375-379] Boundary Conditions 335 



CONDITIONS TO BE SATISFIED AT THE BOUNDARY OF TWO 

 CONDUCTING MEDIA. 



378. The conditions to be satisfied at a boundary at which the current 

 flows from one conductor to another are as follows : 



(i) Since there must be no accumulation of electricity at the boundary, 

 the normal flow across the boundary must be the same whether calculated in 

 the first medium or the second. In other words 



1 dV 



-^ must be continuous, 



r dn 



where denotes differentiation along the normal to the boundary. 



on 



(ii) The tangential force must be continuous, or else the potential 

 would not be continuous. Thus 



must be continuous, 



OS 



where ~- denotes differentiation along any line in the boundary. 



OS 



These boundary conditions are just the same as would be satisfied in an 

 electrostatical problem at the boundary between two dielectrics of inductive 



capacities equal to the two values of - . Thus the equipotentials in this 



electrostatic problem coincide with the equipotentials in the actual current- 

 problem, and the lines of force in the electrostatic problem correspond with 

 the lines of flow in the current problem. 



Clearly these results could be deduced at once from the differential equation (312) on 

 passing to the limit and making r become discontinuous on crossing a boundary. 



Refraction of Lines of Flow. 



379. Let any line of flow cross the boundary between two different 

 conducting media of specific resistances r lt r 2 , making angles 1} e 2 with the 

 normal at the point at which it meets the boundary in the two media 

 respectively. The lines of flow satisfy the same conditions as would be 

 satisfied by electrostatic lines of force crossing the boundary between two 



dielectrics of inductive capacities , , so that we must have (cf. equa- 

 tion (71)) 



COt 1 = COt 2 . 



T"l T 2 



Hence r x tan e : = r 2 tan e 2 , 



expressing the law of refraction of lines of flow. 



