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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.

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

/

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

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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