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this statement.

331. Induction by Cur-

rents (J. J. T., 374). Since

a current through a solenoid

produces a magnetic field

equivalent to that of a mag-

net, the same induction effects

will be produced by inserting

a helix conveying a current

into the long coil (Fig. 189)

as by thrusting in the magnet.

Let the circuit P include a

battery and a key, and the cir-

cuit S a galvanometer. The

former is called the primary,

and the latter the secondary.

If the current through P is kept constant while the

coil is moved about, then when P approaches S an E.M.F.

376 ELECTRICITY AND MAGNETISM.

is generated in S tending to send a current in the opposite

direction to that round P ; while if ^,is moved away from

S 9 the E.M.F. induced in S is in the direction of the cur-

rent round P. These electromotive forces in S act only

so long as P is moving. If P is kept fixed while S is

moved, the results are the same.

Next, let P be in a fixed position near S with the key

open. Then on closing the key in P the galvanometer

needle will be deflected. This deflection is not a perma-

nent one, but the needle oscillates and finally returns to

its initial position of rest, indicating the passage of a

sudden discharge through the galvanometer. The direc-

tion of this momentary current is opposite to that through

P. On opening the key another similar momentary cur-

rent passes through $, but in the same direction as through

P. Thus the starting or stopping of a current in P is

accompanied by the induction of another current in a

neighboring circuit jS.

The sudden increase of the current in P produces an

opposite current in $, and the sudden decrease of the

current in P produces a current through S in the same

direction as through P.

If while P remains inside of S, or coaxial with it, a bar

of soft iron is placed within it, there is an increase of mag-

netic flux through both P and $, and the E.M.F. generated

in 8 is in the same direction as that produced by closing

the key in P, moving P toward S, or increasing the cur-

rent through P. The withdrawal of the iron produces

the opposite effects to its insertion in the coil.

The law of the direction and magnitude of the E.M.F.

generated inductively by another current is the same as

that given in the last article. When the magnetic flux

changes, an E.M.F. is produced equal to the rate of change

ELECTHOMACXETIC INDUCTION. 377

in the magnetic flux passing through the circuit. The

positive direction of the E.M.F. and of the flux through

the circuit are related to each other as are the rotation and

translation of a right-handed screw.

332. Faraday's Ring. Faraday wound upon an iron

ring two coils of wire P and S (Fig. 190). When a bat-

tery and a key were included in the circuit P and a gal-

vanometer in AS', whenever the -=====- ***

circuit of P was closed or opened

a momentary current was pro-

duced in the closed circuit 8.

In this experiment the iron is

the medium through which the

induction between P and S takes

place. The current through P magnetizes the iron ring as

a closed magnetic circuit. The starting of the current in

the circuit P sends magnetic lines through S and produces

in it an inverse current; the stopping of the primary cur-

rent withdraws lines and produces a direct current through

the secondary. A larger deflection of the galvanometer

will be produced by the first closing of the primary cir-

cuit than by opening it, or by closing it a second time

unless the current be reversed. The reason is that the

ring forms a closed magnetic circuit, and its retentivity or

remanence is so great that only a small part of the lines

of force drop out when the magnetizing current ceases to

flow. But if the current through the primary be reversed,

all the lines will be taken out and will be put in again the

other way round. Hence, a large induction will take

place in S. A closed magnetic circuit is not well adapted,

therefore, to produce induction effects by merely opening

and closing the primary circuit.

378 ELECTRICITY AND MAGNETISM.

The relation between P and $ is a mutual one. If S is

made the primary, induced electromotive forces will be

generated in P as the secondary. The Faraday ring with

its two coils of wire is the type of the modern transformer

for alternating currents.

333. An Inductive System a Conservative System.

It will be instructive to look at a system of two circuits,

or of a circuit and a magnet, as a conservative system.

The action between the parts of the system always tends

to maintain unchanged the number of lines of force

threading through the circuits. Thus in Fig. 188 the

approach of the magnet to the coil increases the mag-

netic flux through it, and the induced current is in a

direction to send a counter flux through the coil so as to

keep'the magnetic induction through it constant. In Fig.

190 the primary current produces a magnetic flux in the

ring, and the current induced in the secondary produces

a magnetic flux in the other direction round the ring ; that

is, the induced current opposes the change in the flux.

After the magnetizing current has produced a steady

magnetic flux through the iron, the opening of the primary

induces a secondary current in the same direction round

the ring as the primary, and this tends to maintain the

flux of induction through the iron unchanged. The same

principle may be applied to two coils without iron. There

is no exception to the law that the induced currents are

always in a direction to conserve the magnetic flux through

the circuit in which the induction takes place. This law

means that the magnetic flux through a circuit does not

change abruptly a property of the magnetic field anal-

ogous to inertia in matter.

ELECTROMAGNETIC INDUCTION. 379

334. Lenz's Law (J. J. T., 432; Max., II., 177).-

When induced currents are produced by the motion of a

conductor in a magnetic field, the circuit is acted on by

a mechanical force. Lenz's law is that the direction of

this force always tends to stop the motion which gives rise

to it. Lenz's law is a particular case of the property of

conservation described in the last article. Every action

on an electromagnetic system, which involves a transfor-

mation of energy, sets up reactions tending to preserve

unchanged the state of the system.

Let E be the E.M.F. generated, I the induced current,

X the mechanical force parallel to the axis of x, and u the

velocity of the circuit in the direction x ; then the work

done on the circuit is Xu, and this is represented by the

electrical activity, or the product of the current and the

E.M.F. ; hence

Xu = EL

An example of Lenz's law is afforded by a coil revolving

in a magnetic field. The mechanical action of the field on

the current induced in the coil produces a couple tending

to stop the rotation. The oscillations of the coil of a

d'Arsonval galvanometer (292) subside quickly when the

coil is short-circuited. The galvanometer is then a mag-

neto-electric machine, and the currents induced in the

closed coil bring it to rest.

335. Arago's Rotations. When a magnet is suspended

horizontally over a copper disk and the disk is rotated,

induced currents are produced in it. These give rise to

a force opposing the rotation. Since the force between

the disk and the magnet is a mutual one, a couple acts on

the magnet and turns it, if it is free to move, in the same

380

ELECTRICITY AND MAGNETISM.

Fig. 191.

direction as the disk. Or if the magnet is spun round a

vertical axis and the disk is movable, it is dragged after

the magnet. These motions are called Arago's rotations ;

they were discovered by Arago, but

were first explained by Faraday. In-

duced currents flow in closed circuits

through the disk, and the action be-

tween them and the magnet tends to

stop the disk ; or if the magnet oscil-

lates, the induced currents damp its

motions. Thus in Fig. 191, if the

needle ab oscillates over the disk as it

moves in the direction of the arrows, a current is induced

on the M side which repels the needle, and one on the N

side attracting it ; or the current under it flows from the

centre toward the circumference if a is a N-seeking pole.

336. Other Examples of Lenz's Law. Let a

copper cube or cylinder be suspended between the

pointed poles of a powerful electromagnet (Fig.

192). The cube may be set rotating by twisting

the thread and releasing it. When the electro-

magnet is excited the cube is instantly brought

to rest; it begins to spin again as soon as the

current is cut off, and is again arrested on closing

the circuit. This resistance to motion in a mag-

netic field is sometimes called magnetic friction.

In another experiment a disk of copper is made

to rotate rapidly between the poles of an electro- R(T |92

magnet (Fig. 193). When the magnet is excited

the disk appears to meet with a sudden resistance. Fou-

cault found that if it is forced to rotate it is heated by

the induced currents flowing in it. These induced currents

ELECT RC MAGNETIC INDUCTION. 381

in masses of metal are often called Foucault currents.

There is a pair of eddy currents in the part of the disk

passing the poles; and these currents, as in Arago's rota-

tions, hold the disk back.

The drag due to eddy currents is

proportional to the speed and to the

square of the magnetic field ; for the

force is proportional to the field and

the current, and the current is pro-

portional to the field and the speed.

Wlien the field is constant the force

is therefore proportional to the speed

- . . Fig.. 193.

or rotation.

The principle is employed to produce damping in rota-

tory meters. A copper disk, attached to the shaft to which

is connected a dial train, rotates between the poles of fixed

magnets. The drag on the copper disk keeps the speed

proportional to the torque.

337. Coefficient of Mutual Induction. The preced-

ing examples of induction by currents all belong to the

class of mutual induction between two circuits. If we cal-

culate the E.M.F. generated by mutual induction, we shall

find that it contains a factor depending on the relative

position of the two circuits, the number of turns of wire

in each, and the reluctance of their common magnetic

circuit.

For definiteness take the case of Faraday's ring (Fig.

190). Let JVi and N* be the number of turns of wire on

P and S respectively. By Art. 322 the magnetic flux

through the helix P is

where / is the current in amperes and & is the reluctance

382 ELECTRICITY AND MAGNETISM.

of the iron ring (324). When the current is passed

through the ooil P, if all the magnetic lines run through

$, the total number of lines cut by the N turns in the

secondary will be

The quantity 47rZVjJV 2 /&, due to the passage of one C.G.S.

electromagnetic unit of current through the primary coil

P, is called the coefficient of mutual induction M. Usually

the magnetic flux through the secondary is somewhat less

than that through the primary. The coefficient M then

denotes the number of lines of force common to the two

coils due to one C.G.S. unit of current through the pri-

mary, multiplied by the number of turns of wire in the

secondary. The practical unit of mutual inductance is

the henry. It is equal to 10 C.G.S. units as calculated

above. Applied to mutual induction, it is the induction

in .the secondary when the E.M.F. induced is one volt,

while the inducing current in the primary varies at the

rate of one ampere per second.

Let dl/dt be the rate at which the current varies in the

primary. Then

E=-M-dI/dt.

E will be in volts if M is in henrys, / in amperes, and t in

seconds.

338. Self-induction. Joseph Henry discovered that

a current through a helix with parallel spirals of wire acts

inductively on its own circuit, producing what he called

the extra current. No spark is produced when such a

circuit is closed, but a bright spark breaks across the gap

when the circuit is opened. The effects are not very

marked unless the helix contains an iron core.

Even a single circuit is a conservative system as regards

ELECTROMAGNETIC INDUCTION. 383

the magnetic flux through it. When the current magnet-

izes the core, the effect is the same as if a magnet had

been plunged into the helix ; that is, the induced E.M.F.

is a counter E.M.F. tending to prevent the flux of mag-

netic induction through the circuit. The result is that

the current in such a circuit does not reach its maximum

value abruptly, but only after a short interval depending

on the value of the coefficient of self-induction, or simply

the inductance. When the circuit is opened the induced

E.M.F. is direct and tends to prolong the current, or to

ivsist the diminution in the magnetic flux.

Let there be N turns of wire on the coil. Then

The total cutting by the N spirals, if all the lines pass

through them, is

This expression divided by the interval required for the

change of flux to take place is the E.M.F. of self-induc-

tion. The lines cut when 10 amperes (one C.G.S. unit)

pass through the coil is the value of the inductance L, or

The value of the induced E.M.F. depends on the rate

of change of magnetic flux ; and since the self-induction

prevents the current from reaching its steady value at

once, during this variable state the rate of increase is not

uniform ; it is better therefore to express the inductance

differently. If di/dt is the rate at which the current

changes value, where i is its instantaneous value, then the

induced E.M.F. is

e = L- di/dt.

The unit of inductance, the henry, is the inductance in a

884 ELECTRICITY ANI> MAGNETISM.

circuit when the E.M.F. induced in this circuit is one volt,

while the inducing current varies at the rate of one ampere

per second.

339. Growth of Current in Inductive Circuits. -

When a constant E.M.F. is impressed on a circuit possess-

ing self-induction, the current does not attain its permanent

value instantly. Daring the variable stage its value is not

given by the simple application of Ohm's law ; the induc-

tance is another quality of the circuit, besides its resistance,

which determines the instantaneous value of the current.

This inductance is a property of a circuit in virtue of

which the passage of a current through it is accompanied

by the absorption of energy in the form of a magnetic

field. If no other work is done, part of the energy flow-

ing out from the source is converted into heat, and the

rest is stored in the .ether as the potential energy of the

magnetic field. This storage of energy goes on while

the current is rising from nothing to its steady value.

The energy so stored is equal to %LT\ where I is the

steady value of the current given by Ohm's law. 1 The

work represented by this energy is done by the current

against the E.M.F. of self-induction. A circuit has large

self-induction, therefore, when a relatively large quantity

of energy is stored in its field while the current is rising to

its final value.

The student should note that the inductance L is a con-

1 The induced E.M.F. is L -, and the work clone in the element of time dt is

dt

L idt t or Lidi. If this expression is integrated between the limits and /, the

dt

whole work done, or the energy stored in the magnetic field when the current

reaches its greatest value /, is

/

flAdi = \LI\

ELECTROMAGNETIC INDUCTION. 385

stant for any given form of circuit only when this circuit

consists of non-magnetic material and is surrounded by a

non-magnetic medium. If it contains iron, then L changes

with the value of the magnetic flux, because the reluctance

is dependent on the permeability, and the permeability

changes with the degree of magnetization of the iron.

34O. Helmholtz's Equation. The equation first given

by von Helmholtz expresses the value of the current in

an inductive circuit at any time t after a constant E.M.F.

has been applied to it. If E is the impressed E.M.F., R

the resistance of the circuit, i the value of the current at

any time t after closing the circuit, then the effective

E.M.F. required to produce the current i is Ri, and we

have the equation

E=Ri + l.

at

The impressed E.M.F. is equal to the sum of the induced

and effective electromotive forces. The solution l of this

equation, if L is constant, is

Divide the equation of electromotive forces by K and

or / - i = 7^ t - ( w here T = LIE) .

Whence . f-^.

Integrating, - = log (/ i) -j- constant (= log 7, for when t is zero i is

zero, and log /+ constant = 0).

Hence -1 = log (/- ) -log 1 = log L=* t

and =-' - **

386

ELECTRICITY AND MAGNETISM.

After t seconds, therefore, the current falls short of its

maximum value by a quantity le Rt/L , The quotient of

the inductance by the resistance L/ftis called the "time-

constant " of the circuit. It is the time required for the

current to reach 0.632 of its final value ; for when T

(or L/R) equals t, Et/L becomes unity. Then

-i -1

Substitute for e its value 2.7183 and the expression equals

0.632. If, for example, L were 2 henrys and R 1 ohm,

the time-constant would be two seconds ; or in two seconds

the current would rise to 0.632 of its final value. This

retardation in the growth of the current is due to the fact

that it has to create magnetic fields. Energy is stored up

in these fields, and the resistance to the work done on

them is manifested as an opposing electromotive force.

As this opposition dies away, the ef-

fective electromotive force increases,

and the current rises to the value

given by Ohm's law.

t

341. Energy stored in a Mag-

netic Field. Let M (Fig. 194) be

a large electromagnet, B a storage

battery, L an incandescent lamp of a

normal voltage equal to that of the

battery, and K a circuit-breaker.

The circuit is divided between the

electromagnet and the lamp ; and

since the former is of very low resistance in comparison

with the latter, when the current reaches its steady state

most of it will go through the coils of the magnet. The

lump is non-inductive; on closing the circuit, the self-

Fig. 194.

ELECTHOMAGXETIC INDUCTION. 387

induction of the electromagnet acts against the current,

like a large resistance, and sends most of it round through

the lamp. It accordingly lights up at first, but quickly

becomes dim, as the current grows to its steady value

through M.

On breaking the circuit and cutting off the battery

entirely, the lamp again flashes up brightly. The lamp

and the electromagnet are then together on a closed

circuit. The energy stored in the magnetic field, as a

strain in the ether about it, is converted into electric

energy, and a reverse current through the lamp lights it

for an instant. This example illustrates not only self-

induction, but the storage of energy in the ether about an

electromagnet.

Fig. 195.

While the iron core in a helix greatly increases the self-

induction, it would be a mistake to assume that the induc-

tion may not be very appreciable without it. A steady

direct current was sent through an electrodynamo meter E

and a coil of wire AB without an iron core (Fig. 195).

The current was such that the potential difference between

the terminals AB was 27 volts. The direct current was

then replaced by an alternating current of the same mean

square value as indicated by the electrod} 7 namometer (301).

The energy expended on the coil AB was then the same as

before, since none was absorbed by iron as heat by reason of

hysteresis. The energy lost was all converted into heat, and

was equal to PR or "27 I watts. But with an alternating

current the potential difference between the terminals of

388 ELECTRICITY AND MAGNETISM.

AB was 100 volts. A pressure of 100 volts was necessary

to force the same current through a coil that required

only 27 volts for a direct current. The difference must

be ascribed to the self-induction of the coil.

342. The Induction Coil. An induction coil is com-

monly employed to obtain transient electric flashes of high

E.M.F. in rapid succession. In modern terms it is a step-

up transformer, with open magnetic circuit. About an

iron core, consisting of a bundle of fine iron wires to avoid

the production of induced currents through the mass of

metal in the core, is wound a primary coil of comparative^

few turns of stout wire ; outside of this, and as carefully

insulated from it as possible, is the secondary composed of a

very large number of turns of fine wire. In Spottiswood's

great coil, which gave a 42J-inch spark, the secondary con-

tained 280 miles of wire wound in 340,000 turns.

The primary must be provided with a circuit-breaker if

the coil is to be used with direct currents. It is commonly

made automatic by a vibrating device actuated by the core

and similar to that of a vibrating electric bell.

In large coils the secondary is wound in flat spirals, and

these are slipped on over the primary and separated from

one another by insulating rings. The difference of poten-

tial between adjacent turns of wire is then not so large as

when the entire coil is wound in layers from end to end,

and it is easier to maintain the insulation. The ratio of

the transformation of the electromotive force is nearly the

same as the ratio between the number of turns of wire on

the primary and secondary.

343. Action of the Coil. While the quantity of elec-

tricity flowing through the closed secondary coil is the

ELECTROMAGNETIC INDUCTION.

389

same at u make v and " break " of the primary, still the

E.M.F. induced in breaking the circuit is so much higher

than in making it that the inductive effects are chiefly

those belonging to the former. This result is brought

about largely by the condenser.

On closing the primary circuit the counter E.M.F. due to

self-induction reduces the time rate of change of the cur-

ivnt on which the E.M.F. induced in the secondary

depends ; but on breaking the primary circuit the self-

J\ I\J\ A J\ AJ\ A J\A J\AJ\ AJ\ \r~ F

U V U \J U iTlJlrVI \r\J\nj\r\J\r\J

Fig. 196.

induction of the primary generates a direct E.M.F. which

tends to prolong the current and prevent its abrupt fall to

zero by sparking across the break. The condenser is added

for the purpose of suppressing this spark and aiding in

the abrupt descent of the primary current to zero. Fig.

196 represents the essential parts of an induction coil ;

PP is the primary and SS the secondary. The circuit is

automatically opened at the point b by the attraction of

the core on the small mass of soft iron F, which is mounted

on a spring.

The condenser C is joined to the points h and m on

opposite sides of the break. When the primary circuit

390 ELECTRICITY AND MAGNETISM.

is opened at b, the extra current flows into the condenser;

but as there is a complete discharge circuit for the con-

denser back through the primary in the opposite direction

to the battery current, the condenser discharge thus aids

in demagnetizing the core, or in rapidly reducing the mag-

netic flux by actually producing a negative one.

Lord Rayleigh has shown 1 that the best results are

secured when the capacity of the condenser is just great

enough to absorb a charge at a rate equal to the full deliv-

ery of the primary current during the time the break-

points are separating beyond the residual sparking-distance.

The condenser then causes an electric recoil in the current

and returns the charge stored up as an equal current in the

reverse direction through the primary, thus doubling the

change in the flux and doubling the induced E.M.F. and

current in the secondary; for the removal of all the lines

of force in one direction and the insertion of them in the

reverse direction amounts to a double diminution of them.

The conditions are then those described by the word

resonance. The current through the primary is rendered

oscillatory by means of the condenser. Instead of absorb-

ing the energy represented by the spark when no con-

denser is used, the condenser stores the energy momentarily

and then returns it to the primary, and by mutual induc-

tion to the secondary, to be expended there as a longer

spark or a greater current.

344. Discharges in Partial Vacua. Many remark-

able luminous effects, which are but imperfectly under-

stood, are produced when the discharges from an induction

coil pass through residual gases under a low pressure in

glass vessels. Such discharges will pass through the air

i Phil. Mag., 1870, p. 428; Fleming's Alter. Current Trans., Vol. I., p. 383.

ELECTRON. LCXETIC INDUCTION. 391

left in receivers exhausted by a good mechanical air-pump,

331. Induction by Cur-

rents (J. J. T., 374). Since

a current through a solenoid

produces a magnetic field

equivalent to that of a mag-

net, the same induction effects

will be produced by inserting

a helix conveying a current

into the long coil (Fig. 189)

as by thrusting in the magnet.

Let the circuit P include a

battery and a key, and the cir-

cuit S a galvanometer. The

former is called the primary,

and the latter the secondary.

If the current through P is kept constant while the

coil is moved about, then when P approaches S an E.M.F.

376 ELECTRICITY AND MAGNETISM.

is generated in S tending to send a current in the opposite

direction to that round P ; while if ^,is moved away from

S 9 the E.M.F. induced in S is in the direction of the cur-

rent round P. These electromotive forces in S act only

so long as P is moving. If P is kept fixed while S is

moved, the results are the same.

Next, let P be in a fixed position near S with the key

open. Then on closing the key in P the galvanometer

needle will be deflected. This deflection is not a perma-

nent one, but the needle oscillates and finally returns to

its initial position of rest, indicating the passage of a

sudden discharge through the galvanometer. The direc-

tion of this momentary current is opposite to that through

P. On opening the key another similar momentary cur-

rent passes through $, but in the same direction as through

P. Thus the starting or stopping of a current in P is

accompanied by the induction of another current in a

neighboring circuit jS.

The sudden increase of the current in P produces an

opposite current in $, and the sudden decrease of the

current in P produces a current through S in the same

direction as through P.

If while P remains inside of S, or coaxial with it, a bar

of soft iron is placed within it, there is an increase of mag-

netic flux through both P and $, and the E.M.F. generated

in 8 is in the same direction as that produced by closing

the key in P, moving P toward S, or increasing the cur-

rent through P. The withdrawal of the iron produces

the opposite effects to its insertion in the coil.

The law of the direction and magnitude of the E.M.F.

generated inductively by another current is the same as

that given in the last article. When the magnetic flux

changes, an E.M.F. is produced equal to the rate of change

ELECTHOMACXETIC INDUCTION. 377

in the magnetic flux passing through the circuit. The

positive direction of the E.M.F. and of the flux through

the circuit are related to each other as are the rotation and

translation of a right-handed screw.

332. Faraday's Ring. Faraday wound upon an iron

ring two coils of wire P and S (Fig. 190). When a bat-

tery and a key were included in the circuit P and a gal-

vanometer in AS', whenever the -=====- ***

circuit of P was closed or opened

a momentary current was pro-

duced in the closed circuit 8.

In this experiment the iron is

the medium through which the

induction between P and S takes

place. The current through P magnetizes the iron ring as

a closed magnetic circuit. The starting of the current in

the circuit P sends magnetic lines through S and produces

in it an inverse current; the stopping of the primary cur-

rent withdraws lines and produces a direct current through

the secondary. A larger deflection of the galvanometer

will be produced by the first closing of the primary cir-

cuit than by opening it, or by closing it a second time

unless the current be reversed. The reason is that the

ring forms a closed magnetic circuit, and its retentivity or

remanence is so great that only a small part of the lines

of force drop out when the magnetizing current ceases to

flow. But if the current through the primary be reversed,

all the lines will be taken out and will be put in again the

other way round. Hence, a large induction will take

place in S. A closed magnetic circuit is not well adapted,

therefore, to produce induction effects by merely opening

and closing the primary circuit.

378 ELECTRICITY AND MAGNETISM.

The relation between P and $ is a mutual one. If S is

made the primary, induced electromotive forces will be

generated in P as the secondary. The Faraday ring with

its two coils of wire is the type of the modern transformer

for alternating currents.

333. An Inductive System a Conservative System.

It will be instructive to look at a system of two circuits,

or of a circuit and a magnet, as a conservative system.

The action between the parts of the system always tends

to maintain unchanged the number of lines of force

threading through the circuits. Thus in Fig. 188 the

approach of the magnet to the coil increases the mag-

netic flux through it, and the induced current is in a

direction to send a counter flux through the coil so as to

keep'the magnetic induction through it constant. In Fig.

190 the primary current produces a magnetic flux in the

ring, and the current induced in the secondary produces

a magnetic flux in the other direction round the ring ; that

is, the induced current opposes the change in the flux.

After the magnetizing current has produced a steady

magnetic flux through the iron, the opening of the primary

induces a secondary current in the same direction round

the ring as the primary, and this tends to maintain the

flux of induction through the iron unchanged. The same

principle may be applied to two coils without iron. There

is no exception to the law that the induced currents are

always in a direction to conserve the magnetic flux through

the circuit in which the induction takes place. This law

means that the magnetic flux through a circuit does not

change abruptly a property of the magnetic field anal-

ogous to inertia in matter.

ELECTROMAGNETIC INDUCTION. 379

334. Lenz's Law (J. J. T., 432; Max., II., 177).-

When induced currents are produced by the motion of a

conductor in a magnetic field, the circuit is acted on by

a mechanical force. Lenz's law is that the direction of

this force always tends to stop the motion which gives rise

to it. Lenz's law is a particular case of the property of

conservation described in the last article. Every action

on an electromagnetic system, which involves a transfor-

mation of energy, sets up reactions tending to preserve

unchanged the state of the system.

Let E be the E.M.F. generated, I the induced current,

X the mechanical force parallel to the axis of x, and u the

velocity of the circuit in the direction x ; then the work

done on the circuit is Xu, and this is represented by the

electrical activity, or the product of the current and the

E.M.F. ; hence

Xu = EL

An example of Lenz's law is afforded by a coil revolving

in a magnetic field. The mechanical action of the field on

the current induced in the coil produces a couple tending

to stop the rotation. The oscillations of the coil of a

d'Arsonval galvanometer (292) subside quickly when the

coil is short-circuited. The galvanometer is then a mag-

neto-electric machine, and the currents induced in the

closed coil bring it to rest.

335. Arago's Rotations. When a magnet is suspended

horizontally over a copper disk and the disk is rotated,

induced currents are produced in it. These give rise to

a force opposing the rotation. Since the force between

the disk and the magnet is a mutual one, a couple acts on

the magnet and turns it, if it is free to move, in the same

380

ELECTRICITY AND MAGNETISM.

Fig. 191.

direction as the disk. Or if the magnet is spun round a

vertical axis and the disk is movable, it is dragged after

the magnet. These motions are called Arago's rotations ;

they were discovered by Arago, but

were first explained by Faraday. In-

duced currents flow in closed circuits

through the disk, and the action be-

tween them and the magnet tends to

stop the disk ; or if the magnet oscil-

lates, the induced currents damp its

motions. Thus in Fig. 191, if the

needle ab oscillates over the disk as it

moves in the direction of the arrows, a current is induced

on the M side which repels the needle, and one on the N

side attracting it ; or the current under it flows from the

centre toward the circumference if a is a N-seeking pole.

336. Other Examples of Lenz's Law. Let a

copper cube or cylinder be suspended between the

pointed poles of a powerful electromagnet (Fig.

192). The cube may be set rotating by twisting

the thread and releasing it. When the electro-

magnet is excited the cube is instantly brought

to rest; it begins to spin again as soon as the

current is cut off, and is again arrested on closing

the circuit. This resistance to motion in a mag-

netic field is sometimes called magnetic friction.

In another experiment a disk of copper is made

to rotate rapidly between the poles of an electro- R(T |92

magnet (Fig. 193). When the magnet is excited

the disk appears to meet with a sudden resistance. Fou-

cault found that if it is forced to rotate it is heated by

the induced currents flowing in it. These induced currents

ELECT RC MAGNETIC INDUCTION. 381

in masses of metal are often called Foucault currents.

There is a pair of eddy currents in the part of the disk

passing the poles; and these currents, as in Arago's rota-

tions, hold the disk back.

The drag due to eddy currents is

proportional to the speed and to the

square of the magnetic field ; for the

force is proportional to the field and

the current, and the current is pro-

portional to the field and the speed.

Wlien the field is constant the force

is therefore proportional to the speed

- . . Fig.. 193.

or rotation.

The principle is employed to produce damping in rota-

tory meters. A copper disk, attached to the shaft to which

is connected a dial train, rotates between the poles of fixed

magnets. The drag on the copper disk keeps the speed

proportional to the torque.

337. Coefficient of Mutual Induction. The preced-

ing examples of induction by currents all belong to the

class of mutual induction between two circuits. If we cal-

culate the E.M.F. generated by mutual induction, we shall

find that it contains a factor depending on the relative

position of the two circuits, the number of turns of wire

in each, and the reluctance of their common magnetic

circuit.

For definiteness take the case of Faraday's ring (Fig.

190). Let JVi and N* be the number of turns of wire on

P and S respectively. By Art. 322 the magnetic flux

through the helix P is

where / is the current in amperes and & is the reluctance

382 ELECTRICITY AND MAGNETISM.

of the iron ring (324). When the current is passed

through the ooil P, if all the magnetic lines run through

$, the total number of lines cut by the N turns in the

secondary will be

The quantity 47rZVjJV 2 /&, due to the passage of one C.G.S.

electromagnetic unit of current through the primary coil

P, is called the coefficient of mutual induction M. Usually

the magnetic flux through the secondary is somewhat less

than that through the primary. The coefficient M then

denotes the number of lines of force common to the two

coils due to one C.G.S. unit of current through the pri-

mary, multiplied by the number of turns of wire in the

secondary. The practical unit of mutual inductance is

the henry. It is equal to 10 C.G.S. units as calculated

above. Applied to mutual induction, it is the induction

in .the secondary when the E.M.F. induced is one volt,

while the inducing current in the primary varies at the

rate of one ampere per second.

Let dl/dt be the rate at which the current varies in the

primary. Then

E=-M-dI/dt.

E will be in volts if M is in henrys, / in amperes, and t in

seconds.

338. Self-induction. Joseph Henry discovered that

a current through a helix with parallel spirals of wire acts

inductively on its own circuit, producing what he called

the extra current. No spark is produced when such a

circuit is closed, but a bright spark breaks across the gap

when the circuit is opened. The effects are not very

marked unless the helix contains an iron core.

Even a single circuit is a conservative system as regards

ELECTROMAGNETIC INDUCTION. 383

the magnetic flux through it. When the current magnet-

izes the core, the effect is the same as if a magnet had

been plunged into the helix ; that is, the induced E.M.F.

is a counter E.M.F. tending to prevent the flux of mag-

netic induction through the circuit. The result is that

the current in such a circuit does not reach its maximum

value abruptly, but only after a short interval depending

on the value of the coefficient of self-induction, or simply

the inductance. When the circuit is opened the induced

E.M.F. is direct and tends to prolong the current, or to

ivsist the diminution in the magnetic flux.

Let there be N turns of wire on the coil. Then

The total cutting by the N spirals, if all the lines pass

through them, is

This expression divided by the interval required for the

change of flux to take place is the E.M.F. of self-induc-

tion. The lines cut when 10 amperes (one C.G.S. unit)

pass through the coil is the value of the inductance L, or

The value of the induced E.M.F. depends on the rate

of change of magnetic flux ; and since the self-induction

prevents the current from reaching its steady value at

once, during this variable state the rate of increase is not

uniform ; it is better therefore to express the inductance

differently. If di/dt is the rate at which the current

changes value, where i is its instantaneous value, then the

induced E.M.F. is

e = L- di/dt.

The unit of inductance, the henry, is the inductance in a

884 ELECTRICITY ANI> MAGNETISM.

circuit when the E.M.F. induced in this circuit is one volt,

while the inducing current varies at the rate of one ampere

per second.

339. Growth of Current in Inductive Circuits. -

When a constant E.M.F. is impressed on a circuit possess-

ing self-induction, the current does not attain its permanent

value instantly. Daring the variable stage its value is not

given by the simple application of Ohm's law ; the induc-

tance is another quality of the circuit, besides its resistance,

which determines the instantaneous value of the current.

This inductance is a property of a circuit in virtue of

which the passage of a current through it is accompanied

by the absorption of energy in the form of a magnetic

field. If no other work is done, part of the energy flow-

ing out from the source is converted into heat, and the

rest is stored in the .ether as the potential energy of the

magnetic field. This storage of energy goes on while

the current is rising from nothing to its steady value.

The energy so stored is equal to %LT\ where I is the

steady value of the current given by Ohm's law. 1 The

work represented by this energy is done by the current

against the E.M.F. of self-induction. A circuit has large

self-induction, therefore, when a relatively large quantity

of energy is stored in its field while the current is rising to

its final value.

The student should note that the inductance L is a con-

1 The induced E.M.F. is L -, and the work clone in the element of time dt is

dt

L idt t or Lidi. If this expression is integrated between the limits and /, the

dt

whole work done, or the energy stored in the magnetic field when the current

reaches its greatest value /, is

/

flAdi = \LI\

ELECTROMAGNETIC INDUCTION. 385

stant for any given form of circuit only when this circuit

consists of non-magnetic material and is surrounded by a

non-magnetic medium. If it contains iron, then L changes

with the value of the magnetic flux, because the reluctance

is dependent on the permeability, and the permeability

changes with the degree of magnetization of the iron.

34O. Helmholtz's Equation. The equation first given

by von Helmholtz expresses the value of the current in

an inductive circuit at any time t after a constant E.M.F.

has been applied to it. If E is the impressed E.M.F., R

the resistance of the circuit, i the value of the current at

any time t after closing the circuit, then the effective

E.M.F. required to produce the current i is Ri, and we

have the equation

E=Ri + l.

at

The impressed E.M.F. is equal to the sum of the induced

and effective electromotive forces. The solution l of this

equation, if L is constant, is

Divide the equation of electromotive forces by K and

or / - i = 7^ t - ( w here T = LIE) .

Whence . f-^.

Integrating, - = log (/ i) -j- constant (= log 7, for when t is zero i is

zero, and log /+ constant = 0).

Hence -1 = log (/- ) -log 1 = log L=* t

and =-' - **

386

ELECTRICITY AND MAGNETISM.

After t seconds, therefore, the current falls short of its

maximum value by a quantity le Rt/L , The quotient of

the inductance by the resistance L/ftis called the "time-

constant " of the circuit. It is the time required for the

current to reach 0.632 of its final value ; for when T

(or L/R) equals t, Et/L becomes unity. Then

-i -1

Substitute for e its value 2.7183 and the expression equals

0.632. If, for example, L were 2 henrys and R 1 ohm,

the time-constant would be two seconds ; or in two seconds

the current would rise to 0.632 of its final value. This

retardation in the growth of the current is due to the fact

that it has to create magnetic fields. Energy is stored up

in these fields, and the resistance to the work done on

them is manifested as an opposing electromotive force.

As this opposition dies away, the ef-

fective electromotive force increases,

and the current rises to the value

given by Ohm's law.

t

341. Energy stored in a Mag-

netic Field. Let M (Fig. 194) be

a large electromagnet, B a storage

battery, L an incandescent lamp of a

normal voltage equal to that of the

battery, and K a circuit-breaker.

The circuit is divided between the

electromagnet and the lamp ; and

since the former is of very low resistance in comparison

with the latter, when the current reaches its steady state

most of it will go through the coils of the magnet. The

lump is non-inductive; on closing the circuit, the self-

Fig. 194.

ELECTHOMAGXETIC INDUCTION. 387

induction of the electromagnet acts against the current,

like a large resistance, and sends most of it round through

the lamp. It accordingly lights up at first, but quickly

becomes dim, as the current grows to its steady value

through M.

On breaking the circuit and cutting off the battery

entirely, the lamp again flashes up brightly. The lamp

and the electromagnet are then together on a closed

circuit. The energy stored in the magnetic field, as a

strain in the ether about it, is converted into electric

energy, and a reverse current through the lamp lights it

for an instant. This example illustrates not only self-

induction, but the storage of energy in the ether about an

electromagnet.

Fig. 195.

While the iron core in a helix greatly increases the self-

induction, it would be a mistake to assume that the induc-

tion may not be very appreciable without it. A steady

direct current was sent through an electrodynamo meter E

and a coil of wire AB without an iron core (Fig. 195).

The current was such that the potential difference between

the terminals AB was 27 volts. The direct current was

then replaced by an alternating current of the same mean

square value as indicated by the electrod} 7 namometer (301).

The energy expended on the coil AB was then the same as

before, since none was absorbed by iron as heat by reason of

hysteresis. The energy lost was all converted into heat, and

was equal to PR or "27 I watts. But with an alternating

current the potential difference between the terminals of

388 ELECTRICITY AND MAGNETISM.

AB was 100 volts. A pressure of 100 volts was necessary

to force the same current through a coil that required

only 27 volts for a direct current. The difference must

be ascribed to the self-induction of the coil.

342. The Induction Coil. An induction coil is com-

monly employed to obtain transient electric flashes of high

E.M.F. in rapid succession. In modern terms it is a step-

up transformer, with open magnetic circuit. About an

iron core, consisting of a bundle of fine iron wires to avoid

the production of induced currents through the mass of

metal in the core, is wound a primary coil of comparative^

few turns of stout wire ; outside of this, and as carefully

insulated from it as possible, is the secondary composed of a

very large number of turns of fine wire. In Spottiswood's

great coil, which gave a 42J-inch spark, the secondary con-

tained 280 miles of wire wound in 340,000 turns.

The primary must be provided with a circuit-breaker if

the coil is to be used with direct currents. It is commonly

made automatic by a vibrating device actuated by the core

and similar to that of a vibrating electric bell.

In large coils the secondary is wound in flat spirals, and

these are slipped on over the primary and separated from

one another by insulating rings. The difference of poten-

tial between adjacent turns of wire is then not so large as

when the entire coil is wound in layers from end to end,

and it is easier to maintain the insulation. The ratio of

the transformation of the electromotive force is nearly the

same as the ratio between the number of turns of wire on

the primary and secondary.

343. Action of the Coil. While the quantity of elec-

tricity flowing through the closed secondary coil is the

ELECTROMAGNETIC INDUCTION.

389

same at u make v and " break " of the primary, still the

E.M.F. induced in breaking the circuit is so much higher

than in making it that the inductive effects are chiefly

those belonging to the former. This result is brought

about largely by the condenser.

On closing the primary circuit the counter E.M.F. due to

self-induction reduces the time rate of change of the cur-

ivnt on which the E.M.F. induced in the secondary

depends ; but on breaking the primary circuit the self-

J\ I\J\ A J\ AJ\ A J\A J\AJ\ AJ\ \r~ F

U V U \J U iTlJlrVI \r\J\nj\r\J\r\J

Fig. 196.

induction of the primary generates a direct E.M.F. which

tends to prolong the current and prevent its abrupt fall to

zero by sparking across the break. The condenser is added

for the purpose of suppressing this spark and aiding in

the abrupt descent of the primary current to zero. Fig.

196 represents the essential parts of an induction coil ;

PP is the primary and SS the secondary. The circuit is

automatically opened at the point b by the attraction of

the core on the small mass of soft iron F, which is mounted

on a spring.

The condenser C is joined to the points h and m on

opposite sides of the break. When the primary circuit

390 ELECTRICITY AND MAGNETISM.

is opened at b, the extra current flows into the condenser;

but as there is a complete discharge circuit for the con-

denser back through the primary in the opposite direction

to the battery current, the condenser discharge thus aids

in demagnetizing the core, or in rapidly reducing the mag-

netic flux by actually producing a negative one.

Lord Rayleigh has shown 1 that the best results are

secured when the capacity of the condenser is just great

enough to absorb a charge at a rate equal to the full deliv-

ery of the primary current during the time the break-

points are separating beyond the residual sparking-distance.

The condenser then causes an electric recoil in the current

and returns the charge stored up as an equal current in the

reverse direction through the primary, thus doubling the

change in the flux and doubling the induced E.M.F. and

current in the secondary; for the removal of all the lines

of force in one direction and the insertion of them in the

reverse direction amounts to a double diminution of them.

The conditions are then those described by the word

resonance. The current through the primary is rendered

oscillatory by means of the condenser. Instead of absorb-

ing the energy represented by the spark when no con-

denser is used, the condenser stores the energy momentarily

and then returns it to the primary, and by mutual induc-

tion to the secondary, to be expended there as a longer

spark or a greater current.

344. Discharges in Partial Vacua. Many remark-

able luminous effects, which are but imperfectly under-

stood, are produced when the discharges from an induction

coil pass through residual gases under a low pressure in

glass vessels. Such discharges will pass through the air

i Phil. Mag., 1870, p. 428; Fleming's Alter. Current Trans., Vol. I., p. 383.

ELECTRON. LCXETIC INDUCTION. 391

left in receivers exhausted by a good mechanical air-pump,

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