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FWLec7 - A F Peterson Notes on Electromagnetic Fields Waves...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 10/04 Fields & Waves Note #7 Gauss’ Law in Differential Form Objectives: Use the integral form of Gauss’ Law in a limiting case to motivate the divergence operation and obtain the law in its differential form. Consider examples that illustrate the calculation and significance of divergence. The integral form of Gauss’ Law relates the total flux out of a closed surface to the enclosed charge. Since it essentially involves only the average values of the flux and charge, it is less useful than we might like. In fact, as observed in the examples of Note #6, the integral form of Gauss’ Law can only be used to solve for the electric field when the problem under consideration has substantial symmetries. In the following, we use the integral form of Gauss’ Law to derive the equivalent differential law. The differential form of Gauss’ Law can be applied at points and therefore may have broader applicability for our study of electromagnetic fields. Application of Gauss’ Law to a small volume Consider the point ( x 0 , y 0 , z 0 ) and a small rectangular volume of dimension D x by D y by D z (Figure 1), having surface S . If the field and charge density are assumed to be slowly varying with respect to the dimensions D x , D y , and D z , the left-hand side of Gauss’ Law D dS dv S v V = ÚÚ ÚÚÚ r (7.1) applied to S can be expressed as the sum of six integrals, one over each of the six faces of the box in Figure 1. Since the dS vector captures the outward normal component of the D -field over each face, and since the D -field is slowly varying (and can be assumed to be constant over each face), these integrals can be written approximately as D dS D y z D x z D x y D y z D x z D x y S x x x x y y y y z z z z x x x y y y z z z @ + + - - - ÚÚ = + = + = + = = = 0 0 0 0 0 0 D D D D D D D D D D D D D D D (7.2) In a similar manner, the right-hand side of Gauss’ Law can be approximated by r r v V v x x y y z z dv x y z ÚÚÚ @ = = = 0 0 0 , , D D D (7.3)
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