Transcript Document
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 = dddz
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 = r2sindrdd
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