ECE-305-Lecture16

# ECE-305-Lecture16 - Engineering Electro-Magnetics...

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Unformatted text preview: Engineering Electro-Magnetics Engineering Electro-Magnetics ECE-305, Lecture 16 Dennis McCaughey, Ph.D. 23 March 2010 03/23/2010 Dennis McCaughey, ECE305, Spring 2010 2 Stokes Theorem: Stokes Theorem: ( 29 Applying the curl to one of the incremental surfaces right hand normal to the surface The subscript on indicates the path is the perimeter of an incremental area . Thus S N S S d S N d S d S ∆ ∆ ∆ • ∇× ∆ ⇒ ∆ • ∆ ∫ ∫ H L H L H L B Ñ Ñ ( 29 ( 29 ( 29 ( 29 or Every interior wall is covered once in each direction All interior contributions will cancel Only the outside perimeter is left N N S S a d a S d d ∆ ∇× • • ∇× • ∆ = ∇× • ∆ • • ⇒ • • ≡ ∇× • ∫ ∫ ∫ H H L H H S H L H S B B Ñ Ñ Stokes theorem allows us to determine the point form of Ampere’s circuital law from its integral form and vice versa Stokes theorem equates a line integral to a surface integral ( 29 s d I d • = = ∇ × • ∫ ∫ H L H S Ñ 03/23/2010 Dennis McCaughey, ECE305, Spring 2010 3 Example (Line Integral) Example (Line Integral) 1 2 3 2 6 sin 18 sin cos 6 sin 18 sin cos sin sin 6 sin 18 sin cos sin 6 sin 18 si r r r r r r r r d d d d H a H a H a r a r a H r H H r d dra rd a r d a d H dr H rd H r d d r dra a r r d a a r dr r θ φ θ φ φ θ φ θ φ θ φ φ φ φ θ φ φ θ φ θ θ φ θ θ φ φ θ φ θ φ φ • = • + • + • = + + = + = = = = + + • = + + • = • + • = + ∫ ∫ ∫ ∫ H L H L H L H L H H L H L H L Ñ ( 29 ( 29 ( 29 2 2 2 n cos 6 sin 18 sin cos paths 1 4,0 0.1 , 2 4, 0.1 ,0 0.3 3 4,0 0.1 , 0.3 r d H dl r dr H dl H dl r d r r r θ φ θ φ φ φ θ φ φ θ π φ θ π φ π θ π φ...
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ECE-305-Lecture16 - Engineering Electro-Magnetics...

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