Equipotential Curves and Surfaces, fyc. 31 



A case of this kind occurs when the electrodes lie on the straight line 

 where the surface of the liquid meets the plane vertical side of the vessel 

 containing it. 



Electrical Distribution with Linear Electrodes. 



Next consider the case in which the electrodes are parallel straight 

 lines, extending throughout the liquid. With one electrode, the potential 

 at any point at a distance r from it is 



C+A log r. 



When there are several electrodes, the potential is 



C + Alogr+Blogr 1 + &c. 



If the currents flowing into the liquid at the several electrodes are all 

 equal, the potential Is 



C+A(logr+logr 1 +&c); 



and for an equipotential surface we have 



r r' r" . . . =c T r ' r" . . . 



where dashes below the letter indicate negative electrodes. 



This equation is the same as that obtained for a plane conductor of 

 small thickness, which has been already considered ; and the forms of the 

 equipotential surfaces are cylindrical. The disk of tinfoil may be con- 

 sidered as a particular case of the solid with electrodes throughout the 

 whole thickness when that thickness is supposed to be small. 



For two linear electrodes with equal and opposite currents, the equi- 

 potential surfaces will be circular cylinders, and the lines of flow will 

 also be circular cylinders cutting them at right angles. 



The particular cases of circular and other disks which have already 

 been considered will give the sections of the equipotential surfaces for 

 the corresponding cases with linear electrodes in the liquid, and will 

 completely determine them. 



Supposing the surface of the liquid to be of unlimited extent, and 

 that a current enters by one line electrode, and leaves by equal currents 

 at two negative electrodes, the forms of the cylindrical surfaces will be 

 given by the equation 



r 2 = cr 1 r 2 . 



When the positive electrode is at the middle point of one side of the 

 rectangular box used in the experiments, and an equal negative electrode 

 is within the box and near the same side, as in Case 10, the forms of the 

 equipotential surfaces will be given by the equation 



r*=zcr % r % ' 3 



r v r 2 are the distances of the point from the negative electrode and from 

 its image. 



This equation will give the form of the surfaces accurately only when 



