162164] 



STEADY FLOW IN CONDUCTORS. 



327 



163. Boundary Condition. Refraction of Lines of 

 Flow. In passing from one homogeneous conductor to another, 

 A may be discontinuous, and since the current must be continuous, 

 we must have at the surface 



q l cos fan*) + q 2 cos (q z n 2 ) = 0, 

 (6) A^ cos (F^) + \ 2 F 2 cos (F z n^ = 0, 



or 



The boundary condition (6) has a simple geometrical meaning. 

 Since the derivatives of V are discontinuous only on crossing the 

 surface, we have the derivatives in any direction t tangent to the 

 surface, dV/dt the same on both sides of the surface. If l be the 

 acute angle made by the current line with the normal on one side 

 of the surface, 2 the acute angle on the other, resolving along the 

 normal, 



(7) A! J^ cos 0! = \ 2 F 2 cos 2 . 



Resolving along the tangent plane, since this component is 

 continuous, 



(8) F l sin 0! = F 2 sin <9 2 . 



Dividing the second of these equations by the first, we obtain 

 tan 0i tan 9 



(9) 



X, 



or the line of flow is refracted on passing the surface, so that the 

 tangents of the angles of incidence and refraction are in the ratio 

 \J\a dependent only on the media. The law of refraction is 

 different from the optical law, in which we have the sine instead 

 of the tangent, and in the case of the 

 tangent law we do not have the phe- 

 nomenon of total reflection, since the 

 tangent takes all values from zero to 

 infinity. 



164. Systems of Conductors. 



All the statements heretofore made 

 are true for the flow of heat, if V 

 represent the temperature, but whereas 

 in the case of heat in passing from one 

 conductor to another the temperature 



FIG. 67 a. 



