduction current Displacement current © Mamdouh Hassan Abbas,2012

36. Ampere’s Law and the Continuity Equation  Continuity equation is satisfied © Mamdouh Hassan Abbas,2012

37. Ampere’s Law and the Continuity Equation  © Mamdouh Hassan Abbas,2012

38. Ampere’s Law and the Continuity Equation  Displacement current is the type of current that flows between the plates of a capacitor.  Displacement current is the mechanism which allows electromagnetic waves to propagate in a non-conducting medium. © Mamdouh Hassan Abbas,2012

39. Ampere’s Law and the Continuity Equation  Current in a capacitor © Mamdouh Hassan Abbas,2012

40. Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

41. Retarded Potentials  The retarded potential formula describes the potentials due to time varying currents or charges distributions. © Mamdouh Hassan Abbas,2012

42. Retarded Potentials

43. © Mamdouh Hassan Abbas,2012 Retarded Potentials

44. © Mamdouh Hassan Abbas,2012 Retarded Potentials

45. © Mamdouh Hassan Abbas,2012 Retarded Potentials

46. © Mamdouh Hassan Abbas,2012 Retarded Potentials

47. © Mamdouh Hassan Abbas,2012 Retarded Potentials

48. © Mamdouh Hassan Abbas,2012 Retarded Potentials

49. © Mamdouh Hassan Abbas,2012 Retarded Potentials

50. © Mamdouh Hassan Abbas,2012 Retarded Potentials

51. © Mamdouh Hassan Abbas,2012 Retarded Potentials

52.  Modeling Source Observer (Origin) © Mamdouh Hassan Abbas,2012

53. Retarded Potentials  Retarded potentials Variable Constant © Mamdouh Hassan Abbas,2012