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By Robert Plonsey and Robert E. Collin
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Example text
131) + * In rectangular coordinates 'V'A = a. 'V·A. + au'VtAy a. , v'A = 'VV • A - V X V X A. The simple expansion in rectangull1r coordinates is possible only because the orientations of the unit vectors are independent of position. CHAPTER 2 ELECTROSTATICS ThiS"chapter develops the basic properties of the electrostatic field in vacuum. The law of Coulomb for the force between two point charges is the experimental basis for the work of this chapter. The electric field is defined as the force exerted on a unit positive charge and leads to its establishment as a fundamental entity.
The most general vector field will have both a nonzero divergence and a nonzero curl. We shall show that this field can always be considered as the sum of a solenoidal and an irrotational field. This statement is essentially the content of Helmholtz's theorem. Another way of stating the Helmholtz theorem is that a vector field is completely specified by its divergence and curl. The latter constitute the source and vortex source of the field, respectively. Before proceeding to the general case we shall treat the two special cases mentioned above first.
Y - y:)2 + (z - ZD2]JoS (192) • ;=1 In any physical problem the sources will be confined to some finite region. It is often of interest to compute the potential set up at distances which are very large compared with the extent of the source region. 92) in powers of y:' which are small compared with x, y, z. Then by Taylor's theorem, x:, z: where the index 0 means that the quantities in the parentheses are evaluated for = = = O. 92) leads to cp(x,Y,z) = 4~r n 2: ':=1 Q. -1 QiX: n 2: + ~ 2: QiZi + ..