(2) Vector Analysis
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Transcript of (2) Vector Analysis
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Electromagnetics (I)
Chapter 2Vector Analysis
Vector algebra
Coordinate systems
Vector calculus
Electromagnetics (I)
Maxwells EquationsThe governing equations for electromagnetic energy involves vectorand scalar calculation.
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Electromagnetics (I)
Vector Algebra
Electromagnetics (I)
Vector Addition and Subtraction
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Electromagnetics (I)
Product of Vector
Multiplication by a scalar
Dot (scalar) product
Cross (vector) product
Electromagnetics (I)
Product of Vector Multiplication by a scalar
Dot (scalar) product
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Electromagnetics (I)
Product of Vector
Electromagnetics (I)
Product of Vector Cross (vector) productA new product perpendicular to the plane containing the twoVectors.
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Electromagnetics (I)
Product of Three Vectors
Electromagnetics (I)
Vector Triple Product
2
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Electromagnetics (I)
Orthogonal Coordinate Systems
Electromagnetics (I)
Metric Coefficients
A conversion factor , h i , is defined and used to convert a differential change of the coordinates, du i , into a change in length, dl , where u i may not be a length and thus a conversion factor is needed.
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Electromagnetics (I)
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Differential Length change, Area, and Volume
Electromagnetics (I)
Orthogonal Coordinate Systems
(r, , z) (r, , )
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Electromagnetics (I)
Cartesian Coordinates
Electromagnetics (I)
Cartesian Coordinates
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Electromagnetics (I)
Cylindrical Coordinates
Electromagnetics (I)
Cylindrical Coordinates
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Electromagnetics (I)
Spherical Coordinates
Electromagnetics (I)
Spherical Coordinates
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Electromagnetics (I)
Coordinate TransformationCylindrical coordinateCartesian coordinate
where
Electromagnetics (I)
Coordinate TransformationChange of variables
Cylindrical coordinateCartesian coordinate
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Electromagnetics (I)
Coordinate TransformationChange of variables
Spherical coordinateCartesian coordinate
Electromagnetics (I)
Coordinate TransformationSpherical coordinateCartesian coordinate
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Electromagnetics (I)
Coordinate TransformationSpherical coordinateCartesian coordinate
)where
A sin
similarly,
Electromagnetics (I)
Metric Coefficients
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Electromagnetics (I)
Differential Length change, Area, and Volume
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Electromagnetics (I)
Basic Orthogonal Coordinate Systems
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Electromagnetics (I) Integrals Containing Vector Functions
Line IntegralsIntegral of scalar functions
Integral of vector function (scalar line integral)
Cartesian coordinate
Cartesian coordinate
Electromagnetics (I)
Scalar Line Integral
(i) In Cartesian coordinates.
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Electromagnetics (I)
Scalar Line Integral
(ii) In cylindrical coordinates.
Electromagnetics (I)
Scalar Surface Integral
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Electromagnetics (I)
Scalar Surface Integral
Electromagnetics (I)
Vector Calculus
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Electromagnetics (I)
Gradient
Electromagnetics (I)
Gradient
For Cartesian coordinate
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Electromagnetics (I)
Example 2-16
The electrostatic field intensity E is derivable as the negative gradient of a scalar electric potential V; that is, E = V.
(a)
(b)
Electromagnetics (I)
Vector FieldsField : directed field lines flux lines or streamlines
(i) Direction of the vector field: directed line or curve.
(ii) Magnitude of the field: density of the lines.
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Electromagnetics (I)
Divergence
Electromagnetics (I)
Expression for Divergence
Similarly,
front surface
back surface
(expanded as Taylor series)
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Electromagnetics (I)
Expression for Divergence
Recall
A = a x A x + a y A y + a z A z
Divergence of a vector field
Electromagnetics (I)
Divergence of a Vector Field General orthogonal coordinates
Cartesian coordinates
Spherical coordinates
Cylindrical coordinates
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Electromagnetics (I)
Divergence Theorem
Recall
Electromagnetics (I)
Example 3-20
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Electromagnetics (I)
Vector Calculus
Electromagnetics (I)
For Cartesian coordinate
Gradient
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Electromagnetics (I)
Divergence
Divergence of a vector field
Electromagnetics (I)
Divergence of a Vector Field General orthogonal coordinates
Cartesian coordinates
Spherical coordinates
Cylindrical coordinates
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Electromagnetics (I)
Divergence Theorem
Recall
Electromagnetics (I)
Curl
(Vortex source can cause a circulation of a vector field around the field)
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Electromagnetics (I)
Expression for Curl
Electromagnetics (I)
Expression for Curl
Similarly,
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Electromagnetics (I)
Expression for Curl
Curl of a vector field in Cartesian coordinates
Curl of a vector field in general orthogonal curvilinear coordinates
Electromagnetics (I)
General orthogonal coordinates
Cartesian coordinates
Spherical coordinates
Cylindrical coordinates
Expression for Curl
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Electromagnetics (I)
Basic Orthogonal Coordinate Systems
Electromagnetics (I)
Example 2-21
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Electromagnetics (I)
Stokes Theorem
where
Electromagnetics (I)
Divergence Theorem
Recall
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Electromagnetics (I)
Example 2-22
For surface integral
Electromagnetics (I)
Example 2-22
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Electromagnetics (I)
Two Null Identities Identity I
Electromagnetics (I)
Two Null Identities Identity II
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Electromagnetics (I)
Identity II
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Vector Field Classification
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Electromagnetics (I)
Helmholtzs Theorem
Electromagnetics (I)
Helmholtzs Theorem
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Electromagnetics (I)
Helmholtzs Theorem
Electromagnetics (I)
Helmholtzs Theorem
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Electromagnetics (I)
Example