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Unformatted text preview: Prof. David R. Jackson ECE Dept. Spring 2011 Notes 13 ECE 2317 ECE 2317 Applied Electricity and Magnetism Applied Electricity and Magnetism Divergence  Physical Concept Divergence  Physical Concept r < a : 2 2 3 C/m ˆ 4 4 3 3 encl S r v v r D n dS Q r D r r D π ρ π ρ ⋅ = = = ∫ Ñ Start by considering a sphere of uniform volume charge density The electric field is calculated using Gauss's law: x y z ρ v = ρ v a r r > a : 2 2 3 3 2 C/m 4 4 3 3 r v v r r D a a D r π ρ π ρ = = Divergence  Physical Concept (cont.) Divergence  Physical Concept (cont.) x y z ρ v = ρ v a r 2 ˆ 4 r S D ndS r D ψ π = ⋅ = ∫ 3 4 3 v a ψ π ρ = ( r > a ) ( r < a ) Recall that 3 4 3 v r ψ π ρ = 2 C/m 3 v r r D ρ = 2 3 2 C/m 3 v r a D r ρ = Divergence  Physical Concept (cont.) Divergence  Physical Concept (cont.) ( r < a ) ( r > a ) Flux through a spherical surface: Hence More flux lines are added as the radius increases (as long as we stay inside the charge). ˆ S D n dS ψ ∆ ∆ = ⋅ ∫ Observation: ∆ V ∆ S Divergence  Physical Concept (cont.) Divergence  Physical Concept (cont.) ˆ S D n dS ψ ∆ ∆ = ⋅ ∫ The net flux out of a small volume ∆ V inside the charge is not zero. Divergence is a mathematical way of describing this. 1 ˆ lim V S div D D n dS V ∆ → ∆ ≡ ⋅ ∆ ∫ Ñ ∆ V Definition of divergence: Gauss’s Law  Differential Form Gauss’s Law  Differential Form Note: The limit exists independent of the shape of the volume (proven later). A point with a positive divergence acts as a “source.” A point with a negative divergence acts as a “sink.” 1 ˆ lim V S div D D n dS V ∆ → ∆ ≡ ⋅ ∆ ∫ Ñ ∆ V ρ v ( r ) ( 29 ˆ encl v S D n dS Q r V ρ ∆ ⋅ = ≈ ∆ ∫ Ñ ( 29 ( 29 ( 29 ( 29 1 lim v V v div D r r V V r ρ ρ ∆ → = ∆ ∆ = Apply divergence definition to a small volume inside a region of charge: Gauss’s Law  Differential Form Gauss’s Law  Differential Form Gauss's law: Hence Gauss’s Law  Differential Form (cont.) Gauss’s Law  Differential Form (cont.) ( 29 ( 29 v div D r r ρ = The electric Gauss law in point form: This is one of Maxwell’s equations . Example Example Choose ∆ V to be small sphere of radius r : x y z ρ v = ρ v a r 1 ˆ lim...
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This note was uploaded on 08/05/2011 for the course ECE 2317 taught by Professor Staff during the Spring '08 term at University of Houston.
 Spring '08
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