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EE2030: Electromagnetics (I)
Text Book: - Sadiku, Elements of Electromagnetics, Oxford University
References: - William Hayt, Engineering Electromagnetics, Tata McGraw Hill
Part 1:
Vector Analysis
1-3
Vector Addition
Associative Law:
Distributive Law:
1-4
Rectangular Coordinate System
1-5
Point Locations in Rectangular Coordinates
1-6
Differential Volume Element
1-7
Summary
1-8
Orthogonal Vector Components
1-9
Orthogonal Unit Vectors
1-10
Vector Representation in Terms of Orthogonal Rectangular Components
1-11
Summary
1-12
Vector Expressions in Rectangular Coordinates
General Vector, B:
Magnitude of B:
Unit Vector in the Direction of B:
1-13
Example
1-14
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)
1-15
The Dot Product
Commutative Law:
1-16
Vector Projections Using the Dot Product
B • a gives the component of Bin the horizontal direction
(B • a) a gives the vector component of B in the horizontal direction
Projection of a vector on another vector
1-18
Operational Use of the Dot Product
Given
Find
where we have used:
Note also:
1-19
Cross Product
1-20
Operational Definition of the Cross Product in Rectangular Coordinates
Therefore:
Or…
Begin with:
where
Vector Product or Cross Product
1-22
Cylindrical Coordinate Systems
1-23
Cylindrical Coordinate Systems
1-24
Cylindrical Coordinate Systems
1-25
Cylindrical Coordinate Systems
1-26
Differential Volume in Cylindrical Coordinates
dV = dddz
1-27
Point Transformations in Cylindrical Coordinates
1-28
Dot Products of Unit Vectors in Cylindrical and Rectangular Coordinate Systems
1-29
Transform the vector, into cylindrical coordinates:
Example
Start with:
Then:
Finally:
Example: cont.
1-31
Spherical Coordinates
1-32
Spherical Coordinates
1-33
Spherical Coordinates
1-34
Spherical Coordinates
1-35
Spherical Coordinates
1-36
Spherical Coordinates
Point P has coordinatesSpecified by P(r)
1-37
Differential Volume in Spherical Coordinates
dV = r2sindrdd
1-38
Dot Products of Unit Vectors in the Spherical and Rectangular Coordinate Systems
1-39
Example: Vector Component Transformation
Transform the field, , into spherical coordinates and components
Constant coordinate surfaces- Cartesian system
1-40
If we keep one of the coordinate variables constant and allow theother 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.
1-41
Constant coordinate surfaces- cylindrical 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 therectangular system.
1-42
Constant coordinate surfaces- Spherical system
Orthogonal surfaces in spherical coordinate system can be generated as r=constant θ=constant Φ=constant
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.
r=constant is a sphere with its centre at the origin,
Differential elements in rectangularcoordinate systems
1-43
1-44
Differential elements in Cylindricalcoordinate systems
1-45
Differential elements in Sphericalcoordinate systems
1-46
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.
Surface integrals
1-47
Volume integrals
1-48
DEL Operator
1-49
DEL Operator in cylindrical coordinates:
DEL Operator in spherical coordinates:
Gradient of a scalar field
1-50
The gradient of a scalar field V is a vector that represents themagnitude and direction of the maximum space rate of increase of V.
For Cartesian Coordinates
For Cylindrical Coordinates
For Spherical Coordinates
Divergence of a vector
1-51
In Cartesian Coordinates:
In Cylindrical Coordinates:
In Spherical Coordinates:
Gauss’s Divergence theorem
1-52
Curl of a vector
1-53
1-54
Curl of a vector In Cartesian Coordinates:
In Cylindrical Coordinates:
In Spherical Coordinates:
Stoke’s theorem
1-56
Laplacian of a scalar
1-57
Laplacian of a scalar
1-58