Examples 325 



2. A network is formed of uniform wire in the shape of a rectangle of sides 2a, 3a, 

 with parallel wires arranged so as to divide the internal space into six squares of sides a, 

 the contact at points of intersection being perfect. Shew that if a current enter the 

 framework by one corner and leave it by the opposite, the resistance is equivalent to that 

 of a length 121a/69 of the wire. 



3. A fault of given earth-resistance develops in a telegraph line. Prove that the 

 current at the receiving end, generated by an assigned battery at the signalling end, is 

 least when the fault is at the middle of the line. 



4. The resistances of three wires BC, CA, AB, of the same uniform section and 

 material, are a, b, c respectively. Another wire from A of constant resistance d can make 

 a sliding contact with BC. If a current enter at A and leave at the point of contact 

 with BC, shew that the maximum resistance of the network is 



(a + b + c) d 

 a + b + c + 4d' 

 and determine the least resistance. 



5. A certain kind of cell has a resistance of 10 ohms and an electromotive force of 

 85 of a volt. Shew that the greatest current which can be produced in a wire whose 

 resistance is 22 '5 ohms, by a battery of five such cells arranged in a single series, of 

 which any element is either one cell or a set of cells in parallel, is exactly '06 of an 

 ampere. 



6. Six points A, A', B, B', C, C' are connected to one another by copper wire whose 

 lengths in yards are as follows : A A' = 16, BC = B'C = 1, BC' = B'C' = 2, AB = A'B'= 6, 

 AC' = A'C' 8. Also B and B' are joined by wires, each a yard in length, to the terminals 

 of a battery whose internal resistance is equal to that of r yards of the wire, and all the 

 wires are of the same thickness. Shew that the current in the wire A A' is equal to that 

 which the battery would maintain in a simple circuit consisting of 31r + 104 yards of 

 the wire. 



7. Two places A, B are connected by a telegraph line of which the end at A is 

 connected to one terminal of a battery, and the end at B to one terminal of a receiver, 

 the other terminals of the battery and receiver being connected to earth. At a point C 

 of the line a fault is developed, of which the resistance is r. If the resistances of A C, CB 

 be p, q respectively, shew that the current in the receiver is diminished in the ratio 



r(p + q) : qr + rp+pq, 

 the resistances of the battery, receiver and earth circuit being neglected. 



8. Two cells of electromotive forces e 1 , e% and resistances r it r% are connected in 

 parallel to the ends of a wire of resistance R. Shew that the current in the wire is 



and find the rates at which the cells are working. 



9. A network of conductors is in the form of a tetrahedron PQRS ; there is a battery 

 of electromotive force E in PQ, and the resistance of PQ, including the battery, is R. 

 If the resistances in QR, RP are each equal to r, and the resistances in PS, RS are each 

 equal to \r, and that in QS = f r, find the currents in each branch. 



10. A, B, C, D are the four junction points of a Wheatstone's Bridge, and the 

 resistances c, /3, b, y in AB, BD, AC, CD respectively are such that the battery sends no 

 current through the galvanometer in BC. If now a new battery of electromotive force E 

 be introduced into the galvanometer circuit, and so raise the total resistance in that 

 circuit to a, find the current that will flow through the galvanometer. 



