a direction at right angles both to the direction of the field and to the direction of the current. This fact underlies the operation of all electric motors.

350. The motor and dynamo rules. A convenient rule for determining whether the wire ab (Fig. 302) will move forward or back in a given case may be obtained as follows: If the field of a magnet alone is represented by Fig. 303, and that due to the current* alone by Fig. 304, then the resultant field when the current-bearing wire is placed between the poles of the magnet is that shown in Fig. 305; for the strength of the

FIG. 304. Field of FIG. 305. Field of magnet
current alone
and current

field above the wire is now the sum of the two separate fields, while the strength below it is their difference. Now Faraday thought of the lines of force as acting like stretched rubber bands. This would mean that the wire in Fig. 305 would be pushed down. Whether the lines of force are so conceived or not, the motor rule may be stated thus:

A current in a magnetic field tends to move away from the side on which its lines are added to those of the field.

The dynamo rule follows at once from the motor rule and Lenz's law. Thus, when a wire is moved through a magnetic field the current induced in it must be in such a direction as

* The cross in the conductor of Fig. 304, representing the tail of a retreating arrow, is to indicate that the current flows away from the reader. A dot, representing the head of an advancing arrow, indicates a current flowing toward the reader.

to oppose the motion; therefore the induced current will be in such a direction as to increase the number of lines on the side toward which it is moving.

351. Strength of the induced E.M.F. The strength of an induced E.M.F. is found to depend simply upon the number of lines of force cut per second by the conductor, or, in the case of a coil, upon the rate of change in the number of lines of force which pass through the coil. The strength of the current which flows is then given by Ohm's law; that is, it is equal to the induced E.M.F. divided by the resistance of the circuit. The number of lines of force which the conductor cuts per second may always be determined if we know the velocity of the conductor and the strength of the magnetic field through which it moves. For it will be remembered that, according to the convention of § 270, a field of unit strength is said to contain one line of force per square centimeter, a field of 1000 units strength 1000 lines per square centimeter, etc. In a conductor which is cutting lines at the rate of 100,000,000 per second there is an induced E.M.F. of 1 volt.* The reason why we used a coil of 500 turns instead of a single turn in the experiment of § 346 was that by thus making the conductor in which the current was to be induced cut the lines of force of the magnet 500 times instead of once, we obtained 500 times as strong an induced E.M.F., and therefore 500 times as strong a current for a given resistance in the circuit.

352. Currents induced in rotating coils. Let a 400- or 500-turn coil of No. 28 copper wire be made small enough to rotate between the poles of a horseshoe magnet, and let it be connected into the circuit of a galvanometer, precisely as in § 346. Starting with the coil in the position of Fig. 306, (1), let it be rotated suddenly clockwise (looking down from above) through 180°. A strong deflection of the galvanometer will be observed. Let it be rotated through the next 180° back to the starting point. An opposite deflection will be observed.

*This may be considered as the scientific definition of the volt, convenience alone having dictated the legal definition given in § 334.

(1)

(2)

The arrangement is a dynamo in miniature. During the first half of the revolution (see Fig. 306, (2)) the wires on the right side of the loop were cutting the lines of force in one direction, while the wires on the left side were cutting them in the opposite direction. A current was being generated down on the right side of the coil and up on the left side (see dynamo rule). It will be seen that both currents flow around the coil in the same direction. The induced current is strongest when the coil is in the position shown in Fig. 306, (2), because there the lines of force are being cut most rapidly. Just as the coil is moving into or out of the position shown in Fig. 306, (1), its edges are moving parallel to the lines of force, and hence no current is induced, since no lines of force are being cut. As the coil moves through the last 180° of its revolution both sides are cutting the same lines of force as before, but they are cutting them in an opposite direction; hence the current generated during this last half is opposite in direction to that of the first half. *

FIG. 306. Direction of currents induced in a coil rotating in a magnetic field

QUESTIONS AND PROBLEMS

1. Can the number of lines of force within a closed coil of wire be increased or decreased without the lines being cut by the wire? Explain. 2. Under what conditions may an electric current be produced by a magnet?

3. How many lines of force must be cut per second to induce 10 volts? 4. If a coil of wire is rotated about a vertical axis in the earth's field, an alternating current is set up in it. In what position is the coil wher the current changes direction?

* A laboratory experiment on the principles of induction should be performed at about this point. See, for example, Experiment 36 of the authors' Manual.

5. State Lenz's law, and show how it follows from the principle of the conservation of energy.

6. A coil is thrust over the S pole of a magnet. Is the direction of the induced current clockwise or counterclockwise as you look down upon the pole?

7. A ship having an iron mast is sailing east. In what direction is the E.M.F. induced in the mast by the earth's magnetic field? If a wire is brought from the top of the mast to its bottom, no current will flow through the circuit. Why?

8. A current is flowing from top to bottom in a vertical wire. In what direction will the wire tend to move on account of the earth's magnetic field?

DYNAMOS

353. A simple alternating-current dynamo. The simplest form of commercial dynamo consists of a coil of wire so arranged as to rotate continuously between the poles of a powerful electromagnet

(Fig. 307).

In order to make the magnetic field in which the conductor is moved as strong as possible, the coil is wound upon an iron core C. This greatly increases the total number of lines of magnetic force which pass between N and S, for instead of an air path the core offers an iron path, as shown in Fig. 308.

R

MC

FIG. 307. Drum-wound armature

The rotating part, consisting of the coil with its core, is called the armature. One end of the coil is attached to the insulated metal ring R, which is attached rigidly to the shaft of the armature and therefore rotates with it, while the other end of the coil is attached to a second ring R'. The brushes b and b', which constitute the terminals of the external circuit, are always in contact with these rings.

As the coil rotates, an induced alternating current passes through the circuit. This current reverses direction as often

as the coil passes through the position shown in Fig. 308, that is, the position in which the conductors are moving parallel to the lines of force; for at this instant the conductors which were moving up begin to move down, and those which were moving down begin to move up. The current reaches its maximum value when the coils are moving through a position 90° farther on, for then the lines of force are being cut most rapidly by the conductors on both sides of the coil. These facts are graphically represented by the curve of E.M.F.'s (Fig. 309).

354. The multipolar alternator. For most commercial purposes it is found desirable to have 120 or more alternations of current per second. This could not be attained easily with two-pole machines like those

FIG. 309. Curve of alternating electromotive force

sketched in Figs. 307 and 308. Hence commercial alternators are usually built with a large number of poles alternately N and S, arranged around the circumference of a circle in the manner shown in Fig. 310. These poles are excited by a direct current. The dotted lines represent the direction of the lines of force through the iron. It will be seen that the coils which are passing beneath N poles have induced currents set up in them the direction of which is opposite to that of the currents which are induced in the conductors which are passing beneath the S poles. Since, however, the direction of winding of the armature coils changes between each two poles, all the inductive effects of all the poles are added in the coil and constitute at any instant one single current