L06 - Lecture 6 Divergence, Gauss Law in Differential Form...

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LECTURE 6 slide 1 Lecture 6 Divergence, Gauss Law in Differential Form Sections: 3.4, 3.5, 3.6, 3.7 Homework: D3.6, D3.7, D3.8, D3.9; 3.15, 3.17, 3.19, 3.21, 3.25, 3.27
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LECTURE 6 slide 2 flux is the net normal flow of the vector field F through a surface Flux – 1 flux is positive when flow is outward and negative when flow is inward cos SS dF d s α Ψ= = ∫∫ Fs total flux implies integration over a closed surface cos dFd s = ww F is called the density of the flux
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LECTURE 6 slide 3 Flux – 2 flux lines show the direction and density of the flux ds n a F high density low density
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LECTURE 6 slide 4 Total Flux The flux through a closed surface is a measure of the field’s sources in the enclosed volume S d Ψ =⋅ ∫∫ Fs w 0 Ψ = source (positive flux) sink (negative flux) no source (zero net flux) 0 Ψ> 0 Ψ < S Qd Ds w
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LECTURE 6 slide 5 Total Flux: Example A – 1 Compute the flux of the vector field 2 xy z x zy z x y = −− Da a a through the surface of a cube centered at the origin and with sides equal to 2 units each and parallel to the Cartesian axes. (1 ) ) ) ( 1 ) () xx x x SS S dd ydz dydz =− =+ = Ψ= = ⋅ − + ∫∫ Ds D a D a w ) ) ) ( 1 ) yy y y dxdz dxdz = +⋅ + D a ) ) ) ( 1 ) zz z z dxdy dxdy = + D a x y z 2 units
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LECTURE 6 slide 6 Total Flux: Example A – 2 11 2 22 ( ) ( ) yz xz xy zdydz z dxdz xydxdy xydxdy =− Ψ= + + + ∫∫ 8 40 33 3 ⎡⎤ ⎛⎞ Ψ= ⋅ + ⎜⎟ ⎢⎥ ⎝⎠ ⎣⎦
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LECTURE 6 slide 7 x a x Δ y Δ z Δ 00 0 (, , ) 2 x x Fx y z Δ + 0 , ) 2 x x y z Δ 0 , ) 2 y y y z Δ + 0 , ) 2 y y y z Δ 0 , ) 2 z z y
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L06 - Lecture 6 Divergence, Gauss Law in Differential Form...

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