Examples 329 



32. A network is formed by a system of conductors joining every pair of a set of 

 n points, the resistances of the conductors being all equal, and there is an electromotive 

 force in the conductor joining the points J. 1} A 2 . Shew that there is no current in any 

 conductor except those which pass through A l or A 2 , and find the current in these 

 conductors. 



33. Each member of the series of n points AI, A 2 ... A n is united to its successor 

 by a wire of resistance p, and similarly for the series of n points lt B 2 ... B n . Each 

 pair of points corresponding in the two series, such as A r and B r , is united by a wire 

 of resistance R. A steady current i enters the network at AI and leaves it at B n . Shew 

 that the current at AI divides itself between A 1 A 2 and AiBim the ratio 



sinha + sinh(n l)a + sinh(w-2)a : sinh a -f sinh (?i l)a sinh(n 2) a, 



where cosh a = =^ . 



H 



34. An underground cable of length a is badly insulated so that it has faults 

 throughout its length indefinitely near to one another and uniformly distributed. The 

 conductivity of the faults is l/p r per unit length of cable, and the resistance of the 

 cable is p per unit length. One pole of a battery is connected to one end of a cable 

 and the other pole is earthed. Prove that the current at the farther end is the same 

 as if the cable were free from faults and of total resistance 



Y 



35. Two parallel conducting wires at unit distance are connected by n + l cross pieces 

 of the same wire, so as to form n squares. A current enters by an outer corner of the 

 first square, and leaves by the diagonally opposite corner of the last. Shew that, if 

 the resistance is that of a length %n + a n of the wire, 



36. A, B are the ends of a long telegraph wire with a number of faults, and C is 

 an intermediate point on the wire. The resistance to a current sent from A is R when 

 C is earth connected, but if C is not earth connected the resistance is 8 or T according 

 as the end B is to earth or insulated. If R', S', T denote the resistances under similar 

 circumstances when a current is sent from B towards A, shew that 



37. The inner plates of two condensers of capacities C, C' are joined by wires of 

 resistances R, R to a point P, and their outer plates by wires of negligible resistance 

 to a point Q. If the inner plates be also connected through a galvanometer, shew that 

 the needle will suffer no sudden deflection on joining P, Q to the poles of a battery, 



38. An infinite cable of capacity and resistance K and R per unit length is at zero 

 potential. At the instant t =0 one end is suddenly connected to a battery for an 

 infinitesimal interval and then insulated. Shew that, except for very small values of t, 

 the potential at any instant at a distance x from this end of the cable, will be pro- 

 portional to 



