CHAPTER III. LAW OF ELECTRO-MAGNETIC INDUCTION. 11. If an electric conductor moves relatively to a mag- netic field, an E.M.F. is induced in the conductor which is proportional to the intensity of the magnetic field, to the length of the conductor, and to the speed of its motion perpendicular to the magnetic field and the direction of the conductor ; or, in other words, proportional to the number of lines of magnetic force cut per second by the conductor. As a practical unit of E.M.F., the volt is defined as the E.M.F. induced in a conductor, which cuts 108 = 100,000,000 lines of magnetic force per second. If the conductor is closed upon itself, the induced E.M.F. produces a current. A closed conductor may be called a turn or a convolution. In such a turn, the number of lines of magnetic force cut per second is the increase or decrease of the number of lines inclosed by the turn, or n times as large with n turns. Hence the E.M.F. in volts induced in n turns, or con- volutions, is n times the increase or decrease, per second, of the flux inclosed by the turns, times 10~8. If the change of the flux inclosed by the turn, or by n turns, does not take place uniformly, the product of the number of turns, times change of flux per second, gives the average E.M.F. If the magnetic flux, 4>, alternates relatively to a number of turns, n — that is, when the turns either revolve through the flux, or the flux passes in and out of the turns, the total flux is cut four times during each complete period or cycle, twice passing into, and twice out of, the turns. LAW OF ELECTRO-MAGNETIC INDUCTION. 17 Hence, if N= number of complete cycles per second, or the frequency of the flux 3>, the average E.M.F. induced in n turns is, £&vg, = 4 « 3> N 10 ~ 8 volts. This is the fundamental equation of electrical engineer- ing, and applies to .continuous-current, as well as to alter- nating-current, apparatus. 12. In continuous-current machines and in many alter- nators, the turns revolve through a constant magnetic field ; in other alternators and in induction motors, the mag- netic field revolves ; in transformers, the field alternates with respect to the stationary turns. Thus, in the continuous-current machine, if n = num- ber of turns in series from brush to brush, = flux inclosed per turn, and N = frequency, the E.M.F. induced in the machine is E = 4«4>7V10~8 volts, independent of the num- ber of poles, of series or multiple connection of the arma- ture, whether of the ring, drum, or other type. In an alternator or transformer, if n is the number of turns in series, $ the maximum flux inclosed per turn, and JV the frequency, this formula gives, £avg = 4 « 4> JVW ~ 8 volts. Since the maximum E.M.F. is given by, — •^maz. = £ ^avg we have ^"max. = 27r»7V710-8VOltS. And since the effective E.M.F. is given by, — we have £es. = = 4.44 n 4>^10- 8 volts, which is the fundamental formula of alternating-current induction by sine waves. 18 AL TERN A TING-CURRENT PHENOMENA, 13. If, in a circuit of n turns, the magnetic flux, , inclosed by the circuit is produced by the current flowing in the circuit, the ratio — flux X number of turns X 10~8 current . is called the inductance, L, of the circuit, in henrys. The product of the number of turns, n, into the maxi- mum flux, , produced by a current of / amperes effective, or / V2 amperes maximum, is therefore — n® =Z/V2 108; and consequently the effective E.M.F. of self-inductance is: E = V2 =' 2 TT NLI volts. The product, x = 2 vNL, is of the dimension of resistance, and is called the reactance of the circuit ; and the E.M.F. of self-inductance of the circuit, or the reactance voltage, is E = Ix, and lags 90° behind the current, since the current is in phase with the magnetic flux produced by the current, and the E.M.F. lags 90° behind the magnetic flux. The E.M.F. lags 90° behind the magnetic flux, as it is propor- tional to the change in flux ; thus it is zero when the mag- netism is at its maximum value, and a maximum when the flux passes through zero, where it changes quickest. GRAPHIC REPRESENTA TION, 19