Henry S. (Henry Smith) Carhart.
Physics for university students (Volume 2) online
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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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