If T is the time occupied by a single revolution, and S is the surface of the disc, we have @T: expression therefore becomes = 2π and S = Tа2 the above Suppose that the disc has a surface of one square metre, and let it revolve at the rate of 10 turns per second about a horizontal axis in the magnetic meridian; then if H 0.2, the electromotive force equals = E = 104 x 0.2 x 10 = 2 × 10a C.G.S. = 2 × 10-4 volts. If the circuit had a resistance of only 10-4 ohms, the strength of the current would be 2 amperes. N In like manner, if an arc of any form (Fig. 277) be made to turn uniformly about the north pole of a magnet, the strength of the pole being m, the arc will, in each unit of time, cut a flux equal to 2mw, and in a circuit of resistance, R, will give a constant current of strength Let us suppose that at A and at C the arc is in contact by means of two brushes with the surface of the magnet, or with a conducting surface rigidly connected therewith; the same current will be produced going from C to A by the arc, whether the arc is moved about the magnet in the direction of the hands of a watch, or, the arc being stationary, the magnet is moved about its axis in the opposite direction. This latter effect is often referred to as unipolar induction. FIG. 277.. 310. Variable Period-Establishment of a Current.-Let E be the total electromotive force of the battery, R the total resistance, L the coefficient of self-induction of the circuit, and C=E|R the ultimate strength of the current when it has become uniform.. While the current is increasing in strength from o to C, the strength at each instant satisfies the equation (§ 296)— since E and R are both constant. Rearranging this expression, we get E d R -c) E where the left-hand side is the differential of log, (−c). If = R we count time from the instant of closing the circuit, the strength of the current o when t = o. Putting C for the current after any assigned interval of time, t, and integrating on the left between the limits o and C, and on the right between o and t, we get or, passing from logarithms to exponentials and rearranging the expression Where e is the base of natural logarithms, the never-ending decimal 2.71828. . . The self-induction and resistance of the circuit are permanent characteristics of it, and their quotient L/R determines the behaviour of the circuit in all cases of varying currents. It is commonly called the time-constant of the circuit, and we shall denote it by the symbol λ, as in the last formula. This formula shows that when t is great, that is, when the circuit has been closed for a long time, the strength of the current is equal to C=ER, for e raised to a very high negative power is sensibly equal to nothing. In order to trace the growth of the current, we may consider what fraction of its ultimate strength, C, it has acquired at the end of various intervals from making contact for instance, suppose to have in succession the values λ, 2λ, 3λ . . . . and let C1, C2, C3 . . . . denote the corresponding resistance, the magnetic forces at any point would be always equal and in opposite directions, that there would therefore be no inductive action behind the plate, which would thus play the part of an absolute screen. Experiment shows, in fact, that the magnetic force is lessened, and the more so, the better the plate conducts, and the more rapidly the changes of magnetic force occur. CHAPTER XXVII. SPECIAL CAses of indUCTION. ALTERNATING CURRENTS AND ELECTRICAL OSCILLATIONS. 320. Currents varying Harmonically.-Consider the case of a coil wound in a narrow groove, and rotating uniformly about a vertical diameter in a uniform magnetic field like that of the earth (Fig. 284). Let H be the horizontal component of the magnetic field, and S the area enclosed by the coil : then, when the plane of the coil is perpendicular to the meridian, the magnetic flux through the coil is HS; and, when the coil has been turned through an angle a from this position, or when its axis makes an angle a with the meridian, the magnetic flux is Q = HS cos a. If w be the velocity of rotation, and ☎ the period, or the duration of one complete revolution, we have at time t, measured from an instant at which the axis of the coil is in the meridian The electromotive force due to the varia tion of the flux through the coil is given at each instant by the corresponding value of constant, we may write Hence, seeing that H and S are The magnetic flux through the coil varies most rapidly when its instantaneous value is zero; this is when the plane of the coil is parallel to the meridian, and therefore a is a right angle. At this instant, therefore, the induced electromotive force is a maximum, and if we denote this by E,, we have for the electromotive force at any time t E = E sin a = E sin wt = sin 2πί T In such a case the electromotive force is said to vary harmonically. A variation of this kind is expressed graphically by the ordinates of the sine-curve OAB (Fig. 285). In this figure the distance OB represents half the period 7, and the maximum ordinate represents E, the amplitude. If self-induction produced no effect, the current would be proportional at every instant to the existing electromotive force, or CER. The maximum strength of current would be ER, and the current at any instant would be E. 2πt that is, the current would also vary harmonically in the same period as the electromotive force, and might be represented by a similar sine-curve, having the same period and the same nodes as the curve OAB. 321. Effect of Self-Induction.—From what has been said already (§§ 310, 311) as to the way in which self-induction retards the rise or fall of the current when a steady electromotive force is suddenly applied or removed, it is easy to infer that, in the case we are now considering, the changes of current-strength will lag behind those of electromotive force, and that the maximum strength of current |