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1-1 EE2030: Electromagnetics (I) Text Book: - Sadiku, Elements of Electromagnetics, Oxford University References: - William Hayt, Engineering Electromagnetics, Tata McGraw Hill

# 1-1 EE2030: Electromagnetics (I) Text Book: - Sadiku, Elements of Electromagnetics, Oxford University References: - William Hayt, Engineering Electromagnetics,

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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

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

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: 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

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