326 Steady Currents in Linear Conductors [CH. ix 



11. A cable AB, 50 miles in length, is known to have one fault, and it is necessary to 

 localise it. If the end A is attached to a battery, and has its potential maintained 

 at 200 volts, while the other end B is insulated, it is found that the potential of B when 

 steady is 40 volts. Similarly when A is insulated the potential to which B must be raised 

 to give A a steady potential of 40 volts is 300 volts. Shew that the distance of the fault 

 from A is 19'05 miles. 



12. A wire is interpolated in a circuit of given resistance and electromotive force. 

 Find the resistance of the interpolated wire in order that the rate of generation of heat 

 may be a maximum. 



13. The resistances of the opposite sides of a Wheatstone's Bridge are a, a' and 6, b' 

 respectively. Shew that when the two diagonals which contain the battery and galvano- 

 meter are interchanged, 



E_E^_ (a-a'}(b-V}(G-R) 

 C C'~ aa'-bV 



where C and C' are the currents through the galvanometer in the two cases, G and R are 

 the resistances of the galvanometer and battery conductors, and E the electromotive force 

 of the battery. 



14. A current C is introduced into a network of linear conductors at A, and taken 

 out at B, the heat generated being HI. If the network be closed by joining A, B by a 

 resistance r in which an electromotive force E is inserted, the heat generated is H 2 - 

 Prove that 



2?i a.!^-l 



C 2 r ~* E*~ 



15. A number N of incandescent lamps, each of resistance r, are fed by a machine of 

 resistance R (including the leads). If the light emitted by any lamp is proportional to 

 the square of the heat produced, prove that the most economical way of arranging the 

 lamps is to place them in parallel arc, each arc containing n lamps, where n is the integer 

 nearest to 



16. A battery of electromotive force E and of resistance B is connected with the two 

 terminals of two wires arranged in parallel. The first wire includes a voltameter which 

 contains discontinuities of potential such that a unit current passing through it for a 

 unit time does p units of work. The resistance of the first wire, including the voltameter, 

 is R: that of the second is r. Shew that if E is greater than p (B + r)/r, the current 

 through the battery is 



17. A system of 30 conductors of equal resistance are connected in the same way as 

 the edges of a dodecahedron. Shew that the resistance of the network between a pair of 

 opposite corners is of the resistance of a single conductor. 



18. In a network PA, PB, PC, PD, AB, BC, CD, DA, the resistances are a, 0, y, 8, 

 y+8, 5+ a, a-fjS, /3+y respectively. Shew that, if AD contains a battery of electromotive 

 force E, the current in BC is 



where 



