By P. Hammond
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Extra info for Applied Electromagnetism
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. 24) The solution of this equation proceeds in the same way except that there is no "particular integral" to correspond to the right-hand side of Pois son's equation. This discussion also applies to the equations of the vector potential A. We shall deal in the next chapter with the process of solving these equations. The discussion in this chapter is included first to show why the potentials are introduced into the theory, and, secondly, how we can find the potentials either by integration of the sources or by the solution of a differential equation with known boundary values.
4. φ This can be done because the divergence of A had not been defined as yet. 7 that a vector is fully defined when both its divergence and curl are defined. 21) shows these equations to be Poisson's equa tions and their'solution can therefore be written by comparison with eqn. e. afieldof zero divergence, can be obtained from a vector potential, since the relationship of eqn. 38) implies that div В = 0 because of the vector identity div curl A = 0. 46) This can easily be illustrated by writing div curl A as V .
Test for curl: (a) "curl meter", (b) curl, (c) curl, (d) no curl. ELECTROMAGNETIC FIELDS AND THEIR SOURCES 47 rotation will, of course, depend on the position of the axis of the paddle wheel. Curl is a vector and the axis of the paddle wheel defines the direc tion of the curl. The sign of the vector is determined by choosing a righthanded set of axes. A field which has no divergence but only curl sources, like the mag netic field of eqn. 26), which arises from current sources, is said to be solenoidal This name has, of course, been taken from the magnetic field of a solenoid which is of this type.