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by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus

Fundamentals of Electromagnetics for Teaching and Learning: A Two-Week Intensive Course for Faculty in Electrical-, Electronics-, Communication-, and Computer- Related Engineering Departments in Engineering Colleges in India. by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus

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by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus

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  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 2 Maxwell’s Equations in Integral Form 2.1 The line integral 2.2 The surface integral 2.3 Faraday’s law 2.4 Ampere’s circuital law 2.5 Gauss’ Laws 2.6 The Law of Conservation of Charge 2.7 Application to static fields

  4. Instructional Objectives 8.Evaluate line and surface integrals 9. Apply Faraday's law in integral form to find the electromotive force induced around a closed loop, fixed or revolving, for a given magnetic field distribution 10. Make use of the uniqueness of the magnetomotive force around a closed path to find the displacement current emanating from a closed surface for a given current distribution 11. Apply Gauss’ law for the electric field in integral form to find the displacement flux emanating from a closed surface bounding the volume for a specified charge distribution within the volume 12. Apply Gauss’ law for the magnetic field in integral form to simplify the problem of finding the magnetic flux crossing a surface

  5. Instructional Objectives (Continued) 13. Apply Gauss' law for the electric field in integral form, Ampere's circuital law in integral form, the law of conservation of charge, and symmetry considerations, to find the line integral of the magnetic field intensity around a closed path, given an arrangement of point charges connected by wires carrying currents 14. Apply Gauss’ law for the electric field in integral form to find the electric fields for symmetrical charge distributions 15. Apply Ampere’s circuital law in integral form, without the displacement current term, to find the magnetic fields for symmetrical current distributions

  6. 2.1 The Line Integral (EEE, Sec. 2.1; FEME, Sec. 2.1)

  7. a2 a1 The Line Integral: Work done in carrying a charge from A to B in an electric field:

  8. (Voltage between A and B)

  9. In the limit , = Line integral of E from A to B. = Line integral of E around the closed path C.

  10. 2-9 If then is independent of the path from A to B (conservative field)

  11. 2-10 2-10 along the straight line paths, from (0, 0, 0) to (1, 0, 0), from (1, 0, 0) to (1, 2, 0) and then from (1, 2, 0) to (1, 2, 3).

  12. From (0, 0, 0) to (1, 0, 0), From (1, 0, 0) to (1, 2, 0), 2-11

  13. From (1, 2, 0) to (1, 2, 3),

  14. In fact,

  15. Review Questions 2.1. How do you find the work done in moving a test charge by an infinitesimal distance in an electric field? 2.2. What is the amount of work involved in moving a test charge normal to the electric field? 2.3. What is the physical interpretation of the line integral of E between two points A and B? 2.4. How do you find the approximate value of the line integral of a vector field along a given path? How do you find the exact value of the line integral? 2.5. Discuss conservative versus nonconservative fields, giving examples.

  16. Problem S2.1. Evaluation of line integral around a closed path in Cartesian coordinates

  17. Problem S2.2. Evaluation of line integral around a closed path in spherical coordinates

  18. 2.2 The Surface Integral (EEE, Sec. 2.2; FEME, Sec. 2.2)

  19. a 2-18 The Surface Integral Flux of a vector crossing a surface: Flux = (B)(DS) Flux = 0

  20. 2-19 = Surface integral of B over S.

  21. 2-20 = Surface integral of B over the closed surface S. D2.4 (a)

  22. (b)

  23. (c)

  24. (d) From (c),

  25. Review Questions 2.6. How do you find the magnetic flux crossing an infinitesimal surface? 2.7. What is the magnetic flux crossing an infinitesimal surface oriented parallel to the magnetic flux density vector? 2.8. For what orientation of an infinitesimal surface relative to the magnetic flux density vector is the magnetic flux crossing the surface a maximum? 2.9. How do you find the approximate value of the surface integral of a vector field over a given surface? How do you find the exact value of the surface integral? 2.10. Provide physical interpretation for the closed surface integrals of any two vectors of your choice.,

  26. Problem S2.3. Evaluation of surface integral over a closed surface in Cartesian coordinates

  27. 2.3 Faraday’s Law (EEE, Sec. 2.3; FEME, Sec. 2.3)

  28. Faraday’s Law

  29. Voltage around C, also known as electromotive force (emf) around C (but not really a force), = Magnetic flux crossing S, = Time rate of decrease of magnetic flux crossing S,

  30. Important Considerations (1)Right-hand screw (R.H.S.) Rule The magnetic flux crossing the surface S is to be evaluated toward that side of S a right-hand screw advances as it is turned in the sense of C.

  31. (2)Any surface S bounded by C The surface S can be any surface bounded by C. For example: This means that, for a given C, the values of magnetic flux crossing all possible surfaces bounded by it is the same, or the magnetic flux bounded by C is unique.

  32. (3)Imaginary contour C versus loop of wire There is an emf induced around C in either case by the setting up of an electric field. A loop of wire will result in a current flowing in the wire. (4)Lenz’s Law States that the sense of the induced emf is such that any current it produces, if the closed path were a loop of wire, tends to oppose the change in the magnetic flux that produces it.

  33. Thus the magnetic flux produced by the induced current and that is bounded by C must be such that it opposes the change in the magnetic flux producing the induced emf. (5)N-turn coil For an N-turn coil, the induced emf is N times that induced in one turn, since the surface bounded by one turn is bounded N times by the N-turn coil. Thus

  34. where  is the magnetic flux linked by one turn.

  35. D2.5 (a)

  36. 2-35 Lenz’s law is verified.

  37. 2-36 2-36 (b)

  38. 2-37 2-37 (c)

  39. E2.2 Motional emf concept conducting rails conducting bar

  40. This can be interpreted as due to an electric field induced in the moving bar, as viewed by an observer moving with the bar, since

  41. where is the magnetic force on a charge Q in the bar. Hence, the emf is known as motional emf.

  42. Review Questions 2.11. State Faraday’s law. 2.12. What are the different ways in which an emf is induced around a loop? 2.13. Discuss the right-hand screw rule convention associated with the application of Faraday’s law. 2.14. To find the induced emf around a planar loop, is it necessary to consider the magnetic flux crossing the plane surface bounded by the loop? Explain. 2.15. What is Lenz’ law? 2.16. Discuss briefly the motional emf concept. 2.17. How would you orient a loop antenna in order to receive maximum signal from an incident electromagnetic wave which has its magnetic field linearly polarized in the north-south direction?

  43. Problem S2.4. Induced emf around a rectangular loop of metallic wire falling in the presence of a magnetic field

  44. Problem S2.5. Induced emf around a rectangular metallic loop revolving in a magnetic field

  45. 2.4 Ampére’s Circuital Law (EEE, Sec. 2.4; FEME, Sec. 2.4)

  46. Ampére’s Circuital Law

  47. 2-46 = Magnetomotive force (only by analogy with electromotive force), = Current due to flow of charges crossing S, = Displacement flux, or electric flux, crossing S,

  48. = Time rate of increase of displacement flux crossing S, or, displacement current crossing S, Right-hand screw rule. Any surface S bounded by C, but the same surface for both terms on the right side.

  49. Three cases to clarify Ampére’s circuital law (a)Infinitely long, current carrying wire No displacement flux

  50. (b)Capacitor circuit (assume electric field between the plates of the capacitor is confined to S2)

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