By J. E. Parton, S. J. T. Owen, M. S. Raven (auth.)
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Extra resources for Applied Electromagnetics
This must not be confused with p BP= a OB =b b r= a cosec a! 4 Field due to a line charge PL coulombs per metre. THE ELECTRIC FIELD 25 used as a coordinate in the cylindrical polar system or with the same symbol commonly used for resistivity. We wish to ftnd the electric fteld strength Eat P due to this charged line. La P P 21te 0 a P this is the only (radial, normal) fteld component, for the axial fteld strength Ez = 0. The fteld has cylindrical symmetry. 3 Field due to a surface charge Ps coulombs/m 2 The electric fteld strength at any point P due to such a surface S may be found by integration E= J surface Ps dS A_ 2 a, "f'T(;e 0 r where a, is a unit vector along r, the position vector of P with respect to dS.
DS has to be taken over six sides f D · dS = f + f + f + f + f + f front S f= front f+f front back back left DfrontAx Ay = Dz right top aDz + 12"Az- Ax Ay az aDz aDz = - AxAyAz=-A(vol) az bottom az 32 APPLIED ELECTROMAGNETICS Repeating with left, right and top, bottom in pairs we can write finally J. dS= (oDx oD::) yD. :l(vol) 8 oDx oDy oDz r fD. :l(vol) v The left-hand side is simply div D hence div D = V. D =p v Maxwell's first equation (differential form) If we integrate the equations we obtain fD.
This is often used as an alternative criterion for the defmition of a conservative field. Obviously, if Fhas been derived from a scalarS by the gradient, then from the above div curl F == 0. This is, however, true for a general vector F. The components of curl Fare the circulations per unit area as the area tends to a point and are oriented perpendicular to the vectors ax, ay, Oz. The vector curl F is associated with closed paths in a vector field, and has no sources or sinks associated with it, in which case the divergence of curl F == 0.