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 2 Maxwell’s Equations In Integral Form
The line integral
The surface integral
Faraday’s law
Ampere’s circuital law
Gauss’ Laws
The Law of Conservation of Charge

3. Instructional Objectives
5. 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
6. Apply Ampere's circuital law in integral form to find the magnetomotive force induced around a closed path, for a given current and/or electric field distribution
7. 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

4. The Line Integral
(FEME, Sec. 2.1; EEE6E, Sec. 2.1)

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

6. (Voltage between
A and B)

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

8. is independent of the path from A to B (conservative field) If then
2-7 If
then
is independent of
the path from A to B
(conservative field)

9. along the straight line paths,
2-8
Ex. For find
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).

10. From (0, 0, 0) to (1, 0, 0),
From (1, 0, 0) to (1, 2, 0),

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

12. In fact,

13. 1.5-
The Surface Integral
(FEME, Sec. 2.2; EEE6E, Sec. 2.2) 12

14. Flux = (B)(DS) Flux = 0 The Surface Integral
2-13
The Surface Integral
Flux of a vector crossing a surface:
Flux = (B)(DS)
Flux = 0 a

15. = Surface integral of B over S.
2-14
= Surface integral of B over S.

16. = Surface integral of B
over the closed surface S.
D2.4
(a)

17. (b)

18. (c)

19. (d) From (c),

20. Faraday’s Law
(FEME, Sec. 2.3; EEE6E, Sec. 2.3)

21. Faraday’s Law

22. 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,

23. 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.

24. (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.

25. (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.

26. Thus the magnetic flux produced by the
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

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

28. y
y dec. wt
y inc.
wB0 wt
–wB0
Lenz’s law is verified.

29. 2-28
(b)

30. 2-29
(c)

31. Motional emf concept
conducting
rails
conducting bar

32. 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

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

34. Ampére’s Circuital Law
(FEME, Sec. 2.4; EEE6E, Sec. 2.4)

35. Ampére’s Circuital Law

36. = Magnetomotive force (only by analogy with electromotive force),
2-35
= Magnetomotive force (only by analogy with electromotive force),
= Current due to flow of charges crossing S,
= Displacement flux, or electric flux, crossing S,

37. = 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.

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

39. (b) Capacitor circuit (neglect fringing of field in the capacitor)

40. (c) Finitely long wire

41. Uniqueness of

42. Displacement current emanating from a closed surface = – (current due to flow of charges emanating from the same closed surface)

43. D2.9 (a) Current flowing from Q2 to Q3.
2-42
D2.9 (a) Current flowing from Q2 to Q3.

44. (b) Displacement current emanating from the
2-43
(b) Displacement current emanating from the
spherical surface of radius 0.1 m and centered at Q1.
(c) Displacement current emanating from the spherical
surface of radius 0.1 m and centered at Q3.

45. Interdependence of Time-Varying Electric and Magnetic Fields

46. Hertzian Dipole

47. Radiation from Hertzian Dipole

48. Gauss’ Laws
(FEME, Secs. 2.5, 2.6; EEE6E, Sec. 2.5)

49. Gauss’ Law for the Electric Field
Displacement flux emanating from a closed surface S = charge contained in the volume bounded by S = charge enclosed by S. r

50. Gauss’ Law for the Magnetic Field
Magnetic flux emanating from a closed surface S = 0.

51. P2.21 Finding displacement emanating from a surface
enclosing charge
(a)
Surface of cube bounded by

52. (b)
Surface of the volume x > 0, y > 0, z > 0, and (x2 + y2 + z2) < 1.

53. P2.23

55. The Law of Conservation of Charge (FEME, Sec. 2.5; EEE6E, Sec. 2.6)

56. Law of Conservation of Charge
Current due to flow of charges emanating from a closed surface S
= Time rate of decrease of charge enclosed by S.
r(t)

57. (1) (2) (3) (4) Summarizing, we have the following:
2-56
Summarizing, we have the following:
Maxwell’s Equations
(1)
(2)
(3)
(4)

58. (5) Law of Conservation of Charge
(4) is, however, not independent of (1), whereas (3) follows from (2) with the aid of (5).
(5)

59. (Ampére’s Circuital Law)
Example:
(Ampére’s Circuital Law)
(Gauss’ Law for the electric field and symmetry considerations)

60. (Law of Conservation of Charge)

61. The End