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If we write Poisson's or Laplace's equation in one or two cartesian coordi- nates, we imply that the potential does not vary at all with the omitted coordinates. This might conceivably be strictly true in a limited region of an electrostatic system under very special conditions; but we could hardly be sure of it a priori unless the system had an infinite transla- tional symmetry along the omitted coordinates. This in turn means that in a system for which only two cartesian coordinates were used, all boundaries would be infinite cylinders (of course, not necessarily circular); while in a system for which only one cartesian coordinate was used, all boundaries would be infinite planes.
It should be remembered that the expansion can be continued to terms of higher than the first degree in ratios x'/R, etc. 96) become stronger, only the lowest nonvanishing term of the expansion eventually remains significant. For a sufficiently distant observer, the system is then entirely character- ized as a point charge if q ^ 0; as a dipole if q = 0 but p ^ 0; and as a multipole of higher order if some higher-degree term in the expansion is the first to exist. Sometimes it is convenient at moderate distances to retain more than the first nonzero term of the expansion, so that, for example, a system having finite values of both q and p is considered a combination of a point charge and a dipole.
The problem of ascertaining the electric field at every point of a system of charges is straightforward and elementary in principle so long as all the charges in the system are given. If the system is finite, the potential may be evaluated by performing the scalar integration rr_ 1 fdq all charges where r is the distance from the charge element dq to the point of observa- tion. From U the field is obtained immediately. Even if the system is infinite, the process is only slightly more difficult, and the field may be found from the vector integral all charges The problem encountered in practice is usually much more subtle.