391-394] 



Special Problems 



345 



As an illustration, we may examine the case of a straight wire of variable 

 cross-section S. Let us imagine that at small distances along its length we 

 take cross-sections of infinitely small thickness, and make these into perfect 

 conductors. The resistance between two such sections at distance ds apart, 



w i\\ be ~- , where 8 is the cross-section of either. Thus a lower limit to 



o 



the resistance is supplied by the formula 



?cb 



393. Again, if we replace parts of the conductor by insulators, so causing 

 the current to flow in given channels, the resistance of the whole is increased, 

 and in this way we may be able to assign an upper limit to the resistance 

 of a conductor. 



394. As an instance of a conductor to the resistance of which both 

 upper and lower limits can be assigned, let us consider the case of a 

 cylindrical conductor AB terminating in an infinite 



conductor G of the same material. This example is 

 of practical importance in connection with mercury 

 resistance standards. The appropriate analysis was 

 first given by Lord Rayleigh, discussing a parallel 

 problem in the theory of sound. 



Let I be the length and a the radius of the tube. 

 To obtain a lower limit to the resistance, we imagine 

 a perfectly conducting plane inserted at B. The resistance then consists of 

 the resistance to this new electrode at B, plus the resistance from this with 



the infinite conductor G. The former resistance is 2 , the latter, by 388, 

 is -T- , so that a lower limit to the whole resistance is 



. 

 ira 2 4,a' 



which is the resistance of a length I -f -j- of the tube. 



To obtain an upper limit to the resistance, we imagine non-conducting 

 tubes placed inside the main tube AB, so that the current is constrained to 

 flow in a uniform stream parallel to the axis of the main tube until the 

 end B is reached. After this the current flows through the semi-infinite 

 conductor G as directed by Ohm's Law. 



The resistance of the tube AB is, as before, ^r . To obtain the resist- 



TTGT 



ance of the conductor (7, we must examine the corresponding electrostatic 

 problem. If / is the total current, the flow of current per unit area over 



