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
1Fundamentals 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
2Program 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
3Maxwells 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
5Instructional 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
6Instructional 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.
22To find C, use (1) or (2).
23Review 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?
24Problem S1.1. Performing several vector
algebraic manipulations
25Problem S1.1. Performing several vector
algebraic manipulations (continued)
26- 1.2 Cartesian
- Coordinate System
- (EEE, Sec. 1.2 FEME, Sec. 1.2)
27Cartesian 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.
30Vector 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 -
-
341-33
- (2) Differential Length Vector (dl)
35dl 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,
-
40Review 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?
41Problem 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)
43Cylindrical Coordinate System
44Spherical Coordinate System
451-44
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.
461-45
- 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
-
471-46
481-47
491-48
(d)
50- Conversion of vectors between coordinate systems
-
511-50
- 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)
521-51
531-52
54 55Differential 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
56Review 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.
57Problem S1.3. Determination of the equality of
vectors specified in cylindrical and spherical
coordinates
58Problem 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)
64D1.10 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.
67cylindrical
spherical
68(Position vector)
691-68
- \ Direction lines are straight lines emanating
radially from the origin. For the line passing
through (1, 2, 3),
70Review 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.
71Problem 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)
73Sinusoidal 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
76Linear 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
78Sum 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.
80Circular Polarization
81Example
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
831-82
Example
841-83
- 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.
-
851-84
-
- (c) At (2, 1, 1), f p (1) p.
-
- (d) At (1, 3, 0.2) f p (0.2) 0.2p.
86Review 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?
87Problem 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
90D1.13(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)
911-90
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
93Electric Field of a Point Charge
(Coulombs Law)
94Constant 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
95E1.2
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
971-96
- 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
991-98
- (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)
1001-99
- E1.3 Finitely-Long Line Charge
1011-100
1-100
1021-101
- Infinite Plane Sheet of Charge
- of Uniform Surface Charge Density
1031-102
1041-103
1051-104
106Solving, we obtain
(b)
(a)
(d)
(c)
107Review 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.
108Problem S1.7. Determination of conditions for
three point charges on a circle to be in
equilibrium
109Problem 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)
1111-110
- 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 1141-113
- Magnetic field due to a current element
- (Biot-Savart Law)
a
Note
B right-circular to the axis of the current
element
115 116(No Transcript)
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 1201-119
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
1231-122
(a)
Point Charge
Current Element
1241-123
- (b) Line Charge Line Current
1251-124
- Then,
- (c) Sheet Charge Sheet Current
126Review 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?
127Review 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.
128Problem S1.9. Finding parameters of an
infinitesimal current element that produces a
specified magnetic field
129Problem 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,
-
132D1.21
Find E for which acceleration experienced by q
is zero, for a given v.
(a)
1331-132
1-132
(b)
(c)
134- For a given E, to find B,
- One force not sufficient. Two forces are needed.
1351-134
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)
1371-136
Alternatively,
138(No Transcript)
139Review 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.
140Problem S1.11. Finding the electric and magnetic
fields from three forces experienced by a test
charge
141The End