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47), leads to the hypothesis that eqn. f. = 1 Я . 48) 7 dt where Ψ = JJ D . f. is measured. This hypothesis led Maxwrell to the idea of radia­ tion by electromagnetic waves. To illustrate the result let us examine a very simple case in which there is an electricfieldin the ^-direction and a magneticfieldin the ^-direction as shown in Fig. 13. E. кк FIG. 13. A plane electromagnetic wave. 28 APPLIED ELECTROMAGNETISM allei to the xy-plane. Thus õEjdx, dEjdy, дН\дх, and õHjõy are all zero and the only variation is along the axis of z.

2) should not be forgotten. 2. Scalar and Vector Products In the previous chapter we used the scalar (or dot) product of two vec­ tors in such relationships as W = J F . dr. Such a product produces a scalar quantity, for instance force . distance = work done. In rectangular coordinates A . 3) where the subscripts denote the components of the vectors. ELECTROMAGNETIC FIELDS AND THEIR SOURCES 37 We also briefly introduced the vector (or cross) product of two vectors. This can be illustrated by the relationship torque = force x lever arm and is illustrated in Fig.

50 APPLIED ELECTROMAGNETISM (4) A vector field may have divergence sources and curl sources within a region. It has then an irrotational and also a solenoidal part and can be expressed in terms of the gradient of a scalar potential plus the curl of a vector potential (Fig. 9). FIG. 9. Field with divergence and curl sources. 6 to obtain a for­ mal proof of the uniqueness of a vector field in terms of its sources. 49) curl F = с. 50) We wish to prove that F can be uniquely specified in terms of these sources.

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