a resistance of 13 ohms; what is the resistance of 440 yards of wire 0.8 mm. in diameter made of the same material? 34. The resistance at o° of a column of mercury I metre in length and I sq. mm. in cross-section is called a "Siemen's Unit." Find the value of this unit in terms of the ohm. 35. Copper wire of No. 20 on the new standard wiregauge has a resistance of 0.026 ohm per metre, and weighs 5.84 gm. per metre: what is the résistance of a metre of No. 32 S. W. G. copper wire weighing 0.524 gm.? 36. Compare the resistances of two wires, one of which weighs 20.5 gm. and is 4.5 metres long, while the other weighs 82 gm. and is 18 metres long. 37. A trough 2 cm. deep and 2.5 cm. broad is cut in a wooden board. The trough is 2 metres long and is half filled with mercury: find the resistance between its ends. 38. What length of platinum wire 1 mm. in diameter is required in order to make a 1-ohm resistance coil? 39. A wire m metres in length and 1/nth of a millimetre in diameter is found to have a resistance r: what is the specific resistance of the material of which it is made? 40. Find the resistance of an iron wire ABC which consists of two parts, the first (AB) being 60 cm. long and 1 mm. in diameter, and the second (BC) being 3 metres long and 1.6 mm. in diameter. 41. The poles of a battery are connected by a wire whose resistance is equal to that of the battery, and the poles of a second exactly similar battery are connected by a wire of the same weight and material, but three times as long as the first. Compare the currents in the two cases. 42. Find the length of an iron wire one-twentieth of an inch in diameter which will have the same resistance as a copper wire one-sixtieth of an inch in diameter and 720 yards long, the conducting power of copper being six times that of iron. 43. Express in microhms (or millionths of an ohm) the resistance of a strip of silver 10 cm. long, 0.5 cm. broad, and 0.1 mm. thick. 44. A wire made of platinoid (German-silver containing a small percentage of tungsten) is found to have a resistance of 0.203 ohm per metre. The cross-section of the wire is o·016 sq. cm.: express the specific resistance of platinoid in microhms. 45. The density of aluminium is 2.7: what is the resistance of an aluminium wire 1 metre long and weighing 1 gm.? 46. The central conductor of the Bessbrook and Newry electrical tramway is made of steel having a specific resistance of 0.0000121 ohm, and costing £7: Ios. per ton: high conductivity copper of 0.0000016 ohm specific resistance would have cost £84 per ton. The cross-section of the steel conductor was 8.817 sq. cm.: calculate the cross-section of a copper conductor of the same resistance, and show that it would have cost half as much again as the steel conductor. The Tangent Galvanometer.-A tangent galvanometer consists of a circular coil of wire placed with its plane in the magnetic meridian, and at the centre of which is suspended a small magnetic needle. The force exerted by a current flowing through the coil upon a magnet pole placed at (or very near to) its centre is directly proportional to the current-strength, the total length of the coil and the strength of the pole, and is inversely proportional to the square of the distance between each element of the circuit and the pole. Let be the radius of the coil, n the number of turns, and m the pole-strength of the magnet: then if the current C is expressed in terms of the C.G.S. electro magnetic unit (as defined on p. 5), the force exerted upon each pole of the magnet is equal to 2πnr × Cm/r2 =2πnCm/r. Suppose the needle to be deflected through an angle 8 from the magnetic merid It is ian NS (see Fig. 6) so as couple due to the equal 2πnCm xl cos 8/1. The earth's field acts upon each pole with a force mH1 in the direc Fig. 6. tion of the arrows: the arm of this directive couple is PQ=/ sin 8, and its moment is mH x 7 sin 8. For equilibrium these moments must be equal, or The quantity rН/2πn, by which the tangent of the deflection has to be multiplied in order to obtain the current-strength, may be called the reduction-factor of the galvanometer: but we shall use this term to denote the value of the quantity k in the equation C = k. tan 8, 1 The letter H is the symbol usually adopted for representing the horizon, tal intensity of the earth's magnetic force. where the current C is understood to be expressed in ampères instead of C.G.S. units. 47. A current of 0.85 ampère flows through a tangent galvanometer consisting of three turns of wire, each of 30 cm. diameter: what is the strength of the field due to the circular current at its centre? If the value of H at the place is 0.17, through what angle will the needle be deflected? The strength of the field produced by the current is measured by the force which would be experienced by a unit magnetic pole, and this by the preceding paragraph is 2πnС/г. Since an ampère is one-tenth of the C.G.S. unit of current, the required strength of field is 2′′ × 3 × 0.085/15=0.1068. By equation (6) Referring to the table of tangents we see that since tan 39°=0.629, the angular deflection will be nearly 39°. 48. The same current is sent through two concentric circular wires of 1 and 3 decimetres radius respectively, and a small magnet is suspended at their common centre. The currents flow in opposite directions: compare their joint effect upon each pole of the magnet with that produced by a single coil of 2 decimetres radius traversed by the same current. The force due to the coil of I dcm, radius is f=2πCm/10, and that due to the coil of 3 dcm. radius is ƒ'=2πCm/30. Their joint effect (since the forces act in opposite directions) is The force due to the coil of 2 dcm. radius is F'= 2πCm/20. [This is not the ratio of the deflections. If the deflections are d and d' respectively, then tan d: tan d' = 4:3.] 49. Find the strength of a current which produces a deflection of 45° in a tangent galvanometer consisting of a single copper rod bent into a circle of 40 cm. diameter. [H=0.17.] 50. A current of 0.04 ampère flows through a circular coil of wire consisting of 3 turns, each 2 dcm. in diameter what force will be exerted by the current upon a magnetic pole of strength 10 placed at the centre of the coil? K B 51. A battery is connected up in series with a known resistance R and a tangent galvanometer of resistance G (Fig. 7). The deflection of the galvanometer needle is a: on increasing the known resistance to R' the deflection falls to a.' Show that the internal resist ance of the battery is given by G the equation R' tan a' R tan a R Fig. 7. B= - G. 52. Calculate the internal resistance of a Grove's cell from the following data, obtained by the method of Ex. 50: Resistance of Galvanometer=0.102 ohm. (1) R 2 ohms, a = 53°. (2) R'=5 ohms, a' = 30°. 53. What does the equation in Ex. 50 reduce to when the second deflection indicates a current of one-half the original strength? A current from a battery of 4 Daniell cells was sent through a resistance-box and a tangent galvanometer of negligible resistance. With a resistance of 58 ohms a deflection of 55° was obtained, and when the resistance was increased to 131 ohms the deflection fell to 3510 Find the resistance of the battery, given that tan 55° = 2 tan 351°. |