On the Forms of the Lines of Electric Force. 303 



results arrived at are as follows : — The state of affairs in the 

 wire is governed, as is well known, by the magnitude of the 



quantity a\f — , where a is the radius of the wire, //,, p its 



permeability and resistivity, and t j- the frequency. When 



this is large we have the " skin-effect " well developed. Let 

 us suppose for simplicity that p alone is altered. Then 

 starting with the limiting case of very great conductivity and 

 very thin skin, we find the region of eddy amounts to a 

 quarter of each half wave-length, or say 45° in the argument 

 of the periodic vectors. Further, as we should expect from 

 the smallness of the dissipation of energy, the flow is outward 

 in the eddy and onward everywhere As the resistance of 

 the wire is increased the extent of the eddy-region at first 

 diminishes, reaches a minimum, and then increases again. 

 In a case which I have worked out numerically the minimum 

 is 18°. The anomalous flow is now partly outward and partly 

 backward. As the resistance of the wire becomes very great 

 the length of the eddy again approaches the limiting value of 

 45°, but the flow is now everywhere inward; it is backward 

 in the eddy, forward elsewhere. 



In the course of the investigation I have obtained the 

 equations to the lines of force and of Poynting flux in the 

 plane of the wires, inside and out. In order to show the 

 properties of the curves I have plotted them in an exaggerated 

 form, i. e. using constants of a different order of magnitude 

 from those actually occurring in the physical cases. 



§2. Values of Electric and Magnetic Vectors. 



Take first the case of a single ivire in a dielectric of per- 

 mittivity K and permeability unity. Let Z, R be the longi- 

 tudinal and radial components of electric force, H the magnetic 

 force in circles round the wire. Then, measuring z along the 

 direction of propagation and r outwards from the axis of the 

 wire, the differential equations to be satisfied are 



BH 4ttR _3R 



- ir = • or K ^r; 



■ ll^-f'-^i >> ■ ■ ■ « 



BR _ BZ BH _ ?3H 



"dz ~dr ~ P ~dt ' 01 ^t 



according as we are dealing with wire or dielectric. 



