Transcript Document

Engineering Electromagnetics
Chapter 1:
Vector Analysis
1-1
Vector Addition
Associative Law:
Distributive Law:
1-2
Rectangular Coordinate System
1-3
Point Locations in Rectangular Coordinates
1-4
Differential Volume Element
1-5
Summary
1-6
Orthogonal Vector Components
1-7
Orthogonal Unit Vectors
1-8
Vector Representation in Terms of
Orthogonal Rectangular Components
1-9
Summary
1-10
Vector Expressions in Rectangular
Coordinates
General Vector, B:
Magnitude of B:
Unit Vector in the
Direction of B:
1-11
Example
1-12
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)
1-13
The Dot Product
Commutative Law:
1-14
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
1-15
Operational Use of the Dot Product
Given
Find
where we have used:
Note also:
1-16
Cross Product
1-17
Operational Definition of the Cross Product in
Rectangular Coordinates
Begin with:
where
Therefore:
Or…
1-18
Circular Cylindrical Coordinates
Point P has coordinates
Specified by P(z)
z
y
x
1-19
Orthogonal Unit Vectors in Cylindrical
Coordinates
1-20
Differential Volume in Cylindrical
Coordinates
dV = dddz
1-21
Summary
1-22
Point Transformations in Cylindrical
Coordinates
1-23
Dot Products of Unit Vectors in Cylindrical and
Rectangular Coordinate Systems
1-24
Example
Transform the vector,
into cylindrical coordinates:
1-25
Transform the vector,
into cylindrical coordinates:
Start with:
1-26
Transform the vector,
into cylindrical coordinates:
Then:
1-27
Transform the vector,
into cylindrical coordinates:
Finally:
1-28
Spherical Coordinates
Point P has coordinates
Specified by P(r)
1-29
Constant Coordinate Surfaces in Spherical
Coordinates
1-30
Unit Vector Components in Spherical
Coordinates
1-31
Differential Volume in Spherical Coordinates
dV =
r2sindrdd
1-32
Dot Products of Unit Vectors in the Spherical
and Rectangular Coordinate Systems
1-33
Example: Vector Component Transformation
Transform the field,
, into spherical coordinates and components
1-34
Summary Illustrations
1-35