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

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Fundamentals of Electromagneticsfor Teaching and
LearningA 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

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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, 2009 Workshop for Master Trainer
Faculty Sponsored byIUCEE (Indo-US Coalition for
Engineering Education)Infosys Campus, Mysore,
KarnatakaJune 22 July 3, 2009
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• Module 2
• Maxwells Equations
• in Integral Form
• 2.1 The line integral
• 2.2 The surface integral
• 2.3 Faradays law
• 2.4 Amperes circuital law
• 2.5 Gauss Laws
• 2.6 The Law of Conservation of Charge
• 2.7 Application to static fields

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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

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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 Amperes circuital law in integral
  form, without
• the displacement current term, to find the
  magnetic fields
• for symmetrical current distributions

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• 2.1 The Line Integral
• (EEE, Sec. 2.1 FEME, Sec. 2.1)

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• The Line Integral
• Work done in carrying a charge from A to B in an
  electric field

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(Voltage between A and B)
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• In the limit ,

Line integral of E from A to B.
Line integral of E around the closed path C.
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2-9
If then
is independent of the path from A to
B (conservative field)
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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).
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2-11

• From (0, 0, 0) to (1, 0, 0),
• From (1, 0, 0) to (1, 2, 0),

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• From (1, 2, 0) to (1, 2, 3),

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• In fact,

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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.

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Problem S2.1. Evaluation of line integral around
a closed path in Cartesian coordinates
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Problem S2.2. Evaluation of line integral around
a closed path in spherical coordinates
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2.2 The Surface Integral (EEE, Sec. 2.2 FEME,
Sec. 2.2)
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2-18
The Surface Integral Flux of a vector crossing a
surface
Flux (B)(DS)
Flux 0
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2-19
Surface integral of B over S.
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2-20
Surface integral of B over the closed surface
S.
D2.4 (a)
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(b)
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(c)
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• (d) From (c),

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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.,

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Problem S2.3. Evaluation of surface integral over
a closed surface in Cartesian coordinates
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• 2.3 Faradays Law
• (EEE, Sec. 2.3 FEME, Sec. 2.3)

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• Faradays Law

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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,
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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.
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• (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.

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• (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) Lenzs 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.

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• 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

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where ? is the magnetic flux linked by one turn.
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D2.5
(a)
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2-35
Lenzs law is verified.
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2-36
2-36
(b)
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2-37
2-37
(c)
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E2.2 Motional emf concept
conducting rails
conducting bar
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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
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• where
• is the magnetic force on a charge Q in the bar.
  Hence, the emf is known as motional emf.

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Review Questions

• 2.11. State Faradays 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
  Faradays 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?

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Problem S2.4. Induced emf around a rectangular
loop of metallic wire falling in the presence of
a magnetic field
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Problem S2.5. Induced emf around a rectangular
metallic loop revolving in a magnetic field
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• 2.4 Ampéres Circuital Law
• (EEE, Sec. 2.4 FEME, Sec. 2.4)

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• Ampéres Circuital Law

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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,
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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.
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• Three cases to clarify Ampéres circuital law
• (a) Infinitely long, current carrying wire

No displacement flux
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(b) Capacitor circuit (assume electric field
between the plates of the capacitor is confined
to S2)
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• (c) Finitely long wire

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• Uniqueness of

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Displacement current emanating from a closed
surface (current due to flow of charges
emanating from the same closed surface)
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2-53
D2.9 (a) Current flowing from Q2 to Q3.
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2-54

• (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.

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• Interdependence of Time-Varying Electric and
  Magnetic Fields

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• Hertzian Dipole

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Radiation from Hertzian Dipole
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Review Questions

• 2.18. State Amperes circuital law.
• 2.19. What is displacement current? Compare and
  contrast
• displacement current with current due to
  flow of
• charges, giving an example.
• 2.20. Why is it necessary to have the
  displacement current
• term on the right side of Amperes
  circuital law?
• 2.21. Is it meaningful to consider two different
  surfaces
• bounded by a closed path to compute the
  two different
• currents on the right side of Amperes
  circuital law to
• find the line integral of H around the
  closed path?
• 2.22. When can you say that the current in a wire
  enclosed by
• a closed path is uniquely defined? Give
  two examples.

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Review Questions (Continued)

• 2.23. Give an example in which the current in a
  wire enclosed
• by a closed path is not uniquely
  defined.
• 2.24. Discuss the relationship between the
  displacement
• current emanating from a closed surface
  and the current
• due to flow of charges emanating from
  the same closed
• surface.
• 2.25. Discuss the interdependence of time-varying
  electric
• and magnetic fields through Faradays
  law and
• Amperes circuital law, and, as a
  consequence, the
• principle of radiation from a wire
  carrying time-varying
• current.

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Problem S2.6. Finding the displacement current
emanating from a closed surface for a given
current density J
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Problem S2.7. Finding the rms value of current
drawn from a voltage source connected to a
capacitor
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• 2.5 Gauss Laws
• (EEE, Sec. 2.5 FEME, Secs. 2.5, 2.6)

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• 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
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• Gauss Law for the Magnetic Field
• Magnetic flux emanating from a closed
  surface S  0.

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

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• (b)
• Surface of the volume x gt 0, y gt 0, z gt 0,
  and (x2 y2 z2) lt 1.

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• P2.23

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Review Questions

• 2.26. State Gauss law for the electric field.
• 2.27. How do you evaluate a volume integral?.
• 2.28. State Gauss law for the magnetic field.
• 2.29. What is the physical interpretation of
  Gauss law for the
• magnetic field.

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Problem S2.8. Finding the displacement flux
emanating from a surface enclosing charge
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Problem S2.9. Application of Gauss law for the
magnetic field in integral form
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2.6 The Law ofConservation of Charge(EEE, Sec.
2.6 FEME, Sec. 2.5)
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• Law of Conservation of Charge

r(t)
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2-74

• Summarizing, we have the following
• Maxwells Equations

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

(5)
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E2.3
(Ampéres Circuital Law)
(Gauss Law for the electric field and symmetry
considerations)
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(Law of Conservation of Charge)
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Review Questions

• 2.30. State the law of conservation of charge..
• 2.31. How do you evaluate a volume integral?.
• 2.32. Summarize Maxwells equations in integral
  form for
• time-varying fields.
• 2.33. Which two of the Maxwells equations are
  independent?
• Explain..

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Problem S2.10. Combined application of several
of Maxwells equations in integral form
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2.6 Application to Static Fields(EEE, Sec. 2.7)
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Review Questions

• 2.34. Summarize Maxwells equations in integral
  form for
• static fields.
• 2.35. Are static electric and magnetic fields
  interdependent?
• 2.36. Discuss briefly the application of Gauss
  law for the
• electric field in integral form to
  determine the electric
• field due to charge distributions.
• 2.37. Discuss briefly the application of Amperes
  circuital
• law in integral form for the static case
  to determine the
• magnetic field due to current
  distributions.

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Problem S2.11. Application of Gauss law for the
electric field in integral form and symmetry
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Problem S2.12. Application of Amperes circuital
law in integral form and symmetry
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The End