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315. Magnetic Induction and Magnetic Force. If

a long iron bar be placed in a uniform magnetic field par-

allel to the lines of force, the lines of force in the bar are

called lines of magnetic induction. They will be parallel to

the axis of the bar in the portions distant from the ends.

If a narrow crevasse perpendicular to the lines of induc-

tion be made in the bar, the flux of force in the crevasse

continues as a flux of induction in the iron. In the air the

flux may be considered indifferently as induction or force.

Lines of induction are consequently continuous throughout

the magnetic circuit. Near the ends of the bar the lines

of induction have not the same direction as the lines of

force of the uniform field. The poles induced in the bar

produce lines of force running counter to the lines of

induction in the iron. In some seolotropi-c substances the

axis of magnetization does not coincide with the lines of

force of the impressed magnetic field.

316. Curv.es of Magnetization. When an iron bar or

ring is subjected to a gradually increasing magnetizing

force, the flux of induction through it increases at first

slowly, then very rapidly, and after this very slowly. The

ratio between 6t> and (%" decreases toward a constant quan-

tity, which equals unity in the limit. If the magnetizing

force ftf be plotted horizontally and the induction ver-

tically, the resulting curve represents the successive stages

in the magnetization of iron. In Fig. 177 a is the curve

for mild steel, b for wrought iron, and c for cast. iron.

360

ELECTRICITY AXJ) MAGNETISM,

If the ratio of c& to (J6 were constant, the curve of mag-

netization would be a straight line. Since the curve is

concave toward the horizontal axis, except for very small

values of #?, /JL is not a constant, but decreases with

increase of induction.

Fig. 177.

Beyond the bend of the curve the iron is said to be

approaching saturation. For good soft iron this stage is

reached when equals from 16,000 to 18,000 lines per

square centimetre, with fc> from 50 to 200.

317. Hysteresis. If the magnetization is carried

through a complete cycle by increasing the magnetizing

force by successive steps from zero to some definite value,

decreasing it from that value by small steps through zero

to an equal value in the other direction, and then again

reducing it to zero and completing the cycle, the curve

EL ECTR OMA G NE TISM.

361

connecting *> and ,'V will not be the same with decreasing

values of <%' as with increasing ones (Fig. 178). The

induction tfi lags behind the magnetizing force. Thus,

when ilt-? is reduced to zero from its maximum positive

value, , ( (3 has the value Ob, and &6 must be given a nega-

r

^

,

^==

^

r

^^~

a

/

/

/

^

/

/

'V

f

f

/

1

/

c

1

II

f

/

/

1

/

'('

/

]

/

6000

^

/

/

^

&

-i *-

^

.

^

\

i

!

r ;

(

f. I?

3.

>

i

t

<

>

i

if>

3\s

live value equal to 0<: before the induction becomes zero.

So when <%^ returns from its maximum value in the other

direction to zero, the induction decreases only to the value

Oe. r \ his phenomenon of the lag of the induction behind

the magnetizing force Ewing has called magnetic hysteresis.

The result of plotting the corresponding values of & and

(V through a complete cycle is a curve enclosing an area,

and this area represents the heat lost per cubic centimetre

in the iron in carrying it through a single cycle. 1

1 Ewiug's Jfag. Ind. in Iron and other Metals,

362 ELECTRICITY AND MAGNETISM.

318. Remanence and Coercive Force. A cyclic mag-

netization curve, exhibiting hysteresis, serves among other

things to give definiteness to the terms remanence or reten-

tivity and coercive force. The residual value of 6B when

B8 is reduced to zero is Ob (Fig. 178). This value is the

remanence. It depends on the quality of the iron and the

limit to which 68 has been pushed. The figure relates to

a closed magnetic circuit consisting of a ring. The value

of EfS required to reduce this residual induction to zero,

viz., Oc, is the measure of the coercive force. Mechanical

vibration applied by external forces has the effect of dimin-

ishing residual magnetism, coercive force, and hysteresis.

If the iron in thin plates be carried rapidly through suc-

cessive cycles of magnetization by alternating currents, a

vibration will be set up in the plates unless they are rigidly

clamped together. Any vibration resulting from this

cause absorbs energy and increases the area of the hys-

teresis curve.

319. Law of the Magnetic Circuit. - The idea of a

magnetic circuit in a vague form is older than that of an

electric circuit, for it appears to go back to Euler in 1761.

Later Joule * asserted that the resistance to induction is

proportional to the' length of a closed magnetic circuit ;

and Faraday insisted that the lines of magnetic force are

always closed curves. He also made the very apt com-

parison of an electromagnet with open magnetic circuit to

a voltaic cell immersed in an electrolyte of poor conduc-

tivity. The low permeability of the air corresponds to the

low conductivity of the electrolyte.

Maxwell gave mathematical expression to Faraday's

1 Reprint of Sci. Papers, Vol. I., p. 34.

EL ECTROMA GLVT/> V. 363

ideas. He says: "In isotropic media the magnetic induc-

tion depends on the magnetic force in a manner which

exactly corresponds with that in which the electric current

depends on the electromotive force." l

But the first -definite expression of the law of the mag-

netic circuit in the form of an equation, like the equation

expressing Ohm's law, was given by Rowland in 1873 ; he

says expressly that it u is similar to the law of Ohm." :

In 1883 Bosanquet introduced the term "magnetomotive

force," corresponding to electromotive force in the electric

circuit. We may then write

Magnetomotive force

Magnetic flux = - .

Magnetic reluctance

Before attempting to write a more definite equation for

the magnetic circuit, it is necessary to introduce certain

general propositions which determine the magnetomotive

force.

320. Rotation of a Closed Circuit in a Magnetic

Field Conceive a current of / C.G.S. units flowing

through the half circle

abed (Fig. 1T9), and let

there be a unit magnetic

pole at the centre P.

Then the field produced

at P by the current urges the pole in a direction normal

to the plane of the ring. The circuit is urged by an

equal force in the opposite direction. Let be be unit

length of the curve. Then by Ampere's law of the recip-

rocal mechanical action between a magnet and a current,

which has been experimentally demonstrated, we have

1 Elec. and Mag., Vol. II., p. 51.

* Phil. May., Vol. XL VI., August, 1873.

364 ELECTRICITY AND MAGNETISM.

(286) the force at P due to the current I in the length be

of the conductor equal to II f. Hence, the work done in

rotating the arc be through 360 about the axis ad against

this force is fxbcx 2 < 7rr / . But this is f times the area of

that portion of the spherical surface generated by be during

the rotation. Hence, the entire work done against the

magnetic reaction between the whole semi-circumference

and the unit pole at the centre, for one revolution, is the

product of / and the surface of the sphere whose radius

is r, or

W=f X rrr- = J X 4-Trr 2 = irl.

r-

Since 4?r lines of force radiate from unit pole, and all of

these are cut by the semi-circle during one rotation around

the axis ad, it follows that the work done is the product

of the whole number of lines cut by the conductor and the

strength of the current flowing through it.

Suppose further that the rotation takes place in a period

t, that R is the resistance of the conductor between the

points a and c?, and E the potential difference between

the same points. Then from the law of conservation of

energy the whole electrical work done is the sum of the

energy spent in heat and the work done in rotating the

conductor in the magnetic field. We may therefore write

Elt = I-Rt + 4-7T I

as the energy equation.

Therefore, J=ZK+ ,

T/

V 4-7T

j

and j _ t

~R~

This is an expression for the current in the form of Ohm's

ELECTROMA GXETISM.

365

law. It shows that there is generated by the rotation an

E.M.F. equal to \irlt. But this fraction is the rate at

which the 4vr lines of force from the unit pole are cut

by the conductor. The E.M.F. generated by a conductor

cutting across lines of magnetic force is, therefore, the rate

at which they are cut.

These two propositions we have derived from Ampere's

law and the conservation of energy applied to a particular

case. While the method is not a perfectly general one,

the results are of general application. In estimating the

numl^er of lines cut or the rate of cutting them, attention

must be paid to the direction in which they are cut, and the

algebraic sum must be taken in all cases.

321. Force at a Point due to a Straight Current of

Indefinite Length (Th., 335). Let ab (Fig. 180) be a

portion of the straight conductor conveying a

current of strength /, and let P be a point at a

distance r from it. Then if we imagine a unit

pole at P, and if the conductor be carried round

it at the constant distance r, or the pole round

the conductor at the same distance, all the lines

of force from the pole will be cut once. Hence,

the work done will be 4?r/. If the field pro-

duced by the current at the point P is c\-?, the

work done is the product of the field intensity

and the distance '2-rrr, or 2-Trr $t>. Hence

a

Fig. ISO.

or 8g = 2J/r.

If the current is in amperes, then the force in dynes at the

point is

366 ELECTRICITY AND MAGNETISM.

322. Force within a Helix. Let AB (Fig. 181)

represent a section through the axis of a long helix, and

let unit pole be at the point P.

Let there be n turns of wire in

A B a length of one centimetre par-

allel to the axis of the helix, each

turn carrying a current 1. Then

if the unit pole be carried along

the axis from P to P', a distance of one centimetre, each

of the 4-7T lines of force from this pole will be cut by n

turns of wire. Hence, the whole number of lines cut will

be 4?m, and the work done ^nrnl. Since the distance

moved is one centimetre, the force is numerically equal

to the work, or

If the current is in amperes,

This is the value of the field at points distant from the

ends of the helix. At the ends only half as many lines

would be cut by a movement of one centimetre, and the

field is only 27rw7/10.

If the helix or solenoid forms a closed curve, so that

there are no ends to the helix, the field along the magnetic

axis will be everywhere the same.

323. Magnetomotive Force. The electromotive force

in a circuit is the work required to carry unit quantity of

electricity entirely round the circuit (186). So the mag-

netomotive force is the work done" in carrying a unit pole

once round the magnetic circuit. If L is the length of

the solenoid, the work done will be L times the strength

of field or kirnlL, if the field is uniform. If it be not

ELECTROMA GNET1SM.

367

uniform, then the magnetomotive force is the "line-inte-

gral " of the field intensity round the whole magnetic

circuit. Now nL is the entire number of turns of wire in

the solenoid. Let this be denoted by N; then the mag-

netomotive force is

if the current is expressed in amperes. NI is called the

amperr-turit*. The magnetomotive force in a long solenoid

is, therefore, 1.257 times the ampere-turns.

324. Reluctance (Th., 369). - - The magnetic reluc-

tance of a bar of iron is "its resistance to lines of force."

It may be calculated from its length, its sectional area, and

its permeability, just as the electrical

resistance of a conductor may be cal-

culated from its length, its cross-sec-

tion, and its specific conductivity.

Let the length of the bar be I cms.,

its section S square cms., and its

permeability /-i. Then its reluctance is

&t> =1/1*8.

Let us apply this formula to the case

of the closed circuit of an electro-

magnet (Fig. 182). It is made up of

two parts, the core and the armature. Let the lengths,

sections, and permeabilities be denoted by l v , and L , Si and

$>, and /A! and ^ respectively. Then the reluctance of

the whole circuit is

Fig. 182.

368

ELECTRICITY AND MAGNETISM.

325. Law of the Magnetic Circuit applied. When

the magnetic circuit is not closed, the lines of induction

must be forced across the air-gap be-

tween the faces of the iron parts of the

circuit. Suppose the armature removed

a short distance Z ;5 from the poles (Fig.

183). Then the length of the circuit

is thereby increased 2Z 3 cms., and ad-

ditional reluctance is introduced equal

to 2 ; />S'3, where S 3 is the cross-section

of the air traversed by the induction.

The permeability of the air is unity,

and does not appear in the expression.

We may therefore write for the flux

Fig. 183.

of magnetic induction

10

L 2*3

M*& *

where /is expressed in amperes.

While this expression is simple in theory it is rendered

difficult of application because /JL, unlike specific conduc-

tivity, is not a constant, but is a function of the magneti-

zation or induction in the iron. In applying the formula

to any particular magnetic circuit it is necessary to know

the curve of magnetization or the quality of iron used, and

to ascertain from it or from tables the values of JJL corre-

sponding to the degree of saturation which it is desired to

use. When this has been determined the formula gives

the number of ampere-turns of excitation required. For

open magnetic circuits an allowance must be made for

leakage of lines of force through the air between parts of

the magnet. This leakage requires excitation, but con-

tributes nothing to the purpose for which the magnet is

ELECTROMAGNETIC. 369

designed. The allowance for it must be estimated from

experience with the particular form of magnet employed.

The electromagnets of dynamos are designed by a process

similar to this.

326. Motion in Electromagnetic Systems. When-

ever any part of an electromagnetic system is movable, for

example, the armature of an electromagnet, the tendency

is always to move in a direction

to reduce the magnetic reluc-

tance and so to increase the mag-

netic flux. When the armature

approaches the poles, the air-gap V|

is shortened, the reluctance is

diminished, and more lines of

induction traverse the magnetic

circuit. So when any change

tends to occur in the configura-

tion of the parts of an electromagnetic system, it is always

such as to make the magnetic flux a maximum.

The same law may be applied to the dynamic action

between conductors conveying currents. Their relative

movements are in a direction to make the flux of magnetic

lines around them a maximum. Hence, two circuits tend

to move toward coincidence. Each is urged to a position

that will make the lines of force common to the two as

numerous as possible. Similar statements hold with respect

to a magnet and a circuit. When a bar magnet and a

helix come into a relative position where the middle point

of the former coincides with the mean plane of the latter,

the lines of force of the two are identical in direction

through the helix, and the position is one of stable

equilibrium (Fig. 184).

370 ELECTRICITY AND MAGNETISM.

327. Superficial Magnetization by Electric Dis-

charges. Steel needles or small steel rods may be mag-

netized by the passage of an electric discharge around

them, or even at right angles to their length. It has long

been known that lightning flashes sometimes magnetize

hard steel. If a Leyden jar be discharged through a strip

of tin foil across which lies a sewing-needle, the needle

Avill be magnetized by the discharge. Better results will

be obtained by surrounding the needle with an open helix

of rubber-covered wire and discharging through it. It was

with simple means like these that Joseph Henry discovered

the oscillatory character of the Ley den-jar discharge.

Anomalous results have sometimes been observed in the

relation of the poles to the direction of the discharge around

the needles or rods, the poles being turned in the direction

opposite to what the rule would lead one to expect. This

result is due to the oscillatory discharge combined with the

superficial character of the magnetism imparted. If small

steel rods, magnetized by electric discharges, be examined

by removing the external portions with acid, it will be

found that the magnetized part is confined to a thin shell,

the underlying parts remaining unmagnetized. If a second

discharge succeeds the first in the opposite direction, it will

reduce the external magnetism to zero if the magnetism of

half the shell is reversed. Two shells of equal magnetic

moment will then be superposed in opposite senses. If

therefore the reverse discharge have more than half the

magnetizing effect of the first, the resultant magnetism

will be apparently "anomalous;" but it is accounted for

by the direct and reverse discharges, and does not con-

stitute an exception to the law of magnetization.

Fig. 185 contains the curves obtained from two glass-

hard steel rods, 6 cms. long and 1.8 mms. in diameter, mag-

netized by ten successive discharges of a small Leyden jar

EL ECTEOMA GNETISM.

371

20

Fig. 185.

all in the same direction. 1 The relation of the two magnet-

izing coils was such that the first reverse oscillation was

more powerful with B than with

A. The data for these curves

were obtained by removing suc-

ve portions of the outside

with acid and measuring the 10

magnetic moments after each re-

moval. Moments are plotted as

ordinates. and decreasing weights 5

as abscissas. The moment of B

at first increases to a maximum,

and then decreases parallel to the

.i-ciirve. B had a thin external

shell magnetized in a sense op-

posite to that of the underlying portions. When this had

all been removed, the magnetic moment was a maximum.

PROBLEMS.

1. An iron bar 50 cms. long and 3 cms. in diameter was magnet-

ized to 15,780 lines per square centimetre, when fj. equaled 800.

Find the reluctance and the total induction through the bar.

2. A ring of soft iron 20 cms. in diameter and 3 sq. cms. sectional

area is wound uniformly with a magnetizing helix. Find the number

of ampere-turns required to magnetize to 13,640 lines per square

centimetre, with// equal to 2,200 ; what will be the total induction ?

3. A straight wire carries a current of 10 amperes ; find the

force in dynes on a pole of strength 20 at a distance of 5 cms.

from the wire.

4. A conductor is bent into a circle of 15 cms. radius ; find the

current through it which will deflect a short magnet at its centre 45

if the horizontal intensity of the earth's field is 0.25.

~>. An electric motor is wound with 128 wires on the outside of

the armature; the total magnetic flux through it is 1,250,000 lines ;

find the work done in ergs in one revolution when a current of 50

amperes flow r s through each wire; also find the power in kilowatts

when there are 960 revolutions per minute.

372 ELECTRICITY AND MAGNETISM.

CHAPTER XXIV.

ELECTROMAGNETIC INDUCTION.

328. Faraday's Discovery. It has been seen that

Oersted's discovery led speedily to the discovery of mag-

netization by electric currents, and to the mechanical action

between conductors conveying them. Faraday completed

this correlated group of electromagnetic phenomena by

discovering in 1831 the laws of the electromagnetic induc-

tion of currents, or the laws under which induced currents

are produced by means of other currents or by magnets. 1

These discoveries are of great interest, and it is of the

utmost importance that the student should familiarize

himself with the laws of induced currents, and should

connect them with the phenomena and laws developed in

the last three chapters.

Induced electromotive forces and currents are those

produced by the action of magnets and other currents.

Strictly only electromotive forces are induced; currents

flow as a consequence when the circuit in which the elec-

tromotive force is generated is closed. But the electromo-

tive force may still be induced whether the circuit is closed

or not.

All modern methods of producing large currents for

commercial purposes by dynamo machines, and all induc-

tion coils and alternate current transformers, are based on

electromagnetic induction.

1 Maxwell's Elec. and Mag., Vol. II., p. 163.

ELECTROMAGNETIC INDUCTION.

373

Fig. 186

329. Induction by Magnets. Let a coil of insulated

wire of many turns be connected to a sensitive galva-

nometer (Fig. 186), and thrust

into it the pole B of a bar mag-

net. The galvanometer will in-

dicate a transient current, which

will continue to flow only dur-

ing the motion of the magnet.

If the magnet be withdrawn

from the coil a transient in-

duced current will flow in the

reverse direction.

When the magnet enters the

coil it carries with it its lines of

force, and they are therefore cut

across by the spirals of the coil. Now it will be seen in

Art. 320 that the reasoning there employed is independent

of the electro-

motive force E.

Hence, this may

be made equal

to zero, and the

conclusion still

holds that the

E.M.F. gener-

ated by cutting

across lines of

force is equal to the rate at which they are cut by the con-

ductor. For most cases it is better to express the E.M.F.

induced as the rate of change of the magnetic induction

through an electric circuit.

If a coil of fine wire be wound around the armature of

a magnet (Fig. 187), then when the armature is in contact

Fig. 187.

3T4

ELECTRICITY AND MAGNETISM.

with the poles the flux of induction through the coils is

a maximum. When it is pulled away the magnetic flux

through the armature and the coil decreases rapidly, and a

direct E.M.F. is generated. This experiment illustrates

Faraday's method of producing electric currents by the

aid of magnetism.

330. Direction and Value of an Induced Electromo-

tive Force. The numerical value of an induced elec-

tromotive force in C.G.S. units may be expressed as

follows :

The E.M.F. induced

is equal to the rate of

change of the number of

^ lines of force threading

through the circuit.

^ If- d<& is the change

in the magnetic flux

through the circuit tak-

ing place in the short

time dt, the induced

E.M.F. is

E= -d^/dt.

The minus sign indicates that a direct E.M.F. corresponds

to a decrease in the flux of induction. It is to be noted

that number of lines of force, magnetic induction, and

magnetic flux are all equivalent expressions.

The direction of the induced E.M.F. Faraday deter-

mined by experiment, but it can be deduced from con-

siderations with which we are already familiar. Let the

magnet NS (Fig. 188) be thrust into the helix. Then if

an E.M.F. is generated and a current circulates through

the coil, the energy of the current must be derived from the

ELECT ROIL 1 KXETIC INDUCTION.

375

work done in moving the magnet. There must therefore

be a resistance opposing this movement ; this resistance is

due to the helix considered as a magnetic shell, and the cur-

rent must flow around it in a direction to make a Npole of

the side entered by the JVpole of the magnet. Its direction

is therefore against the motion of watch-hands as indicated

by the arrows. If the observer looks along the positive

direction of the lines of force, a current flowing with

watch-hands is said to be direct; if opposite to watch-

hands, it is indirect. Therefore we have the following law

relating to the direction of the induced E.M.F. :

An increase in the number of lines of force threading

throuf/h a helix produce* <in indirect E.M.F. , while a decrease

in tli>' niiinl'tr of lint'* />r./<t<'es a direct E.M.F.

The minus sign in the expression above corresponds to

a long iron bar be placed in a uniform magnetic field par-

allel to the lines of force, the lines of force in the bar are

called lines of magnetic induction. They will be parallel to

the axis of the bar in the portions distant from the ends.

If a narrow crevasse perpendicular to the lines of induc-

tion be made in the bar, the flux of force in the crevasse

continues as a flux of induction in the iron. In the air the

flux may be considered indifferently as induction or force.

Lines of induction are consequently continuous throughout

the magnetic circuit. Near the ends of the bar the lines

of induction have not the same direction as the lines of

force of the uniform field. The poles induced in the bar

produce lines of force running counter to the lines of

induction in the iron. In some seolotropi-c substances the

axis of magnetization does not coincide with the lines of

force of the impressed magnetic field.

316. Curv.es of Magnetization. When an iron bar or

ring is subjected to a gradually increasing magnetizing

force, the flux of induction through it increases at first

slowly, then very rapidly, and after this very slowly. The

ratio between 6t> and (%" decreases toward a constant quan-

tity, which equals unity in the limit. If the magnetizing

force ftf be plotted horizontally and the induction ver-

tically, the resulting curve represents the successive stages

in the magnetization of iron. In Fig. 177 a is the curve

for mild steel, b for wrought iron, and c for cast. iron.

360

ELECTRICITY AXJ) MAGNETISM,

If the ratio of c& to (J6 were constant, the curve of mag-

netization would be a straight line. Since the curve is

concave toward the horizontal axis, except for very small

values of #?, /JL is not a constant, but decreases with

increase of induction.

Fig. 177.

Beyond the bend of the curve the iron is said to be

approaching saturation. For good soft iron this stage is

reached when equals from 16,000 to 18,000 lines per

square centimetre, with fc> from 50 to 200.

317. Hysteresis. If the magnetization is carried

through a complete cycle by increasing the magnetizing

force by successive steps from zero to some definite value,

decreasing it from that value by small steps through zero

to an equal value in the other direction, and then again

reducing it to zero and completing the cycle, the curve

EL ECTR OMA G NE TISM.

361

connecting *> and ,'V will not be the same with decreasing

values of <%' as with increasing ones (Fig. 178). The

induction tfi lags behind the magnetizing force. Thus,

when ilt-? is reduced to zero from its maximum positive

value, , ( (3 has the value Ob, and &6 must be given a nega-

r

^

,

^==

^

r

^^~

a

/

/

/

^

/

/

'V

f

f

/

1

/

c

1

II

f

/

/

1

/

'('

/

]

/

6000

^

/

/

^

&

-i *-

^

.

^

\

i

!

r ;

(

f. I?

3.

>

i

t

<

>

i

if>

3\s

live value equal to 0<: before the induction becomes zero.

So when <%^ returns from its maximum value in the other

direction to zero, the induction decreases only to the value

Oe. r \ his phenomenon of the lag of the induction behind

the magnetizing force Ewing has called magnetic hysteresis.

The result of plotting the corresponding values of & and

(V through a complete cycle is a curve enclosing an area,

and this area represents the heat lost per cubic centimetre

in the iron in carrying it through a single cycle. 1

1 Ewiug's Jfag. Ind. in Iron and other Metals,

362 ELECTRICITY AND MAGNETISM.

318. Remanence and Coercive Force. A cyclic mag-

netization curve, exhibiting hysteresis, serves among other

things to give definiteness to the terms remanence or reten-

tivity and coercive force. The residual value of 6B when

B8 is reduced to zero is Ob (Fig. 178). This value is the

remanence. It depends on the quality of the iron and the

limit to which 68 has been pushed. The figure relates to

a closed magnetic circuit consisting of a ring. The value

of EfS required to reduce this residual induction to zero,

viz., Oc, is the measure of the coercive force. Mechanical

vibration applied by external forces has the effect of dimin-

ishing residual magnetism, coercive force, and hysteresis.

If the iron in thin plates be carried rapidly through suc-

cessive cycles of magnetization by alternating currents, a

vibration will be set up in the plates unless they are rigidly

clamped together. Any vibration resulting from this

cause absorbs energy and increases the area of the hys-

teresis curve.

319. Law of the Magnetic Circuit. - The idea of a

magnetic circuit in a vague form is older than that of an

electric circuit, for it appears to go back to Euler in 1761.

Later Joule * asserted that the resistance to induction is

proportional to the' length of a closed magnetic circuit ;

and Faraday insisted that the lines of magnetic force are

always closed curves. He also made the very apt com-

parison of an electromagnet with open magnetic circuit to

a voltaic cell immersed in an electrolyte of poor conduc-

tivity. The low permeability of the air corresponds to the

low conductivity of the electrolyte.

Maxwell gave mathematical expression to Faraday's

1 Reprint of Sci. Papers, Vol. I., p. 34.

EL ECTROMA GLVT/> V. 363

ideas. He says: "In isotropic media the magnetic induc-

tion depends on the magnetic force in a manner which

exactly corresponds with that in which the electric current

depends on the electromotive force." l

But the first -definite expression of the law of the mag-

netic circuit in the form of an equation, like the equation

expressing Ohm's law, was given by Rowland in 1873 ; he

says expressly that it u is similar to the law of Ohm." :

In 1883 Bosanquet introduced the term "magnetomotive

force," corresponding to electromotive force in the electric

circuit. We may then write

Magnetomotive force

Magnetic flux = - .

Magnetic reluctance

Before attempting to write a more definite equation for

the magnetic circuit, it is necessary to introduce certain

general propositions which determine the magnetomotive

force.

320. Rotation of a Closed Circuit in a Magnetic

Field Conceive a current of / C.G.S. units flowing

through the half circle

abed (Fig. 1T9), and let

there be a unit magnetic

pole at the centre P.

Then the field produced

at P by the current urges the pole in a direction normal

to the plane of the ring. The circuit is urged by an

equal force in the opposite direction. Let be be unit

length of the curve. Then by Ampere's law of the recip-

rocal mechanical action between a magnet and a current,

which has been experimentally demonstrated, we have

1 Elec. and Mag., Vol. II., p. 51.

* Phil. May., Vol. XL VI., August, 1873.

364 ELECTRICITY AND MAGNETISM.

(286) the force at P due to the current I in the length be

of the conductor equal to II f. Hence, the work done in

rotating the arc be through 360 about the axis ad against

this force is fxbcx 2 < 7rr / . But this is f times the area of

that portion of the spherical surface generated by be during

the rotation. Hence, the entire work done against the

magnetic reaction between the whole semi-circumference

and the unit pole at the centre, for one revolution, is the

product of / and the surface of the sphere whose radius

is r, or

W=f X rrr- = J X 4-Trr 2 = irl.

r-

Since 4?r lines of force radiate from unit pole, and all of

these are cut by the semi-circle during one rotation around

the axis ad, it follows that the work done is the product

of the whole number of lines cut by the conductor and the

strength of the current flowing through it.

Suppose further that the rotation takes place in a period

t, that R is the resistance of the conductor between the

points a and c?, and E the potential difference between

the same points. Then from the law of conservation of

energy the whole electrical work done is the sum of the

energy spent in heat and the work done in rotating the

conductor in the magnetic field. We may therefore write

Elt = I-Rt + 4-7T I

as the energy equation.

Therefore, J=ZK+ ,

T/

V 4-7T

j

and j _ t

~R~

This is an expression for the current in the form of Ohm's

ELECTROMA GXETISM.

365

law. It shows that there is generated by the rotation an

E.M.F. equal to \irlt. But this fraction is the rate at

which the 4vr lines of force from the unit pole are cut

by the conductor. The E.M.F. generated by a conductor

cutting across lines of magnetic force is, therefore, the rate

at which they are cut.

These two propositions we have derived from Ampere's

law and the conservation of energy applied to a particular

case. While the method is not a perfectly general one,

the results are of general application. In estimating the

numl^er of lines cut or the rate of cutting them, attention

must be paid to the direction in which they are cut, and the

algebraic sum must be taken in all cases.

321. Force at a Point due to a Straight Current of

Indefinite Length (Th., 335). Let ab (Fig. 180) be a

portion of the straight conductor conveying a

current of strength /, and let P be a point at a

distance r from it. Then if we imagine a unit

pole at P, and if the conductor be carried round

it at the constant distance r, or the pole round

the conductor at the same distance, all the lines

of force from the pole will be cut once. Hence,

the work done will be 4?r/. If the field pro-

duced by the current at the point P is c\-?, the

work done is the product of the field intensity

and the distance '2-rrr, or 2-Trr $t>. Hence

a

Fig. ISO.

or 8g = 2J/r.

If the current is in amperes, then the force in dynes at the

point is

366 ELECTRICITY AND MAGNETISM.

322. Force within a Helix. Let AB (Fig. 181)

represent a section through the axis of a long helix, and

let unit pole be at the point P.

Let there be n turns of wire in

A B a length of one centimetre par-

allel to the axis of the helix, each

turn carrying a current 1. Then

if the unit pole be carried along

the axis from P to P', a distance of one centimetre, each

of the 4-7T lines of force from this pole will be cut by n

turns of wire. Hence, the whole number of lines cut will

be 4?m, and the work done ^nrnl. Since the distance

moved is one centimetre, the force is numerically equal

to the work, or

If the current is in amperes,

This is the value of the field at points distant from the

ends of the helix. At the ends only half as many lines

would be cut by a movement of one centimetre, and the

field is only 27rw7/10.

If the helix or solenoid forms a closed curve, so that

there are no ends to the helix, the field along the magnetic

axis will be everywhere the same.

323. Magnetomotive Force. The electromotive force

in a circuit is the work required to carry unit quantity of

electricity entirely round the circuit (186). So the mag-

netomotive force is the work done" in carrying a unit pole

once round the magnetic circuit. If L is the length of

the solenoid, the work done will be L times the strength

of field or kirnlL, if the field is uniform. If it be not

ELECTROMA GNET1SM.

367

uniform, then the magnetomotive force is the "line-inte-

gral " of the field intensity round the whole magnetic

circuit. Now nL is the entire number of turns of wire in

the solenoid. Let this be denoted by N; then the mag-

netomotive force is

if the current is expressed in amperes. NI is called the

amperr-turit*. The magnetomotive force in a long solenoid

is, therefore, 1.257 times the ampere-turns.

324. Reluctance (Th., 369). - - The magnetic reluc-

tance of a bar of iron is "its resistance to lines of force."

It may be calculated from its length, its sectional area, and

its permeability, just as the electrical

resistance of a conductor may be cal-

culated from its length, its cross-sec-

tion, and its specific conductivity.

Let the length of the bar be I cms.,

its section S square cms., and its

permeability /-i. Then its reluctance is

&t> =1/1*8.

Let us apply this formula to the case

of the closed circuit of an electro-

magnet (Fig. 182). It is made up of

two parts, the core and the armature. Let the lengths,

sections, and permeabilities be denoted by l v , and L , Si and

$>, and /A! and ^ respectively. Then the reluctance of

the whole circuit is

Fig. 182.

368

ELECTRICITY AND MAGNETISM.

325. Law of the Magnetic Circuit applied. When

the magnetic circuit is not closed, the lines of induction

must be forced across the air-gap be-

tween the faces of the iron parts of the

circuit. Suppose the armature removed

a short distance Z ;5 from the poles (Fig.

183). Then the length of the circuit

is thereby increased 2Z 3 cms., and ad-

ditional reluctance is introduced equal

to 2 ; />S'3, where S 3 is the cross-section

of the air traversed by the induction.

The permeability of the air is unity,

and does not appear in the expression.

We may therefore write for the flux

Fig. 183.

of magnetic induction

10

L 2*3

M*& *

where /is expressed in amperes.

While this expression is simple in theory it is rendered

difficult of application because /JL, unlike specific conduc-

tivity, is not a constant, but is a function of the magneti-

zation or induction in the iron. In applying the formula

to any particular magnetic circuit it is necessary to know

the curve of magnetization or the quality of iron used, and

to ascertain from it or from tables the values of JJL corre-

sponding to the degree of saturation which it is desired to

use. When this has been determined the formula gives

the number of ampere-turns of excitation required. For

open magnetic circuits an allowance must be made for

leakage of lines of force through the air between parts of

the magnet. This leakage requires excitation, but con-

tributes nothing to the purpose for which the magnet is

ELECTROMAGNETIC. 369

designed. The allowance for it must be estimated from

experience with the particular form of magnet employed.

The electromagnets of dynamos are designed by a process

similar to this.

326. Motion in Electromagnetic Systems. When-

ever any part of an electromagnetic system is movable, for

example, the armature of an electromagnet, the tendency

is always to move in a direction

to reduce the magnetic reluc-

tance and so to increase the mag-

netic flux. When the armature

approaches the poles, the air-gap V|

is shortened, the reluctance is

diminished, and more lines of

induction traverse the magnetic

circuit. So when any change

tends to occur in the configura-

tion of the parts of an electromagnetic system, it is always

such as to make the magnetic flux a maximum.

The same law may be applied to the dynamic action

between conductors conveying currents. Their relative

movements are in a direction to make the flux of magnetic

lines around them a maximum. Hence, two circuits tend

to move toward coincidence. Each is urged to a position

that will make the lines of force common to the two as

numerous as possible. Similar statements hold with respect

to a magnet and a circuit. When a bar magnet and a

helix come into a relative position where the middle point

of the former coincides with the mean plane of the latter,

the lines of force of the two are identical in direction

through the helix, and the position is one of stable

equilibrium (Fig. 184).

370 ELECTRICITY AND MAGNETISM.

327. Superficial Magnetization by Electric Dis-

charges. Steel needles or small steel rods may be mag-

netized by the passage of an electric discharge around

them, or even at right angles to their length. It has long

been known that lightning flashes sometimes magnetize

hard steel. If a Leyden jar be discharged through a strip

of tin foil across which lies a sewing-needle, the needle

Avill be magnetized by the discharge. Better results will

be obtained by surrounding the needle with an open helix

of rubber-covered wire and discharging through it. It was

with simple means like these that Joseph Henry discovered

the oscillatory character of the Ley den-jar discharge.

Anomalous results have sometimes been observed in the

relation of the poles to the direction of the discharge around

the needles or rods, the poles being turned in the direction

opposite to what the rule would lead one to expect. This

result is due to the oscillatory discharge combined with the

superficial character of the magnetism imparted. If small

steel rods, magnetized by electric discharges, be examined

by removing the external portions with acid, it will be

found that the magnetized part is confined to a thin shell,

the underlying parts remaining unmagnetized. If a second

discharge succeeds the first in the opposite direction, it will

reduce the external magnetism to zero if the magnetism of

half the shell is reversed. Two shells of equal magnetic

moment will then be superposed in opposite senses. If

therefore the reverse discharge have more than half the

magnetizing effect of the first, the resultant magnetism

will be apparently "anomalous;" but it is accounted for

by the direct and reverse discharges, and does not con-

stitute an exception to the law of magnetization.

Fig. 185 contains the curves obtained from two glass-

hard steel rods, 6 cms. long and 1.8 mms. in diameter, mag-

netized by ten successive discharges of a small Leyden jar

EL ECTEOMA GNETISM.

371

20

Fig. 185.

all in the same direction. 1 The relation of the two magnet-

izing coils was such that the first reverse oscillation was

more powerful with B than with

A. The data for these curves

were obtained by removing suc-

ve portions of the outside

with acid and measuring the 10

magnetic moments after each re-

moval. Moments are plotted as

ordinates. and decreasing weights 5

as abscissas. The moment of B

at first increases to a maximum,

and then decreases parallel to the

.i-ciirve. B had a thin external

shell magnetized in a sense op-

posite to that of the underlying portions. When this had

all been removed, the magnetic moment was a maximum.

PROBLEMS.

1. An iron bar 50 cms. long and 3 cms. in diameter was magnet-

ized to 15,780 lines per square centimetre, when fj. equaled 800.

Find the reluctance and the total induction through the bar.

2. A ring of soft iron 20 cms. in diameter and 3 sq. cms. sectional

area is wound uniformly with a magnetizing helix. Find the number

of ampere-turns required to magnetize to 13,640 lines per square

centimetre, with// equal to 2,200 ; what will be the total induction ?

3. A straight wire carries a current of 10 amperes ; find the

force in dynes on a pole of strength 20 at a distance of 5 cms.

from the wire.

4. A conductor is bent into a circle of 15 cms. radius ; find the

current through it which will deflect a short magnet at its centre 45

if the horizontal intensity of the earth's field is 0.25.

~>. An electric motor is wound with 128 wires on the outside of

the armature; the total magnetic flux through it is 1,250,000 lines ;

find the work done in ergs in one revolution when a current of 50

amperes flow r s through each wire; also find the power in kilowatts

when there are 960 revolutions per minute.

372 ELECTRICITY AND MAGNETISM.

CHAPTER XXIV.

ELECTROMAGNETIC INDUCTION.

328. Faraday's Discovery. It has been seen that

Oersted's discovery led speedily to the discovery of mag-

netization by electric currents, and to the mechanical action

between conductors conveying them. Faraday completed

this correlated group of electromagnetic phenomena by

discovering in 1831 the laws of the electromagnetic induc-

tion of currents, or the laws under which induced currents

are produced by means of other currents or by magnets. 1

These discoveries are of great interest, and it is of the

utmost importance that the student should familiarize

himself with the laws of induced currents, and should

connect them with the phenomena and laws developed in

the last three chapters.

Induced electromotive forces and currents are those

produced by the action of magnets and other currents.

Strictly only electromotive forces are induced; currents

flow as a consequence when the circuit in which the elec-

tromotive force is generated is closed. But the electromo-

tive force may still be induced whether the circuit is closed

or not.

All modern methods of producing large currents for

commercial purposes by dynamo machines, and all induc-

tion coils and alternate current transformers, are based on

electromagnetic induction.

1 Maxwell's Elec. and Mag., Vol. II., p. 163.

ELECTROMAGNETIC INDUCTION.

373

Fig. 186

329. Induction by Magnets. Let a coil of insulated

wire of many turns be connected to a sensitive galva-

nometer (Fig. 186), and thrust

into it the pole B of a bar mag-

net. The galvanometer will in-

dicate a transient current, which

will continue to flow only dur-

ing the motion of the magnet.

If the magnet be withdrawn

from the coil a transient in-

duced current will flow in the

reverse direction.

When the magnet enters the

coil it carries with it its lines of

force, and they are therefore cut

across by the spirals of the coil. Now it will be seen in

Art. 320 that the reasoning there employed is independent

of the electro-

motive force E.

Hence, this may

be made equal

to zero, and the

conclusion still

holds that the

E.M.F. gener-

ated by cutting

across lines of

force is equal to the rate at which they are cut by the con-

ductor. For most cases it is better to express the E.M.F.

induced as the rate of change of the magnetic induction

through an electric circuit.

If a coil of fine wire be wound around the armature of

a magnet (Fig. 187), then when the armature is in contact

Fig. 187.

3T4

ELECTRICITY AND MAGNETISM.

with the poles the flux of induction through the coils is

a maximum. When it is pulled away the magnetic flux

through the armature and the coil decreases rapidly, and a

direct E.M.F. is generated. This experiment illustrates

Faraday's method of producing electric currents by the

aid of magnetism.

330. Direction and Value of an Induced Electromo-

tive Force. The numerical value of an induced elec-

tromotive force in C.G.S. units may be expressed as

follows :

The E.M.F. induced

is equal to the rate of

change of the number of

^ lines of force threading

through the circuit.

^ If- d<& is the change

in the magnetic flux

through the circuit tak-

ing place in the short

time dt, the induced

E.M.F. is

E= -d^/dt.

The minus sign indicates that a direct E.M.F. corresponds

to a decrease in the flux of induction. It is to be noted

that number of lines of force, magnetic induction, and

magnetic flux are all equivalent expressions.

The direction of the induced E.M.F. Faraday deter-

mined by experiment, but it can be deduced from con-

siderations with which we are already familiar. Let the

magnet NS (Fig. 188) be thrust into the helix. Then if

an E.M.F. is generated and a current circulates through

the coil, the energy of the current must be derived from the

ELECT ROIL 1 KXETIC INDUCTION.

375

work done in moving the magnet. There must therefore

be a resistance opposing this movement ; this resistance is

due to the helix considered as a magnetic shell, and the cur-

rent must flow around it in a direction to make a Npole of

the side entered by the JVpole of the magnet. Its direction

is therefore against the motion of watch-hands as indicated

by the arrows. If the observer looks along the positive

direction of the lines of force, a current flowing with

watch-hands is said to be direct; if opposite to watch-

hands, it is indirect. Therefore we have the following law

relating to the direction of the induced E.M.F. :

An increase in the number of lines of force threading

throuf/h a helix produce* <in indirect E.M.F. , while a decrease

in tli>' niiinl'tr of lint'* />r./<t<'es a direct E.M.F.

The minus sign in the expression above corresponds to

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