ECE-305-Lecture8

# ECE-305-Lecture8 - Engineering Electro-Magnetics...

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Engineering Electro-Magnetics Engineering Electro-Magnetics ECE-305, Lecture 8 Dennis McCaughey, Ph.D. 28 February, 2010

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02/18/2010 Dennis McCaughey, ECE305, Spring 2010 2 Homework Homework None Exam Monday October 5
02/18/2010 Dennis McCaughey, ECE305, Spring 2010 3 Example 3.5 (Divergence Theorem) Example 3.5 (Divergence Theorem) x y z x =1 y =2 z =3 2 xy a x 2 x 2 a y 2 2 has no component 0 on the surfaces: x y xya x a z d = + = D D D S { } { } , , 0 1, 0 2, 0 and , , 0 1, 0 2, 3 x y z x y z x y z x y z = = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 3 2 3 2 0 0 0 0 3 2 3 2 0 0 0 0 3 2 0 0 Surfaces in the plane 0, , 1, , 0 2 2 3 0 4 0 Surfaces in the plane 12 x x x x S x x x x yz d dz dy y z a dz dy y z a dz dy a a dz dy ya a dz dy y xz d = - + = - + = = - = - D S D D D Ñ ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 3 1 3 1 0 0 0 0 3 1 3 1 2 2 0 0 0 0 ,0, 2 0 , , y y y y S y y y y dz dx x z a dz dx x z a dz dx x a a dz dx x a a = - + = - + = S D D Ñ

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02/18/2010 Dennis McCaughey, ECE305, Spring 2010 4 Example 3.5 Continued Example 3.5 Continued ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 3 2 1 0 0 0 2 2 2 0 = 2 3 0 4 0 1 0 12 x y v xya x a xy x y x y dz dy y dx = = + ∇• = + = + ∇• = - - - D D D
02/18/2010 Dennis McCaughey, ECE305, Spring 2010 5 Reminder: Expressions for Reminder: Expressions for d d L L Rectangular: Cylindrical: Spherical: sin x y z z r d dxa dya dza d d a d a dza d dra rd a r d a ρ φ θ ρ φ θ φ = + + = + + = + + L L L

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02/18/2010 Dennis McCaughey, ECE305, Spring 2010 6 Circular Path Circular Path ( 29 0 1 2 0 1 0 2 1 0 1 0 2 Work done moving a positive charge around the x axis Cylindrical coordinates: 2 0 2 force is always at right angles to the path L L L E a a d d a W Q a d Q a a d ρ φ π πε ρ ρ φ = = = = - = - = E L L
02/18/2010 Dennis McCaughey, ECE305, Spring 2010 7 Radial Path Radial Path ( 29 0 0 0 0 0 2 Work done moving a positive charge from to Cylindrical coordinates: 2 2 ln 2 2 L b L a b L a b L L a E a a a b d d a W Q a d Q a a d d b Q Q a ρ πε ρ πε = = = = - = - = - = - E L L b > a means the work done is negative

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## This note was uploaded on 09/13/2011 for the course ECE 305 taught by Professor Staff during the Spring '08 term at George Mason.

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ECE-305-Lecture8 - Engineering Electro-Magnetics...

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