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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

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Vector Addition

Associative Law:

Distributive Law:

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Rectangular Coordinate System

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Point Locations in Rectangular Coordinates

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Differential Volume Element

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Summary

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Orthogonal Vector Components

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Orthogonal Unit Vectors

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Vector Representation in Terms of Orthogonal Rectangular Components

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Summary

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Vector Expressions in Rectangular Coordinates

General Vector, B:

Magnitude of B:

Unit Vector in the Direction of B:

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Example

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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)

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The Dot Product

Commutative Law:

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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

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Operational Use of the Dot Product

Given

Find

where we have used:

Note also:

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Cross Product

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Operational Definition of the Cross Product in Rectangular Coordinates

Therefore:

Or…

Begin with:

where

Vector Product or Cross Product

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Cylindrical Coordinate Systems

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Cylindrical Coordinate Systems

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Cylindrical Coordinate Systems

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Cylindrical Coordinate Systems

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Differential Volume in Cylindrical Coordinates

dV = dddz

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Point Transformations in Cylindrical Coordinates

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Dot Products of Unit Vectors in Cylindrical and Rectangular Coordinate Systems

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Transform the vector, into cylindrical coordinates:

Example

Start with:

Then:

Finally:

Example: cont.

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Spherical Coordinates

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Spherical Coordinates

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Spherical Coordinates

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Spherical Coordinates

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Spherical Coordinates

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Spherical Coordinates

Point P has coordinatesSpecified by P(r)

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Differential Volume in Spherical Coordinates

dV = r2sindrdd

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Dot Products of Unit Vectors in the Spherical and Rectangular Coordinate Systems

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Example: Vector Component Transformation

Transform the field, , into spherical coordinates and components

Constant coordinate surfaces- Cartesian system

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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.

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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.

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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

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Differential elements in Cylindricalcoordinate systems

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Differential elements in Sphericalcoordinate systems

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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

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Volume integrals

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DEL Operator

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DEL Operator in cylindrical coordinates:

DEL Operator in spherical coordinates:

Gradient of a scalar field

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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

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In Cartesian Coordinates:

In Cylindrical Coordinates:

In Spherical Coordinates:

Gauss’s Divergence theorem

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Curl of a vector

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Curl of a vector In Cartesian Coordinates:

In Cylindrical Coordinates:

In Spherical Coordinates:

Stoke’s theorem

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Laplacian of a scalar

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Laplacian of a scalar

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