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
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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
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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 φ ρ ρ φ θ 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 ρ ρ φ π ρ π ρ φ ρ πε ρ ρ φ ρ πε ρ ρ ρ φ πε ρ = = = = - = - = E L L
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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 ρ ρ ρ ρ ρ ρ ρ πε ρ ρ ρ πε ρ ρ ρ πε ρ ρ ρ ρ πε ρ πε = = = = - = - = - = - E L L b > a means the work done is negative This means the external source that is moving the charge is receiving energy This is completely taken care of by the limits of intergration. Do not “play” with signs on dL
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02/18/2010 Dennis McCaughey, ECE305, Spring 2010 8 Definition of Potential Difference and
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