EE2030: Electromagnetics (I)
Text Book:
- Sadiku, Elements of Electromagnetics, Oxford University
References:
- William Hayt, Engineering Electromagnetics, Tata McGraw Hill
1-1
Part 1:
Vector Analysis
Vector Addition
Associative Law:
Distributive Law:
1-3
Rectangular Coordinate System
1-4
Point Locations in Rectangular Coordinates
1-5
Differential Volume Element
1-6
Summary
1-7
Orthogonal Vector Components
1-8
Orthogonal Unit Vectors
1-9
Vector Representation in Terms of
Orthogonal Rectangular Components
1-10
Summary
1-11
Vector Expressions in Rectangular
Coordinates
General Vector, B:
Magnitude of B:
Unit Vector in the
Direction of B:
1-12
Example
1-13
Vector Field
We are accustomed to thinking of a specific vector:
A vector field is a function defined in space that has magnitude
and direction at all points:
where r = (x,y,z)
1-14
The Dot Product
Commutative Law:
1-15
Vector Projections Using the Dot Product
B • a gives the component of B
in the horizontal direction
(B • a) a gives the vector component
of B in the horizontal direction
1-16
Projection of a vector on another
vector
Operational Use of the Dot Product
Given
Find
where we have used:
Note also:
1-18
Cross Product
1-19
Operational Definition of the Cross Product in
Rectangular Coordinates
Begin with:
where
Therefore:
Or…
1-20
Vector Product or Cross Product
Cylindrical Coordinate Systems
1-22
Cylindrical Coordinate Systems
1-23
Cylindrical Coordinate Systems
1-24
Cylindrical Coordinate Systems
1-25
Differential Volume in Cylindrical
Coordinates
dV = dddz
1-26
Point Transformations in Cylindrical
Coordinates
1-27
Dot Products of Unit Vectors in Cylindrical and
Rectangular Coordinate Systems
1-28
Example
Transform the vector,
into cylindrical coordinates:
Start with:
Then:
1-29
Example: cont.
Finally:
Spherical Coordinates
1-31
Spherical Coordinates
1-32
Spherical Coordinates
1-33
Spherical Coordinates
1-34
Spherical Coordinates
1-35
Spherical Coordinates
Point P has coordinates
Specified by P(r)
1-36
Differential Volume in Spherical Coordinates
dV = r2sindrdd
1-37
Dot Products of Unit Vectors in the Spherical
and Rectangular Coordinate Systems
1-38
Example: Vector Component Transformation
Transform the field,
, into spherical coordinates and components
1-39
Constant coordinate surfacesCartesian system
 If we keep one of the coordinate
variables constant and allow the
other two to vary, constant
coordinate surfaces are generated in
rectangular, cylindrical and
spherical coordinate systems.
 We can have infinite planes:
X=constant,
Y=constant,
Z=constant
 These surfaces are perpendicular to x, y and z axes respectively.
1-40
Constant coordinate surfacescylindrical system
 Orthogonal surfaces in cylindrical
coordinate system can be generated as
ρ=constnt
Φ=constant
z=constant
 ρ=constant is a circular cylinder,
 Φ=constant is a semi infinite plane with
its edge along z axis
 z=constant is an infinite plane as in the
rectangular system.
1-41
Constant coordinate surfacesSpherical system
 Orthogonal surfaces in spherical
coordinate system can be generated
as
r=constant
θ=constant
Φ=constant
 r=constant is a sphere with its centre at the origin,
 θ =constant is a circular cone with z axis as its axis and origin at
the vertex,
 Φ =constant is a semi infinite plane as in the cylindrical system.
1-42
Differential elements in rectangular
coordinate systems
1-43
Differential elements in Cylindrical
coordinate systems
1-44
Differential elements in Spherical
coordinate systems
1-45
Line integrals
 Line integral is defined as any integral that is to be evaluated
along a line. A line indicates a path along a curve in space.
1-46
Surface integrals
1-47
Volume integrals
1-48
DEL Operator
 DEL Operator in cylindrical coordinates:
 DEL Operator in spherical coordinates:
1-49
Gradient of a scalar field
 The gradient of a scalar field V is a vector that represents the
magnitude and direction of the maximum space rate of increase of V.
 For Cartesian Coordinates
 For Cylindrical Coordinates
 For Spherical Coordinates
1-50
Divergence of a vector
 In Cartesian Coordinates:
 In Cylindrical Coordinates:
 In Spherical Coordinates:
1-51
Gauss’s Divergence theorem
1-52
Curl of a vector
1-53
Curl of a vector
 In Cartesian Coordinates:
 In Cylindrical Coordinates:
 In Spherical Coordinates:
1-54
Stoke’s theorem
1-56
Laplacian of a scalar
1-57
Laplacian of a scalar
1-58