1-1 ee2030: electromagnetics (i) text book: - sadiku, elements of electromagnetics, oxford...
TRANSCRIPT
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EE2030: Electromagnetics (I)
Text Book: - Sadiku, Elements of Electromagnetics, Oxford University
References: - William Hayt, Engineering Electromagnetics, Tata McGraw Hill
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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
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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
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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:
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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
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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,
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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.
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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:
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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
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Divergence of a vector
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In Cartesian Coordinates:
In Cylindrical Coordinates:
In Spherical Coordinates:
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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:
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Stoke’s theorem
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Laplacian of a scalar
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Laplacian of a scalar
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