325_Sp2011_3_Flux,Gauss_Law,and_Divergence

325_Sp2011_3_Flux,Gauss_Law,and_Divergence - © Copyright...

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Unformatted text preview: © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin Electric Field Flux, Gauss’s Law, Divergence 1 3. Gauss’s law, electric flux, and divergence EE325 Mikhail Belkin © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin Electric Field Flux, Gauss’s Law, Divergence 2 Another way to look at electric fields Electric field streamlines look like “electric field flow lines” where positive charges are sources of the electric field and negative charges as sinks for the electric field. We expect then that the total electric field ‘flow’ or FLUX into or out of the surface that encloses an assembly of charges is only determined by the amount of the charges enclosed Net electric field flow ~ (+q+(-q))=0 Net electric field flow ~ +q © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin Electric Field Flux, Gauss’s Law, Divergence 3 A mathematical concept of Flux • imagine a fluid flowing through a rectangular pipe, with velocity constant across the cross section of the pipe – The flow density or “flux density” of fluid flowing through the pipe is the amount of liquid passing per second per unit area • total liquid “flux” through the surface is Φ =(flux density)*(total area normal to the flow) – what if you choose a surface not perpendicular to the flow? • we need to find the “projection” of the surface onto a plane normal to the direction of flow: S ⊥ =S*cos( α ) where α is the angle between liquid flow direction and the normal to surface S. α α Flow direction Flow direction © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin Electric Field Flux, Gauss’s Law, Divergence 4 “directed” surface elements • We can generalize the concept of a “flux” through a surface for any vector field F in this way: 1. We need to associate a direction with a surface area element: this would be a unit vector normal to the surface element (directed “outwards”) 2. Define a flux of a vector field using the projection between the surface and the vector field F: surface flux F dS Φ = • ∫ r r vector field F F r dS r flux F dS θ = Φ = cos θ dS r F r θ • Total flux through the surface: © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin Electric Field Flux, Gauss’s Law, Divergence 5 Flux of an electric field from a point charge • recall that for a single point charge Q fixed at the origin • what is the “flux” due to this charge crossing the surface of a sphere centered on the charge? ( 29 2 4 ˆ o Q E r r πε = r ( 29 ( 29 ( 29 2 4 ˆ ˆ sin sphere of constant radius R o Q r Rd R d r R θ θ φ πε = • ∫ R R sin θ d φ z y x θ φ R d θ ( 29 2 4 ˆ sphere of constant radius R o Q flux r dS R πε Φ = • ∫ r ( 29 ( 29 2 2 2 2 2 4 sin o Q R d d R π π π φ θ θ πε = ∫ ∫ 123 1 442 443 o Q ε = i.e., the charge enclosed / ε o d φ R sin θ ©...
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This note was uploaded on 04/03/2012 for the course EE 325 taught by Professor Brown during the Fall '08 term at University of Texas.

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325_Sp2011_3_Flux,Gauss_Law,and_Divergence - © Copyright...

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