CHAPTER III 
LAW OF ELECTROMAGNETIC INDUCTION 

13. If an electric conductor moves relatively to a magnetic 
field, an e.m.f. is generated in the conductor which is propor- 
tional 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 by the e.m.f. 
generated in a conductor, which cuts 10^ = 100,000,000 lines of 
magnetic flux per second. 

If the conductor is closed upon itself, the 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 generated in n turns, or convolutions, 
is n times the increase or decrease, per second, of the flux inclosed 
by the turns, times 10~^. 

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, $, 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. 

Hence, if / = number of complete cycles per second, or the 
frequency of the flux, $, the average e.m.f. generated in n turns is 

Eavg. = 4 71$/ 10-« volts. 

This is the fundamental equation of electrical engineering, 
and applies to continuous-current, as well as to alternating- 
current, apparatus. 

16 


LAW OF ELECTROMAGNETIC INDUCTION 17 

14. In continuous-current machines and in many alternators, 
the turns revolve through a constant magnetic field; in other 
alternators and in induction motors, the magnetic field revolves; 
in transformers, the field alternates with respect to the sta- 
tionary turns; in other apparatus, alternation and rotation occur 
simultaneously, as in alternating-current commutator motors. 

Thus, in the continuous-current machine, if n = number of 
turns in series from brush to brush, $ = flux inclosed per turn, 
and / = frequency, the e.m.f. generated in the machine is 
E = 4/i$/10~^ volts, independent of the number of poles, of 
series or multiple connection of the armature, 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 /the frequency, 
this formula gives 

Eavg. = 4:7l^f lO'S VOltS. 

Since the maximum e.m.f. is given by 
we have 

^max. = 2 7rW$/ 10-8 volts. 

And since the effective e.m.f. is given by 

Emax. 


E. 


eff. — 


V2 


we have 

Eeff. = V2 wn^f 10-^ 

= 4.44 nf^ 10-« volts, 

which is the fundamental formula of alternating-current induc- 
tion by sine waves. 

15. If, in a circuit of n turns, the magnetic flux, <E>, inclosed 
by the circuit is produced by the current in the circuit, the 
ratio, 

flux X number of turns X 10"^ 
current ' 

is called the inductance, L, of the circuit, in henrys. 

The product of the number of turns, n, into the maximum 
flux, $, produced by a current of / amperes effective, or 7^/2 
amperes maximum, is therefore 

n$ = LIV2 108; 


18 ALTERNATING-CURRENT PHENOMENA 

and consequently the effective e.m.f. of self-induction is 

E = V2 7rn$/10-8 
= 2 wfLI volts. 

The product, x = 2 7r/L, is of the dimension of resistance, 
and is called the inductive reactance of the circuit; and the e.m.f. 
of self-induction 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 proportional to the rate of change in 
flux; thus it is zero when the magnetism does not change, at its 
maximum value, and a maximum when the flux changes quick- 
est, which is where it passes through zero.