Lecture 4 - 4 Divergence and curl Expressing the total...

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4 Divergence and curl Expressing the total charge Q V contained in a volume V as a 3D volume integral of charge density ρ ( r ) we can express Gauss’s law examined during the last few lectures in the general form ± S D · d S = ² V ρdV. This equation asserts that the Fux of displacement D = ± o E over any closed surface S equals the net electrical charge contained in the enclosed volume V — only the charges included within V a±ect the Fux of D over surface S , with charges outside surface S making no net contribution to the surface integral ³ S D · d S . Gauss’s law stated above holds true everywhere in space over all sur- faces S and their enclosed volumes V , large and small. Application of Gauss’s law to a small volume Δ V = Δ x Δ y Δ z sur- rounded by a cubic surface Δ S of six faces, leads, in the limit of van- ishing Δ x , Δ y , and Δ z , to the di±erential form of Gauss’s law expressed in terms of a divergence operation to be reviewed next: ( x,y,z + Δ z ) x y z ( ) ( x ) ( x,y y,z ) Given a su²ciently small volume Δ V x Δ y Δ z , we can assume that ² Δ V ρdV ρ Δ x Δ y Δ z. 1
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Again under the same assumption ± S D · d S ( D x | 2 - D x | 1 y Δ z +( D y | 4 - D y | 3 x Δ z +( D z | 6 - D z | 5 x Δ y with reference to displacement vector components like D x | 2 shown on cubic surfaces depicted in the margin. Gauss’s law demands the equality of the two expressions above, namely (after dividing both sides by Δ x Δ y Δ z ) x y z ( x,y,z ) 5 2 1 4 3 6 D x | 2 - D x | 1 Δ x + D y | 4 - D y | 3 Δ y + D z | 6 - D z | 5 Δ z ρ, in the limit of vanishing Δ x , Δ y , and Δ z . In that limit, we obtain ∂D x ∂x + y ∂y + z ∂z = ρ, which is known as diferential Form oF Gauss’s law .
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This note was uploaded on 05/05/2010 for the course ECE ece 442 taught by Professor Kim during the Spring '10 term at University of Illinois at Urbana–Champaign.

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Lecture 4 - 4 Divergence and curl Expressing the total...

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