342 Steady Currents in continuous Media [CH. x 



If the conducting medium extends to infinity, or is bounded entirely by 

 the two electrodes, the transformations will be identical with those already 

 discussed for two conductors at different potentials ( 386). If the medium 



dV 

 has also boundaries at which -^ = 0, the procedure must be slightly different. 



We must try to transform the two electrodes into lines V = constant, and the 

 other boundaries into lines U = constant, so that the whole of the medium 

 becomes transformed into the interior of a rectangle in the U, V plane. 



Let U+iV=f(x + iy) 



be a transformation which gives the required value for V over both electrodes, 



3F 

 and gives -^ = over the boundary of a conductor. Then V will be the 



potential at any point, the lines V= constant will be the equipotentials, and 

 the lines U = constant, the orthogonal trajectories of the equipotentials, will 

 be the lines of flow. 



At any point the direction of the current is normal to the equipotential 

 through the point, and of amount 



But -^ is equal to -^ , where -=- denotes differentiation in the equipotential. 

 dn ds ds 



Thus the current flowing across any piece PQ of an equipotential 



Cds 



! 



P 



--=-ds = -(U Q - Up), 

 r ds r v 



If P, Q are any two points in the conductor, a path from P to Q can be 

 regarded as made up of a piece of an equipotential PN, and a piece of a line 

 of flow NQ. The flow across NQ is zero, that across PN is 



This is accordingly the total flow across PQ, and since U N U Q) it may 

 be written as 



\(U Q -U. P ). 



390. As an illustration, let us suppose that the conducting plate is a 

 polygon, two or more edges being the electrodes. We can transform this 

 into the real axis in the f-plane by a transformation of the type 



ll-ft-o^tf-a,)*' 1 (319), 



and this real axis has to be transformed into a rectangle formed (say) by the 



