By P. Hammond
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Additional info for Applied Electromagnetism
5 we specified a non-conservative solenoidal field by its curl sources. It is important to notice that a vector field derived solely from curl sources has no divergence, since div curl F = V . V x F = 0. 48) We have already discussed this in the last section in connection with the vector potential A. Consideration of eqns. 48) suggests that the divergence and curl sources are independent of each other and that the complete specification of a general vectorfieldshould include both types of sources.
In order to make full use of the field concept we shall need a means of relating the field to the sources. 18) and curl H = J. 32) Unfortunately these relationships are not very useful as they stand. Both of them relate the local variation of the field to the source density at that point, whereas in general we wish to find thefielddue to sources elsewhere. Often we shall wish to calculate the field at a place free of sources, for instance in the air gap of an electrical machine. 63) and curl H = 0.
19) div grad V = V . 21) is the desired relationship. 21) is called Poisson's equation and it relates the variation of the potential at a field point to the charge density at the same point. The integral form of the relationship can be t Called "del squared". ELECTROMAGNETIC FIELDS AND THEIR SOURCES 43 obtained from eqn. 6) in the previous chapter, K--fi—-i-flU*. 22) 4же0г Ane0jjj r We write do9 to remind ourselves that the integration has to be carried out over the source coordinates. 22) is the solution of eqn.