Electromagnetic field (1)
Course Description
 Electric charge. Charge distributions. Coulomb's law. The Field
concept. Electric field. The field for different charge distributions.
Electric flux and flux density. The divergence. Gauss's law and the
divergence theorem. Potential difference. Potential of different
charge distributions. Potential gradient. Relationship between electric
field and Potential. The electric dipole. The energy in electrostatic
field. Dielectrics. The nature of dielectrics materials. Boundary
conditions. Capacitance. Current.
Conductors, Resistance,
Continuity equation. Relaxation time. Poisson's equation and
Laplace's equation. Solution of Laplace's equation under different
conditions. Magnetic field in Vacuum.
 The Boit- Savart Law. Basic laws of magnetic field. Gauss's theorem for
the magnetic field. Ampere's circuital law. Curl concept, Curl of the
magnetic field. Vector magnetic potential. Stocke's theorem.
Maxwell's equations for static fields.
 References:
 William H. Hayt. J.R. "Engineering Electromagnetic"
Course Outlines
 Ch1:- Vector Analysis
 Ch2:- Coulomb’s Law and Electric Field Intensity.
 Ch3:-Electric Flux Density, Gauss’ Law, and Divergence.
 Ch4:- Energy and Potential.
 Ch5:- Conductors, Dielectrics and Capacitance
 Ch6:- Poisson and Laplace's equation
Marks Distribution
 Assignments – 10%
 Quizzes – 20%
 Class Participation – 10%
 Mid Term – 20%
 Final – 40%
Introduction
 A field is a function that specifies a particular quantity everywhere in
a region.
 Electromagnetics (EM) is a branch of physics or electrical engineering
in which electric and magnetic phenomena are studied
 Electromagnetics: is the study of the effect of charges at rest and
charges in motion.
 Some special cases of electromagnetics:
 Electrostatics: charges at rest
 Magneto-statics: charges in steady motion (DC)
 Electromagnetic waves: waves excited by charges
varying motion
in time-
Why do we learn Engineering Electromagnetics?
 Electric and magnetic field exist nearly everywhere.
Applications
 Electromagnetic principles find
application in various disciplines such
as microwaves, x-rays, antennas,
electric machines, plasmas, etc.
Applications
 Electromagnetic fields are used in
induction heaters for melting,
forging, annealing, surface
hardening, and soldering
operation.
 Electromagnetic devices include
transformers, radio, television,
mobile phones, radars, lasers, etc.
Applications
Transrapid Train
• A magnetic traveling field moves
the vehicle without contact.
• The speed can be continuously
regulated by varying the
frequency of the alternating
Introduction
Electric field
Produced by the presence of
electrically charged particles,
and gives rise to the electric
force.
Magnetic field
Produced by the motion of
electric charges, or electric
current, and gives rise to the
magnetic force associated
with magnets.
Introduction
Introduction
 A quick look at these equations shows that
we shall be dealing with vector quantities.
 Vector analysis is a mathematical tool with
which electromagnetic concepts are most
conveniently
expressed
and
best
comprehended.
 A quantity can be either a scalar or a
vector.
Introduction
 A scalar is a quantity that has only magnitude.
 Quantities such as time, mass, distance, temperature,
entropy, electric potential, and population are scalars.
 (small letters: x,y,z,….)
 A vector is a quantity that has both magnitude and
direction.
 Vector
quantities
include
velocity,
displacement, and electric field intensity.
 (Capital letters: X,Y,Z, A,G,….)
force,
Vector Algebra
AB BA
A  (B + C)  ( A  B) + C
A  B  A  ( B )
A 1
 A
n n
AB  0  A  B
Coordinate System
 A coordinate system defines a set of reference
directions. In a 3D space, a coordinate system
can be specified by the intersection of 3 surfaces
at each and every point in space.
 The origin of the coordinate system is the
reference point relative to which we locate every
other point in space.
Coordinate System
 A position vector defines the position of a point in
space relative to the origin. These three
reference directions are referred to as coordinate
directions or base vectors, and are usually taken
to be mutually perpendicular (orthogonal) . In
this class, we use three coordinate systems:
Cartesian
cylindrical
Spherical
Rectangular Coordinate System
 Rectangular Coordinate System In Cartesian or
rectangular coordinate system a point P is
represented by coordinates (x,y,z) All the three
coordinates represent the mutually perpendicular
plane surfaces
 The range of coordinates are

-∞< x< ∞
 -∞< y< ∞
-∞< z< ∞
Rectangular Coordinate System
• Differential surface units:
dx  dy
dy  dz
dx  dz
• Differential volume unit :
dx  dy  dz
Vector Components and Unit Vectors
R PQ ?
r  xyz
r  xa x  ya y  za z
a x , a y , a z : unit vectors
R PQ  rQ  rP
 (2a x  2a y  a z )  (1a x  2a y  3a z )
 a x  4a y  2a z
Vector Components and Unit Vectors
 For any vector B,
:
B  Bxa x  By a y + Bz a z
B  Bx2  By2  Bz2  B
aB 
Magnitude of B
B
B

2
2
2
Bx  By  Bz
B
Unit vector in the direction of B
Vector Components and Unit Vectors
 Example
 Given points M(–1,2,1) and N(3,–3,0), find RMN and aMN.
R MN  (3a x  3a y  0a z )  (1a x  2a y  1a z )
 4a x  5a y  a z
a MN
R MN 4a x  5a y  1a z


R MN 42  (5) 2  (1) 2
 0.617a x  0.772a y  0.154a z
The Dot Product
 Given two vectors A and B, the dot product, or scalar product, is
defines as the product of the magnitude of A, the magnitude of
B, and the cosine of the smaller angle between them:
A  B  A B cos AB
 The dot product is a scalar, and it obeys the commutative law:
A B  B A
 For any two vectors Aand B
A  Axa x  Ay a y + Az a z
B  Bxa x  By a y + Bz a z
A  B  Ax Bx  Ay By + Az Bz
The Dot Product
 Example
The three vertices of a triangle are located at A(6,–1,2),
B(–2,3,–4), and C(–3,1,5). Find: (a) RAB; (b) RAC; (c) the angle
θBAC at vertex A; (d) the vector projection of RAB on RAC.
R AB  (2a x  3a y  4a z )  (6a x  a y  2a z )  8a x  4a y  6a z
B
R AC  (3a x  1a y  5a z )  (6a x  a y  2a z )  9a x  2a y  3a z
R AB  R AC  R AB R AC cosBAC
 BAC
A
C
(8a x  4a y  6a z )  (9a x  2a y  3a z )
R AB  R AC

 cos BAC 
R AB R AC
(8)2  (4)2  (6)2 (9)2  (2) 2  (3) 2

62
116 94
 0.594
  BAC  cos 1 (0.594)  53.56
The Dot Product
 One of the most important applications of the dot product
is that of finding the component of a vector in a given
direction.
• The scalar component of B in the direction of
the unit vector a is Ba
• The vector component of B in the direction of
the unit vector a is (Ba)a
B  a  B a cosBa  B cosBa
The Dot Product
 Example
The three vertices of a triangle are located at A(6,–1,2),
B(–2,3,–4), and C(–3,1,5). Find: (a) RAB; (b) RAC; (c) the
angle θBAC at vertex A; (d) the vector projection of RAB on
RAC.
R AB on R AC   R AB  a AC  a AC

(9a x  2a y  3a z )

  (8a x  4a y  6a z )
2
2
2
(

9)

(2)

(3)



 (9a x  2a y  3a z )

2
2
2
(

9)

(2)

(3)


62 (9a x  2a y  3a z )

 5.963a x  1.319a y  1.979a z
94
94
The Cross Product
 Given two vectors A and B, the magnitude of the cross
product, or vector product, written as AB, is defines as the
product of the magnitude of A, the magnitude of B, and the
sine of the smaller angle between them.
 The direction of AB is perpendicular to the plane containing
A and B and is in the direction of advance of a right-handed
screw as A is turned into B.
A  B  aN A B sin  AB
 The cross product is a vector, and it
is not commutative:
(B  A )  ( A  B )
ax  a y  az
a y  az  ax
az  ax  a y
The Cross Product
 Example
Given A = 2ax–3ay+az and B = –4ax–2ay+5az, find AB.
A  B  ( Ay Bz  Az By )a x  ( Az Bx  Ax Bz )a y  ( Ax By  Ay Bx )a z
  (3)(5)  (1)(2)  ax   (1)(4)  (2)(5)  a y  (2)(2)  (3)(4)  az
 13a x  14a y  16a z
The Cylindrical Coordinate System
The Cylindrical Coordinate System
The Cylindrical Coordinate System
• Differential surface units:
d   dz
 d  dz
d    d
• Differential volume unit :
d    d  dz
• Relation between the rectangular
and the cylindrical coordinate
systems
x    cos 
y    sin 
zz
  x2  y 2
y
  tan 1
x
zz
The Cylindrical Coordinate System
az
az
?
A  Ax a x  Ay a y + Az a z  A  A a   A a + Az a z
a
ay
a
ax
• Dot products of unit vectors in
cylindrical and rectangular
coordinate systems
A  A  a 
 ( Ax a x  Ay a y + Az a z )  a 
 Axa x  a   Ay a y  a  + Az a z  a 
 Ax cos   Ay sin 
A  A  a
 ( Axa x  Ay a y + Az a z )  a
 Ax a x  a  Ay a y  a + Az a z  a
  Ax sin   Ay cos 
Az  A  a z
 ( Axa x  Ay a y + Az a z )  a z
 Axa x  a z  Ay a y  a z + Az a z  a z
 Az
The Spherical Coordinate System
 Point P has coordinates
Specified by P(r, θ, ø)
 the r coordinate represents
a sphere of radius r
centered at origin . The θ
coordinate represents the
angle made by the cone
with z-axis. The ø coordinate
is the same as cylindrical
coordinate.
 The range of coordinates are 0≤ r < ∞ , 0≤ θ ≤ π , 0≤ ø < 2π
The Spherical Coordinate System
The Spherical Coordinate System
The Spherical Coordinate System
• Differential surface units:
dr  rd
dr  r sin  d
rd  r sin  d
• Differential volume unit :
dr  rd  r sin  d
The Spherical Coordinate System
• Relation between the rectangular and the
spherical coordinate systems
x  r sin  cos 
r
y  r sin  sin 
  cos
1
z  r cos
  tan
1
x2  y 2  z 2 , r  0
z
x y z
2
y
x
• Dot products of unit vectors in spherical and
rectangular coordinate systems
2
2
, 0    180
The Spherical Coordinate System
 Example
Given the two points, C(–3,2,1) and D(r = 5, θ = 20°, Φ = –
70°), find: (a) the spherical coordinates of C; (b) the
rectangular coordinates of D.
r
x 2  y 2 z 2 (3) 2  (2) 2  (1) 2  3.742
  cos
1
z
x2  y 2  z 2
 cos 1
1
 74.50
3.742
y
1 2
  tan  tan
 33.69  180  146.31
x
3
 C ( r  3.742,   74.50,   146.31)
1
 D( x  0.585, y  1.607, z  4.698)
Homework 1
Prop: 4,8,10
 Due: Next lecture