The Stability Theory of Charge Distributions on Conductors

Stability of solution is considered for the charge-density problemof several rigid charged conductors, stability being with respectto the geometrical configuration of the system. Under the analyticalrestrictions usual in potential theory for the surfaces on whichthe charges reside, two theorems are proved establishing thestability: the first for when the potentials of the conductorsare fixed, and the second for when their total charges are fixed.Subsequently, it is shown how the rigidity condition can berelaxed, but a counterexample to stability is described fora situation where the surface analytical conditions are notfulfilled. Finally, an application of the theory is given, inwhich the second of the said theorems is used to close a gapin the theory by the energy method of the electrostatic forceon a conductor.     

A Class of Physically Relevant Insoluble Dirichlet Problems

The three-dimensional exterior Dirichlet problem, whose boundarydata include conditions on lines, is generally insoluble. Thisis shown through the treatment of the axisymmetric case of oneclosed surface and one line segment attached to it. The insolubilityhas implications to electrostatics and, more importantly, theelectrodynamics of continuous media.     

Electrohydrostatic equilibrium and field solutions under space-charge conditions

An examination is made, in terms of electrohydrostatic stresstheory, of the restrictions placed on the solutions of the high-pressurespace-charge equation in order to comply with the assumptionof hydrostatic equilibrium, implicit in the equation's derivation.It is concluded that, except for the known elementary solutions,there are no solutions consistent with the equilibrium. Theimplication is that, except in the simple cases to which theelementary solutions apply, the space-charge situation is necessarilyelectrohydrodynamical and existing non-elementary solutionsare acceptable only insofar as they are valid as approximations.     

A Perturbation Method for Solving Torus Problems in Electrostatics

Considering an axisymmetric electrostatic problem in which thereis just one conductor or dielectric the surface of which isa torus of revolution, a perturbation technique is developedwhich may allow one to proceed to the solution by starting withthe solution of the two-dimensional problem for the same crosssection. The method is applied to a general torus of revolutionwith elliptic section, having as special case the ordinary torus(anchor ring) and, as limiting cases, various ring-shaped laminae.Capacity formulae are obtained, some probably representing newresults, and others whose predictions are compared with resultsthat have been obtained by other methods.     

On integral equations of axisymmetric potential theory

It is shown that two integral equations of the first kind, muchused in, respectively, axisymmetric electrostatics and hydrodynamics,are wrong in the sense that they do not in general possess solutions.A theorem is established giving the precise conditions necessaryfor solutions to exist, but perhaps more important practicallyis the fact, brought out by examples, that the necessary conditionsare far from sufficient. An alternative integral equation inthe electrostatic case is proposed and justified, one havinga similar form and the same computational advantages, but freefrom existence difficulties. The apparent paradox that ‘solutions’are found to problems when the governing equations may not possesssolutions is explained by the fact that these purported solutionsare obtained by numerical or asymptotic analysis, when an approximatingequation possesses a solution, but one which cannot be saidto approximate to the solution of the problem if the equationto which, formally, it approximates cannot, through being meaningless,represent the problem. The arguments are given mainly in theelectrostatic context, but it is shown how they are modifiedto carry over to the hydrodynamical one.