ece107_set3_nbg

ece107_set3_nbg - ECE 107: Electromagnetism Set 3: Review...

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Unformatted text preview: ECE 107: Electromagnetism Set 3: Review of vector analysis Instructor: Prof. Vitaliy Lomakin Department of Electrical and Computer Engineering University of California, San Diego, CA 92093 1 Why do we need vector analysis? Maxwell’s equations Time domain Maxwell’s equations: ∂B ∂t ∂D ∇×H = +J ∂t ∇×E = − ∇⋅D = ρ ∇⋅B = 0 Basic quantities: E(t , r ) − electric field H(t , r ) − magnetic field Vm Am D(t , r ) − electric flux density C m2 B(t , r ) − magnetic flux density webers m2 J − electric current density A m2 ρ − charge density Cm 3 2 Scalar fields Single value function that depends on position Vector fields A vector is given for each position in space Velocity of each atom flow of heat ˆ ˆ ˆ ˆ ˆ A = a A = aA = x Ax + y A y + z Az ˆ a =1 Vector fields A vector is given for each position in space ˆ ˆ ˆ ˆ ˆ A = a A = aA = x Ax + y A y + z Az A = A2 + A2 + A2 x y z ˆ ˆ ˆ C = A + B = x(Ax + B x ) + y(Ay + B y ) + z(Az + B z ) C A B Vector fields Position Vector: From origin to point P Distance Vector: Between two points Scalar and vectors products •  Scalar product •  Vector product ˆ n ⊥ A & B (right-handed rule) 7 Vector relations cartesian cylindrical spherical 8 Relation between the coordinates 9 Gradient •  Gradient –  Gradient applies to a scalar. Its result is a vector of maximum rate of change of the scalar del operator directional derivative –  Del operator –  Operations with the gradient 10 Divergence •  Divergence –  Divergence applies to a vector. Its result is a scalar giving the flux per unit volume. –  Gauss’s theorem: –  11 Curl •  Curl operator –  Curl operator applies to a vector and results in a vector. Curl is the maximal circulation of the vector per unit area –  Stockes’s theorem: –  Vector identities: 12 Laplacian •  Scalar Laplacian operator –  Applies to a scalar and produces a scalar •  Vector Lapacian operator –  Applies to a vector and produces a vector this simple only in the Cartesian coordinates! is valid in any coordinate system with proper expresion for ∇! 13 Gradient, divergence, and curl 14 ...
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This note was uploaded on 04/02/2012 for the course ECE 107 taught by Professor Fullterton during the Winter '07 term at UCSD.

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ece107_set3_nbg - ECE 107: Electromagnetism Set 3: Review...

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