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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 = V ρ dV. This equation asserts that the flux 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 ff ect the flux of D over surface S , with charges outside surface S making no net contribution to the surface integral 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 ff 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, y, z ) ( x + Δ x, y, z ) ( x, y + Δ y, z ) Given a su ffi ciently small volume Δ V = Δ x Δ y Δ z , we can assume that Δ V ρ dV 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 ) Δ y Δ z +( D y | 4 - D y | 3 ) Δ x Δ z +( D z | 6 - D z | 5 ) Δ 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 + D y y + D z z = ρ , which is known as di ff erential form of Gauss’s law .
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