Examples 327 



19. A wire forms a regular hexagon and the angular points are joined to the centre 

 by wires each of which has a resistance - of the resistance of a side of the hexagon. 



Shew that the resistance to a current entering at one angular point of the hexagon and 

 leaving it by the opposite point is 



(+!)(* + 4) 



times the resistance of a side of the hexagon. 



20. Two long equal parallel wires AB, A'B' of length I, have their ends , B' joined 

 by a wire of negligible resistance, while A, A' are joined to the poles of a cell whose 

 resistance is equal to that of a length r of the wire. A similar cell is placed as a bridge 

 across the wires at a distance x from A, A'. Shew that the effect of the second cell is to 

 increase the current in BB' in the ratio 



21. There are n points 1, 2, ... n, joined in pairs by linear conductors. On introducing 

 a current C at electrode 1 and taking it out at 2, the potentials of these are F l5 F 2 , ... F n . 

 If # 12 is the actual current in the direction 12, and #' 12 any other that merely satisfies the 

 conditions of introduction at 1 and abstraction at 2, shew that 



2 (r 12 # 12 # 12 ') = ( F!- F 2 ) (7=2 (r 12 ^ 12 2 ), 

 and interpret the result physically. 



If x typify the actual current when the current enters at 1 and leaves at 2, and y 

 typify the actual current when the current enters at 3 and leaves at 4, shew that 



2 (r 12 * 12 y 12 ) = (X 3 -Xi)C = (Y l - F 2 ) C, 



where the JTs are potentials corresponding to currents x, and the Ps are potentials 

 corresponding to currents y. 



22. A, B, C are three stations on the same telegraph wire. An operator at A knows 

 that there is a fault between A and B, and observes that the current at A when he uses a 

 given battery is i, i' or i", according as B is insulated and C to earth, B to earth, or B 

 and C both insulated. Shew that the distance of the fault from A is 



{ka - k'b +(b- of (ka - Vbfy(k - k'\ 



" " 



where AB=a, BC=b-a, k= 



23. Six conductors join four points A, B, (7, D in pairs, and have resistances 

 a, a, 5, /3, c, y, where a, a refer to BC, AD respectively, and so on. If this network 

 be used as a resistance coil, with J, B as electrodes, shew that the resistance cannot 

 lie outside the limits 



1 1 "l-i , n /1 IN- 1 /I I 1 ) " 



H" 1 

 J ' 



. 24. Two equal straight pieces of wire A Q A n , B Q B n are each divided into n equal parts 

 at the points A l ... A n ^ and B l ... 5 n _ 1 respectively, the resistance of each part and 

 that of A n B n being R. The corresponding points of each wire from 1 to n inclusive 

 are joined by cross wires, and a battery is placed in AiB . Shew that, if the current 

 through each cross wire is the same, the resistance of the cross wire A 8 B 8 is 



