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ECE-305-Lecture6

# ECE-305-Lecture6 - Engineering Electro-Magnetics...

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Engineering Electro-Magnetics Engineering Electro-Magnetics ECE-305, Lecture 6 Dennis McCaughey, Ph.D. 4 February, 2010

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02/04/2010 Dennis McCaughey, ECE305, Spring 2010 2 Homework Homework Problems 3.3, 3.7,3.21 3.23 and 3.27 Due Next Mon Sept 28
02/04/2010 Dennis McCaughey, ECE305, Spring 2010 3 In-Class Problem In-Class Problem ( 29 { } ( 29 { } 2 2 Calculate the field along the z-axis for a surface charge density distributed over the following surfaces: , , , , 0 , , , 0 S a S x y z x a y b z b S x y z x y R z ρ = = = + = E

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02/04/2010 Dennis McCaughey, ECE305, Spring 2010 4 is everywhere either normal or tangental to the surface so that it becomes either 0 or On that portion of the surface where is not zero, is constant Applications of Gauss’s Law: Some Applications of Gauss’s Law: Some Symmetrical Distributions Symmetrical Distributions S Q d = S D S Ñ S D Solution is easy if we are able to choose a close surface with the two conditions: d S D S d S D S S D dS
02/04/2010 Dennis McCaughey, ECE305, Spring 2010 5 If These Conditions Hold If These Conditions Hold and become scalars and The remaing integral becomes S S S S d D dS D dS D S Only a knowledge of the symmetry enables us to chose such a closed surface Remember that the electric field intensity due to a positive point charge is directed radially outward from the point at which it it located

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02/04/2010 Dennis McCaughey, ECE305, Spring 2010 6 Drill 3.1 Drill 3.1 { } 60 point charge at the origin Find the electric flux passing through (a) , , 26 ,0 ,0 2 2 (b) , , 26 ,0 2 , 26 (c) the plane 26 C S r r cm S z cm z cm S z cm μ π π θ φ θ φ ρ φ ρ φ π - = = < < < < = = < < = ± = =
02/04/2010 Dennis McCaughey, ECE305, Spring 2010 7 Drill 3.2 Drill 3.2 ( 29 LB Calculate in rectangular coordinates at the point 2, 3,6 produced by: (a) a point charge 55 at ( 2,3, 6) (b) a uniform line charge 20 / on the -axis (c) a uniform surface charge densit A P Q mC Q mC m x ρ - = - - = D 2 y 120 / on the pane 5 SC C m z m ρ μ = = -

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02/04/2010 Dennis McCaughey, ECE305, Spring 2010 8 Drill 3.3 Drill 3.3 ( 29 2 2 0 0 Given 0.3 / in free space, find (a) at 2, 25 , 90 (b) the total charge within a sphere 3 (c) the total flux leaving the sphere 4 r r a nC m P r r r θ φ = = = = = = D E
02/04/2010 Dennis McCaughey, ECE305, Spring 2010 9 Special Gaussian Surfaces Special Gaussian Surfaces The surface over which Gauss’ law is applied must be closed, but it can be made up of several surface elements The defining conditions for a special gaussian surface are: 1. The surface is closed 2. At each point of the surface is either normal or tangental to the surface so that is becomes or 3. is sectionally constant over that part of the sur S S S S d D dS zero D D S D face where is normal S D

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02/04/2010 Dennis McCaughey, ECE305, Spring 2010 10 Line Charge Revisited Line Charge Revisited ( 29 2 0 0 From previous example, only the radial component is present Use a cylindrical surface where is everywhere normal 0 0 2 obt s s cyl sides top bottom z L s s z D a D f D Q d D dS dS dS D d dz D L ρ ρ ρ ρ φ π φ ρ ρ φ πρ = = = = = = = = + + = = ∫ ∫ D D S Ñ 0 aining 2 in terms of the line charge density 2 and 2 s L L L Q D D L Q L D E ρ ρ ρ πρ ρ ρ πρ ρ πε ρ = = = = =
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