Applied Electromagnetism by P. Hammond

By P. Hammond

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22). But more frequently we shall seek a solu­ tion in afiniteregion, where the potential (or perhaps the normal gradient of potential) is specified on the enclosing surface. In such a case we shall find a solution of the differential equation in terms of arbitrary constants and then obtain the constants from the given boundary condition. Phys­ ically the boundary conditions account for the distant sources. 7 will guarantee that the solution is also unique. If there is no charge density at the place at which the variation of V is being considered, we have Laplace's equation V2V = 0.

Hence the net outflow in the x-direction is given t The symbol ρ is used also for resistivity. The context generally makes clear which quantity is under discussion. ELECTROMAGNETIC FIELDS A N D THEIR SOURCES 41 (DX + Ë^JL δχ] ду δζ - Dx ду dz = ^L δχ òyòz. 11) by Thus the outflow per unit volume in the x-direction is dDx\dx. The total outflow of D in all directions per unit volume is called the divergence of D, or more briefly, div D. Thus dDy dD2 ,. ^ dDx div D = — - + —ί + —Ξ. 12) дх ду dz This can be written in terms of the vector operator V by noting that dDx ох dDy ду dDz dz f.

It will become easier to resist the temptation when in later chapters we deal with applications of the theory. But even at this early stage it is worth while to take a de­ tached look at the mathematics and ask what physical information there may be in the various symbols and equations. We remind ourselves that the subject starts with electricity as a source of mechanical force defined by the inverse square law. The primary task of an electrical engineer is always to determine the sources, be they charges or currents.

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