46. Induced Currents.-When a change occurs in the number of lines of force passing through a closed coil a current is produced in the coil. These currents are called Induced Currents. They may be produced by any means that causes a change in the number of lines of force passing through the coil, as by the approach or withdrawal of a magnet, a solenoid, or by an increase or decrease of a current in a circuit adjacent to the coil.

The Laws of Induced Currents are as follows:

1. An increase in the number of lines of force passing through a closed coil or helix produces a current flowing through the coil in the opposite direction from the motion of the hands of a watch. A decrease in the number of lines of force produces a current in the same direction as the motion of the hands of a watch.

2. The electromotive force of the induced current is proportional to the rate of increase or decrease in the number of lines of force passing through the coil.

ELECTRICAL MEASUREMENTS.

47. Resistance.-The Laws of Resistance are as follows: 1. The resistance of a conductor varies directly with its length. 2. Inversely with the cross-section area or, if the conductor be a round wire or rod, inversely with the square of the diameter. 3. Directly with the "specific resistance" of the material. NOTES. (a) The resistance also depends upon the temperature of the conductor, being in general greater for higher temperatures (chief exception, carbon).

(b) In the C. G. S. system the specific resistance of a substance is the resistance between opposite faces of a cube 1 cm. on each edge made of the substance.

The following table gives, in micro-ohms (thousandths of an ohm), the specific resistance of some of the metals and a few other substances:

(c) In general good conductors of heat are good conductors of electricity. Liquids, except liquid metals which conduct as metals, conduct only by electrolysis. Gases are almost perfect insulators, except under conditions which allow convection currents.

Algebraically the laws of resistance may be expressed as follows, where R is the resistance, L is length, D is diameter, and r is specific resistance:

Tables of actual resistance and of relative resistivity will be found at the end of this book.

LXVI. RESISTANCE.

1. If a wire 6 meters long and 1 mm. in diameter has a resistance of 2 ohms, what will be the resistance of a wire 4 m. long and of diameter 2 mm.?

2. By reference to the table of resistances determine the length of a copper wire 1 mm. in diameter, having a resistance of 80 ohms.

3. The same for a German-silver wire.

4. If a No. 16 brass wire of a certain length has a resistance of 1 ohm, how many times as long must a No. 12 wire be to have the same resistance?

5. By reference to the table of Specific Resistances above, determine the relative resistivity of copper and iron.

6. How many cm. of iron wire are required to have the same resistance as 50 cm. of silver wire?

7. By Ohm's Law find the current which with an E. M. F. of 1.08 volts will flow through 2 m. of No. 30 copper wire. (See tables.)

8. What number of wire should be used if 50 meters of it are required to deliver a 4-ampere current when the E. M. F. is 5 volts?

9. Two wires of same length and material have resist

ances 5 and 8 ohms respectively. The first is .5 mm. in diameter. Find the diameter of the second.

10. From the specific resistance of copper calculate the résistance of 2 m. of No. 14 copper wire.

11. Compare the resistance of one meter of copper wire No. 18 with that of one meter of platinum wire No. 22.

12. The resistance of a coil of wire was found to be 90 ohms. A piece of the wire 2 meters long had a resistance of .6 ohm. How long was the wire of the coil?

13. Find the length of an iron wire No. 9 to have a resistance of 1000 ohms.

14. What number of copper wire has a resistance nearest to one ohm to the meter ?

15. A cable is 6 miles long and has a total resistance when unbroken of 48 ohms. How far from one end is a break such that the resistance of one part is 22.8 ohms?

48. Effect of Temperature on Resistance.-The temperaturecoefficient of resistance is the increase of resistance per ohm per degree of rise of temperature. Or, where K is the coefficient, v the resistance at temperature t1, and r2 the resistance at temperature ta,

The value of K for most pure metals is about .004 through ordinary ranges of temperature. For alloys it is much less, being only about .00044 for German silver. For this reason German silver is often used for the coils of resistance-boxes.

LXVII. TEMPERATURE AND RESISTANCE.

1. Find the resistance at 50° C. of a copper wire whose resistance at 15° is 2 ohms.

2. The same for a German-silver wire.

3. Determine the value of the temperature-coefficient of resistance from the following record of an experiment with a coil of copper wire:

Resistance of coil in oil-bath at 3°.3, 2.76 ohms.

4. How many degrees must the temperature of a copper wire be raised that its resistance may be increased from 10 ohms to 10.7 ohms?

5. Compare the resistance of one meter of No. 14 copper wire at 30° with that of one meter of No. 18 wire at 0°.

49. The Measurement of Resistance.-Six methods will be considered: Substitution, Differential Galvanometer, Resistance of Divided Circuits, Fall of Potential, Wheatstone's Bridge, (a) 1000 mm. wire bridge and (b) box bridge, and the measurement of Internal Resistances.

The student will do well to draw a diagram for each circuit as described, using the conventional symbols for batteries, galvanometers and resistance-boxes.

LXVIII. SUBSTITUTION.

1. A battery, galvanometer, resistance-box, and a wire whose resistance is to be measured are connected in series. When the box reads 5 ohms the galvanometer shows a deflection of 40°. The wire is then removed, and it is found necessary to increase the box resistance to 8.2 ohms to reduce the galvanometer-reading to 40°. Find the resistance of the wire.

2. If the wire of Prob. 1 was 38 cm. long, how much of it must be used to make a resistance-coil of one ohm?

3. Two coils supposed to be each of 10 ohms resistance are placed in series with a standard resistance-box, a galvanometer, and a battery, and the box adjusted till a deflection of 45° is obtained. The two coils were then removed and the box readings were increased 20.2 ohms before a deflection of 45° was again obtained. What was the aver

age per cent of error of the coils?

4. In a certain circuit it is found that 50 cm. of silver wire may be replaced by either 46 cm, of copper wire or by

7.74 cm. of iron wire of the same diameter. Determine the resistances of copper and iron as compared with silver. 5. If platinum has six times the resistance of silver, how many cm. of No. 14 silver wire are required to offer the same resistance as 10 cm. of No. 12 platinum wire?

50. Differential Galvanometer for Measurement of Resistance.—A differential galvanometer is one having two equal coils so wound as to tend to turn a needle hung within them in opposite directions. The coils are usually adjusted to be exactly alike in all respects. Equal currents in the two coils will then balance, and no deflection be observed.

LXIX. DIFFERENTIAL GALVANOMETER.

1. The current from a battery is divided so that one portion flows through a wire whose resistance is to be tested and one coil of a differential galvanometer, and the other portion through a standard resistance-box and the other coil. Write an equation to express by letters the value of the current (Ohm's Law) in each portion, and show, when no deflection occurs, what two quantities are equal.

2. With apparatus like that of the preceding problem, if the resistance-box reads 11 ohms, what is the resistance of the wire?

51. Divided Circuits.—A consideration of Fig. 60 will show that the current flowing in AB before division into the branches

1, 2, and 3 must be the same as in CD after reuniting. Let this current be C and the currents in 1, 2, and 3 be respectively C1, C2, and Cs. Thence we have Kirchhoff's Law,