2O4 PROPAGATION OF ELECTRICITY. 



In this case the electrical displacement is no longer parallel to 

 the electrical force. 



217. CONDUCTORS IN Two DIMENSIONS. The preceding con- 

 siderations apply to unlimited media. It is clear that nothing is 

 altered if we limit the conducting medium by a surface formed 

 entirely by the lines of flow of the unlimited system. For a limited 

 conductor placed in an insulating medium, the external surface, 

 whatever it may be, is always parallel to the lines of flow, and 

 therefore, if the medium is isotropic, it is perpendicular to the equi- 

 potential surfaces. 



This is the case of the propagation of electricity in a thin plate, 

 which may be regarded as a conductor of two dimensions. We may 

 determine by experiment the locus of points which have the same 

 potential, by the condition that no current flows through a conducting 

 wire one end of which is in connection with a fixed point in the 

 plane. The results obtained by experiment are in complete accor- 

 dance with those deduced from Fourier's formula, and furnish 

 a fresh confirmation of the analogy between the two orders of 

 phenomena. 



In both cases we may suppose the propagation to take place 

 either with or without loss in the surrounding medium. 



If there is no loss in the external medium, Poisson's equation 

 for any point outside the electrodes reduces to 



and, for any point in the outside of the plate, we have 



It is easily seen that this problem merges into that of the problem 

 of equilibrium in the case of a cylindrical distribution (132 et seq.) 

 We have seen there that in a plane traversed perpendicularly at points 

 Aj, A 2 , A 3 ..... by parallel lines of the densities A p A 2 , A 3 . . . , the 

 potential at a point P, at distances r v r^ r z ... from these lines, has 

 the value 



The flows of force which, in a layer of the thickness e comprised 



