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 - PowerPoint PPT Presentation

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Title: 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


1
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

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, 2009 Workshop for Master Trainer
Faculty Sponsored byIUCEE (Indo-US Coalition for
Engineering Education)Infosys Campus, Mysore,
KarnatakaJune 22 July 3, 2009
3
Maxwells Equations
d


E
d
l


B
d
S
ò
ò
dt
S
C
Charge density
Electric field intensity
Magnetic flux density
d




B
d
S

0
ò
ò
ò
ò
H
d
l

J
d
S

D
d
S
dt
S
S
C
S
Current density
Magnetic field intensity
Displacement flux density
4
  • Module 1
  • Vectors and Fields
  • 1.1 Vector algebra
  • 1.2 Cartesian coordinate system
  • 1.3 Cylindrical and spherical coordinate systems
  • 1.4 Scalar and vector fields
  • 1.5 Sinusoidally time-varying fields
  • 1.6 The electric field
  • 1.7 The magnetic field
  • 1.8 Lorentz force equation

5
Instructional Objectives
  • Perform vector algebraic operations in Cartesian,
    cylindrical, and spherical coordinate systems
  • Find the unit normal vector and the differential
    surface at a point on the surface
  • Find the equation for the direction lines
    associated with a vector field
  • Identify the polarization of a sinusoidally
    time-varying vector field
  • Calculate the electric field due to a charge
    distribution by applying superposition in
    conjunction with the electric field due to a
    point charge
  • Calculate the magnetic field due to a current
    distribution by applying superposition in
    conjunction with the magnetic field due to a
    current element

6
Instructional Objectives (Continued)
  • 7. Apply Lorentz force equation to find the
    electric and magnetic fields, for a specified set
    of forces on a charged particle moving in the
    field region

7
  • 1.1 Vector Algebra
  • (EEE, Sec. 1.1 FEME, Sec. 1.1)
  • In this series of PowerPoint presentations, EEE
    refers to
  • Elements of Engineering Electromagnetics, 6th
    Edition,
  • Indian Edition (2006), and FEME refers to
    Fundamentals of Electromagnetics for
    Engineering, Indian Edition (2009).
  • Also, all D Problems and P Problems are from
    EEE.

8
  • (1) Vectors (A) vs. Scalars (A)
  • Magnitude and direction Magnitude only Ex
    Velocity, Force Ex Mass, Charge

9
  • Unit Vectors have magnitude
  • unity, denoted by symbol a
  • with subscript. We shall use
  • the right-handed system
  • throughout.
  • Useful for expressing vectors in terms of their
    components.

10
  • (3) Dot Product is a scalar
  • A
  • A B AB cos a
  • B
  • Useful for finding angle between two vectors.

11
  • (4) Cross Product is a vector
  • A
  • A B AB sin a
  • B
  • is perpendicular to both A and B.
  • Useful for finding unit vector perpendicular to
    two vectors.

an
12
  • where
  • (5) Triple Cross Product
  • in general.

13
  • (6) Scalar Triple Product
  • is a scalar.

14
  • Volume of the parallelepiped

15
  • D1.2 (EEE) A 3a1 2a2 a3
  • B a1 a2 a3
  • C a1 2a2 3a3
  • (a) A B 4C
  • (3 1 4)a1 (2 1 8)a2
  • (1 1 12)a3
  • 5a2 12a3

16
  • (b) A 2B C
  • (3 2 1)a1 (2 2 2)a2
  • (1 2 3)a3
  • 4a1 2a2 4a3
  • Unit Vector

17
  • (c) A C 3 1 2 2 1 3
  • 10
  • (d)
  • 5a1 4a2 a3

18
  • (e)
  • 15 8 1 8
  • Same as
  • A (B C) (3a1 2a2 a3) (5a1 4a2
    a3)
  • 3 5 2 (4) 1 1
  • 15 8 1
  • 8

19
  • P1.5 (EEE)
  • D B A ( A D B)
  • E C B ( B E C)
  • D and E lie along a straight line.

20
  • What is the geometric interpretation of this
    result?

21
  • E1.1 Another Example
  • Given
  • Find A.

22
To find C, use (1) or (2).
23
Review Questions
  • 1.1. Give some examples of scalars.
  • 1.2. Give some examples of vectors.
  • 1.3. Is it necessary for the reference vectors
    a1, a2, and a3
  • to be an orthogonal set?
  • 1.4. State whether a1, a2, and a3 directed
    westward,
  • northward, and downward, respectively, is
    a right-
  • handed or a left-handed set.
  • 1.5. State all conditions for which A B is
    zero.
  • 1.6. State all conditions for which A B is
    zero.
  • 1.7. What is the significance of A B C 0?
  • 1.8. What is the significance of A (B C) 0?

24
Problem S1.1. Performing several vector
algebraic manipulations
25
Problem S1.1. Performing several vector
algebraic manipulations (continued)
26
  • 1.2 Cartesian
  • Coordinate System
  • (EEE, Sec. 1.2 FEME, Sec. 1.2)

27
Cartesian Coordinate System
28
  • Cartesian Coordinate System

29
  • Right-handed system
  • xyz xy
  • ax, ay, az are uniform unit vectors, that is, the
    direction of each unit vector is same everywhere
    in space.

30
Vector drawn from one point to another From
P1(x1, y1, z1) to P2(x2, y2, z2)
31

32
  • P1.8 A(12, 0, 0), B(0, 15, 0), C(0, 0, 20).
  • (a) Distance from B to C
  • (b) Component of vector from A to C along vector
    from B to C
  • Vector from A to C
  • Unit vector along vector from B to C

33
  • (c) Perpendicular distance from A to the line
    through B and C

34
1-33
  • (2) Differential Length Vector (dl)

35
dl dx ax dy ay dx ax f (x) dx ay
Unit vector normal to a surface
36
  • D1.5 Find dl along the line and having the
    projection dz on the z-axis.
  • (a)
  • (b)

37
  • (c) Line passing through (0, 2, 0) and (0, 0,
    1).

38
  • (3) Differential Surface Vector (dS)
  • Orientation of the surface is defined uniquely
    by the normal an to the surface.
  • For example, in Cartesian coordinates, dS in any
    plane parallel to the xy plane is

39
  • (4) Differential Volume (dv)
  • In Cartesian coordinates,

40
Review Questions
  • 1.9. What is the particular advantageous
    characteristic
  • associated with unit vectors in the
    Cartesian coordinate
  • system?
  • 1.10. What is the position vector?
  • 1.11. What is the total distance around the
    circumference of a
  • circle of radius 1 m? What is the total
    vector distance
  • around the circle?
  • 1.12. Discuss the application of differential
    length vectors to
  • find a unit vector normal to a surface
    at a point on the
  • surface.
  • 1.13. Discuss the concept of a differential
    surface vector.
  • 1.14. What is the total surface area of a cube of
    sides 1 m?
  • Assuming the normals to the surfaces to
    be directed
  • outward of the cubical volume, what is
    the total vector
  • surface area of the cube?

41
Problem S1.2. Finding the unit vector normal to
a surface and the differential surface vector, at
a point on it
42
  • 1.3 Cylindrical and Spherical Coordinate Systems
  • (EEE, Sec. 1.3 FEME, Appendix A)

43
Cylindrical Coordinate System
44
Spherical Coordinate System
45
1-44
Cylindrical and Spherical Coordinate Systems
Cylindrical (r, f, z) Spherical (r, q,
f) Only az is uniform. All three unit ar
and af are vectors are nonuniform. nonuniform.
46
1-45
  • x r cos f x r sin q cos f
  • y r sin f y r sin q sin f
  • z z z r cos q
  • D1.7 (a) (2, 5p/6, 3) in cylindrical coordinates

47
1-46
  • (b)

48
1-47
  • (c)

49
1-48
(d)
50
  • Conversion of vectors between coordinate systems

51
1-50
  • P1.18 A ar at (2, p/6, p/ 2)
  • B aq at (1, p/3, 0)
  • C af at (3, p/4, 3p/2)

52
1-51
53
1-52
  • (a)
  • (b)

54
  • (c)
  • (d)

55
Differential length vectors
  • Cylindrical Coordinates
  • dl dr ar r df af dz az
  • Spherical Coordinates
  • dl dr ar r dq aq r sin q df af

56
Review Questions
  • 1.15. Describe the three orthogonal surfaces that
    define the
  • cylindrical coordinates of a point.
  • 1.16. Which of the unit vectors in the
    cylindrical coordinate
  • system are not uniform? Explain.
  • 1.17. Discuss the conversion from the cylindrical
    coordinates
  • of a point to its Cartesian coordinates.
  • 1.18. Describe the three orthogonal surfaces that
    define the
  • spherical coordinates of a point.
  • 1.19. Discuss the nonuniformity of the unit
    vectors in the
  • spherical coordinate system.
  • 1.20. Discuss the conversion from the cylindrical
    coordinates
  • of a point to its Cartesian coordinates.

57
Problem S1.3. Determination of the equality of
vectors specified in cylindrical and spherical
coordinates
58
Problem S1.4. Finding the unit vector tangential
to a curve, at a point on it, in spherical
coordinates
59
  • 1.4 Scalar and Vector Fields
  • (EEE, Sec. 1.4 FEME, Sec. 1.3)

60
  • FIELD is a description of how a physical quantity
    varies from one point to another in the region of
    the field (and with time).
  • (a) Scalar fields
  • Ex Depth of a lake, d(x, y)
  • Temperature in a room, T(x, y, z)
  • Depicted graphically by constant magnitude
    contours or surfaces.

61
  • (b) Vector Fields
  • Ex Velocity of points on a rotating disk
  • v(x, y) vx(x, y)ax vy(x, y)ay
  • Force field in three dimensions
  • F(x, y, z) Fx(x, y, z)ax Fy(x, y, z)ay
  • Fz(x, y, z)az
  • Depicted graphically by constant magnitude
    contours or surfaces, and direction lines (or
    stream lines).

62
  • Example Linear velocity vector field of points
    on a rotating disk

63
  • (c) Static Fields
  • Fields not varying with time.
  • (d) Dynamic Fields
  • Fields varying with time.
  • Ex Temperature in a room, T(x, y, z t)

64
D1.10 T(x, y, z, t)
(a)
Constant temperature surfaces are elliptic
cylinders,
65
  • (b)
  • Constant temperature surfaces are spheres
  • (c)
  • Constant temperature surfaces are
    ellipsoids,

66
  • Procedure for finding the equation for the
    direction lines of a vector field

The field F is tangential to the direction line
at all points on a direction line.
67
  • Similarly

cylindrical
spherical
68
  • P1.26 (b)

(Position vector)
69
1-68
  • \ Direction lines are straight lines emanating
    radially from the origin. For the line passing
    through (1, 2, 3),

70
Review Questions
  • 1.21. Discuss briefly your concept of a scalar
    field and
  • illustrate with an example.
  • 1.22. Discuss briefly your concept of a vector
    field and
  • illustrate with an example.
  • 1.23. How do you depict pictorially the
    gravitational field of
  • the earth?
  • 1.24. Discuss the procedure for obtaining the
    equations for
  • the direction lines of a vector field.

71
Problem S1.5. Finding the equation for direction
line of a vector field, specified in spherical
coordinates
72
  • 1.5 Sinusoidally
  • Time-Varying Fields
  • (EEE, Sec. 3.6 FEME, Sec. 1.4)

73
Sinusoidal function of time
74
  • Polarization is the characteristic which
    describes how the position of the tip of the
    vector varies with time.
  • Linear Polarization
  • Tip of the vector
  • describes a line.
  • Circular Polarization
  • Tip of the vector
  • describes a circle.

75
  • Elliptical Polarization
  • Tip of the vector
  • describes an ellipse.
  • (i) Linear Polarization
  • Linearly polarized in the x direction.

Direction remains along the x axis
Magnitude varies sinusoidally with time
76
Linear polarization
77

Direction remains along the y axis
Magnitude varies sinusoidally with time
Linearly polarized in the y direction.
If two (or more) component linearly polarized
vectors are in phase, (or in phase opposition),
then their sum vector is also linearly
polarized. Ex
78
Sum of two linearly polarized vectors in phase
(or in phase opposition) is a linearly polarized
vector
1-77
79
  • (ii) Circular Polarization
  • If two component linearly polarized vectors are
  • (a) equal in amplitude
  • (b) differ in direction by 90
  • (c) differ in phase by 90,
  • then their sum vector is circularly polarized.

80
Circular Polarization
81
Example
82
  • (iii) Elliptical Polarization
  • In the general case in which either (i) or (ii)
    is not satisfied, then the sum of the two
    component linearly polarized vectors is an
    elliptically polarized vector.
  • Example

83
1-82
Example
84
1-83
  • D3.17
  • F1 and F2 are equal in amplitude ( F0) and
    differ in direction by 90. The phase difference
    (say f) depends on z in the manner 2p z  (3p
    z) p z.
  • (a) At (3, 4, 0), f p (0) 0.
  • (b) At (3, 2, 0.5), f p (0.5) 0.5 p.

85
1-84
  • (c) At (2, 1, 1), f p (1) p.
  • (d) At (1, 3, 0.2) f p (0.2) 0.2p.

86
Review Questions
  • 1.25. A sinusoidally time-varying vector is
    expressed in
  • terms of its components along the x-,
    y-, and z- axes.
  • What is the polarization of each of the
    components?
  • 1.26. What are the conditions for the sum of two
    linearly
  • polarized sinusoidally time-varying
    vectors to be
  • circularly polarized?
  • 1.27. What is the polarization for the general
    case of the sum
  • of two sinusoidally time-varying
    linearly polarized
  • vectors having arbitrary amplitudes,
    phase angles, and
  • directions?
  • 1.28. Considering the seconds hand on your analog
    watch
  • to be a vector, state its polarization.
    What is the
  • frequency?

87
Problem S1.6. Finding the polarization of the
sum of two sinusoidally time-varying vector fields
88
  • 1.6 The Electric Field
  • (EEE, Sec. 1.5 FEME, Sec. 1.5)

89
  • The Electric Field
  • is a force field acting on charges by virtue of
    the property of charge.
  • Coulombs Law

90
D1.13(b) From the construction, it is
evident that the resultant force is directed away
from the center of the square. The magnitude of
this resultant force is given by
Q2/4pe0(2a2)
Q2/4pe0(4a2)
Q2/4pe0(2a2)
91
1-90
92
  • Electric Field Intensity, E
  • is defined as the force per unit charge
    experienced by a small test charge when placed in
    the region of the field.
  • Thus
  • Units

Sources Charges Time-varying magnetic field
93
Electric Field of a Point Charge
(Coulombs Law)
94
Constant magnitude surfaces are spheres centered
at Q. Direction lines are radial lines emanating
from Q.
E due to charge distributions (a) Collection of
point charges
95
E1.2
Electron (charge e and mass m) is displaced from
the origin by D (ltlt d) in the x-direction and
released from rest at t 0. We wish to obtain
differential equation for the motion of the
electron and its solution.
96
  • For any displacement x,
  • is directed toward the origin,
  • and

97
1-96
  • The differential equation for the motion of the
  • electron is
  • Solution is given by

98
  • Using initial conditions and
    at t 0, we obtain
  • which represents simple harmonic motion about the
    origin with period

99
1-98
  • (b) Line Charges
  • Line charge
  • density, rL (C/m)
  • (c) Surface Charges
  • Surface charge
  • density, rS (C/m2)
  • (d) Volume Charges
  • Volume charge
  • density, r (C/m3)

100
1-99
  • E1.3 Finitely-Long Line Charge

101
1-100
1-100
102
1-101
  • Infinite Plane Sheet of Charge
  • of Uniform Surface Charge Density

103
1-102
104
1-103
105
1-104
  • D1.16
  • Given

106
Solving, we obtain
(b)
(a)
(d)
(c)
107
Review Questions
  • 1.29. State Coulombs law. To what law in
    mechanics is
  • Coulombs law analogous?
  • 1.30. What is the value of the permittivity of
    free space?
  • What are its units?
  • 1.31. What is the definition of electric field
    intensity?
  • What are its units?
  • 1.32. Describe the electric field due to a point
    charge.
  • 1.33. Discuss the different types of charge
    distributions.
  • How do you determine the electric field
    due to a charge
  • distribution?
  • 1.34. Describe the electric field due to an
    infinitely long line
  • charge of uniform density.
  • 1.35. Describe the electric field due to an
    infinite plane sheet
  • of uniform surface charge density.

108
Problem S1.7. Determination of conditions for
three point charges on a circle to be in
equilibrium
109
Problem S1.8. Finding the electric field due to
an infinite plane slab charge of specified charge
density
110
  • 1.7 The Magnetic Field
  • (EEE, Sec. 1.6 FEME, Sec. 1.6)

111
1-110
  • The Magnetic Field
  • acts to exert force on charge when it is in
    motion.
  • B Magnetic flux density vector
  • Alternatively, since charge in motion constitutes
    current, magnetic field exerts forces on current
    elements.

112
  • Units of B
  • Sources Currents
  • Time-varying electric field

113
  • Ampères Law of Force

114
1-113
  • Magnetic field due to a current element
  • (Biot-Savart Law)

a
Note
B right-circular to the axis of the current
element
115
  • E1.4

116
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117
  • Current Distributions
  • (a) Filamentary Current
  • I (A)
  • (b) Surface Current
  • Surface current density, JS (A/m)

118
  • (c) Volume Current
  • Density, J (A/m2)

119
  • P1.44

120
1-119
121
  • For infinitely long wire,

122
  • Magnetic Field Due to an Infinite Plane Sheet of
    Uniform Surface Current Density
  • This can be found by dividing the sheet into
    infinitely long strips parallel to the current
    density and using superposition, as in the case
    of finding the electric field due to an infinite
    plane sheet of uniform surface charge density.
    Instead of going through this procedure, let us
    use analogy. To do this, we first note the
    following

123
1-122
(a)
Point Charge
Current Element
124
1-123
  • (b) Line Charge Line Current

125
1-124
  • Then,
  • (c) Sheet Charge Sheet Current

126
Review Questions
  • 1.36. How is magnetic flux density defined in
    terms of force
  • on a moving charge? Compare the magnetic
    force on a
  • moving charge with electric force on a
    charge.
  • 1.37. How is magnetic flux density defined in
    terms of force
  • on a current element?
  • 1.38. What are the units of magnetic flux
    density?
  • 1.39. State Amperes force law as applied to
    current elements.
  • Why is it not necessary for Newtons
    third law to hold
  • for current elements?
  • 1.40. Describe the magnetic field due to a
    current element.
  • 1.41. What is the value of the permeability of
    free space?
  • What are its units?

127
Review Questions (continued)
  • 1.42. Discuss the different types of current
    distributions.
  • How do you determine the magnetic flux
    density due to
  • a current distribution?
  • 1.43. Describe the magnetic field due to an
    infinitely long,
  • straight, wire of current.
  • 1.44. Discuss the analogies between the electric
    field due to
  • charge distributions and the magnetic
    field due to
  • current distributions.

128
Problem S1.9. Finding parameters of an
infinitesimal current element that produces a
specified magnetic field
129
Problem S1.10. Finding the magnetic field due to
a specified current distribution within an
infinite plane slab
130
  • 1.8 Lorentz Force Equation
  • (EEE, Sec. 1.7 FEME, Sec. 1.6)

131
  • Lorentz Force Equation
  • For a given B, to find E,

132
D1.21
Find E for which acceleration experienced by q
is zero, for a given v.
(a)
133
1-132
1-132
(b)
(c)
134
  • For a given E, to find B,
  • One force not sufficient. Two forces are needed.

135
1-134

provided , which means v2 and v1
should not be collinear.
136
  • P1.54 For v v1 or v v2, test charge
    moves with constant velocity equal to the initial
    value. It is to be shown that for
  • the same holds.

(1)
(2)
(3)
137
1-136
Alternatively,
138
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139
Review Questions
  • 1.45. State Lorentz force equation.
  • 1.46. If it is assumed that there is no electric
    field, the
  • magnetic field at a point can be found
    from the
  • knowledge of forces exerted on a moving
    test charge
  • for two noncollinear velocities.
    Explain.
  • 1.47. Discuss the determination of E and B at a
    point from the
  • knowledge of forces experienced by a
    test charge at that
  • point for several velocities. What is
    the minimum
  • number of required forces? Explain.

140
Problem S1.11. Finding the electric and magnetic
fields from three forces experienced by a test
charge
141
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