Electromagnetic Field Theory: A Problem Solving Approach Part 6. Electromagnetic field theory is often the least popular course in the electrical engineering curriculum. Heavy reliance on vector and integral calculus can obscure physical phenomena so that the student becomes bogged down in the mathematics and loses sight of the applications. This book instills problem solving confidence by teaching through the use of a large number of worked problems. To keep the subject exciting, many of these problems are based on physical processes, devices, and models. This text is an introductory treatment on the junior level for a two-semester electrical engineering. | Flux and Divergence 25 b Figure 1-16 Infinitesimal volumes used to define the divergence of a vector in a cylindrical and b spherical geometries. Again because the volume is small we can treat it as approximately rectangular with the components of A approximately constant along each face. Then factoring out the volume A V rAr A Az in 7 r Ar Ar r Ar r Ar rAr W 8 rA J Az 26 Review of Vector Analysis lets each of the bracketed terms become a partial derivative as the differential lengths approach zero and 8 becomes an exact relation. The divergence is then _ feA-dS ia ix ia u ad V- A lim ---------- rAr ---- - 9 Ar- o AV r dr r d dz A4- 0 Az- O b Spherical Coordinates Similar operations on the spherical volume element A V r2 sin 0 Ar Afl A in Figure 1-166 defines the net flux through the surfaces 4 fA-dS is . r Ar 2A r Ar-r2Ar J r2 Ar sin 0 A0 -Ae sin 0 r sin 0 0 r2 g Ar g r sin 0 A p The divergence in spherical coordinates is then AdS V A lim s . T_ Ar- O A V Ae- o A4--0 2T r2Ar 7 Ae sin 0 H--------r 11 r dr r sin 0 90 r sin 0 dtf 1-4-4 The Divergence Theorem If we now take many adjoining incremental volumes of any shape we form a macroscopic volume V with enclosing surface S as shown in Figure 1-17a. However each interior common surface between incremental volumes has the flux leaving one volume positive flux contribution just entering the adjacent volume negative flux contribution as in Figure 1-176. The net contribution to the flux for the surface integral of 1 is zero for all interior surfaces. Nonzero contributions to the flux are obtained only for those surfaces which bound the outer surface S of V. Although the surface contributions to the flux using 1 cancel for all interior volumes the flux obtained from 4 in terms of the divergence operation for Flux and Divergence 27 b Figure 1-17 Nonzero contributions to the flux of a vector are only obtained across those surfaces that bound the outside of a volume a Within the volume the flux leaving one incremental .
