IN MEMORIAM
FLOR1AN CAJORI
v-<y
N OT ES
ELECTRICITY AND MAGNETISM.
DESIGNED AS A COMPANION TO SILVANUS P. THOMPSON'S
ELEMENTARY LESSONS.
BY
J. B. MURDCCK,
LIEUTENANT U. S. NAVY.
MACMILLAN & CO,
1891.
COPYRIGHT, 1883,
BY MACMILLAN & CO.
/1$
PREFACE.
THE design of this small volume of notes is to supple-
ment the instruction given in Prof. Thompson's admirable
Elementary Lessons by such explanations and additional
instruction as an experience with two classes of cadets in
this institution has shown to be necessary. In general,
notes have been made on separate paragraphs in the les-
sons, but it was thought best to treat many subjects inde-
pendently in order to present them more connectedly, and
it is hoped that the notes may thus be of some service by
themselves. The endeavor has been made to trace the
theory of the dynamo machine and of electric motors from
the primary laws of electro-magnetic induction, and de-
scriptions of several of the more important dynamo ma-
chines have been added, chiefly to illustrate the various
applications of the general principle underlying all. As
Prof. Thompson's treatise has come to be largely used in
colleges and high schools, demonstrations by the aid of
calculus, most of which have been used in the course in
this institution, have been given to replace the geometrical
proofs of the lessons if desired.
J. B. MURDOCK.
U. S. NAVAL ACADEMY,
Annapolis, Md., July 10, 1883.
NOTE. Reference have been made in the text to the figures in the
Elementary Lessons, as well as to those in this volume. The latter range
from i to 38, and higher numbers are to be understood as referring to
those in the Elementary Lessons.
M304876
INDEX TO NOTES.
I. GALVANOMETERS.
NOTE PAGE
1. Tangent galvanometer I
2. Sine galvanometer 3
3. Mirror galvanometer 4
4. Differential galvanometer 4
5. Ballistic galvanometer 6
II. THEORY OF POTENTIAL.
6. Illustration of meaning of potential 8
7. Difference between work and potential 10
8. Positive and negative work ri
9. Positive electricity always flows from a high to a low
potential _i I
10. Units of potential and work 12
11. Formula for electrostatic potential 13
12. Zero potential 14
13. Difference of potential 14
14. Electric force 15
15. Law of inverse squares 15
16. Capacity . , 16
17. Unit of capacity 17
18. Electric force exerted by a charged plate 17
19. Dimensions of units 18
20. Attracted disc electrometer 20
21. Absolute electrometer. . 20
vi INDEX TO NOTES.
III. THEORY OF MAGNETIC POTENTIAL.
NOTE PAGE
22. Magnetic field 23
23. Method of mapping a field by lines of force 24
24. Equipotential surfaces 26
25. Lines of force due to a single pole 26
26. Lines of force due to a current 28
27. Magnetic potential 29
28. Tubes of force. ... 30
29. Intensity of magnetization 31
30. Solenoidal magnets 32
31. Potential due to a magnetic shell 33
32. Equipotential surfaces and lines of force of a magnetic
shell 35
33. Work done in moving a pole near a shell 36
34. Equivalent magnetic shells 37
35. Potential due to a closed voltaic circuit 39
36. Work done in moving a circuit near a pole 40
37. Intensity of field due to a voltaic circuit 42
38. Position of equilibrium of a circuit and magnet 43
39. Mutual potential of two circuits 44
40. Conversion of units. Tables of absolute and practical
units 45
41. Determination of the horizontal component of the
earth's magnetism 48
IV. MEASUREMENTS AND FORMULAS.
42. Solenoids. Ampere's theory of magnetism 54
43. Best arrangement of cells 5
44. Long and short coil galvanometers 57
45. Divided circuits 5&
46. Shunts 59
47. KirchhofFs laws 61
48. Fall of potential 62
49. Wheatstone's bridge 64
50. Proof of theory of bridge by Kirchhoff's laws 65
INDEX TO NOTES vii
NOTE PAGE
51. Measurement of electromotive force 66
52. Measurement of internal resistance 68
53. Measurement of the capacity of a condenser 70
54. Determination of the ohm 71
55. Practical electromagnetic units of heat and of work. . . 72
V. ELECTRIC LIGHTS.
56. The voltaic arc 75
57. Arc lamps 77
58. Incandescent lamps 79
VI. ELECTRO-MAGNETIC INDUCTION.
59. Induction currents produced by currents 82
60. Determination of the induced electromotive force 84
61. Practical rule for the direction of induced currents. ... 86
62. Lenz's law 87
63. Mutual induction of two circuits 88
64. Self induction 88
65. Helmholtz's equations 89
66. Induction coil 90
VII. DYNAMO MACHINES.
67. General principles. ... 93
68. Electromotive force 97
69. Efficiency IO2
70. Electromotive force in circuit 105
71. Siemens' machine 106
72. Gramme machine 107
73. Brush machine 109
74. Edison machine 114
75. Alternate current machines 1 15
VIII. ELECTRIC MOTORS.
76. General principles II?
77. Electric transmission of power to a distance 118
vin INDEX TO NOTES.
NOTE PAGE
78. Theory of electric motors ng
79. Modifications of theory in practice 125
80. Peltier effect 127
81. Secondary batteries 128
IX. TELEGRAPHY AND TELEPHONY.
82. The Morse alphabet 130
83. American system of telegraphy 131
84. Faults. 134
85. Simultaneous transmission 134
86. Blake's transmitter 137
87. Telephone exchanges 138
NOTES
ON
ELECTRICITY AND MAGNETISM.
I. GALVANOMETERS.
(Thompson's Electricity, pages 163-171.)
i. Tangent Galvanometer ( 199).
To derive the formula in g 200.
fleeted by the current to an
angle with the meridian and
be in equilibrium. The sum
of the moments of the forces
acting on it, taken around its
axis, is then zero. The moment
tending to bring it back into
the meridian is the force mH
into the arm BC, or mlHsm B,
m being the strength of mag-
net pole and / the distance be-
tween the poles. The moment
of the deflecting couple is fm
x DE, or fml cos 0, f being
the deflecting force of the cur-
rent. If now the magnet be
so small that in all positions its
poles may be considered to be
at the centre of the coil/ =
Let the magnet be de-
H
^ @ 195). If the coil has n turns, each exerting this
NOTES ON
force on a pole at the centre, f = - . Substituting
this value of / and equating the moments of the forces
around A
27tnC
mtffsm 0= ml cos Q
.-. C = /f tan 0.
This formula shows not only that currents vary as the
tangent of the angle of deflection, but gives the value of
the currents in the absolute units of current defined in
g 196. The practical unit of current, the ampere, being
only iV of the absolute unit, the result must be multi-
plied by ten to obtain the current in amperes from any
observed deflection. The fraction - depending on the
construction of the instrument, is called the " reduction
factor " of the galvanometer. It is generally furnished by
the maker, but may be readily determined by electrolysis.
If an electrolytic cell and a galvanometer are in the same
circuit, as the current is the same throughout, the value
of C in amperes, as given by the tangent galvanometer,
may be equated with that from the equation on page 177.
Thus
L. //tan 6
zt zitn
.'. Reduction factor = - -.
Hzt tan 6
The demonstration given above shows that the deflec-
tion is independent of the strength of pole, and it is not,
therefore, necessary to magnetize the needle strongly. As
the demonstration involves a value of/ which is true only
when the pole is at the centre of the coil, the magnet
ELECTRICITY AND MAGNETISM. 3
must be so small, relatively to the radius, that neither of
its poles should in any position depart widely from the
centre. As the lines of force due to a current in a coil
pass through its plane perpendicularly, as shown in Fig.
86, the dimensions of the needle and the accuracy of the
instrument may be increased by using two coils with the
needle midway on their common axis. The lines of force
of the two coils then act together so that they are sensi-
bly parallel in the region in which the needle moves,
many of them passing through both coils perpendicularly.
2. Sine Galvanometer ( 201).
It is possible to construct a sine galvanometer to meas-
ure currents in absolute units, but the ordinary form of
the instrument is not intended for absolute but for rela-
tive values only. The coil is placed parallel to the needle
before the observation, and when a deflection has been
produced by the passage of a current, the coil is rotated
and the attempt made to bring it once more parallel to
the needle. Every movement of the coil produces a
further displacement of the needle and the two are brought
into the same vertical plane only by careful adjustment.
When the coil and needle are parallel, the moment of the
earth's directive force is, as before, mlH sin 6, but the
deflecting influence of the current, acting always at right
angles, is/;/
Equating f=Hs'mB.
In the same galvanometer, /is always a function of C,
and hence the current varies as the sine of the angle of
deflection. Thus knowing the deflection that a current
of known strength produces
C : C' : : sin : sin 0'.
No current can produce a deflection of more than 90
NOTES ON
in a tangent galvanometer or in one in which the coil is
fixed. In a sine galvanometer, however, as the coil is
always kept parallel to the needle it exerts the same de-
flecting force in all positions. If now the current is of
such a strength that equilibrium is attained at 90 from
the original position of the needle, a stronger current
would deflect it still farther, and it would be impossible to
obtain equilibrium.
3. Mirror Galvanometer ( 202).
The supposition is generally made that with the mirror
galvanometer the currents vary directly as the scale read-
ings, but this is true only within limits. The needle being
small and the coil large, the current is proportional to the
tangent of the deflection, but as the deflection is read by
the movement of a spot of light on a tangent scale, the
current would be proportional to the reading, if it did
not follow from the laws of reflection that the spot of light
moved over twice the angle that the mirror moved.
Calling the observed deflections d and d' t the true ratio is
C : O : : tan - . tan .
In assuming that the currents are proportional to the
readings, the supposition is
C : C' : : r : r' : : tan d : tan d' .
If tan d : tan d' : : tan . tan , as it is sensibly
with small values of d and d' t the deflections may be taken
to be proportional to the strengths of the currents. In the
mirror galvanometer, therefore, the currents are propor-
tional to the readings if the deflections are small.
4. Differential Galvanometer ( 203).
The differential galvanometer is used not to measure
ELECTRICITY AND MAGNETISM.
currents, but to indicate that two are either equal or un-
equal. As the greater the resistance a current has to
flow through, the smaller the current is, by changing the
resistances in the circuit of either of the two coils of the
galvanometer the currents may ba made equal. It af-
fords, therefore, an easy method of comparing resistances.
Fig. 2.
In the figure it is seen that if the key is depressed to make
contact with EE, the currents pass around the needle in
opposite directions. In the figure one current is farther
from the needle than the other, but great care is taken in
making the instrument to have the coils similarly placed
and of equal resistance, so that equal currents will flow
through them, producing also equal but opposite effects
on the magnet. If there is no deflection, the currents are
equal, or C C ;
E E
but C =
and C =
But since a = b, x = ft, if there is no deflection of the
needle when the key is pressed. By having R adjusta-
ble, x may be determined.
NOTES ON
5. Ballistic Galvanometer ( 204).
When a current is of very short duration, it may be
supposed to exert an impulsive force on a galvanometer
needle, especially if the latter is heavy, as it would then,
by virtue of its inertia, fail to move until all the varying
impulses of the transient current had been given it. The
effect is then, practically, that produced by a momentary
impulse, and the needle will move with a velocity pro-
portional to the quantity passing, the force exerted in any
case varying as the current and the time the current
lasts, or as the quantity if the time is small, and will vibrate
through a certain arc, coming to rest, and then under
the directive action of the earth's magnetism will make a
return oscillation acquiring the same velocity as that
originally given by the current. In this oscillation the
effective force tending to bring it to rest varies as the
sine of the angle of deflection, as it does also in the case
of a simple pendulum, and the relations deduced from the
latter may therefore be applied without perceptible error
to the vibrating magnet.
The velocity acquired by the pendu-
lum in falling through the arc D B is
equal to that it would acquire in falling
from C to B ; hence from the laws of
falling bodies
v = V
But BC = AD (i -cos 6)=2 AD sin 2 1
Fig- 3- calling AD, I
= V 4 Ig sin 2
v = 2 sn
But as the velocity is, proportional in the galvanometer to
ELECTRICITY AND MAGNETISM. 7
the quantity passing, the quantity varies as the sine of
half the angle of deflection. If the deflection is noted
when a known quantity is discharged through a ballistic
galvanometer, any other quantity may be determined by
taking the ratio of the sines of half the respective deflec-
tions.
NOTES ON
II. THEORY OF POTENTIAL.
(Thompson's Electricity, pages 190-208.)
6. Potential ( 237).
An explanation of the term potential must precede
any definition. To illustrate, suppose a weight of a pound
be moved by the hand. In order to lift it to a higher
level the muscles have to be called on to do work. If the
pound is lifted ten feet, it is only by the expenditure often
foot pounds of work. If after being lifted, it is placed on
a shelf, the work done on it is evidently in the form of
potential energy, and may be recovered if the weight is
allowed to fall, when it will do ten foot pounds of work.
Strictly speaking, the weight on the shelf possesses energy
by virtue of the work done on it, but if this work had not
been done visibly, the weight might be said to possess
energy by virtue of its mass and the potential or height
to which it had been raised. A stone on the top of a prec-
ipice is thus said to have energy by virtue of its potential,
disregarding th2 question of how much work is required
to lift it to its position. Let a weight of two pounds be
now lifted to the same shelf, at a height or potential of
ten feet. If allowed to fall it would do twenty foot
pounds of work, although falling from the same height or
potential as did the other weight which performed only ten
foot pounds in its descent. The work has been doubled, al-
though the potential is the same. The work done, either on
the weight in lifting it to the shelf, or by the weight in falling,
is evidently the product of the weight into the potential.
ELECTRICITY AND MAGNETISM. 9
If a common bar magnet be taken in the hand and
moved near a powerful fixed magnet, work will be done.
If like poles are near each other, a force of repulsion is
exerted, and the muscles are called upon to do work in
bringing the magnet nearer to the fixed magnet, moving
it against this force. If the bar magnet be now suspended
so as to be unable to turn end for end, it will when re-
leased by the hand fly away from the fixed magnet under
the influence of the force of repulsion, doing work in its
movement. The work done against the magnetic forces
in bringing the bar magnet nearer the fixed magnet be-
comes potential energy, and is available as kinetic when-
ever the restraining force of the hand is removed. Here,
as in the case of the weight, the potential energy is
derived from the work previously done in moving the
magnet, but it is simpler to say that the magnet possesses
energy by virtue of the potential to which it is raised.
If another magnet of double the strength be moved,
twice as much work will have to be expended on it in
bringing it to the position occupied by the first, and it
will have twice the potential energy. As it has been
brought to the same potential the double work is due to
its being of twice the strength. In this case, therefore, work
is the product of strength of pole and magnetic potential.
Suppose a unit quantity of positive electricity be moved
near a larger quantity also positive. A force of repulsion
exists between them, and if they be brought nearer
together, the unit quantity will if released fly away from
the larger quantity, doing work. It evidently possesses
potential energy, or does work by virtue of its potential.
If a charge of two units be brought up twice the work
will be done, and there will be twice the potential energy.
Here work is evidently the product of quantity of elec-
tricity and electrostatic potential.
10 NOTES ON
7. Difference between Work and Potential.
As in each of the three cases examined the work
done or the potential energy possessed is the product of
potential and some other factor, if that factor be known
the potential may be obtained by measuring the work.
Potential is therefore measured by work, but is not work.
In several places in " Thompson's Electricity " the state-
ment is made that " potential is the work." The difference
between them may appear more clearly from a recapitu-
lation of the relations already traced.
Work done in lifting weight
Gravitation Potential = TJ . . . _. . . ., f
Weight lifted.
Magnetic Potential =
Work done in moving magnet pole
Strength of pole.
Electrostatic Potential =
Work done in moving quantity of electricity
Quantity moved.
Having thus traced the general analogies, in further
consideration electrostatic potential alone need be con-
sidered. From the last equation it is seen that if unit
quantity be moved, the potential is numerically equal to
the work done in moving it. As the work done in moving
from zero potential is the measure of the potential energy
acquired, the electrostatic potential at a point equals
the potential energy possessed by unit quantity of posi-
tive electricity at that point, and is measured by the
work that must be spent in bringing unit quantity of
positive electricity up to the point from an infinite dis-
tance. The infinite distance enters the definition from
the fact that there the potential is zero. This is evident
ELECTRICITY AND MAGNETISM. II
from the fact that potential is measured by work ; work
is the product of force into distance, over which it acts,
and force = JQL . If r is infinite, the force is zero, and no
r *
work is done in moving unit quantity.
8. Positive and Negative Work.
As in any case in which a positive unit is repelled, a
negative unit is attracted, the work which in the first case
is necessary to move the unit against the force of repulsion
would in the second be done in preventing the movement
of the unit under the forces of attraction. The two cases
are evidently diametrically opposite, and the work done is
therefore considered as positive when it is done on the
the unit, negative when done by the unit moving freely.
It becomes necessary, therefore, to specify the positive
unit in the definition, that the nature of the work and con'
sequently the sign of the potential may be known.
9. Positive Electricity always flows from a High to a Low
Potential.
Potential energy always tends to run down to a mini-
mum. A weight acted on by gravity will fall to the
earth if not prevented ; a magnet pole placed near another
similar pole possesses potential energy and tends to move
away into a position in which its potential energy is
less ; a unit of electricity placed near a similar quantity
is likewise repelled and moves so as to decrease its poten-
tial. In any electrified region the relative potential is
therefore indicated by the direction in which a unit of
positive electricity tends to move, and the distribution of
potential may be examined by conceiving a positive unit
to be moved throughout the neighborhood of the electrified
12 NOTES ON
bodies, noting whether it is necessary to do work to move
it or to restrain its movement. Let this positive unit be
approached to a quantity of positive electricity. Work
must be done to move it, and if left free it will fly away,
moving to decrease its potential. It has evidently been
moved into a region of higher potential. If approached
to a negative unit, work must be done to restrain its
movement. If free to move it will move toward the
negative unit, and is moving into a region of lower
potential. Generally speaking, a body positively electri-
fied is at a positive potential, and one negatively electri-
fied at a negative, but there are many exceptions.
Suppose a cylinder B to be unelectrified and to be
connected with the earth by a wire. There is no flow of
electricity in the wire, and
B is therefore at the same
potential as the earth, as
electricity tends to move
Fig. 4. toward a lower potential,
and the fact of there being no movement shows there is
no difference of potential. Let a positively electrified ball
A be approached, and B becomes electrified by induction
as in the figure. If again connected with the earth, a flow
of positive electricity takes place from the cylinder to the
earth, showing that the potential of B had been raised by
the approach of A. Before being connected with the
earth the second time it was therefore at a positive poten-
tial but was negatively electrified at one end. When in
contact with the earth, with A as in the figure, it is
negatively electrified, but at zero potential.
10. Units of Potential and Work.
Potential is measured by work, and the units of poten-
tial are numerically equal to the units of work. Work is
ELECTRICITY AND MAGNETISM. 13
defined as force acting through distance, and the C. G. S.
unit of work called the erg is the work done in opposing
the force of one dyne through the distance of one centi-
metre. It may be defined after the analogy of foot-pounds
as a dyne-centimetre. If, therefore, the work necessary
to move a unit quantity of electricity, or the work a unit
quantity does in moving, is measured in ergs, it numeri-
cally equals the difference of potential through which the
unit moves.
ii. Electrostatic Potential ( 238).
We can now derive the general formula for electrostatic
potential. From the definitions of potential and work
.00
Work =r fdr t where r is distance j
.0
fdr t
J r
but / = 9 l and Work = - dr.
* r
If q is unity, work measures potential, hence, denoting
potential by V,
The potential at any point due to a quantity q is, there-
fore, numerically equal to the quantity divided by the dis-
tance in centimetres. If other quantities q' ' , q" , etc., were
near, the potential due to them would be , etc. The
potential due to the whole system is then
v=<- + t + C = 2 i.
r r r' r
NOTES ON
If either q, q or q" is negative, or will be nega-
tive, and must be given its proper sign in the summation.
12. Zero Potential ( 239).
Although as shown, the theoretical zero potential exists
at an infinite distance, the potential of the earth at the
place is the practical zero. All electrical manifestations
are dependent on a difference of potential, and the abso-
lute potential is never needed.
13. Difference of Potentials ( 240).
Potential being measured by work done, the difference
of potential between two points is numerically equal to the
number of ergs required to move a positive unit from one
point to the other. It is immaterial what path be followed,
as if all the work done, both positive and negative, be
summed up, it will be equal to that done in moving in
a direct line between the points. Two quantities of elec-
tricity at different potentials may be compared to two
ponds of water at different levels. If the ponds are con-
nected by a pipe, the water in the upper will by virtue of
its height possess potential energy and will run down into
the lower. If no current of water flowed in the pipe it
would indicate that the ponds were at the same level. If
no other means of measuring the difference of level were
available, it could be done by measuring in foot-pounds
the work done by one pound of water in flowing through
the pipe. Similarly electricity flows from a body electri-
fied to a high potential to one at a lower connected with
it, and the difference of potential is measured by the work
in ergs done by unit quantity in flowing from one to the
other.
ELECTRICITY AND MAGNETISM. 15
14. Electric Force ( 241).
When_/ is force exerted on unit quantity
V= \fdr t wdV=fdr .:/ = .
dV
But -j is the rate of change of potential, hence the
average electric force between two points at different po-
tentials is measured by the rate of change of potential per
centimetre.
15. Law of Inverse Squares ( 245, Fig. 98).
Coulomb's observations, without proving exactly that
the 1 >w of inverse squares applied to electric force, so
nearly proved it as to lead one to think that more careful
experimentation, were such possible, would demonstrate
the exactness of the law. Assume, therefore, the law and
trace the results.
Let p be the electric density, or the amount of electricity
per square centimetre of surface. On a sphere removed
from other conductors the density is uniform, and the
quantity on two surfaces varies as the area of the surfaces.
The quantity on the surface AB (Fig. 98, " Thompson ")
is p x Area AB pA.
The quantity on CD is p x Area CD = pC.
Assuming that electric force varies inversely as the
square of the distance, the force exerted on a unit of elec-
tricity at the point P inside the sphere by the quantity on
ABis
The force exerted on the same unit by the quantity on
CD is
1 6 NOTES ON
Cf
Since the tangents drawn to the sphere are equally in-
clined to JSC
C x BP*
A\C\\BP\ CP* , orA = = '
LP*
substituting above/ = /'.
If the sphere be cut up into small cones 2f = 2f ; or in
other words, if electric force varies inversely as the square
of the distance, there should be no resultant force on the
inside of a closed conductor. If it follows any other law
/cannot equal/'. The most careful experiments fail to
detect any force existing, and in corroborating the result
of the above demonstration, confirm the hypothesis made
that electric force follows the law of inverse squares.
1 6. Capacity (3 246).
The capacity of a conductor is by the definition given
a fixed quantity, while "the amount of electricity the con-
ductor can hold," the definition as generally given by be-
ginners, is variable, depending on the potential as well as
on the capacity. An illustration may make the distinction
clearer. If a jar has a volume of one litre, its capacity is
a litre, and it will hold a litre of air at atmospheric press-
ure. If the pressure be doubled, however, the quantity
of air in the jar is also doubled, although the capacity is
the same. "The amount" the jar "will hold" is evi-
dently determined by the capacity and pressure. If the
pressure be unity, that of one atmosphere, the capacity is
then the amount the jar actually holds, but quantity and
capacity are under other conclitious different. With ref-
erence to electricity, the capacity is similarly the charge
ELECTRICITY AND MAGNETISM. I?
the conductor will hold at unit potential, or the charge
which will raise. the potential to unity, and the actual
charge in a conductor is the product of the capacity and
potential.
17. Unit of Capacity ( 247).
In an electrified sphere, as the surface is an equipoten-
tial surface, the charge may be considered as concentrated
at the centre and the potential at the surface is - As
the capacity is equal to the quantity divided by the poten-
tial
Capacity = - = r.
A sphere of one centimetre radius is, therefore, of unit
capacity.
18. Electric Force exerted by a Charged Plate ( 252).
Let a be the radius of the plate and r the radius of the
ring x, x,' x." p is the
density, or the charge
per unit of area.
The quantity on any
small circular element
is p (27trdr). The force
exerted by this quan-
tity on a unit at O is
p(2itrdr)
and the force acting nor-
mal to the plate is
Fig. 5
1 8 NOTES ON
2-itrdr h \ h
P [ vv- ., - = - ) , Since COS =
*
The total force exerted by the plate in a direction normal
to its surface is, therefore
* iitrdr h , C a 2 rdr
' =
Integrating, = 27tph (h
+ r 2 )i I
= 27tph h
' 2 7tp ,
27tp (I COSO').
If O is very near the plate, or if the plate is very large,
6' = 90, and the electric force of a charged plate on a unit
of electricity very near it is 2itp.
Care must be taken not to confuse this with the force
exerted by a sphere as deduced in g 251. If a plate is
charged with positive electricity, and a positive unit is
placed very near it on each side, the force will be one of
repulsion in each case, but if one unit is repelled upward
the other tends to move downward, and if one force is
2it p, the other must be 2rtp. The force changes, there-
fore, by 47tp in passing through any charged surface.
19. Dimensions of Units ( 258).
An important use of the dimensional equations is in the
conversion of units based on one system of fundamental
units of mass, length and time to others based on different
fundamental units. The French use units based on the
metrical system, and although the centimetre-gramme-
second, or C. G. S. system, is now almost universally used
ELECTRICITY AND MAGNETISM. 19
in electrical work, there are many observations made and
recorded in which other units are used. In electrical
work the English have heretofore used the foot-grain-
second system, and it is still used in some government
observatories. It is a matter of the highest importance,
therefore, that the method of converting values expressed
in one system to corresponding values in another should
be thoroughly understood. As the ratio of the centi-
metre to the foot is that of I to 30.48, it is evident that
the ratio of the units of length in the C. G. S. and foot-
grain-second systems is the same. The units of area in
the two systems are the square centimetre and the
square foot, and no one would think of saying that this
ratio was the same as the preceding -, but rather
- 1 . Similarly the ratio between the units of volume
(3048)'
i i 3
is not but . 5T7-. This is exactly the relation
30.48 (30.48)'
shown by the dimensions of area and volume on page
211, they being respectively Z, 2 and D . In simple cases
like the above the change is easily made, but in others,
where the dimensions of the unit are more complex and
the unit itself an unfamiliar one, the dimensions must be
used to calculate the ratio. As an illustration, let it be
required to express in units of potential based on the foot,
grain and second, the difference of electrostatic potential
expressed by 2.7 C. G. S. units of potential.
The dimensions of electrostatic potential are M^ L T~ ,
The C G. S. unit /;;Ai//\i
= \M) \L)
then
Foot-Grain-Sec, unit
_
30.48;
= >7II6
20 NOTES ON
.'. I C. G. S. unit = .7116 Foot-grain-second unit
2. 7 C. G. S, =. 1.92 Foot-grain sec.
The ratios between the different units in the two systems
are given in Note 40.
20. Attracted Disc Electrometers ( 261, p. 215).
Let the difference of potential between two plates be V,
and the distance apart be D. By Note 14 the average
y
electric force between the plates is . As proved in Note
1 8, the electric force changes by ^itp in passing through a
surface, and being zero in the conductor is therefore
y
47tp just outside. Equating, /o = - . The density on
each plate being p, the attraction exerted by the lower plate
on a unit of electricity on the upper one is 2jrp when the
plates are near each other. The upper plate contains,
however, Sp units and the total attraction is iitp x Sp =
21. Absolute Electrometer (p. 216).
Sir William Thompson's absolute electrometer, so named
from giving the potential in absolute units, is an attracted
disc electrometer.
The disc C (see Figure TOO, " Thompson") is held in place
by springs, instead of a counterpoise as shown, and is in
metallic connection with B. When no part of the appa-
ratus is electrified, small weights are placed on C, to bring
ELECTRICITY AND MAGNETISM. 21
it into a standard position such that a small hair attached
to it is seen midway between two dots, as shown in the
figure. The weights are then removed and B and C con-
nected to one of the bodies whose difference of potential
is required and A to the other. The electric force of
attraction between the two plates will act to lower C, but
as the accuracy of the instrument depends on its being in
the plane of B, the plate A is moved up or down until
the force of attraction is such as to bring the movable
plate into this standard position, which is known by see-
ing the hair again midway between the dots. It is now
under the attraction of the electrical forces, in exactly the
same position as when acted upon by the weights, and
the two forces are therefore equal. Substituting, there-
fore, for .Fin the formula of Note 20, 981 times the weight
in grammes required to bring C into the standard position,
and for D the distance in centimetres between A and C t
all quantities in the equation are known and the difference
of potential may be calculated.
Another method is more common, dispensing with the
use of weights. If the difference of potential between two
bodies P and P' is required, one of the bodies is connected
to A, and B and C are then electrified to a high potential.
The plate A is then moved up or down until the plate C
comes into the standard position, the hair showing mid-
way between the dots, and the distance D of A from Cis
noted. Then
Potential of B - Potential of P = D y
The plate A is next disconnected from the first body
and connected with the second. As the difference of
potential between A and Chas now been changed, the
force acting between the plates is different and C is no
22 NOTES ON
longer in the standard position. It is brought there by
raising or lowering A, and when adjusted the distance D'
between the plates is noted r
Then Potential of B Potential of P' = D 'i/ 8 ^
V S.
Subtracting this from the former,
Difference of Potential between P and P' (DD') x
constant of instrument.
It is of course necessary that the potential of B should
be the same in both cases. This is verified by a separate
attracted disc, which is in each case electrified until its
attraction for another disc at a fixed distance brings it
into a standard position. The absolute potential of B is
immaterial, the only requirement being that it should be
the same in each case. If the absolute potential of P is
wished, connect A first to P and then to earth.
ELECTRICITY AND MAGNETISM. 23
III. THEORY OF MAGNETIC POTENTIAL.
(Thompson's Electricity, pages 265-278.)
22. Magnetic Field.
Any region throughout which forces act is called a
" field," but the term is more frequently used in connec-
tion with magnetic than with other forces. A magnetic
field is, therefore, a region in which magnetic effects are
produced. Any movement of a magnet pole can take
place only in a magnetic field, and the term is of use, as
it disregards all ideas of how the field is caused, and con-
siders only the forces and the direction in which they act.
If at any point a line is drawn indicating the direction of the
force at that point, it is called a line of force. This direc-
tion is that shown by a magnet placed at the point, and
may therefore be easily determined by experiment ; but as
any representation of a magnetic field must present the
whole field at once, the determination of the position of
the lines of force by this process would be tedious. The
reasoning in \ 126 leads to an easier though less accurate
method, but one of great utility in enabling clear concep-
tions to be formed. If a magnet is covered by a sheet of
paper and iron filings are sprinkled over the paper they
will on being gently tapped arrange themselves in curves
passing from pole to pole. From the definition of lines of
force, these curves must be the lines of force in the plane
of the paper, and the mind has only to conceive the space
above and below the magnet to be similarly filled to gain
a clear idea of the field. It is necessary, however, to
24 NOTES ON
know not only the direction of the force at any point, but
also its strength, and a correct plotting of the field must
furnish this. Maxwell has shown that if in any part of
their course, the number of lines of force passing through
unit area of a perpendicular plane is proportional to the
strength of the force there, the number passing through
unit area in any other part of the field is in the same pro-
portion to the strength in that part. The closeness of the
lines of force is therefore a measure of the strength of the
forces of the field, or, as more commonly expressed, of the
intensity of the field. By drawing the lines of force
therefore in this way, the strength and direction of the
forces in all parts of the field are indicated. As a south
pole moves always in the opposite direction to that in
which a north pole moves, it is necessary in order to
establish the direction of the force to consider the nature
of the pole acted upon. All investigations in magnetism
are made by considering a north pole free to move, and
the positive direction of the lines of force is therefore
that in which a free north pole moves. This definition
is of great importance in many of the demonstrations
given later.
23. Mapping a Field by Lines of Force.
A magnetic field is of unit intensity when unit pole is
acted upon by a force of one dyne. As by definition
unit pole acts on an equal and similar pole at a distance
of one centimetre with a force of one dyne, it follows
that unit pole causes unit field at unit distance. As in-
tensity of field is measured by the force acting on unit
pole, unit field exists at a greater distance from a more
powerful pole. It is, therefore, unnecessary to consider
the question of distance from the pole producing the field,
but simply bear in mind that the intensity of the field at
ELECTRICITY AND MAGNETISM. 25
any point is measured by the force in dynes acting on a
unit pole at that point. If the pole be of a strength m,
the force with which it is attracted or repelled is ;// times
that experienced by unit pole, or
Force acting: on pole
Intensity of field = = -^~ f -
Strength of pole
The value of H, or the strength of field, is given numer-
ically. Thus the horizontal force of the earth's magnetism
at London being .18, a pole of unit strength is impelled to
move in a horizontal plane by a force of .18 dynes. A
pole of strength 100 would be acted on by a force of
1 8 dynes in a horizontal direction, or by a force of 47
dynes in the line of dip. To represent the field graphic-
ally, recourse is had to Maxwell's demonstration of the
fact that the number of lines cutting unit area in different
points of the field is proportional to the intensity at those
points, and the numerical value of //"is interpreted as the
number of lines per square centimetre of a surface per-
pendicular to the direction of the lines. Thus at London
the earth's horizontal field would be represented by draw-
ing horizontal lines of force in the magnetic meridian,
equidistant and so spaced that they cut a vertical east
and west plane at the rate of .18 per square centimetre
or of one line to every 5.56 + square centimetres. The
positive direction is toward the north. The total field at
London would be represented by lines of force in the di-
rection of the dipping needle, equidistant and spaced so
as to cut a perpendicular plane at the rate of .47 per
square centimetre, or one to every 2.13 square centimeters.
These lines projected intersect at the magnetic pole, and
are, therefore, sensibly parallel within ordinary limits, in
which case the field is said to be uniform.
26 NOTES ON
24. Equipotential Surfaces,
Being surfaces in which no work is done in moving a
unit pole, are necessarily perpendicular to the lines of
force. If not, some component of the force would act, and
work would be done in moving against it. Knowing the
direction of the lines of force, the equipotential surfaces
can be readily drawn by cutting all the lines at right
angles. Like lines of force they may be drawn in any
number required, but it is customary to have them rep-
resent unit difference of potential, and this requires that
they should be so far apart that an erg of work is done
in moving a unit pole from one to the other. The dis-
tance may be readily calculated.
Work = Hx,
but by definition work is unity
I
y . . L
H '
or the distance of the equipotential surfaces is inversely as
the intensity of the field. The field may, therefore, be
represented in this way as accurately as by the lines of
force. Taking the case already considered, the earth's
horizontal field at London would be represented by verti-
cal surfaces, sensibly planes, extending east and west and
5.56 + centimetres apart.
25. Lines of Force due to a Single Pole.
In the case of a free pole of strength in, the number of
lines of force is determined by the fact that there are ;;/
lines intersecting every square centimetre at unit dis-
tance, or that there are m lines cutting every square centi-
metre of a sphere of unit radius. The surface of this
sphere being 471 there are in all ^itm lines of force radial-
ELECTRICITY AND MAGNETISM. 27
ing equally in all directions. The equipotential surfaces
are spheres, and the radii may be found from the for-
mula for magnetic potential, V .which may be here
assumed to be correct. By substituting values for Fdif-
f . , . . f mm
fenng by unity, r is found to be successively ;;/,
, etc. As a free pole can never exist, but is always as-
4
sociated with another of equal but opposite polarity, any
actual field is much more complex, but the cases given
will illustrate the application of the principles traced,
and will give clear ideas of the conventions made which
underly further investigation.
It is possible without changing the number of lines of
force to change their distribution in the field, and as it is
frequently desirable to intensify a certain part of the field,
the method of doing so becomes of importance. A com-
parison of Figures 52 and 53 " Thompson's Electricity,"
shows that the distribution may be greatly changed by
an arrangement of magnet poles in the field, and as iron
near a magnet becomes magnetized by induction, the
field is similarly affected by the introduction of iron. In
this case the iron seems to gather the lines of force in the
vicinity, causing a great number to pass through its sub-
stance. Iron placed near a magnet pole becomes mag-
netized, so that dissimilar poles are adjacent, producing
the state of affairs shown in Fig. 52. Jenkin compares
this peculiarity of iron in concentrating the lines of force
to that of a lens in converging rays of light. It is likewise
possible to screen any part of a magnetic field from in-
duction, by inclosing it in an iron shell. It may be easily
demonstrated by experiment that if an iron ring be placed
between the poles of a horse-shoe magnet, no lines of
28
NOTES ON
force pass through the interior of the ring, but entering
at one side pass through the metal of the ring issuing on
the opposite side. A magnet inside an iron sphere is in-
dependent of all outside influences. By the use of iron
it is, therefore, possible either to concentrate the lines of
force, or to divert them entirely from any desired part of
the field.
26. Lines of Force due to a Current.
As shown in g 191, the lines of force due to a current
are circles perpendicular to the current and having it for
a centre. If the conductor is straight, the circles are all
in parallel planes, and the equipotential surfaces are
planes radiating from the conductor and each containing
it. The number of these planes is such that an erg is
required to move unit pole from one to the other. As
shown below, the intensity of the field at unit distance is
2(7. A pole moving in a circle of unit radius having the
conductor for a centre passes over a distance of 2?r against
a force of 2C, doing 4?r(7 ergs
of work. The number of
equipotential surfaces is, there-
fore, 4-TtC. After the pole had
made one revolution it would
reach the equipotential surface
from which it started, but hav-
ing done 4/rC ergs in its revolu-
tion the numerical value of the
surface would now be 471- C more
than before. It is impossible,
A therefore, to give an absolute
value to an equipotential sur-
. 6- face due to a current.
Let LL be a portion of an infinite rectilinear current, and let
ELECTRICITY AND MAGNETISM. 29
BC be the force exerted by an element of length, dx> of this
current = Cdx. Place a unit pole at A.
The force exerted by the element dx at A is
_ Cdx cos Q ^
Zg*
If <9^ = i, ~A& = i + x 1
i
Cdx.
Vi + x- _ Cdx
ar = t
TT
The total force = F C \ T~a = c T cos 6 ^ 9 -
Loot 1 + *) J'i*
Integrating, ^=Csin0 zC = //.
zC =
To find the intensity at a distance r from the conductor we
have from Note 24 that the intensity is the reciprocal of the dis-
tance between two equipotential surfaces. There being 4rtC
surfaces cutting a circumference of 2rtr, the distance between
them is - and hence
26?
27. Magnetic Potential.
Magnetic potential has already been alluded to in Note
6. The conception is strictly analogous to that of elec-
trostatic potential, and the demonstration given for the
formula for electrostatic potential in Note n, is applica-
ble to magnetic if for q a quantity of electricity, is substi-
tuted m a strength of magnet pole. The same reasoning
leads to the formula of V =. 2
r
3 NOTES ON
Magnetic potential at a point equals the potential
energy possessed by a unit north pole at that point,
and is measured by the work in ergs done in bringing
a unit north pole from infinity to that point.
Zero of magnetic potential exists at an infinite distance
from all mag-nets.
dV
Magnetic force is , or the rate of change of potential
per unit of distance as in Note 14.
The difference of magnetic potential between two
points is measured by the work clone in moving a unit
north pole from one to the other. Wherever work has to
be done in moving a north pole, it would be done in re-
sisting the motion of a south pole. In all investigations of
magnetic potential, force or work, a unit north pole must
always be considered.
28. Tubes of Force.
The conception of tubes of force is frequently of utility.
The lines of force radiating from a pole may be regarded
as forming cones, and any section through a cone would
cut all the lines of force. But as the number of lines of
force is proportional to the intensity, the force on all cross
sections of the cone is therefore the same. By conceiv-
ing the magnetic force to be equal throughout the cone,
existing between, as well as along the lines of force, a
more accurate idea of the field is attained. It is easy to
imagine a field of so slight intensity that a square centi-
metre would not have any lines of force passing through
it. The example of the earth's horizontal field at London,
already referred to, is a case in ooint. One line of force
passes through every five units of area, but the magnetic
forces are felt just as strongly on the four units through
which the line does not pass as on that which it cuts. By
ELECTRICITY AND MAGNETISM. 31
thinking, therefore, of the lines of force as indicating only
the direction and strength of the forces which act between
them as well as in them, this difficulty is overcome.
29. Intensity of Magnetization.
If a bar magnet be broken in half, instead of obtaining
one piece of nortli and the other of south polarity, each is
found to possess both and to be a perfect magnet. How-
ever far the subdivision be carried, the result is the
same, and the ordinary explanation is that the magnet is
an aggregation of magnetized molecules, the magnetic
axes of the molecules being to a greater or less extent
parallel. If this were so the north pole of one molecule
would be counteracted in its magnetic effects by the south
pole of the next, and the only molecules capable of exert-
ing external magnetic effects would be those on the sur-
face, and the effect is exactly the same as would be pro-
duced by a distribution of a magnetic matter or fluid, or
avoiding the idea of a fluid, a distribution of magnetism
over the surface of the magnet. The amount of magnetism
per unit area is called the magnetic density. If the magnet-
ism is regarded as being uniformly distributed throughout
the mass of the magnet, the quotient of the magnetic cur-
rent by the volume is called the intensity of magnetization.
Let p be the magnetic density, a the cross section and
/ the length of the magnet. Then the strength of pole
is m p a
m I p a I
ml
' = Tt
_ Magnetic moment
Volume
The intensity of magnetization and magnetic density
are therefore practically the same, the one presupposing a
32 NOTES ON
uniform distribution of magnetism throughout the mass,
the other a surface distribution.
If the magnetism is due to the bar being situated in a
magnetic field, the intensity of magnetization is equal to
k H, k being what is called a " coefficient of magnetiza-
tion." A few values of k are given in 340 (" Thomp-
son "). Assuming Barlow's value for iron 32.8, the formula,
intensity of magnetization k H indicates that the in-
tensity of magnetization is dependent only on the intensity
of the field ; but there is found to be a limiting value of
magnetization which cannot be exceeded, however power-
ful the field is. This is stated to be for iron 1390 (p. 269,
" Thompson"), and the strongest field that could be util-
ized in magnetizing iron is therefore ^9 = 42.4. The
value of this coefficient is, however, uncertain, and appears
to be much less at a high intensity of magnetization than
at a low
30. Solenoidal Magnets.
A filament of magnetic matter so magnetized that its
strength is the same at every cross section is called a
magnetic solenoid. A long thin bar magnet uniformly
magnetized is called a solenoidal magnet, or simply a
solenoid, in distinction to a magnetic shell. The name
solenoid is also applied to a helix through which a current
passes. (See Note 42.) As the magnet poles are points at
which the magnetism of the magnet may be supposed to
be concentrated, and from which magnetic forces act, the
potential of any point near the magnet is determined by its
distance from the two poles, or V 2 = m ( , j.
The exact position of the poles is difficult to determine,
but is stated in g 122 to be in long thin steel magnets
ELECTRICITY AND MAGNETISM.
33
about rV of the distance from the end. If the poles are
bent to meet, forming a ring, r = r' for all external points,
and there is therefore no potential due to a magnetized
ring.
31. Potential due to a Magnetic Shell.
As defined in \ 107 (" Thompson") a magnetic shell is a
thin sheet so magnetized that the two sides of the sheet
have opposite kinds of magnetism. The demonstration
and use of the expression for the potential due to a mag-
netic shell requires a preliminary definition of a solid
angle, and of the method of measuring it. The solid
angle subtended at any point by a closed curve or surface
is measured by the area of a sphere of unit radius de-
scribed from the point as a centre, intercepted by lines
drawn from all parts of the curve to the point. (See
Fig. 64, " Thompson.") As the areas of similar surfaces
on spheres are as the square of the radii the solid angle,
&> = area on unit sphere, A
_ area on sphere of radius r
_ area on sphere of radius r^
~^\~
To compute the solid angle.
When the closed curve is circular and
the point is in its axis it is necessary only
to compute the area of a zone of one base
on a sphere of unit radius. The area of
the zone formed by the revolution of AD
around AO as an axis is (see Chauvenet's
Geometry, Book IX., Prop. X. Cor. Ill)
Ad x 2 it OA.
But Ad = A O dO r r cos
.'. Area = zitr (r r cos 6) ;
3
Fig. 7.
34
NOTES ON
but if r i, area is solid angle,
.'. GO = 27t (i cos 6).
To calculate the potential.
Let r, and r 2 be the distances from the point Z> to the faces of
the small element ds, ft be the angle between ds and its projection
,D
Fig. 8.
perpendicular to r 1 and p be the magnetic density. The strength
of the shell i being the product of the density and thickness,
- i = pt .-. p = -.
The quantity of magnetism on the small element ds is
dm = ds.p = ds.
(I)
The potential at D due to ds is
- *=<*-). . . (a)
ELECTRICITY AND MAGNETISM. 35
But cos ft = ; - 1 . Substituting this value and that of dm
rfF=-cos/S (3)
r~
But as ds is an infinitely small element, its plane projection
ds cos ft perpendicular to r l is sensibly equal to the area on a
sphere of radius r. The solid angle, therefore, subtended at D
by the element ds is
_ ds. cos ft
f"
Substituting in (3)
dV = deal
V ooi.
32. Equipotential Surfaces and Lines of Force of
a Magnetic Shell.
The computation of the solid angle is, as shown, simple
when the point D is on the axis normal to the shell at its
centre, but when D is oblique the area becomes an ellipse
on a spherical surface of unit radius, and the calculation
is extremely difficult. From the formula, however, for
magnetic potential a few relations are readily deduced.
As at all points where the shell subtends the same solid
angle the potential is the same, any equipotential sur-
face is evidently most remote from the shell on the axis
normal to its centre. As the point of view becomes
oblique it must approach the shell that the solid angle
may be the same, and at all points in the plane of the shell
the solid angle, and consequently the potential, are zero.
The general form of the equipotential surfaces is, there-
fore, that of deep bowls concave to the shell, and most
remote from it on its perpendicular axis. At a point close
to the shell the solid angle is a hemisphere or 2Tt and the
36 NOTES ON
potential 27rz". On the opposite side the potential is 27T/,
or 47T* ergs must be expended in moving unit pole from
one side of the shell to the other. If the equipotential
surfaces indicate unit difference of potential there are,
therefore, ^Tti surfaces. From the equipotential surfaces
the direction of the lines of force may be traced, as they
start from the side of the shell having north polarity and
curve so as to cut each surface at right angles, finally en-
tering the south pole of the shell at right angles.
33. Work Done in Moving Pole near Shell.
Potential being measured by the work done on unit
pole in bringing it up to a point from an infinite distance,
the work done on a pole of strength in is mcoi. It is pos-
sible under the conventions made as to the number of
lines of force to express this in another way. As already
shown, the number of lines of force given off by a pole of
strength ;;/ is ^nm, but as these radiate in all directions,
they are given off throughout a solid angle 47? subtended
at the centre of a sphere. Through any solid angle &? the
number of lines is, therefore, moa. Calling this number N,
the above expression becomes Ni, or the work done in
bringing a pole up to a position near a magnet shell is
measured by the product of the strength of the shell and
the number of lines of force of the pole cut by the shell.
This is evidently a measure of the work done either in
bringing the pole up to the shell or the shell to the pole,
and is, therefore, sometimes called the mutual potential
of the pole and shell. The work done in bringing the
pole from infinity to a point where it intercepts N l lines is
Nil. If now it be moved to another in which it intercepts
N* lines, the work done between the points is
Work done = i (N., - ,V,).
ELECTRICITY AND MAGNETISM. 37
The difference of potential between the points is
i (Ni A T ,)
^ -'
or
magnetic potential being the quotient of work done in
moving pole, by the strength of pole.
The work done may be either positive or negative and
the above expression may, therefore, have either sign. If A 7 !
> A 2 the work done in passing from N\ to A 2 is negative,
or the shell tends to move in such a direction as to include
a minimum number of lines of force. As these pass in the
positive direction, exactly the same relation is expressed
by saying that a magnetic shell in a field tends to place
itself so as to enclose the maximum number of negative
lines of force. If the north pole of a magnet shell is
brought up to the north pole of a magnet, this relation is
readily seen, as the shell will be repelled into a position
in which it will enclose as few lines of force taken in the
positive direction as is possible. If the same face be ap-
proached to a south pole, it is attracted and moves into a
position in which the maximum number of lines cut the
shell in the negative direction.
34. Equivalent Magnetic Shells.
The relations deduced for magnetic shells are of great
service, as they are applicable to the case of a voltaic
circuit in a magnetic field. If a wire carrying a current
be looped into a circle, the lines of force which ordinarily
encircle the conductor combine to act in the same direc-
tion on a pole at a distance from the circuit. Thus in
Fig. 86 (" Thompson "), it is seen that all the lines of force
due to the current pass in the same direction through the
plane of the circuit as do those of a magnetic shell. A
closed voltaic circuit in a magnetic field, as may be readily
38 NOTES ON
shown by experiment, is acted upon as a magnet would
be. It is found that the magnetic effects of the north pole
of a magnet are identical in nature with those of a circuit,
in which the current flows in a direction opposite to that in
which the hands of a watch move. This direction is
known as the negative direction of the current, and the
magnetic effects of a positive pole and of a negative cur-
rent are, therefore, similar. Looking at the other side of
the loop, the current would appear to pass in the direction
in which the hands of a watch move, or in the positive
direction ; but if a north pole be approached to the circuit
from the side on which the current appears to have this
direction it is attracted, showing that a positive current
produces magnetic effects similar to those of a negative
pole. As the direction in which a north pole moves shows
the direction of the lines of force, it is seen from the above
that the lines of force enter that face of the plane of the
circuit in which the current appears to move " with the
sun," or in the positive direction, and emerge from the
other face. As it is a matter of great importance to be
able to connect the direction of the lines of force with that
of the current to which they are due, several rules have
been given, one of the best of which is the comparison
between the direction of rotation of a corkscrew and that
of the motion of its point. If the wrist be rotated in
the right-handed direction, the point advances ; and con-
sidering the motion of the wrist to be that of the current,
the movement of the point corresponds to that of a north
pole, and indicates the direction of the lines of force.
This relation is said to be that of " right-handed cyclical
order," and the direction of the current and of the lines of
force are spoken of as being thus related.
The magnetic action of a voltaic circuit is found to
depend upon the strength of the current, and on the area
ELECTRICITY AND MAGNETISM. 39
of the enclosed surface. It is, therefore, evident that for
every closed circuit, a magnetic shell whose edges coin-
cided in position with the circuit could be substituted, if a
certain relation were established between the units measur-
ing the strength of the current and the strength of the
shell. This relation is that expressed in the definition in
g 195. The absolute electromagnetic unit of current is
that current which in passing through a conductor one
centimetre long, bent so as to be in all parts distant one
centimetre from a unit pole, acts on the pole with a force
of one dyne. By this definition unit current produces
unit field at unit distance. But so does a shell or pole of
unit strength. By expressing, therefore, the current in
these units, the magnetic effect of the current is the same
as that due to a magnetic shell whose edges coincide with
the circuit, and whose strength is equal to that of the cur-
rent. This is called an equivalent magnetic shell, and
all relations hitherto traced for the shell are now applica-
ble to the closed circuit.
35. Potential due to a Closed Voltaic Circuit.
The potential due to a current at a point is therefore
Coo, where Cis the current measured in absolute units,
and GO is the solid angle subtended by the circuit at
the point. As, however, a positive current produces the
same magnetic effects as a negative pole, the sign of the
potential is always the opposite of that of the current, or
The difference of potential between two points is
The magnetic force due to a current at a distance x is
(Note 14)
40 NOTES ON
dx
The potential is iitC on one side of the circuit and
2.TfC on the other, changing by 4?rC. Hence there
are 4;rCequipotential surfaces.
36. Work Done in Moving a Circuit Near a Pole.
This is a problem of the greatest importance, as it under-
lies the action of the dynamo machine. As already
traced (Note 33), the work done in moving a magnetic
shell near a pole, or conversely the pole near the shell, is
Work done = moot = Ni.
Similarly the work done in moving a closed circuit near
a pole is
Work dons = moaC = NC,
N being the number of lines of force of the pole passing
through the circuit. If the circuit be brought up from an
infinite distance to a point where it intersects /V, lines due
to the pole the work is JV } C. If now moved still farther
so that it intersects a greater number, /V 2 , the work done
between the points is
V/ork =
and
Difference of potential = - C(A/2 ~ N $ = - C(oo, - GO,).
If TVa > N\ the work is negative, and the circuit tends
to move therefore in such a manner as to make the num-
ber of lines enclosed a maximum. If a circuit be placed
in a magnetic field so that the lines of the field while
parallel to those of the current pass in the opposite direc-
ELECTRICITY AND MAGNETISM. 41
tion, the circuit will, if free, first turn to bring the lines in
the same direction, and will then move to make the num-
ber enclosed a maximum (Fig. 87, "Thompson").
It may be useful to obtain an expression for the work
done in this last case, as an understanding of the theory
will assist in the comprehension of the working of electric-
motors. Imagine a closed circuit placed in a uniform
field. If the circuit be moved parallel to itself, the num-
ber of lines enclosed is constant, and consequently no work
is done in whatever direction the circuit be moved. If,
however, the coil be rotated on an axis in its plane, it will
enclose a varying number. If be the angle between the
normal to the plane of the coil and the direction of the
lines of force of the field, and the number of lines passing
through the coil when its plane is perpendicular to them
be N t the number enclosed when at an angle 6 is N cos 0.
If the angle be now changed to 6' the number enclosed is
JV cos V , and the work done in passing from one position
to the other is
Work = - C(Ncos 0' - NCOS 6).
Suppose that the coil be rotated. The work done is
easily calculated :
In the first quadrant,
work = - C (Ncos 90 - ^Ycoso ) = CN\
in the second quadrant,
work = - C(A^cos 180 - N cos 90) = CN;
in the third quadrant,
work = C(A^cos 270 A^cos 180) = CN\
and in the fourth quadrant,
work = - C(/Vcoso - A 7 cos 270) = - CN.
In the first half of the revolution, therefore, work equal
to 2 CN has to be done in order to more the coil, but in
42 NOTES ON
the latter half the coil will do the same amount of work.
The potential energy of the coil is, therefore, greatest when
= 1 80, or when the lines of force of the field are parallel
to those of the coil, but in the opposite direction, and if
the coil be then left free to move, it will rotate to make
6 0, doing work equal to 2QV, and then requiring
work to be done on it to cause further movement. On
the supposition already stated that the field is uniform,
N = HA, H being the intensity of the field and A the
area of the coil. The work done by the coil in rotating
through two quadrants may then be expressed as 2CHA,
this also measuring the work done on the coil in the other
two quadrants.
As a resume of the above, we have the rule, that a mov-
able circuit in a magnetic field tends to place itself so
as to enclose the maximum number of lines of force in
right-handed cyclical order.
37. To Calculate the Intensity of the Field due to a
Voltaic Circuit.
The force acting on unit pole, or the intensity of the field, is
by Note 35 the rate of change of po-
tential per unit of length. The in-
tensity of field at a distance x is
The difficulty of calculating the
value of oo makes the general solu-
tion extremely complicated. It is,
however, easy to calculate the inten-
sity of the field at any point on the
P. axis of the circuit, as in that case
GO 2it (i cos 6).
Let x = the distance of the point A from the circuit.
r r= radius of the coil.
ELECTRICITY AND MAGNETISM.
43
Then dV = zrtC.d (L cos 0) cos =
dx
2.x Cr*
At the centre of the circle the force is a maximum, and is
= as in 195. The sign shows that with a posi-
tive current the force is one of attraction.
38. Position of Equilibrium of a Circuit and Magnet.
Consider the magnet as composed of two poles of strength
m and m connected rigidly. -.
The formula for the force in the
field may be written
= sin 3 0.
The forces acting on the two
poles are
and
/I
1
1
\ ^v.
e X & >
/
A B
--- sin
r
Fig. 10.
The resultant force is
~ (sin^ - sin' G').
This is zero when = 0', or when the centre of the magnet
is at the centre of the coil. In any other position there will be
a force acting and the equilibrium will be unstable. If, there-
fore, either the coil or magnet is free to move, the coil will
NOTES ON
place itself so that the middle of the magnet is at its centre.
(See Fig. 87, " Thompson.")
39. Mutual Potential of two Circuits ( 320).
The work done in bringing one circuit up to another, or
the " mutual potential " of the two circuits is, as given in
, cos s
| 320, cc . ss . This expression is one of great
theoretical importance, but its derivation is difficult and
out of place in an elementary treatise.
The work done in moving a circuit near a pole or in a
field has already been shown to be NC, and it is
obviously immaterial whether the lines offeree TV are due
to a pole, a magnetic shell or another circuit. Suppose
two circuits A and B, carrying currents of strength C
and C,' and let N^ be the number of lines of force
due to A enclosed by B, and N* the number due to B
enclosed by A. If B is moved out of the field caused
by A the work done is A^ C If A is now moved so as to
resume its former relative position to B the work done is
N^ C'. The coils are now in the same relative position
as at first and if there are no external magnetic forces, no
work can have been done in moving the system. Hence
Ni C-N n C = O.
If the current in each is of unit strength
.M = N v
or each encloses the same number of the other's lines of
force. Returning to the expression for the work done.
CC ^- ss', and making Cand C' each unity, the number
of lines enclosed by each is - ss', and this number may
ELECTRICITY AND MAGNETISM. 45
be represented by the single symbol J/and is dependent
only on the position and areas of the two coils.
Let the planes of the two circuits be parallel, and the
current flowing in the same direction in each, that in
which the hands of a watch move. The negative sign of
the formula shows that the circuits attract each other, and
this is also evident from Maxwell's rule that a voltaic cir-
cuit free to move always places itself so as to enclose the
greatest possible number of lines of force. The nearer the
circuits, the greater the value of M, and if they become
coincident M would be , or infinite. As, however, r will
o
always have a finite value, the maximum value of M
exists when the coils are touching, or r is a minimum.
As the coils tend to approach or to diminish, r, the
" coefficient of mutual potential " M, always tends to a
maximum. This quantity is hereafter referred to (Note
63) as the " coefficient of mutual induction."
40. Conversion of Units ( 324).
The use of the dimensions of units in passing from one
system to another has been illustrated in Note 19. In
electrical calculations, the most frequent change to be
made is that from the C.G.S. system to the British units
based on the foot, grain, and second. The ratios between
these units are shown in the following table from Jenkin's
" Electricity."
46
NOTES ON
i
a
Number
of C.G.S. 5
Units in j^J
one British bco
Unit (^4). ^
f
Number
of British
Units in
one C.G.S.
Unit (B).
g Mass
o B T T,-.S
| r Length .
.
"O
Time.. .
r
1.1'
|] Force ._
p
1;
["Work
*
60.198
1.7795820
2.2204179
0.01661185
;*'
Quantity ...
?
42.8346
1.631794912.3682051
0.0233456
3
Current.
* or
C
42.8346
1.6317949
2.3682051
0.0233456
^
g
Potential
V
1.40536 0.14778741.8522125
o.7 T i=;6i
i
Resistance
0.03280899 2.5159929 i. 4040071^0.4704!;
w
Capacity
r
30.47945 !l-4S4007i;2.^I!?QQ2Q
0.03280899
r
Strength of Pole...
m
42.8346
1.6317949
2.3682051
0.0233456
gl Magnetic Potential.
V
1.40536
0.1477874 1.8522125
0.711561
g Intensity of Field...
H
0.0461085
2.6637804
1.3362196
21.6880
*
o
c
Current
i or
C
1.40536
0.1477874
1.8522125
0.711561
i
Quantity ... j @
1.40536
0.1477874
I.8522I25
0.711561
-Potential
Electromotive F'ce f
F
E
42.8346
1.6317949
2.3682051
0-0,33456
o
Resistance. . .
E>
Q
Q o
5
Capacity
c
30.47945 IJ^J^^/i
0.0328089912. 5159929
1.4840071
30.47945
The following table, showing the relations between the
practical units in common use, may be convenient for
reference :
i metre = 39.37043 inches = 3.28087 feet.
I kilogramme = 2.20462 avoirdupois pounds.
i kilogrammetre, or i kilogramme raised one metre
per second = 7.23307 foot-pounds per second.
ELECTRICITY AND MAGNETISM. 47
i Force cle cheval or French H. P. = 75 kilogram-
metres per second.
I English H. P. = 33000 foot-pounds per minute.
= 550 " " " second.
:= 76.04 kilogrammetres per second.
= 1.014 force de cheval,
I gramme 981 dynes (980.868 at Paris),
i gramme centimetre, or I gramme raised one centi-
metre in a second = 981 ergs.
I pound avoirdupois = 4.45 x io 5 dynes nearly.
I foot-pound per second = 1.356 x io 7 ergs nearly.
i Volt = io 8 absolute electromagnetic units of potential,
i Ohm io 9 " electromagnetic units of resistance.
Practical Ohrn = .9895 x io 9 absolute units (Lord Ray-
leigh).
I Ampere = nj of an absolute electromagnetic unit of
current.
i Coulomb = iV of an absolute electromagnetic unit of
quantity.
= I ampere per second.
i Farad = io~ 9 absolute electromagnetic units of ca-
pacity.
i Watt (see Note 55).
= IO T ergs = .7373 foot-pounds,
= TJB- English H. P.
I thermal unit = i gramme of water raised i C.
I Joule = io 7 ergs = .24 thermal unit (See Note 55).
Mechanical equivalent of heat = 772 foot-pounds i F.
= 1390- rc.
Same in metric system = 424 kilogrammetres i C.
= 42400 gramme-centimetres i C.
= 4.16 x io 7 ergs.
i Siemens unit = .9536 practical ohm.
48 NOTES ON
i Jacobi = current evolving i Cm '. of mixed gas per
minute at o and 760 mm.
= .095 amperes.
At any place the weight of the gramme is equal to g
dynes. The value of g for any latitude may be found
approximately from the formula
g 980.6056 2.5028 cos 2/1 .000003^,
A being the latitude and h the height above sea level. The
limiting values of g are 978.1 at the equator and 983.1 at
the poles.
41. Determination of the Horizontal Component of the
Earth's Magnetism ( 325 a).
The time of vibration of a particle acted on by a con-
stant force is / = it , / being the time of a half or sim-
ple vibration, and ju the acceleration. The latter is in any
case equal to the moment of the impressed forces divided
by the moment of inertia. When the arcs of vibration
are small this may be applied to a magnet oscillating in a
uniform field and the time of a complete or double oscilla-
tion of a magnet is therefore
K being the moment of inertia. M the magnetic mo-
ment and //the horizontal intensity.
A, To make the observation, a magnet is allowed to
oscillate and the time of vibration is determined as ac-
ELECTRICITY AND MAGNETISM. 49
curately as possible. This is best done by determining the
approximate time of one oscillation, and allowing the
magnet to oscillate a known time. Dividing this time by
the approximate time of one oscillation gives the approxi-
mate number of oscillations. Taking the nearest whole
number to this, and dividing the whole time by it gives
the exact time of one vibration. If the approximate num-
ber of vibrations fell midway between two whole numbers,
the observation would have to be repeated until it was
known with certainty how many oscillations had been made
in the observed time. The oscillations must be of small
amplitude, and in very exact observations must be reduced
to an infinitely small arc. If possible, the magnet should
be supported by a single fibre to avoid torsion, but if a
wire has to be used, the torsion must be allowed for. If
the magnet is of simple form, either bar or cylindrical, the
value of K may be determined from the formulas ing 325^,
but if these do not apply, A' may be determined by observa-
tion of the time of vibration of the magnet, and of the
time, /,, when its moment of inertia is increased by the
addition of a weiht of known moment K l .
From (i) the value ofMffis - . . . (2)
i
When the weight is added, MH = ^ , + ^
or K =
Knowing A" it is possible to compute Mfffrom (i).
In the case of the oscillating magnet, the magnet and
earth acted mutually on each other. In order to obtain
the ratio T-^the force of the magnet must act against that
NOTES ON
of the earth, and this is done by measuring the deflection
from the magnetic meridian that
the magnet will cause in a small
magnetic needle near it. A'S is
the magnet the time of whose
oscillation has been determined,
and it is placed at right angles to
the magnetic meridian, so that
its centre is due north or south of
the small needle at O. Let r be
the distance of either A^or S from
(9, and m -be the strength of N
N 1 ^S and S. Then the force exerted
by S on a south pole in 1 at O is
Fig. ii.
in the direction Oa, which may be taken to represent
it in magnitude and direction. Similarly Oc would
represent the force of attraction of the pole N. The re-
sultant force is Ob:
From similar triangles
Oa : Ob : : So : NS ;
or calling Ob, T, and NS t L,
mm' mm' L
t-:T::r:L .\T- -=.
Let M' = ;'/', the moment of the small magnet, and
M = mL the moment of the large one. Then the
couples acting on the small needle when it has a perma-
nent deflection are ;
, a <*** mm LI' MM'
to deflect, 77 cos = cos 6 = - 3 cos 6 ;
to retain in the meridian, m'l'H sin = M' //sin 0.
Equating these moments and reducing
ELECTRICITY AND MAGNETISM. 51
^ = r 3 tan0 ....... (3)
By combining (2) and (3)
J3, In the Kew Magnetometer the deflecting magnet is
placed east or west of the small magnet instead of north
or south, and the formula is different.
Let r be the distance between the pole s and the centre
;*
Fig. 12.
of the deflecting magnet, L be the length of the deflecting
magnet, m the strength of pole of the deflecting and m' that
of the deflected magnet, and /' the length of the latter. The
mm'
force of repulsion exerted by 5 on j is -- - , r be-
ing great in comparison with L. The force of attraction
of TV on s is - . These forces may be considered
as acting in the same line but in opposite directions, and
the resultant force acting on s is
= mm'
z
r
_L _ \
L\* }
r + - ) I
a/
52 NOTES ON
(5)
The moment of the deflecting couple on the small
magnet is
FT cos = 2AfM' . cos ,
r-?) 2
or since as above, L is small in comparison with r, its
square and higher powers may be neglected, and the
moment of the couple is
*MM' .- !l_cosfl =
4
The moment of the couple tending to retain the small
magnet in the meridian is
M'Hsm 0.
Equating and reducing
^=ir'tan0 (6)
C. This derivation contains many assumptions. A
more rigorously correct formula is that given by Kohl-
rausch,
M i r * tan 9 ~~ ri * tan 6 '
7f ~ a ~ ""f^^C
In which r and ri are the distances between the centres
of the magnets in two successive positions, and and
0' the corresponding angles of deflection. The deflectir. B
magnet is placed east or west of the deflected needle as
in the last case.
ELECTRICITY AND MAGNETISM. 53
The formula of (6) is the one generally used in work with
the Kew Magnetometer, but is true only when r is large in
comparison with /,. The more accurate formula can be readily
derived. From equation (5) :
r'
The moment of the deflecting couple is
zMM'
Fl cos6 = T-5T- Li + - ' ~? )cos
Equating this with M' H sin 0, the moment tending to retain
the couple in the meridian, and reducing,
By repeating the observation by placing the deflecting mag-
net so that its centre is at a distance of r\ from the needle, a
new deflection, 0', is obtained, and
Multiplying (7) by r 5 and (8) by r, 8 , and subtracting the lat
ter from the former,
r* tan - r, 6 tan C' = 2 . ^ (r 1 - r 2 ) ;
H
or,
54 NOTES ON
IV. MEASUREMENTS AND FORMULAE.
42. Solenoids ( 327). Ampere's Theory of Magnetism
< 338).
As stated in \ 327, a spiral coil of wire through which
a current passes is called a solenoid. The definition has
already been given as that of a magnetic filament uni-
formly magnetized, and as a spiral coil carrying a current
exerts the same forces, and is similarly acted upon in a
magnetic field, it is called by the same name. Theo-
retically the turns of the coil should be exactly parallel,
and at right angles to the longitudinal axis, but as this is
impossible the ends of the helix are brought in through
the coil from each end to the centre as in Fig. 116. The
current flowing in these branches exerts an effect equal
and opposite to that due to the longitudinal component of
the spiral, and the resultant effect of a solenoid thus con-
structed is that of a number of parallel turns only, or "*
the theoretical solenoid.
If the helix be free to move it will, when a current is
passed through it, move so as to include the maximum
number of lines of force in the field. In the earth's field, it
therefore assumes the same position as the dipping needle.
The end pointing north is called the north pole of the sole-
noid, as with the ordinary bar magnet. Let a solenoid
be suspended so as to move freely, and a magnet be
brought near it so that the north poles are nearest. The
solenoid will be repelled, but if after repulsion, the south
pole of the magnet is brought up, the north pole of the
solenoid is attracted. Magnetic forces are evidently act-
ing between the coil through which a current is flowing
ELECTRICITY AND MAGNETISM.
and the piece of steel which we call a magnet. Many
other similar effects have been alluded to, and they
suggested to Ampere a theory of magnetism, which ex-
plains very many peculiar relations and is of great prac-
tical utility. He conceived that a magnet was composed
of a great number of molecules, around each of which
flowed an electric current in a constant direction. In an
ordinary unmagnetized bar of steel, these currents lie in
all possible planes, so that their resultant magnetic effect
is zero. If, however, the bar be magnetized, the energy
expended in so doing operates to turn the molecules so
that the currents are now parallel. In looking at the end
of a magnet, the molecu-
lar currents would, as
shown in the figure, coun-
teract each other in the
substance of the magnet,
but the current on the
outer edge of the outer
row of molecules being
unbalanced would cause
a resultant current on the
surface in the direction
opposite to the motion of
the hands of a clock. In
a solenoid a current flows in this direction when looking
at the north pole of the solenoid, and the figure therefore
shows the theoretical condition of the north pole of the
magnet. Looking at the other face the currents appear to
flow in the opposite direction. From these suppositions
follows the rule given on p. 284. The theory explains many
peculiar effects, such as those referred to in \ 112 and
| 113, but is no more than a theory. It seems unques-
tionable that the process of magnetization is attended by
56 NOTES ON
molecular movements, but it is not proved that magnetism
is due to molecular currents. Prof. Hughes has recently
experimented on tempered steel, using an ingenious mod-
ification of the induction balance, and asserts his belief
that each molecule possesses magnetic polarity. In tem-
pered steel the molecules are comparatively fixed, where-
as in soft iron they possess considerable freedom of move-
ment. He thinks the process of magnetization is merely
one of molecular movement, by which the similar poles of
the molecules are brought facing the same way. This
final position is retained by the steel but lost by the iron.
Ampere investigated the mutual action of magnets and
currents by the theory of action at a distance between
currents, but if in Fig. 13 the current in each molecule is
supposed to have one line of force, the aggregation of
molecules would produce the large number of lines all
passing in the same direction that the magnet is found to
possess, and the theory of lines of force due to Faraday
and Maxwell explains all magnetic phenomena as well if
not better than the method of Ampere.
43. Best Arrangement of Cells ( 351).
From general formula, letting w be the total number of
cells, w m n
mE R E * . . (i)
mr
, r R
mr R
4~ ./
t -i
4
n
n m
w m
* Proof without calculus.
In equation (i) + - = 2 , / ' Rr + ( / r.:r _ / R
V \V ^~ V
This is a minimum when the square it contains is zero, or when
in-r mr
In that case R = as before.
w n
ELECTRICITY AND MAGNETISM. 57
This is a maximum when the denominator is a mini-
mum. Differentiating with regard to m and making first
derivative equal zero.
du r R r R m^r mr
dm ~~ iv ~ m* ~ w ~ m' 1 O1 w ' ' n
But is the internal resistance of the battery. Hence
n
the rule, that the best arrangement is secured when the
internal resistance of the battery equals the external
resistance in circuit.
44. Long and Short Coil Galvanometers ( 352).
In the use of a galvanometer it is desirable that it
should produce a readable deflection without greatly re-
ducing the current. If the current is large a single turn
of wire will cause a sufficiently strong field, but if the cur-
rent is small, it is necessary to multiply its effect by pass-
ing it through many turns in order to obtain a good
deflection.
By Note 37 the field at the centre of the coil is
in a galvanometer having n> and - - in another having
only one turn around its needle. The resistance of the
first will be nearly n times that of the second. Let all
the resistance in the circuit external to the galvanometer
be r, and g be the resistance of the galvanometer, and E
the E. M. F., supposed constant. Then
C = ^ and C =
r + g r + ng
\{r is small C= nC' nearly, or the current is reduced
by the high resistance galvanometer in almost the same
58 NOTES ON
ratio that the field is increased. There is, therefore, no
gain, and the great reduction of the current renders the
use of such an instrument inadvisable. If r is large, the
resistance is not increased n times by the introduction of
ng instead of g, and C is <nC', or the field is strength-
ened in a greater ratio than the current is decreased.
There is, therefore, a gain in using a high resistance gal-
vanometer. If r. is very large, a galvanometer of many
turns must be used to obtain any deflection at all.
45. Divided Circuits ( 353).
It is at first difficult to understand how introducing
another resistance in parallel circuit can reduce the total
resistance, but it is to be recollected that the current pos-
sesses a greater number of paths to flow through. By
the law that the current divides proportionally to the con-
ductivities of the branches, goes through one branch
and through the other. Let R be a resistance such
that the same current would pass through it as through
both r and r 1 '. Then
ill _
-?; = + -7 or 7? =
R r r' r + r'
r' r
But - is less than r> and - - is less than r. The
resistance of a divided circuit is, therefore, always less
than that of any of the resistances entering into it. If
there are three conductors, r, r', r", we have
ELECTRICITY AND MAGNETISM. 59
= + ,+ = conductivity,
R r r .r"
and R = - , "T , .
rr + rr' + r r 1
To find the current in each branch : Let C be the total
current, B the battery resistance, O be the current in r',
and C" that in r". Then
The currents are inversely as the resistances through
which they flow. Taking, therefore, the resistances be-
tween A and B t Figure 129,
r' r" r"
also C: C" : : r" : r ' r " or C" -C -7-^-7, . . (2)
r' + r" r' + r"
46. Shunts.
These formulas are of great importance in the use ot
shunts for galvanometers. If a current is so powerful
that there is danger of its injuring the galvanometer coils,
or if it produces a deflection too near 90, the galvanom-
eter may be shunted by introducing a resistance in
parallel circuit, so that less current will pass through
the galvanometer.
Letting C be the
current when the
galvanometer is B
unshunted, C' the
total current when Fiff- *4-
the galvanometer is shunted, C g and C s the currents
60 NOTES ON
through the galvanometer and shunt respectively, G the
resistance of the galvanometer, S that of the shunt, and /?
all other resistance,
* E
But as ~ is less than G, C' is greater than C, or
Lr + O
the introduction of the shunt has increased the total cur-
rent in circuit. The current through the galvanometer is
from (i), Note 45,
c , 5^_ _ E S
>
(3)
j.\. -r ,
G +
ES
' R (G + S) + GS
If the deflections of the galvanometer are proportional
to the currents,
.-. d : d' : : R (G +
These formulas may be simplified if - ~ - be called
u. This proportion is sometimes called the multiplying
power of the shunt. By its use (3), the current through
C'
the galvanometer becomes . The current through
the shunt is C' The resistance of the shunted gal-
f~ *\ (^
variometer -^ - - becomes , and the resistance
G + 5 u
ELECTRICITY AND MAGNETISM. 6 1
necessary to add to the circuit to retain the same
total current is G = G ( U ~ }. This ratio u is
u \ u J
that between the sensibility of the shunted and the un-
shunted galvanometer. Thus, if the resistance of a shunt
which will reduce the sensibility of a galvanometer of
1,000 ohms one hundred times is required,
looo + S _ 1000 i ~
U = IOO = ~ .'. O = I.OI = (jr.
S 99 99
If the current is kept the same by adding a resistance of
G ( - ), C C, and will pass through the galva-
\ u J u
nometer.
47. Kirchhoff's Laws (353).
The application of theae useful laws may be illustrated
by the figure. The fol- B
lowing equations are
derivable. From the
first law, that in any b-
network of wires the
algebraic sum of the
currents meeting at a v
point is zero. Fig. 15.
I
A
At A c = Ci + c^
MB c^ + c^ c.
From the second law, that in any closed circuit the
electromotive force is equal to the sum of the separate
resistances, each multiplied by the strength of current
flowing through it.
In the left hand circuit c (r + b] + c l r l = E.
In the right hand circuit r, r l c-ir* = zero.
62
NOTES ON
48. Fall of Potential ( 357).
Let A and B be the poles of a battery. They will have
different potentials, numerically equal but of opposite
signs, and the battery may be considered to preserve a
constant difference of potential between them. Connect
the points A and B through an external resistance. The
p potential will fall
along this resistance,
and it is required to
find the potential at
any point.
Draw A B to rep-
resent in length the
B value of the external
resistance, and at A
and B erect perpen-
diculars, one positive
and the other nega-
tive, to represent pro-
r- l6 - portionally the poten-
tials at these points.
From Kirchhoff 's second law, in any part of a circuit
of resistance r, E C r. In another part of resistance
r', E = C' r 1 , E and E' representing the difference of
potentials between the ends of the portions of the circuit
considered. If the resistances are both in the same cir-
cuit C = C', hence E \ E' : : r : r', or the differences of
potential in the same circuit are proportional to the resist-
ances through which they act. Assuming one potential
to be zero, the potentials at the other points vary directly
as the resistances separating them from the point of zero
potential, and as in the figure the horizontal line repre-
sents resistance and the ordinate BH potential, it is evi-
ELECTRICITY AND MAGNETISM. 63
dent that ordinates at other points cut off by the line CH
will correctly represent the potential at these points, as
they and they only will satisfy the above proportion.
The line CH represents, therefore, the fall of potential in
the resistance BC. If CH\s prolonged to K the triangles
I~4C and CHB are similar, and since by hypothesis
A K is equal to BH, AC = BC, or C, the point of zero
potential, is midway between the poles, and the line HK
represents the fall of potential along the resistance AB.
It is to be noticed that in the figure the difference or
potential between A and B is BH + AK \ or if Fis the
potential at A and + V at B, the difference of potential is
V - (- F) =2 V. The difference between A and D is
v ( V] = V + v. If the point C is connected with
the earth, as it is already at zero potential, the potential
is not changed anywhere in the circuit. If, however,
another point D t whose potential is + v t is connected to
earth its potential is lowered by the amount v, and as the
battery preserves a constant difference of potential be-
tween A and B, the absolute potential of all points in
the circuit is also lowered by the amount v. The fall
of potential, is, therefore, the same, and is represented by
drawing a line M N through D parallel to HK, and the
potential at any point is the length of the ordinate at that
point intercepted byMN.
The difference of potential between A and B is now
BN+ AM=
( F - -z/ ) + ( F + ?/ ) = 2 Fas before.
If the negative pole A of the battery is connected to
earth, the potential of all points of the circuit is raised
by the amount V, and the line of potential assumes the
position A P. The potential of every point of the circuit
NOTES ON
is now positive, but the differences of potential are the
same as at first.
49. Wheatstone's Bridge ( 358).
Connect the poles of a battery by two resistances, P Q
2
Fig. 17.
and PQ', and at P erect a perpendicular to represent the
difference of potential due to the battery. Then the lines
ZQ and ZQ will represent the fall of potential in the
resistances PQ and PQ'. The potential at any point on
QQ' being the ordinate cut off by the lines ZQ or ZQ, if
a galvanometer be joined to two points TV 7 and M at which
the ordinates A^Tand MY are equal there will be no de-
flection of the needle. But from the figure, since XY'is
parallel to QQ, the triangles ZRX and XNQ are similar
and NQ : RX : : A 7 X : RZ ;
RY::MY:RZ\ but NX = MY
RX: : MQ : RY,
B -.::>,
also
or
MQ
.NQ
A
as in Fig. 130 and % 358. When, therefore, a galvanom-
eter joined to the junctions of two pair of resistances
ELECTRICITY AND MAGNETISM.
through which a current is flowing shows no deflection,
the resistances are proportional to each other.
An important fact somewhat difficult to understand at
first is apparent from Fig. 17. The resistances PNQ and
PMQ' (see also Fig. 130) are unequal, while having the
same electromotive force acting in each. The currents
in the two branches are therefore unequal. Beginners
are liable to regard the balance in Wheatstone's Bridge as
due to an equality of currents ; but this is wrong, the
equality of potentials at the galvanometer terminals being
the condition of balance. This is equivalent to saying that
there is no current through the galvanometer, offering
another method of proof as follows :
50. Proof of Theory of Wheatstone's Bridge by Kirch-
hoff's Laws.
Let the currents in the different branches be represented by
A
Fig. 18.
c z , etc., and the corresponding resistances by r, r lf
5
66 NOTES ON
etc. Let j be the E. M. F. and b the resistance of the battery.
From the first law
At A f i / \
*** ** ^2 ^3 i C '5 . . . .. . . . (I)
B c = fa + d (2)
C* ^4 = c\ + c<> (3)
" D c = c* + c, (4)
By the tin ABC, c a r 3 - c t r t c*, G = . . . . (5)
second j " ADC, c* r- 2 + c<, G d ri = . . . . (6)
law, ( " BbDC, c (b -f r\ + Cl ^ + c t r t - E - O. (7)
Adjust ri,r<i,r 3 and r^ until the galvanometer shows no
deflection, then c$ is zero. The above equations being general,
substitute zero for d, and they become
(i) ci = c a (5) ca r a = d r 4 or c 2 r 3 c 4 r t . (8)
(3) d = fi (6) c* r* = c, r, or c, r, = c, r, . (9)
Dividing (8) by (9) - = - .
When, therefore, the galvanometer shows no deflection, the
arms of the bridge are proportional. The accuracy of the meas-
urement depends, of course, on the sensibility of the galvanom-
eter.
51. Measurement of Electromotive-Force ( 360).*
(a) WHEATSTONE'S METHOD.
Let G be the resistance of the galvanometer, R the remaining
resistance in circuit, and p the resistance necessary to add to
the second batteiy to reduce its deflection to that of the first.
Then for a deflection d
* The memoranda of Notes 51, 52 and 53 are confined strictly to an ex-
planation of the methods of measurements referred to in " Thompson's
Elementary Lessons," 360, 361 and 362 Other methods may be obtained
from any work on electrical measurements.
ELECTRICITY AND MAGNETISM.
E
E'
R + G
R -
\- p H
- G
whence
R 4- G
R -
\- P J
- G
(i)
To bring the deflection to d' , extra resistances have to be
added. Represent these by r and r 1 . For the deflection d 1
,
whence -
R + G + r
R + G
R + p + G + r 1
r'
-
R + p + G
. . (2)
Substituting (ij in (2) it reduces to
- = ; ; or, r' : r : : E' : E.
(t>) CLARK'S METHOD.
Clark's method requires three cells. E furnishes a current
and has the highest
E. M. F., E' is a \ C
standard cell, gener-
ally that of Clark
( 177), and v/ isthe
cell to be tested. If
E" has a higher
E. M. F. than 1.457
two or more Clark's
cells must be used
at E' '. The similar
poles of the cells
are connected to
A. In measuring,
Fig. 19.
E" is first disconnected and the needle of G' is brought to
zero by adjusting P. E" is then connected at A and the slid-
68 NOTES ON
ing piece C is moved along the resistance A B, shifting contact
until G" also shows no deflection.
From Kirchhoff's laws :
At A c + c' + c" K o . . . . (i)
" C K-c" - K' = o . . . . (2)
" B K' - c' -c =o .... (3)
In circuit G'AB c' (r> + G'} + Ka + A"b - E' - o . (4)
A G"C c" (r" + G") + Ka - E" = o . (5)
But when adjustment is secured c' and c" are each zero.
Substituting these values
from (\)c K from (2) K - K' from (3) c = K'
from (4) Ka + K'b = ' ; or, K (a + b) = E' . . . (7)
from (5) Ka - E" . (8)
Combining (7) and (8)
E' : E" : : a + b : a.
(C) QUADRANT ELECTROMETER.
The two poles of a standard cell are connected to the quad-
rants, the same pole being in connection with opposite seg-
ments, as I and 3, Fig. 101. The deflection of the needle is
then noted. The standard cell is then disconnected and the
one to be tested substituted in the same way. From the ratio
of the deflections, the ratio of the electro-motive forces may be
obtained. Care must be taken that the needle is electrified to
the same potential in the two measurements.
52. Measurements of Internal Resistance ( 361).
(a) Connect the battery in circuit with a galvanometer and a
box of resistance coils, the resistances being B, G and R re-
spectively. Note the deflection d in the galvanometer. In-
crease R to R' and note the deflection d' . Then if a tangent
galvanometer is used
ELECTRICITY AND MAGNETISM.
69
tan d:ia.nd'
Reducing,
Jj =
R' tan d' R tan d
- : - r
tan d tan ^
_
Cr.
If G is of no resistance, and the first deflection was taken
with no other appreciable resistance than that of the battery in
circuit
_ R' *an</'
tan d tan d ' '
(c) MANGE'S METHOD.
The cell whose resistance is to be measured is placed in the
bridge as an unknown resistance, and a galvanometer and key
(not in the same branch) are connected as in the figure. The
arrows denoting the direction
of the current, it is at once
evident that a current passes
through the galvanometer all
the time, and continues to do
so unless the resistance in c
and d becomes zero. Every
change of resistance in a, c
or d will affect the current
flowing through G, as will also
the opening or closing of the
key k when any difference of
potential exists between A and B. If on pressing k there is no
change in the galvanometer deflection, it follows that A and B
are at the same potential and that consequently
a : b :: c : d or b ,
c
the common relation of the Wheatstone bridge. If the galva-
nometer is a sensitive one, the deflection will be nearly 90 and as
in that position a change of current produces but little effect on
Fig. 20.
70
NOTES ON
the needle it is necessary to reduce the deflection either by the
use of a magnet or by shunting the galvanometer.
53. Measurement of the Capacity of a Condenser ( 362).
(a) Let V = the known potential,
x the unknown capacity,
C the capacity of the standard condenser.
The original charge in the condenser of unknown capacity
is Vx. On connecting the standard condenser the capacity is
increased, and the quantity being the same the potential falls
to V"
.'. Vx = V (x + C\
and x : x + C : : V : V.
(8) The impulse acting to deflect the needle varies as the
quantity of electricity passing. If the two condensers are
charged from the same cell the potential is equal. Conse-
quently (Note 5),
sini</: sin**/' : : VC : W
:: C: C .
(c) There is apparently some error in stating this method in
"Thompson." It probably refers to a common and quite ac-
curate method applied by Sir Wm. Thomson to the measure-
ment of the capacity of cables.
If the poles of a battery are
connected through a high resist-
ance I?i + J? and a point F is
connected with earth, its poten-
tial becomes zero and points on
either side separated from it by
equal resistances are at equal but
opposite potentials. If the re-
sistances are unequal,
ELECTRICITY AND MAGNETISM. 7 1
Potential at A : Potential at C : : RI : R,
or V, : F 2 : : R, : R,
/=
Charge the t\vo condensers by making contact on opposite
sides of F at such points that their charges just neutralize. If
A and C are the points,
(</) Charge the condenser and discharge it through a galva-
nometer, noting the deflection ; charge it again to the original
potential and allow it to discharge slowly through a very high
resistance. After discharging a definite time T, note the deflec-
tion it will give when connected directly to the galvanometer.
Then
T
Capacity = - ;
2.303^ log
V and v are potentials, but as the ratio only is needed, this can
be obtained from the ratio of the two deflections.
54. Determination of the Ohm ( 364).
If a coil is rotated in a magnetic field a current is in-
duced, which deflects a needle at the centre of the coil.
The force exerted by this current may be shown to be
, and the moment acting on a magnet deflected
%K?JK.
L? VH
through an angle is ^j-^ ml cos 0, where L is the
^K 1\
length of the coil, Fthe velocity of rotation, H the inten-
sity of the field, K the radius of the coil and R its resist-
ance. The moment of the force of the earth's magnetism
72 NOTES ON
tending to bring the deflected magnet into the meridian
is Hml sin 6. Equating these moments when the needle
maintains a steady deflection,
ml cos = Hml sin 6 ;
The value of R is given in absolute units of resistance.
55. Practical Electromagnetic Uuits of Heat ( 367)
and of Work ( 378).
As shown in Note 7, the measure of the work done in
moving a quantity of electricity is the product of the
quantity of electricity by the difference of potential
through which it is moved, or
Work = Quantity ( V, - F 2 ) = Ct x E.
If the current, time, and electromotive force are all
expressed in absolute units, the work is given in ergs.
Substituting for the practical units of current and elec-
tromotive force, the ampere and the volt, their values in
absolute units,
Work = io * x io 8 = io 7 ergs per second.
This equation gives a practical unit of work, correspond-
ing to the practical units of current and electromotive
force referred to, and Dr. Siemens has proposed that it
be called the "watt" This suggestion has been well
received, and the unit is coming into use in electrical
calculations. The value of the watt is io 7 ergs, and it
may be defined as the work done by a current of one am-
ELECTRICITY AND MAGNETISM. 73
pere in a portion of the circuit in which the potential falls
one volt.
I H. P. English = 33000 foot-pounds per minute.
= 550 " " " second.
= 76.04 kilogrammetres per second.
== 76.04 x io 5 gramme-centimetres per
second.
z= 76.04 x 981 x 10* ergs per second.
= 746 x io 7 ergs = 746 watts.
To find, therefore, the work done by an electric cur-
rent in any portion of a circuit, measure the differ-
ence of potential in volts between the ends of the por-
tion considered, multiply it by the current in amperes,
and divide by 746. The quotient is the work in horse-
power. In the above calculation the dimensional equation
of work cannot be used for the change of foot-pounds to
kilogrammetres, as these are statical units, and the dimen-
sions are for dynamical units only.
As heat and work are the same, the heat measured
in ergs given off in any part of a circuit is also CEt, or
substituting the value of E from Ohm's Law,
Heat in ergs C z Rt.
This formula is more generally used, as it gives the
amount of heat developed in a resistance. Substituting
as before the values of the ampere and ohm in absolute
units, the heat is measured in practical units, each of
which is of the value of io 7 ergs. This unit is some-
times called the "joule," and may be defined as the heat
evolved in one second by a current of one ampere in a
resistance of one ohm.
To obtain the value of the joule in terms of the more
74 NOTES ON
common thermal unit, which is the amount of heat nec-
essary to raise one gramme of water one degree centi-
grade, it is necessary to use the mechanical equivalent of
heat, which is 42,400 gramme-centimetres. A gramme of
water in falling 42,400 centimetres acquires sufficient
energy to raise its temperature one degree centigrade if
suddenly stopped. The thermal unit, therefore, equals
42,400 gramme-centimetres or 42,400 x 981 ergs = 4.16
x io 7 ergs, or approximately 4.2 x io 7 erg's. But as
the joule is io 7 ergs, the water-gramme-degree heat unit is
equal to 4.2 joules, or the joule is .238 of the thermal
unit referred to.
The watt and joule are of the same value, but one ex-
presses the energy given off in a circuit in terms of power,
and the other in heat.
ELECTRICITY AND MAGNETISM. 75
V. ELECTRIC LIGHTS.
56. The Voltaic Arc ( 371).
If a powerful electric current is broken at any point,
there is a bright spark at the break, and if the two termi-
nals of the circuit on each side of the break are kept at a con-
stant short distance, a steady light or arc will be produced.
The color of the light varies with the materials between
which the arc is formed. The heat produced is the highest
known, and most substances fuse so readily in the arc
that they cannot be used for electrodes. For this reason
gas carbon, which is practically infusible and of compar-
atively low electrical resistance, is universally used for the
pencils of arc lights. If the image of the arc is thrown
on a screen the greater part of the light is seen to be due
to the carbon points being heated white hot, the arc itself
being generally bluish and less brilliant. If a magnet is
brought near the arc, the interaction of magnets and
currents is well illustrated by the movements of the arc
to one side, and it is even possible to deflect it until it
assumes the position of a blowpipe flame.
The electric light was prevented from coming into every-
day use, so long as the current it required could not be
obtained from any cheaper source than the voltaic battery.
The invention of dynamo machines, by greatly decreasing
the cost of electric energy, rendered the extensive use of
the arc light a possibility. This may be best illustrated
by an example.
A Number 7 Brush machine works sixteen arc lights in
NOTES ON
series. The E. M, F. of the circuit is 839 volts, the inter-
nal resistance of the machine 10.5 ohms, the resistance of
the lights and leading wires 73 ohms and the current 10
amperes. Assuming the electromotive force of a Grove
2
cell to be 1.8 volts and its internal resistance , thenum-
10
her of cells necessary to do the work of the machine may
be calculated.
To obtain the E. M. F. the number of cells necessary is
| = 466 in series. But 466 cells in series have an inter-
I.o
nal resistance of 93.2 ohms, and that the current may be
the same, enough cells must be introduced in arc to re-
duce the battery resistance to that of the machine.
10.5
is nearly nine, and it is therefore necessary to use 466
groups in series, each group containing 9 cells in arc or
4,194 cells in all. This number of cells would cost ten
times as much as the machine, and could not keep up the
current for more than two or three hours.
Calculating in the same way with a constant battery,
assuming the E. M. F. of a gravity cell to be 1.08 volts
and its internal resistance 5 ohms, it will be found neces-
sary to arrange 777 groups in series, each group contain-
ing 369 cells in parallel arc, a total of 286,713 cells, a
greater number in all probability than are in use in the
United States. The invention of the dynamo machine
has therefore not only operated to diminish the cost of the
electric light, but to bring it within the bounds of com-
mercial practicability.
As a general rule, the light given out by an arc varies
as the current. A small current will give no light at all,
and, as stated in 371, a certain electromotive force and
current are necessary for the production of a satisfactory
ELECTRICITY AND MAGNETISM.
77
light. After that point a stronger current causes more
light.
57. Arc Lamps ( 372).
Gas carbon pencils are now used almost exclusively, and are
frequently coated with a thin film of copper to prevent the
oxidation and waste of the carbon before it becomes incandes-
cent. Two general methods are in use for regulating the dis-
tance of the carbons from each other, one using clock-work and
the other regulating directly by the current. The most widely
used of the latter class is the Brush lamp, a plan of which is
given in the figure. The current entering at A, divides at B
into two branches which pass
around the bobbin C in oppo-
site directions, one branch be-
ing a coarse wire of low resist-
ance and in the same circuit
as the carbons, and the other
branch S S being a shunt of
high resistance to the lamp,
connecting the terminals D and
G. Inside the bobbin is a soft
iron armature F, which is at-
tached to the upper carbon.
When a current passes the two
branch circuits on the bobbin
C tend to magnetize it in oppo-
site directions, but the resist-
ances and number of turns in
the two circuits are so propor-
tioned that the magnetic field due to the low resistance branch
is the stronger, and the armature F is therefore attracted, lift-
ing the upper carbon and establishing the arc. Should the
carbons become too widely separated the resistance of the arc
and consequently of the coarse wire circuit on C increases,
diminishing the current in C and increasing that in the shunt S.
NOTES ON
The field due to the shunt is therefore strengthened and that
due to the coarse wire diminished, allowing the armature F to
fall slightly, bringing the carbons nearer together. By the de-
vice of the two opposing fields, due to the coils on C being
wound in opposite directions, the feeding of the lamp is there-
fore done automatically, and the actual distance of the two car-
bons varies but little. In the lamp as constructed, two bobbins
are used in parallel arc, and the armature F clutches the upper
carbon. A plunger moving in glycerine is also attached to the
upper carbon to render the movements less sudden, and the
shunt circuit S S passes around another bobbin, which, by at-
tracting an armature, closes the main circuit, and short circuits
the lamp in case the carbons are broken or the adjustment
does not work. The figure is designed merely to illustrate the
general method.
The Foucault regulator shown in Fig. 138 is more com-
plicated than the Brush. The two carbons are clamped to
rods which are moved by clock-work, the gearing being such
that the positive carbon moves twice as rapidly as the negative.
In this way the arc is kept approximately in the same position,
admitting of focussing in a projecting apparatus. The clock-
work consists mainly of two barrels driven by springs and act-
ing through gearing on the rods carrying the carbons, one bar-
rel separating the carbons and the other bringing them nearer
together. The electromagnet seen in the base of Fig. 138 is
in the main circuit, and its armature works a system of levers,
the last one of which acts by a pawl on two small flies, each
connected through the gearing with one of the barrels. If, then,
the current becomes too strong, the armaiure is attracted, one of
the fiies is released and the corresponding barrel sets the clock-
work in operation, separating the carbons slightly. As they
approach their normal distance apart the current diminishes, and
the armature moves away slightly, pawling the fly. As the car-
bons burn away, the current is diminished still more, the arma-
ture is less attracted, the other fly is released and the clock-
work moves to bring the carbons nearer together. The arc
ELECTRICITY AND MAGNETISM. 79
may be formed at any height by moving the upper carbon by
means of the rod at the top of the regulator.
58. Incandescent Lamps ( 374).
That electric lighting may be of universal utility, lamps
are necessary which give only a moderate light. Arc
lights cannot be made to work with certainty and econ-
omy at low illuminating power, and the incandescent
lamp is therefore coming into use. In this case the light
is given off from a portion of the circuit which is heated
white hot by the passage of the current, and this requires
that that portion shall be of high resistance (g 367) and
practically infusible. After long experimenting, carbon
has been fixed upon as the best material, and it is now
used in all types of incandescent lamps. A small fibre of.
carbon is obtained by heating some vegetable substance
out of contact with the air and driving off all volatile
matter. Edison uses bamboo fibre, Swan cotton thread,
Lane-Fox a grass fibre, and Maxim paper. This fibre is
mounted in a vacuum, on the perfection of which depends
greatly the time the lamp will last, the presence of a small
amount of oxygen insuring the destruction of the fibre by
chemical action. Even in a perfect vacuum the fibre will
eventually give way on account of what is known as the
" Crookes's effect," a transfer of molecules of carbon across
from one heel of the carbon to the other. Alternate cur-
rents by wearing each heel away equally tend to lengthen
the lifetime of the lamp, but in every case the final giving
way of the fibre is a matter of time only. The lifetime
may be greatly prolonged by working the lamp below its
normal power. The life of an Edison lamp, working with
its normal current, is now (1883) probably about 1,000
hours.
8o NOTES ON
As pointed out, the condition that much heat should be
developed in any part of a circuit is, that that part should
be of high resistance. If several lamps were piaced in
series, the resistance of the circuit would be so great that
the current would be insufficient, and they are, therefore,
placed generally in parallel arc, reducing the external re-
sistance to such an extent that each lamp has it full share
of current. Assume a battery or machine giving an elec-
tromotive force of 200 volts at its terminals and having an
internal resistance of five ohms. If two lamps each of 100
ohms resistance, and requiring one ampere to give their
normal light, were placed in the exterior circuit in series,
the current would be = .075 amperes not suffi-
200+ 5
cient to work the lamps at their normal standard. By
placing the lamps, however, in parallel arc the same ma-
chine would, under the same conditions, work twenty
lamps, thus, =.- 20 amperes, or one in each lamp.
l -i +s
That all the lamps may give the same light, the resistances
must be equal, otherwise some will have stronger currents
passing through them than others. Placing a larger num-
ber in parallel arc will still further reduce the resistance,
strengthening the current, but increasing the number of
parts into which the current must divide. Thus in the
case above, the machine cannot work 21 lamps, for
= 20.47 amperes, or only .975 to each lamp.
A most important feature in incandescent lighting is the
change of resistance of carbon by temperature. The re-
sistance of metals increases with the current passed
through them, while that of carbon decreases, the reduc-
ELECTRICITY AND MAGNETISM.
8l
tion in an incandescent lamp being between 40 and 50
per cent. As the good working of any system of lighting
depends on a correct adjustment of resistances, this pecu-
liarity of carbon must be allowed for. The following
measurements made on an Edison lamp at the U. S. Tor-
pedo Station, Newport, illustrate the change of light and
of resistance accompanying a change of current.
Current.
Resistance.
Candles.
.000
134-
Lamp cold.
.114
H0.5
Cherry red.
.203
94-7
Bright red.
.309
86.7
Orange.
.440
8l.8
1.8 Candles.
.680
735
7.2 "
.750
72.8
11.4
.810
70.6
16.4
.890
69.9
2I.O "
It is seen that a current of to" f an ampere produced a
light just perceptible, but that every increase after that
produced a much greater proportionate increase of light.
The lamps are, therefore, more economical of energy the
more light they give, but working with high currents in-
sures the rapid destruction of the lamp. At present the
Edison lamps are the most economical, yielding under
good working conditions about twelve sixteen-candle
lamps per horse power of current (See 3 378).
6
82 NOTES ON
VI. ELECTRO-MAGNETIC INDUCTION.
59. Induction Currents Produced by Currents ( 393).
Take two coils of insulated wire, wound so that one
may be inserted inside the other, and place a battery
in circuit with one and a galvanometer with the other.
There is no communication between the two coils, the
wire being thoroughly insulated, but if the coil in circuit
with the battery is slowly inserted in that in circuit with
the galvanometer (Fig. 147), the latter will show a de-
flection, which is due to what is known as an induced cur-
rent. Experiment shows that this current is not continu-
ous, but exists only when one coil is moved near the other.
If the small coil is inside the larger, no current is ob-
served in the latter so long as the current in the former is
constant, but if it is broken, an induced current is observed
in the outer or " secondary " coil, flowing in the same di-
rection as that formerly existing in the "primary " or
inner coil. If the circuit is closed in the latter while it is
inside the secondary coil, the induced current in the second-
ary is in the opposite direction to that in the primary.
If the primary coil is gradually removed, a current is in-
duced in the secondary in the same direction as that in the
primary. There is no current in the secondary, unless
there is some change either in the strength of the primary
current, or in its position relatively to the secondary coil.
The same effects are produced by moving a magnet in
or near a coil in circuit with a galvanometer. If in this
case the direction of the Amperian currents (see Note 42)
be considered, the induced current in the coil will be in
ELECTRICITY AND MAGNETISM. 03
the opposite or in the same direction as that of the Am-
perian currents, as the magnet is introduced or withdrawn
from the coil. (See Fig. 146.)
Prof. Thompson generally uses the word " direct" as
applied to currents, to represent a positive current, one
flowing in the direction in which the hands of a watch
move, but in explaining these experiments he has used it
in another sense as meaning a current in the same direc-
tion, and confusion is apt to result. Whether a direct or
inverse current exists in a secondary circuit depends not
only on the motion of the primary coil, but on the direc-
tion of the current in it, but his explanation does not in-
clude the latter at all. Reserving the terms direct and
inverse to apply to the positive and negative directions,
the summing up on Page 360 may be expressed thus :
By
means
of a
Momentary currents
in the opposite direc-
tion are induced in the
secondary circuit
Momentary currents
in the same direction
are induced in the sec-
ondary circuit
Magnet
while approaching.
while receding.
Current
while approaching,
or beginning,
or increasing.
while receding,
or ending,
or decreasing.
These rules, when applied to magnets, call for a con-
ception of imaginary currents flowing around them. By
considering the induction of the current to be due to the
movement with or against magnetic forces, all the above
relations may be expressed by the rule given in 394 (i.).
84 NOTES ON
" A decrease in the number of lines of force which pass
through a circuit produces a current round the circuit
in the positive direction (i. e., a 'direct' current); while
an increase in the number of lines of force which pass
through the circuit produces a current in the negative
direction round the circuit" In the application of this
rule, care must be taken to look along the lines of force in
their positive direction, that in which a north pole tends
to move, as a current which appears to be direct when
viewing it from one side is inverse if seen from the other,
and the rule is worthless if misapplied.
60. Determination of the Induced Electromotive Force
( 394, ").
From the foregoing rules, it is seen that a current is
induced in a closed circuit only when there is some
change in the number of magnetic lines of force enclosed
by the circuit. This change may be due either to the
motion of the pole, to that of the circuit, or to the change of
current strength, if the field is due to a current. Each of
these three cases requires the expenditure of energy in
some form, and without such expenditure there is no
induced current. We are therefore led to look on the
energy of the induced current as directly derivable from
the energy expended in producing the change in the field.
Let a current flow from a battery through a coil placed
in a magnetic field. As already shown (Note 36) the
work done in producing any displacement of the coil is
C (TV, A?i), or if dN is the change in the number of
lines of force passing through the circuit in the time dl t
this may be written as C. dN. The current in flowing
through the coil heats it, the amount of heat being (| 36, -
C a A J . dt. If the coil is placed so that the lines offeree of
the field pass through it in the wrong direction, it will
ELECTRICITY AND MAGNETISM. 85
move, doing work which is due to the original energy of
the current in the circuit. Hence
Energy of the current = heating effect + work done ;
or, CE . dt = C* 7?. dt + C . dN
.-.x=cs + ...... o>
CR is the E. M. F. in the coil, while E is that originally
due to the cell. The former is less than the latter by
- , which must also be an E. M. F. and due to the
at
work done by the coil. As a result, therefore, of the
motion of the coil in the field, the E. M. F. originally in
dN
circuit is diminished by the amount 7- . The E. M. F.
remaining in the circuit is
= *- f ......
The induced E. M. F. is therefore measured by the
rate of change in the number of lines of force which pass
through the circuit, and is opposite in direction to that
originally existing, which caused the motion. By increas-
ing this rate, by diminishing dt, or, what is the same thing,
dN
making the velocity of motion greater, - may be made to
equal or exceed E, and the direction of the induced cur-
rent would therefore be the same whether there was any
E. M. F. to be overcome or not. If a battery current
flows, the induced current diminishes it ; if there is no bat-
tery current, the coil would have to be moved by external
agency, and the induced current is in the opposite direc-
86 NOTES ON
tion to a current which would cause the motion. This is
seen directly from (2). If E O
The fact that the induced current acts in an opposite
direction to that causing the motion is easily deduced from
the principle of conservation of energy, for, if a current
flowing in a given direction caused motion of the circuit,
and this motion induced a current in the same direction as
the original, it would increase the motion and consequently
the energy of the system. Lenz's law, given in $ 396, is
therefore a direct result of the conservation of energy.
On the supposition that the original E. M. F. is zero, or
that there is no current flowing when the coil is at rest,
O = C*RM +C. dN ..... (4)
The original energy being zero, the energy of the current
when the coil is moved can be obtained only from the
work done in moving the coil, or
Work done in moving coil = heating effect + work do^p
by the coil.
If the coil does no work, the total energy appears as heat
in the circuit.
dN
The E. M. F. induced in the circuit being the
dN
current is equal to , in which R is the total resistance
in circuit.
61. Practical Rule for Direction of Induced Current
( 395).
From the rule given in $ 186, " Suppose a man swim-
ming in the wire with the current, and that he turns so
ELECTRICITY AND MAGNETISM. 87
as to look along the lines of force in their positive di-
rection, then he and the conducting wire with him will
be urged towards his left," combined with Lenz's Law
(Note 62), the following rule for the direction of the in-
duced current in a conductor is easily deduced. Suppose
.a man swimming in any conductor to turn so as to look
along the lines of force in their positive direction ; then
if he and the conductor be moved toward his left hand
he will be swimming against the current induced by
this motion ; if he be moved toward his right hand the
current will be with him. Through some error this rule
is given incorrectly in $ 395, and differs there from the
rule as given by Prof. Thompson in his Cantor lectures.
62. Lenz's Law ( 396).
In Note 60 it was shown that in accordance with the
principle of the conservation of energy, the induced cur-
rent resulting from any motion of a conductor must be in the
opposite direction to that of the current which would cause
the motion; Lenz deduced this relation independently,
and his statement that " in all cases of electromagnetic
induction the induced currents have such a direction
that their reaction tends to stop the motion which pro-
duces them " is known as Lenz's Law. As an illustration
of the use of the law, suppose a magnet to be inserted in a
hollow coil. The induced currents must be in such a
direction as to oppose the motion. As opposite currents
repel each other, the current induced in the coil will be
opposite in direction to the Amperian current of the
magnet. If the magnet is withdrawn, the withdrawal
would be opposed by a current in the same direction as
the Amperian current, and by the law a current would
therefore be induced in that direction. The same reason-
ing applies to currents.
88 NOTES ON
63. Mutual Induction of Two Circuits ( 397).
In \ 320 and Note 39 it was seen that two circuits
tended to place themselves in such a position as to inclose
as many of each other's lines of force as possible, and the
number inclosed when each carried unit current was
denoted by M. It has since been shown that any move-
ment of either circuit induces a current in the other, and
consequently changes the value of M. This quantity
is therefore called the " coefficient of mutual induction."
By a course of reasoning similar to that in Note 18 it may
be shown that the force just outside the plane of a voltaic
circuit of unit area is4?rC, and if 6" is the area inclosed by
the circuit the total force is ^nCS, which becomes ^nS
when C is of unit strength and tytnS when there are n
turns each of area S. But as the number of lines of force
is numerically equal to the strength of the field, this num-
ber would therefore be 4?r5, and if all these lines passed
through the other circuit, the maximum value of Mis $7tS
when the two circuits are coincident.
64. Self-induction ( 404).
The extra current is a current induced in the same
conductor in which the original current flows. By Lenz's
law, or by the table in Note 59, when a current begins in a
conductor, a momentary current is induced in the opposite
direction, and this phenomenon is noticeable as well in
the original circuit as in another near it. The current
in beginning is, therefore, opposed by an induced current
in the opposite direction, and its increase is made more
gradual, and more time is necessary for it to gain its full
strength. The fact that the primary current is greatly
reduced by the induced current accounts for the fact that
when a circuit is closed, but little of a spark is seen. After
ELECTRICITY AND MAGNETISM. 89
the current attains its normal strength it remains un-
affected by induction, unless acted upon by external causes ;
but if the circuit is broken, the law and table already re-
ferred to indicate a current in the same direction as the
primary, retarding its decrease. As this current prac-
tically reinforces the primary, the spark at breaking the
circuit is much brighter than at making.
The extra current being strictly an induced current is
subject to the general laws of induction. The induced E,
dN
M. F. is therefore -jr. If dN is constant, as it is with
at
any given current, the E. M. F. of the extra current varies
inversely as dt, or the more quickly the circuit is closed
or broken the greater the extra current. The value of L
in 404 determines the other important relation, that the
self-induction varies as the square of the number of turns,
and as the resistance of the coil increases only as the
number of turns, the extra current is much greater the
more turns the coil possesses. This is of great importance
in enabling extra currents of high E. M. F. to be obtained
when wished, or avoided when they would be detrimental.
65. Helmholtz's Equations ( 405),
The current is prevented by its self-induction from obtaining
its full strength immediately. The electromotive force of the
induced current is L . > L being the coefficient of self-
dt
induction, and the current acting in opposition to the primary
T sJ r*
current is . In any interval of time, *//, after the circuit is
R at
closed, the current has a strength of
E L dC dC R .
90 NOTES ON
and the current at a time / from the instant of closing the cir-
cuit is
66. Induction Coil ( 398).
It is important to remember that in the induction coil
there are two circuits, not only independent of each other
but carefully insulated. The primary coil, or the one im-
mediately surrounding the iron core, is of comparatively
few turns, and of low resistance, that a given E. M. F. may
cause as powerful a current as possible and consequently
induce as many lines of force as possible through the core.
If the primary current is made and broken rapidly, this
number of lines is alternately added and subtracted at
very short intervals, and the E. M. F. induced in the second-
ary coil, through the axis of which all the lines of the
primary pass, is therefore very great. From the general
dN
formula E = j- > the induced E. M. F. in the secondary
is evidently increased by using greater battery power in
the primary or by making and breaking the primary cir-
cuit with greater rapidity. The E. M. F. of the secondary
circuit becomes so great that extreme care has to be taken
with the insulation, and parts of the coil at widely dif-
ferent potentials must not be brought near together.
It is noticeable that the quantity of electricity passing
ELECTRICITY AND MAGNETISM.
in the secondary coil is extremely small. This is at once
apparent when it is considered that the energy of the
secondary coil, which may be expressed as C'E', is all
derivable from that of the primary coil CE and cannot
exceed it. If the energy in the two coils is assumed equal,
C : C : : E' : E,
and the enormous increase of E. M. F. in the secondary is,
therefore, attended with a great reduction in the quantity
of electricity passing. As all the energy of the primary
current cannot be transferred to the secondary, this pro-
portion is not strictly correct, but it illustrates the im-
portant point that an induction coil which might kill a
man could not heat a wire red hot, or perform other work
where quantity was necessary.
The condenser is made of sheets of tin foil insulated
from each other by paper soaked in paraffine. Alternate
sheets are connected throughout, so as to form two large
coatings. The action may be understood from the figure.
The current passes from the battery through Wto I the
Fig. 23.
interrupter, and thence through o back to the battery.
The core, on being magnetized, attracts /, breaking
the current at o. If there were no condenser the extra
current would leap across from / to o in a bright spark,
but when the condenser is used it darts into it, charging P
92 NOTES ON
positively and N negatively, but immediately afterwards
the two charges re-combine, the positive charge passing
from P to L WBN t demagnetizing the core and making
the " break " more rapid, and also opposing the current
at the " make." In this way the time dt of the break is
diminished while that of the make is increased, and the
E. M. F. induced in the secondary coil at the former is,
therefore, much the greater. By separating the poles of
the secondary circuit, they may be placed so that the
break or similar current can strike across them while the
make or reverse current cannot.
ELECTRICITY AND MAGNETISM.
93
VII. DYNAMO MACHINES.
67. General Principle of Dynamo Machines ( 407).
The principle underlying all production of electricity by
machines, is that of Note 59, that if a coil of wire is moved
in a magnetic field a current is induced in the coil. The
successive machines have simply been developments of
this fact, improvements having been made either in the
distribution of the lines of force in the field, or in the con-
struction and movement of the coils. In the first machines,
those of Pixii, Saxton and Clarke, permanent steel magnets
were used, but only a portion of the lines due to these
poles were cut by the coils, and
the machines were, therefore, in-
efficient. As a rule, the coils
moved in front of the poles, in-
tercepting the lines passing off
from the poles in one direction
only. The same general prin-
ciple was followed in the Holmes
and Alliance machines, there
being a greater number of mag-
nets and coils, with a poor dispo-
sition of the different parts. The
introduction of Siemens' armature Flg ' 24 A '
in 1857, was a great step in advance. N and 6* are the
poles of several horseshoe magnets bolted together side by
side, and between the opposite poles rotates a soft iron cyl-
inder Con which the wire is coiled. This armature is thus
N
W
94 NOTES ON
placed in the strongest part of the field, the greater number
of lines of force passing directly from N to S through the
core C, and the construction admits of the coils approach-
ing the poles very closely. Very strong currents were ob-
tained from machines in which this armature was used,
but great difficulty was experienced from the heating of
the armature. If a disk of any metal is rotated rapidly
between the poles of a powerful
magnet it becomes greatly heated
by the currents which are induced
in the metal. Tyndall melted
fusible metal in a copper tube
Fig. 24 B. rotated rapidly in a strong field.
The induced currents are in this case in the metal and
not in the coil and are generally known as " Foucault "
currents. The heat evolved depends on the form and
material of the armature and on its velocity, and heat from
this cause has always been a serious objection to the early
form of Siemens' armature. The next step \vas to have
two armatures in the same machine ; one rotating between
the poles of permanent magnets inducing a current which
passed through the coils of a large electromagnet, between
the poles of which the other armature was placed. The
adoption of electro-magnets greatly intensified the field,
and as the current causing them was also generated by
the machine, a great gain of power and efficiency was
secured. The machines of Ladd and Wilde were of this
type. The above were called magneto-electric machines,
as they all depended on permanent magnets to start them,
but in 1867 the permanent magnets were suppressed, the
current from the armature passing through the coils of the
electro-magnet, the " field " coils being in series with the
external circuit and armature. When the machine was
stopped it was found that the cores of the electro-magnets
ELECTRICITY AND MAGNETISM. 95
possessed sufficient residual magnetism to induce a cur-
rent in the armature when it was started again, and this
current once induced, strengthened the electro-magnets
and in turn induced more current. Machines of this type
were called dynamo-electric or simply dynamo machines.
The advantages they possess over the magneto-electric are
greater power, the field being stronger ; and greater econ-
omy, the magnets being of wrought iron instead of steel.
The distinction between magneto and dynamo machines
is now so slight, by the introduction of a variety of new
types, that it is hardly worth preserving. As has already
been shown, the energy of the induced current is deriva-
ble from the energy expended in moving the coil, and Prof.
Thompson in his Cantor lectures has given a broad defi-
nition of a dynamo machine as "a machine for converting
energy in the form of dynamical power into energy in the
form of electric currents by the operation of setting con-
ductors (usually in the form of coils of copper wire) to
rotate in a magnetic field." Accepting this definition, the
theory of the dynamo is best understood by a reference
to the laws of electro-magnetic induction already examined.
The induced electromotive force in a conductor moving in
dN
a magnetic field is E . As an illustration, exam-
ine the case of a coil spinning in a uniform field, and the
application of the formula to dynamos may be considered
later. Suppose a coil, as in Figure 25, rotating on a vertical
axis, the lines of force passing from the reader down
Through the paper perpendicularly. It incloses a maxi-
mum number of lines offeree, and if rotated so that the
right-hand edge comes to the front, while the left-hand
goes behind the paper it will inclose a constantly decreas-
ing number of lines, and a positive current will be in-
duced. The E. M. F. will at first be small, as the rate of
96 NOTES ON
change is small, the edges of the coil moving almost along
the lines of force. The rate will gradually increase until
the coil has moved through one quadrant and is edge on
to the observer, when, as the motion
of the edges is at right-angles to the
lines, the rate, and consequently the
E. M. F., is a maximum. In this
position the coil incloses no lines of
force, and during the second quad-
rant it will move, inclosing an in-
creasing number, and inducing,
therefore, an inverse current. But
the side of the coil now seen is the
Fig. 25. opposite to that in view during the
first quadrant, and the inverse current is, therefore, in the
same absolute direction in the coil as the former direct
current. During the second quadrant the rate and E. M.
F. decrease, becoming a minimum when the coil has com-
pleted a half revolution and is again in the plane of the
paper. On entering the third quadrant, the number of
lines inclosed decreases, and a direct current is induced ;
but as the same side of the coil is presented to the observer
as in the second, the direction of the current is reversed
in the coil. In the fourth quadrant the number of inclosed
lines increases, but the other side of the coil is toward the
observer, so that the absolute direction of the current is
the same as in the third. The general direction of the
current is, therefore, downward in that part of the coil in
front of the paper, and upward in the other half ; but as
regards the coil itself, the direction of the current changes
twice in every revolution, the point of change being
where the circuit incloses the maximum number of lines
of force. By the use of a commutator which shifts its
connections at this point of the revolution, the current
ELECTRICITY AND MAGNETISM. 97
may be made to flow in one direction in the exterior
circuit.
Considering this coil as the armature of a dynamo ma-
chine, it is apparent that the current could be kept in one
direction in the exterior circuit, but would be of varying
strength. If another coil were fixed on the same axis but
at right angles to the first, its E. M. F. would be a max-
imum when that of the first was a minimum, making
the current in the external circuit more nearly uniform.
By increasing the number of coils a practically uniform
current could be obtained, but at the expense of a very
complicated commutator.
68. Electromotive Force.
dN
From the formula E = it is evident, I. That the
at
E. M. F. varies as the rate of change of the field. For a
constant time dt, the rate varies as the number of lines
taken out or introduced, and the field should therefore
be intense. Mere intensity is not, however, enough,
as a coil could be moved in the most intense uniform
field without inducing any current. The field must be so
arranged that the coil either passes from a maximum
positive to maximum negative, or what amounts to the
same thing, that it rotates in a constant field, the lines
being alternately added and subtracted. In this case the
number of lines should be a maximum or the intensity cf
the field as great as possible.
Perfect working in a dynamo requires a constant
change of E. M. F., and consequently a constant rate. If
a large coil revolves between the poles of two bar mag-
nets, and no iron is present to modify the distribution of
the lines of force in the field, the greater part pass direct"
7
NOTES ON
between the poles, and are cut during a small part of the
revolution of the coil, during which time the rate and in-
duced E. M. F. are high, but in other parts of the revolu-
tion the rate is very small. The available E. M. F. is
induced suddenly, but the sudden creation of a current
causes high self-induction and temporary strong extra
currents, which in a dynamo are not only prejudicial but
dangerous, on account of the high E. M. F. they may
have. Idle wire in the armature also reduces the current.
It is therefore desirable to prevent a concentration of the
lines of force in a small part of the field. In a coil rotat-
ing in a uniform field, the advantage of the constant rate
is attained by the change of the number of lines inclosed
in the ratio of the sine of the angle between the plane of
the coil and the direction of the lines, and as a field tends
to become uniform would this advantage be gained.
To secure uniformity and prevent concentration large
pieces of iron called
pole pieces are fre-
quently attached to
the poles, partially
encircling the arma-
ture. By using long
magnets and heavy
pole pieces, the field
may be made nearly
uniform. An impor-
tant modification of
the field arises from
the lines of force due
to the current in the
armature. Thus in
Fig. 26, representing a cross section of a Siemens arma-
ture, A being the end of the commutator and TT the
Fig. 26.
ELECTRICITY AND MAGNETISM. 99
commutator brushes ; the lines of force of the field ordi-
narily pass from N to S in approximately straight lines.
When the armature is in revolution, each coil in succession
has its maximum current when it is in the position C in
the figure, and the effect of the armature current is there-
fore to induce two poles in the rim of the armature at A '
and S'. The poles A 7 and N' may be supposed to form a
resultant pole at N" and 5 and S 1 at S", and the gen-
eral direction of the lines of force of the field is therefore
N" S". As shown in the discussion of the revolving coil,
the commutator brushes should make contact at the
neutral points, at right angles to the lines of force. If the
lines of force of the machine in motion were in the same
position as when at rest the brushes might remain at
TT in a line perpendicular to the lines of force NS. If
left in this position, however, it will be observed that
there is a constant succession of sparks at the brushes,
which evidently do not press at the neutral points. This
sparking may generally be suppressed by rotating the
brushes into positions T'T'. The explanation is simple :
they have been brought into a line PP perpendicular to
the changed direction N" S" of the lines of force of the
field. The stronger the current, the stronger the induced
pole of the armature N' ', and the nearer TV" is the resultant
pole N". The stronger the current, therefore, the more
the brushes must be advanced.
The different types of dynamos vary principally in the
way in which the field is formed. The principal methods
are:
(i.) The magneto in which the field is due to permanent
magnets. (2.) The separately excited dynamo, a separate
machine being used, the current of which passes through
the field magnet coils of the generator. This possesses
the great advantage of having a constant field, and when
100 NOTES ON
several machines are used in one place, one may be
used to actuate the field magnets of all the others. (3.)
The series dynamo, the field coils being in the main cir-
cuit. As the whole current of the machine passes around
the magnets, an intense field is produced, but with the
great disadvantage that any increase of resistance in the
external circuit weakens the field, and consequently the
E. M. F., just when a high E. M. F. is necessary to over-
come the increased resistance. (4.) The shunt dynamo
has the field magnet coils in a shunt of the main circuit.
In this type an increased external resistance sends a
greater current through the magnet coils, causing a more
intense field and a higher E. M. F. By having the resist-
ance in the magnet coils adjustable, a shunt dynamo may
be made to give a practically constant E. M. F., whatever
the external resistance may be. (5.) A mixture of the last
two types has been used for special purposes, and is
known as a series and shunt, or compound dynamo. It is
a shunt dynamo having in addition a number of coils of
wire on its field magnets in the main circuit. In a shunt
dynamo, if the external resistance is suddenly increased,
a greater part of the current flows around the field coils,
inducing a higher E. M. F. To keep this constant the
speed would have to be decreased. In a series dynamo,
however, an increase of external resistance diminishes the
E. M. F., and to keep it constant the speed must be in-
creased. By making the magnet coils partly in series and
partly in a shunt circuit, it is, therefore, possible to keep
the E. M. F. practically constant at a certain speed within
wide changes of the external resistance. (6.) Some
machines have two coils on the armature, one of which
sends a current through the field coils, while the other is
in the external circuit. (7.) Alternate current machines
are machines giving currents first in one direction and
ELECTRICITY AND MAGNETISM. 1OI
then in the opposite. Their field is caused generally by
another machine, but some types send a part of their own
current, rectified by a commutator, through the field coils.
2. The E. M. F. varies as the velocity.
This relation is almost absolutely true in magneto
machines and in others having a constant field, but
in the series dynamo, where the field is itself a function of
the current, the rate of increase of E. M. F. is much
greater than that of increase of velocity up to the point of
saturation of the magnets, beyond which, the field being
constant, the above relation holds. Its correctness is as-
sumed in practice.
A great waste sometimes occurs from the commutator
brushes not being adjustable. As already shown, the
lines of force of the field are distorted by the current, and
this distortion is greater as the velocity is increased. The
brushes must, therefore, be advanced, or they will take off
the current at the wrong time, involving a waste of
energy and causing "sparking," injuring both commutator
and brushes.
3. The E. M. F. varies as the number of turns of wire
in the armature.
dN
The formula E -= is derived from a conception
of the work done in moving a single coil in a magnetic
field. If n coils were moved either one by one or all to-
gether, n times the work would be done, and n times the
E. M. F. induced. Increasing the number of turns in-
creases the internal resistance in the same ratio, but if the
external resistance is large there is a gain by taking more
turns of wire on the armature.
4. The E. M. F. is greatest when the coil cuts the lines
102 NOTES ON
of force at right angles. The rate of change is then
greatest, and hence the electromotive force.
5. The E. M. F. varies as the area of the coil. If the
field were uniform the gain would be directly as the area.
In any case it varies directly as the number of lines of
force inclosed, and there is, therefore, generally speaking,
an advantage in having the coil of large area.
In the five considerations on which the E. M. F. of in-
duction depends, the only variable, after the machine is
made, is the velocity. Several of the types referred to
admit of a partial adjustment to meet changed circum-
stances, but in general a machine should be adapted to
the work expected of it, and should not be expected to be
efficient under very different conditions. Although the
velocity may be easily varied, it cannot be indefinitely in-
creased without mechanical injury.
69. Efficiency.
A dynamo has properly two efficiencies. As it is a
vehicle for the transformation of mechanical into electrical
energy its gross efficiency is the ratio of the current
energy to the mechanical energy actually applied to the
machine. If an engine developing 16 H. P., is working
a dynamo, two H. P. being lost in transmission to the
dynamo, in friction and in overcoming the inertia of the
engine, only 14 H. P. are actually applied to turn the
armature. If in this case the electrical energy developed
by the dynamo, was 10 H. P. the gross efficiency is \\ t
or 71 per cent. A good dynamo possesses an efficiency
of from 90 to 95 per cent, when working under the
most favorable conditions, and therefore far surpasses
any other machine in its capacity for transforming
energy.
ELECTRICITY AND MAGNETISM. 103
The ordinary use of the dynamo is to produce light.
Whatever its use may be, all the electrical energy not
utilized in producing the desired result is practically
wasted. The net efficiency is the ratio of the electrical
energy in the external circuit to the mechanical energy
applied to the armature. If in the above case only 6
H. P. existed in the external circuit, the net efficiency
= -, & 4 -, or 44 per cent. The distribution of the energy in
the circuit is one of the most important problems relating
to dynamos. The total work in circuit is, by Note 55,
C^Rt, R being the total resistance, consisting of internal r,
and external /. The work is then CV + CY, and the
ratio of the work done in the machine to that in the ex-
C'r r
ternal circuit is - = The fraction of the total
C-l
electrical energy in +he external circuit is similarly -^5
= That this proportion should be great, / must be
nearly equal to R, or in other words, the resistance of the
machine must be small compared with that of the circuit.
From the above the important relation is evident, that
the distribution of energy in an electrical circuit is
determined by the distribution of the resistances in cir-
cuit.
The efficiencies may now be calculated. Let a resist-
ance of armature, /that of the field coils, /of the external
circuit, and R be the total resistance. E is the electro-
motive force, and Cthe current in circuit.
The gross efficiency of a series dynamo
Work of current
= H. P. applied '
104 NOTES ON
This by Note 55 is
CV?
_ _
H. P. applied ~ 746 x H. P. applied*
This energy is given off in all parts of the circuit, that
in the armature being C*a (in watts), that in the field
coils Cy, and in the external circuit C 2 /.
The net efficiency is
Work in external circuit-
H. P. applied
and this is
CW
"746 cn
H. P. applied 746 x H. P. applied
The energy wasted as heat in the machine is C 2 (a + f)
and the ratio of energy wasted is / / That this
may be small the internal resistance (a + f) must be
small in comparison with the external. The energy
wasted takes the form of heat, and is thus not only
wasted but directly harmful, as heating of the machine
increases its resistance and thereby increases the ratio of
wasted energy.
In the shunt dynamo the relations are more complex,
as the current in the various branches of the circuit is
different.
Let a = armature resistance and A armature current,
f = field coil " " .F current in field coil,
/ = external " " L " " external
circuit,
R = Total resistance = a + -, s
ELECTRICITY AND MAGNETISM. 105
Work in armature = A-a, in field coils F*f, and in ex-
ternal circuit Ul.
Total electrical energy = A*R = A 1 (a + ^ J .
Gross efficiency =
H. P. applied 746 x H. P. applied
To find the net efficiency, the ratio of the electrical en-
ergy utilized is
'"- r.
(
and by multiplying this into the value of the gross efficiency,
previously obtained, the product is the net efficiency.
The above expression contains only resistances. If L is
measured the net efficiency is evidently
Z,V
746 x H. P. applied
70. Electromotive Force in Circuit.
In using CE to calculate the electrical energy in any
case, E must not be taken as the difference of potential at
the machine terminals. Calling this difference of potential
E', and considering the case of the series dynamo as being
more simple, we have from Kirchhoff 's second law, E' = Cl,
I being the external resistance,
or c = ?- , but from Ohm's Law C = j^
106 NOTES ON
Whenever E' is measured E must be calculated if the
total electrical energy is to be computed. The product
CE' is evidently the energy in the external circuit, and is
less than the total energy by the quantity CV expended
in the machine. The total energy is, therefore, CE' +
C*r = C-(l -f r). If the resistances are all known the
total energy may be calculated from the last formula with-
out any risk of error.
It has been shown by Sir William Thomson that in
shunt dynamos the best results are obtained when the
external resistance is a mean proportional between the
resistance of the magnet coils and that of the armature,
the latter being small in comparison with the resistance
of the magnet coils.
71. Siemens' Machine ( 409, Fig. 151).
This is a shunt dynamo. The armature is similar in
shape to Siemens' armature already described, being a
cylindrical drum, but having several coils coiled on it
lengthwise instead of one. There are as many divisions
of the commutator as there are coils, the divisions being
longitudinal. An eight-coil machine has, therefore, its
commutator ring divided into eight segments, to each of
which connect the ends of two coils. The other ends of
these coils are connected to other commutator divisions,
so that the eight coils are all in a continuous circuit be-
tween the commutator brushes, so wound that in all eight
the current at any given instant flows in the same direc-
tion. In some coils the E. M. F. is greater than in others,
but as there are so many, the total E. M. F. of all in series
varies but slightly from time to time, and the current is,
therefore, practically constant. By placing the commu-
tator brushes opposite each other, they are in contact with
points of the circuit differing most widely in potential, and
ELECTRICITY AND MAGNETISM. 107
a permanent difference of potential is therefore maintained
between the terminals of the machine. The induction of
the current in any one coil is analogous to that in the coil
described in Note 67.
72. The Gramme Machine ( 410, Fig. 153).
The Gramme is generally a series dynamo, although
sometimes separately excited, and sometimes having its field
coils excited by a separate armature coil. The armature
is a ring of soft iron wire, widened till it might be consid-
ered a short hollow cylinder. Around this ring are coiled
a great number of armature coils, as shown in Fig. 152, the
ends of the coils being brought to divisions of the commu-
tator. The commutator consists of a number of plates
radially arranged around the axis of the armature, and
insulated from each other. The commutator divisions
are seen on the right of the armature in Fig. 153, and cor-
respond in number to the armature coils, which are con-
nected through them in one continuous circuit.
The action of the Gramme may be easily understood
from the rules of Note 59. In Fig. 152 the positive direc-
tion of the lines offeree is from A^to S, the lines entering
the ring opposite N, and dividing, running through each
half of the ring to that part opposite S, where they leave
the ring and pass to S. The poles N and 6* cannot be
considered as points, and the lines, therefore, enter the
ring all along its lower portion (as shown in the figure)
and emerge along the upper part. A coil in the position
E" has, therefore, the maximum number passing through
its plane. If, now, the armature is rotated, so that E"
passes towards E, it continually incloses a decreasing
number of lines of force, and a direct current viewed from
N is induced. The E. M. F. varying as the rate of change,
108 NOTES ON
is zero at E" and a maximum at a point opposite S, where
the coil cuts all the lines at right angles. As the rotation
of the armature continues, the coil after leaving E in-
closes an increasing number of lines of force, and the
current is therefore inverse as viewed from N. But
from E to E' the side of the coil viewed is the oppo-
site of that seen from E" to E, and the inverse current
in the quadrant from E to E' is therefore in the same
absolute direction in the coil as the direct from E" to
E. Throughout the half revolution from E" to ',
therefore, the induced current flows in the same direc-
tion, being strongest when the coil is nearest the pole
S. By connecting all the coils in series, the E. M. F.
in the circuit becomes the sum of all in the individual
coils, and as these occupy all possible positions at any
instant, the total electromotive force is constant, the
machine thus yielding an almost absolutely constant
current.
The action during the other half of the revolution
may be traced in the same way. The coil in moving
from E' to N incloses a decreasing number, inducing a
direct current, which is opposite in direction to that
in the quadrant from E to E'. If, therefore, in the latter
the current had flowed away from the point E' towards
E, it would in E'N flow away from E' towards N, and
although the absolute direction of the currents is dif-
ferent they combine to lower the potential of E'. During
the quadrant between N and E" t the coil incloses an in-
creasing number, and consequently has an inverse current
induced, but this inverse current is in the same absolute
direction as the direct in the preceding quadrant. If,
therefore, throughout the upper half of the revolution the
current flows away from E', it will in the lower half of
the revolution flow away from E'also. Throughout the
ELECTRICITY AND MAGNETISM. 109
whole revolution the effect is to raise the potential of E"
and lower that of ', and if brushes touch the commu-
tator at these points they will possess a difference of po-
tential which may be utilized in the production of a cur-
rent through an external circuit. The Gramme machine
has been the subject of much investigation, and its action
has been variously explained. The most general ex-
planation in any case of electromagnetic induction is
that obtained from a consideration of the lines of force,
and this is the one adopted by Prof. Thompson, which
has only been given here in slightly greater detail.
The armature cylinder is made of soft iron wire, both
to facilitate the rapid magnetization and demagnetization,
and to prevent heating from the Foucault currents which
would take place if solid metal were used. The change
of direction of the lines of force of the field by those due
to the current is frequently very marked in the Gramme
machine, M. Breguet having found it necessary to ad-
vance the commutator brushes 70 when working with a
Gramme at 1770 revolutions. As the internal resistance
of the Gramme is generally small, it is specially adapted
for working a single powerful arc light, while the steadi-
ness of its current renders it well adapted for incandescent
lighting.
73. The Brush Machine ( 411).
This machine has received its main development in the
United States, but is now extensively used throughout the
world. It contains many peculiar features, and is distinct-
ly a separate type, although frequently alluded to, espe-
cially by French authorities, as a modification of the
Gramme. The general appearance of the machine is
shown in Fig. 27.
The first noticeable peculiarity is in the disposition of
TIO
NOTES ON
ELECTRICITY AND MAGNETISM. m
the four field magnets, which are placed so that the arma-
ture coils pass between similar poles. The magnets are
oval in cross section, and are furnished with large pole
pieces, approaching very closely on each side to the arma-
ture. The armature is a soft iron disc, with deep circular
furrows cut in its sides to break the continuity of the sur-
face and thus prevent the heating of the metal by the in-
duction of Foucault currents. On the periphery of the
armature of the small machine there are eight coils, the
two coils diametrically opposite being in one, but coiled in
opposite directions (See Fig. 30), so as to act in unison in
the induction of currents. The coils project from the ar-
mature as seen in Fig. 27, the reason assigned being, that
the fanning of the air thus caused prevents overheating.
The commutator consists of four rings each split into
four segments. A cross sec-
tion of one of the rings is as in
Fig. 28, the two ends of one
pair of coils being connected
to the segments marked I, I,
which are insulated from the
segments 2, 2. When the
brushes of the commutator
touch the latter the coils are
cut out of circuit. These cut-
ting out segments in the dif- Fi - 2g -
ferent rings of the commutator are so placed that at every
instant one coil is cut out ; the connections being made
so that a coil is not in circuit in that part of the rev-
olution when no current is being induced in it. Each of
the four brushes presses on the commutator rings of two
coils not adjacent. Numbering the coils on the armature
I, 2, 3 and 4 (Fig. 30) in order, the brushes Z? and B* are in
circuit with coils I and 3 and * and B* with coils 2 and 4.
112
NOTES ON
As the armature revolves each coil successively passes
through all parts of the field. When a coil is midway
between the dissimilar magnet poles, at the highest point
of its revolution, the number of lines of force inclosed is a
maximum, but changes so slowly that for this portion of
the revolution the induced current is but small, and the
coil is, therefore, cut out. As the coil approaches the large
pole pieces and passes between them the rate changes
rapidly. If a piece of soft iron be placed between two
powerful similar magnet poles, the lines of force pass into
it almost parallel on each side, and a coil moved along the
bar cutting them perpen-
dicularly, has a high rate
of change in the number of
lines inclosed, and conse-
_ quently a high electro-
motive force induced.
Thus in Figure 29, as
nearly all the lines of both
poles pass through the soft
iron between a and b, the
coil A in moving with-
in that region experiences
but little change in the
<, number inclosed, but
as it approaches either
end the rate of change is
very great. The electro-
Vl '** motive force is thus in-
duced somewhat suddenly, but the efficacy of the pe-
culiar arrangement of poles for the induction of a high
electromotive force is evident. A comparison between
Figs. 29 and 30 shows this to be nearly the condition ex-
isting in the Brush machine. The latter figure is a plan
ELECTRICITY AND MAGNETISM. 113
of the machine. As each pair of coils is connected to a
separate commutator ring, the study of the connections is
necessary to understand the complete working. In the
figure, L represents a lamp in the external circuit. B l , B*,
B z and B* are the commutator brushes, and the rings are
Fig. 30.
numbered I, 2, 3, 4, as illustrating the way in which the
ends of the coils, taken in regular order around the arma-
ture, are connected at the commutator. The currents in-
duced in the several coils at any one instant will have the
following circuits, coil 4 being supposed to be cut out :
Coil i i, B\ L, X, 13*, 2, B\ M, B\
Coil 2
2, B\
, B l <*> B*,LX, B\
Coil 33, B\ L, X t B\ 2, B\ M, B\ '
These paths are the same except in the armature coils
and the resultant current will have the path
B l <^> B\ Z, X, B\ 2 B*, M, B\
An instant later coil 4 will be in circuit and I cut out.
The resultant current then flows
3, B\ L, X,
\ M, B\ 3.
114 NOTES ON
The E. M. F. in circuit is evidently that due to two coils
in series, and the internal resistance of the machine is
diminished by the fact that there are always two of the
four armature coils in parallel arc.
The gross efficiency of the Brush machine is lower than
that of some others, but it possesses the great advantage
of yielding so high an electromotive force as to be able to
burn forty arc lights in series, a feat which no other
machine can accomplish. As there are only two coils in
series at one time, the resultant electromotive force is far
from constant, and the fluctuations are so great as to
utterly unfit the machine for incandescent lighting or
other purposes requiring a constant current. The E. M. F.
of the largest Brush machine is 2000 volts and the current
about 10 amperes.
74. Edison Machine.
The Edison machine ($ 411) is a shunt dynamo. Its
chief peculiarities are its long cylindrical magnets ending
in remarkably heavy pole pieces almost encircling the
armature, and the peculiar construction of the armature
itself. Theoretical investigation and experiment both
point to long cylindrical magnets as most efficient ; and in
a shunt dynamo, in which there is a perpetual endeavor
for a permanent adjustment of the strength of the field to
the necessities of the case, it is advantageous to have mag-
nets of considerable mass, as the change of field brought
about by a variation in the strength of the field current is
thus made more gradual. The large pole pieces tend to
make the field more uniform, and thus act to secure a
constant rate or a uniform change of electromotive force
throughout the rotation.
Edison calls his large machine a "steam dynamo," the
engine and dynamo being on the same bed-plate. It is
ELECTRICITY AND MAGNETISM. 115
specially designed for use at a central station to supply
power or work incandescent lights throughout a district of
a city. As established in New York, the whole weight of
dynamo and engine is nearly thirty tons, sixteen of which
are in the magnets and pole pieces. The core of the
armature is made up of sheet iron discs, separated from
each other by tissue paper and bolted together. This
prevents heat currents. Instead of wire, the armature
circuit is made of heavy copper bars, each bar being in-
sulated from the next and from the iron core by an air
space. The bars are connected together at each end of
the armature by copper discs, there being half as many
discs at each end as there are bars. Each disc has lugs
formed on it on opposite edges, to which two bars are
connected, and the whole being bolted together, the bars
and discs form one continuous circuit of wonderfully low
resistance, the total armature resistance of a machine sent
to London being .0032 of an ohm. This very low resist-
ance is necessary from the fact that the machine is in-
tended to work 1300 incandescent lights, each of about
137 ohms, in parallel arc. The external resistance would,
therefore, be only .095 ohms. As the number of lamps in
circuit changes, the resistance in the magnet coils, which
are in a shunt of the main circuit, is regulated so as to
keep a practically constant electromotive force, and each
lamp then burns with the same intensity under all condi-
tions. Edison's large machine gives an E. M. F. of no
volts and an ordinary current of 1000 amperes.
75. Alternate Current Machines.
Alternate current machines have been used in Europe
to a considerable extent for incandescent lamps and the
Jablochkoff and other candles. Almost any machine yields
alternate currents if used without a commutator, but most
1 1 6 NOTES ON
alternate current machines have a large number of armature
coils which pass between the poles of a system of opjx>,< <!
magnets, so arranged that the positive lines of force in the
field pass alternately through the coils in opposite direc-
tions, many machines inducing fifteen or twenty currents
in each direction in every revolution. By placing a num-
ber of coils in series, so disposed that the induced cur-
rent in all is in the same direction at any instant, a high
electromotive force may be induced. Many machines
have the connections of the coils adjustable, so that they
may be arranged either in series or in arc, thus per-
mitting an adaptation to the requirements of the external
circuit. An objection to alternate current machines is
that the frequent reversals of current induce extra cur-
rents of so high electromotive force as to be dangerous.
The more coils there are in the machine, the higher the
coefficient of self-induction, and the greater the velocity
the greater the induced electromotive force, so that the
extra current may be much stronger than the normal
current of the machine. This disadvantage exists with
many continuous current dynamos, particularly the Brush,
the electromotive force of which is, as already stated, not
only high but also very variable. Every change of the
normal current induces an extra current, of greater
strength as the revolution of the machine is more rapid.
Alternate current machines are less economical than those
generating a continuous current.
ELECTRICITY AND MAGNETISM. 117
VIII. ELECTRIC MOTORS.
76. General Principles ( 375).
In describing the action of Ritchie's electric motor, Prof.
Thompson abandons the method pursued elsewhere in
his book, that of a consideration of the lines of force in
the field, and adopts another less general explanation,
that of the mutual action of magnet poles. If the core of
the coils CD were of some non-magnetic substance the
description given would not apply, although the motor
would still work. The two coils Cand D in Fig. 141 are
practically one, and this will in any magnetic field tend to
place itself so as to bring its own lines of force in the
same direction as those of the field. The lines of the
field pass from Nio S, and if those due to the current in
the coil are opposite in direction, the coil will tend to
rotate into a position of equilibrium, but just before this is
attained the rotation shifts the connections of the coil in
the mercury cups, the current changes, its lines are again
opposite to those of the field and the coil continues its
rotation through another semicircle. Owing to the con-
tinued shifting of the direction of the current, the coil is
perpetually in unstable equilibrium, and the rotation is
continuous in the endeavor to attain equilibrium. As
shown in Note 36, the work done by a coil in a half rev-
olution is 2CHA, and if the coil is of n turns this becomes
iCnlfA. This experiment well illustrates many of the
principles of electric motors, and particularly that of the
commutator.
NOTES ON
77. Electric Transmission of Power to a Distance
( 376).
As illustrated more fully in Note 78, mechanical energy
may, by the use of a dynamo machine, be converted into
electrical energy, and be transferred into mechanical
energy again by another dynamo at a distance. The
value of this fact depends on circumstances. There are
very many cases in which power thus obtained, due pri-
marily to a water-fall at a distance, would, in spite of the
great loss in transmission, be more economical than the
same power generated from steam on the spot. If in-
candescent lighting becomes an established system in
cities, as started by Edison in New York, the same wires
which work the lamps at night may transmit power by
day, and small motors may be used on the lamp circuits.
The readiness with which electric energy in the form of a
current may be subdivided, offers great advantages for
its distribution in small amounts over the system of dis-
tribution of steam power by a complicated system of
shafting and belting. Another point in which electric
motors may be used is in electric railroads. A stationary
steam engine is vastly more economical than a locomo-
tive, both in wear and tear and in consumption of fuel
and water. Generating the required power by stationary
engines, this economy may be pushed to the utmost by
the adoption of large engines of the most approved type;
and as an electric motor may be made of great power but
of little weight, the injury to rolling stock and road bed
resulting from the use of heavy locomotives would be
prevented. Stationary engines working dynamos may be
placed as needed and the current generated be trans-
mitted through the rails to the motor. Several such roads
have been constructed for short distances, and the ques-
ELECTRICITY AND MAGNETISM. 119
tion of their development is merely one of commercial
economy.
78. Theory of Electric Motors ( 377).
The theory of electric motors first propounded by Jacobi
has of late received mathematical development from
others, and has been the subject of much experiment.
The underlying principle is that referred to in Note 76,
that if a current be passed through a coil, free to move in a
magnetic field, the coil will move into a certain position,
and in moving is capable of doing work. This principle
is the converse of that underlying the action of the dynamo
machine, and it is therefore easy to see that a machine
which will generate currents when worked, will, on the
other hand, work when a current is sent through it, the
rotation as a motor being in the opposite direction to that
as a generator of current. By using two machines in the
same circuit, the current generated by one will cause the
other to work. The commercial value of the fact is deter-
mined by the cost of the power given off by the second
machine.
If C be the current in circuit and E the E. M. F. of the
generator not the difference of potential at the terminals
of the generator (See Note 70) the electrical work of the
generator is CE. The motor in rotating backwards gen-
erates a current in the circuit in the opposite direction to
that of the generator, thus diminishing the current in
circuit. This follows directly from the principle of con-
servation of energy. If is the back electromotive force
of the motor, the electrical work of the motor is Ce. The
mechanical work given off by the motor can never equal
this, as the motor is not a perfect vehicle for the trans-
mission of electrical into mechanical energy, but in calcu-
120 NOTES ON
lation the motor may be assumed to be perfect, and correc-
tions applied to the final results.
The ratio of the work of the motor to that of the gener-
ator is
Ce _ _
CE~ E '
or the return is the ratio of the electromotive forces of the
two machines. If these are exactly similar and working
with fields of equal intensity, a condition not holding in
practice, the ratio of s to E is that of the velocities of the
two machines.
As the motor generates an inverse current, the current
in circuit is
7? being the total resistance.
The energy of the generator is expended partly as heat
in the circuit, and partly as work in the motor
or CE = C*R + work ..... (3)
Work = CE - C*R.
Differentiating for a maximum,
C4)
The maximum work is, therefore, done by the motor
when its velocity of rotation is such as to reduce the cur-
rent in circuit to one-half that due under Ohm's Law to
the electromotive force of the generator and the resistance
in circuit. This is the case when s = h E. The return
is then by (i), - = *.
ELECTRICITY AND MAGNETISM.
121
From these equations the conditions of economy are
deducible. If the only consideration is that the motor
should do the most work, equation (4) gives the condition
that its back electromotive force should be one-half that
of the generator. If, however, the governing condition is
that the motor should work as economically as possible,
(i) indicates that the electromotive force of the motor
should nearly equal that of the generator. The return is,
then, greater than one-half, but (2) shows that the current
is reduced, that CE, the work of the generator, is also di-
minished and that consequently a greater proportion of
a smaller amount of work is transmitted. The governing
consideration is whether the motor should do as much work
as possible, regardless of cost, or work with the greatest
economy, regardless of the amount of work done.
These conditions are of the greatest importance, but are
somewhat difficult to reconcile. Prof. Thompson has de-
vised a graphic illustration A K D
which presents them very
clearly. Draw AB to represent
E, the electromotive force of
the generator, and on it con-
struct a square, ADCB. On
AS lay off from B, BF\.o repre-
sent proportionally e, the elec-
tromotive force of the motor,
and on BF construct a square
BLGF. Through G draw
FH parallel to C and KL parallel to AB.
Then the work of the generator is CE ^ ^-^
y?
work of the motor is Ce =
R
122
NOTES ON
But e (E e) is the area of GLCH, and E (E - ) is
the area of AFHD. These areas are, therefore, marked
as "electric energy expended" and "useful work," and
the ratio of the two areas is the return. From the
construction the point G will always fall on the diagonal
BD, approaching D more nearly as F approaches
A. The area GLCH corresponding to the useful work
will, therefore, always be inscribed within the triangle
BCD, and will have its maximum value when it is a square
as in Fig. 31. G is then midway between .Z?and D, and
the area of GLCH is one-half that of AFHD. From
similar triangles, e is also one-half of E. This demonstrates
the case of maximum work ; that of maximum efficiency
A K D is evident from Fig. 32.
The lettering and construc-
l_l tion are the same, but the
value of 8 has been increased.
The ratio of GLCH to
AFHD is greater, although
each area is less than in the
preceding case. A greater
ratio of a less amount of en-
ergy is thus shown to be
Fig. y ^ transmitted. In the last
figure the area GLCH is by geometry equal to AKGF,
and the square KDHG, the difference between them,
represents, therefore, the factor C*R, or the loss by heat.
The smaller this loss the greater the return. Any desired
return may be calculated by making GHCL the required
fraction of ADHF. If this is to be 90 per cent. KDHG
must be -fa of ADHF t and this may be secured by making
ELECTRICITY AND MAGNETISM. 123
DH h of DC, or what is the same thing BF '-ft of AS.
The geometrical construction, therefore, gives the same
result as that already obtained, that the return is the ratio
of the electromotive forces, or that should be -ft.
r,
The work done by the generator being CE the question
arises whether it is best to increase the energy by using
a stronger current or a higher electromotive force. The
loss by heat is C*R, and this by (2) is equal to
R
If now E and e are both increased by the addition of the
same numerical quantity, the difference (E ), and con-
sequently the loss by heat, is the same. Calling the new
values E' and ' the amount of work done is easily calcu-
lated. The work done by the generator is now
E'(E'-e') E' (E - e}
and that by the motor,
g'("-O e' (E -
R R
.... (6)
The original work of the generator was . (7)
and of the motor "" (8)
As E' and ' are greater respectively than E and e, a
comparison of (5) with (7) and (6) with (8) shows that
more work has been done by the generator and more
by the motor, while the loss by heat was the same.
There is, therefore, clearly an economy in using high
124 NOTES ON
electromotive forces both in generator and motor. The
use of high electromotive force is, however, more dan-
gerous and necessitates better insulation.
The above sets forth the general conditions of trans-
mission of energy by electricity. A few deductions are
easily made. Solving (3)
E
2R
Equating (9) and (2),
E T ^/~ r
From (10) and (i)
_ _ _ _ (JO)
Return = =
..... (")
(n) shows that the return is not independent of the
amount of work done by the motor, the return diminish-
ing, other things equal, as the work done increases. If,
however, the return is wished to be the same, it may
be secured by making the work done vary inversely as
the resistance through which it is transmitted.
Denoting the return, or , by K, the following equa-
tions are readily deduced.
Work of generator = E ^ E ~ E ^ = (i - K} . ~ . (12)
Work of motor = KCE = A'(i - K} . . . . (13)
K
ELECTRICITY AND MAGNETISM. 125
Loss in heat = (E R g) * = (i - Kf . ^ . . . . (14)
These equations, due to Marcel Deprez, show that work
and loss by heat remain constant whatever the resistance
E*
in circuit may be, if E is made to vary as y ' R, that -=?
may be constant. By an increase, therefore, of the elec-
tromotive force of the generator, the same amount of
work may be transmitted to a greater distance.
79. Modifications of Theory in Practice.
The preceding demonstration is entirely theoretical,
and although the general conditions hold, they are much
modified in practice, the modifications, moreover, being
all unfavorable for economical results. Among the
causes of error are the following : no dynamo machine or
motor is a perfect device for transmitting energy. The
work CE done by the first dynamo is less than that re-
ceived by it, and the motor is unable to transfer the whole
quantity Cs into mechanical energy. If /'"represents the
gross efficiency of the generator and /that of the motor,
CE
the work done on the generator is , and that done by
I 1
the motor is/. Cs. The commercial return is, therefore,
Mechanical 'work done by motor fCs
Power applied to generator ' CE *'* ' E '
F,
Assuming for both generator and motor the gross effi-
ciency of 90 per cent., the maximum commercial return
is .81 of the power applied, the work done being then very
small, or .9 of the theoretical return in the preceding equa-
tions. Another loss is that caused by leakage. If the
126 NOTES ON
insulation is not perfect, leakage occurs all along the line,
and the current at the motor is less than the current at
the generator. This loss is greater as the electromotive
force is raised. If the motor is a self-exciting dynamo,
the weaker current causes a weaker field than that of the
generator, so that if the two machines are exactly similar,
the ratio of the electromotive forces are not the same as
the ratio of the velocities. There may be other causes
operating, but these are sufficient to show that in no case
can the theoretical return be fully realized. Marcel De-
prez has at different times made experiments in transmit-
ting power to a distance. The most careful up to this
date were made in Paris on March 4, 1883, when he suc-
ceeded in obtaining from a motor 4.439 H. P. (French),
which had been transmitted through a resistance of 160
ohms, corresponding to about ten miles of ordinary tele-
graph wire. The power applied to the generator was
12.267 H. P., but the electrical work CE of the gener-
ator was only 9.751 H. P. Gramme machines of high
resistance were used, the electromotive force of the
generator being 2,480 volts and that of the motor 1,779.
The electrical return was, therefore, 71.7 per cent., the
commercial return 36.2 per cent.
That the electric transmission of energy may be a suc-
cess, a generator is necessary which with a constant
speed gives a constant electromotive force, whatever
changes may take place in the external circuit. If a gen-
erator is to work one hundred motors, it must be able to
work them all at once or a few at a time, without running
risk of injury from sudden changes of the number in use.
Machines, or combinations of machines, for this purpose
have been invented, one type of which, compound dynamos
(Note 68, i), has already been referred to. It is also nec-
essary that the motors should move at the same speed
ELECTRICITY AND MAGNETISM. 127
whether doing work or not. Improvements in these two
points will make the transmission of power a commercial
success.
Go. Peltier Effect ( 380).
This effect is caused when a current flows from one
metal to another, and is independent of the resistance. As
stated on page 343, the heating varies directly as the cur-
rent, and the junction which is heated by a current in one
direction is cooled if the direction of the current is re-
versed. Letting P be the heat in joules produced at a
junction of two metals per second by a current of one
ampere :
Total heat in joules = C*R PC
= C(CR P).
In the last equation the quantity in brackets and the
term CR are both electromotive forces. P must, there-
fore, be also an electromotive force measured in volts,
although commonly called the " coefficient of the Peltier
effect."
By carefully measuring the change of temperature at a
junction with the current alternately in opposite directions :
//, = CV? + PC
H* = C*R - PC
Hi - H, = 2PC orP = Hl ~ H<i .
2 C
If the current is one ampere,
Hi and H^ are measured in joules, or in ergs x io 7 ,
hence the numerical value of P may readily be found in
ergs.
128 NOTES ON
The equation in the Elementary Lessons was written
before the joule had come into use as a practical unit of
heat, but is the same as the above, since the joule is equal
to .24 of a water-gramme-degree centigrade thermal
unit.
81. Secondary Batteries ( 415).
It has been found that lead dioxide is highly electro-
negative to metallic lead, the difference of potential between
the two in dilute sulphuric acid being about 2.7 volts.
These two substances are used in the secondary battery,
partly on account of the high difference of potential they
possess, but mainly on account of the facility with which
the lead dioxide may be formed by electrolysis. The Plante
cell consists of two plates of lead in dilute sulphuric acid.
If a current is passed through the cell, the liquid is decom-
posed, hydrogen is evolved on the kathode and oxygen
on the anode. The latter unites chemically with the lead,
forming lead dioxide, and this being, as stated, highly
electro negative to lead, if the original source of electricity
be removed and the secondary cell short circuited a cur-
rent will flow through the cell in the opposite direction to
that of the charging current, and will in time deoxidize
the negative plate. The cell is then discharged. The
process of charging is, therefore, merely one of polariza-
tion, and the effect of the current which it is of the most
importance to avoid in the ordinary cell is the basis of the
utility of the secondary battery. Charged in this way,
however, a Plante" cell yields but little current. In prac-
tice the cell is charged as above, discharged and then
charged in the opposite direction, and this alternate
charging in opposite directions in time renders both plates
spongy or cellular in texture, enabling the oxygen given
up at the anode to more readily enter into combina-
ELECTRICITY AND MAGNETISM. 1 29
tion with the lead and forming a dioxide layer of greater
thickness.
The Faure and later secondary batteries are similar in
principle to the Plants, but have the plates at the begin-
ning coated with lead oxide. When, then, a current is
passed the anode is oxidized to lead dioxide, and the
kathode deoxidized to metallic lead by the hydrogen
evolved. The cell does not need the preliminary treat-
ment of the Plante", but is ready for use immediately after
the first charging, but the chemical action is much more
complex, and the cell is probably not so durable.
The term " storage of electricity " is frequently used in
connection with secondary batteries, but is not strictly
accurate, as the portion of the energy of the charging cur-
rent which is stored in the cell is in the form of energy of
chemical separation, and is again transformed into elec-
trical energy when the circuit is closed. The utility of the
secondary battery arises from the fact that the chemical
action when the circuit is open, is not great, and that the
cell may be used after an interval of a few days from the
time it was charged. The objection to its use is that, in
in the first place, only a portion of the electrical energy of
the current can be stored in the cell as chemical energy,
and secondly, that chemical action does take place to a
certain extent when the cell is not in use, and that it can-
not, therefore, store energy indefinitely. The chief dete-
rioration arises from the formation of lead sulphate. A
cell, moreover, wears out eventually and becomes prac-
tically useless. There are many ways in which secondary
batteries may be of service, particularly in connection
with dynamos in electric lighting, but the anticipations of
their sphere of usefulness entertained shortly after the in-
troduction of Faure 's battery were exaggerated.
13 NOTES ON
IX. TELEGRAPHY AND TELEPHONY.
82. The Morse Alphabet ( 425).
The alphabet printed on p. 397 is the international
alphabet used in Europe and, in fact, everywhere except
in the United States and Canada, where the code origi-
nally introduced by Morse is still in use. The international
is probably the better, as it is more easy to distinguish
combinations of letters and avoid mistakes, but it is ex-
tremely difficult to make any change in a code, however
faulty it may be, when it has once come into use. The
alphabet used in America is as follows :
A T
B U
C--- V
D W
E - X
F Y - - - -
Q 2
H & -
T ___ _ 2
L 4
M 5
N 6
O - 7
p 8
Q 9
R _ _ _ o
S
ELECTRICITY AND MAGNETISM. 131
83. American System of Telegraphy* ( 426).
The European system is known as the "open circuit"
system, the current flowing only when the key is depressed.
Many inconstant cells like the Leclanche" may, therefore,
be used. In America, however, the current flows continu-
ously when no messagejs passing. When an operator
wishes to telegraph he first breaks the circuit by a switch
attached to the key, and then makes the signals, the cir-
cuit being closed when the key is depressed. When not
Fig. 33 KEY.
telegraphing the switch must be closed or no signals can
be made by any other operator on the line. The general
appearance of the American key is shown in Fig. 33. The
key is fastened to the table by the screws B and L, the
former being insulated from the metal of the key, the
latter in connection with it. One wire is connected to
the metal of the key, generally at L, and the other clamped
by B. The switch moves horizontally, and when pushed
towards the left in the figure, makes contact with B
* Only a mere outline of the closed circuit system is here given. Full
information may be found in books on telegraphy, the best being prob-
ably Prescott's u Electricity and the Electric Telegraph."
132
NOTES ON
and connects it with L. When pushed to the right the
circuit is open, and is closed only when the key is depressed
and contact made with the head of the screw B.
The general arrangement of apparatus at a way station
is shown in Fig. 34. The current entering by the line
wire on the right first passes through the key A", the switch
Fig. 34-
in the position shown touching the head //"of the screw B
(Fig. 33), and closing the circuit. From the key it passes
to the relay R, entering at the binding post A, passing
around the electromagnet^/, and issuing at B, passing into
the line to the next station. This current is furnished
either by a powerful battery at one end of the line, or by a
battery at each end, acting in the same direction. In
front of the electromagnet M is a vibrating lever of iron or
one furnished with an iron armature, pivoting at the point
P. When the current passes, this lever touches the stop
D and closes the local circuit DXLSYP through the
ELECTRICITY AND MAGNETISM.
133
"sounder" S. When the line current ceases, the lever V
is drawn back by the spring and contact at D is broken.
Whenever, therefore, an operator at any station opens his
switch and signals, every relay on the line works, and
each relay works a "sounder" through the intervention of
its local battery. As the current always runs in the same
35- RELAY.
direction, the relay works for every signal from which evei
way it may come. In the open-circuit system a relay is
necessary for messages in each direction. The Western
Union relay is shown in Fig. 35,
The printing receiver or embosser is but little used in
America, messages being read by sound. The "sounder"
Fig. 36. SOUNDER,
consists of two electromagnets which attract an armature
134 NOTES ON
whenever the local current passes. The armature is at-
tached to a lever, which makes a sharp click by striking
against a stop whenever the armature moves. After a
little practice the operator can read the message easily.
84. Faults ( 427).
Formulas may be easily worked out for determining
the position of a fault, on the supposition that the resistance
of the fault is itself constant. In practice, this is seldom
the case, and never so in submarine cables, as the cur-
rent escaping at the fault causes electrolysis of the sea
water, either depositing chloride of copper over the fault,
or clearing away such deposit according as the current is
positive or negative. The exact determination of the po-
sition of faults requires, therefore, great skill in making
the tests and good judgment in interpreting the results
obtained.
85. Simultaneouss Transmission ( 428).
This method generally requires the use of a polarized
relay. That of Siemens is probably the most easily un-
derstood. The simple form of the relay is shown in Fig.
37. 5 is the south pole of a steel magnet bent at a right
angle. The lever aD is of soft iron pivoted at D and is
of south polarity. Attached to the north pole of the steel
magnet are two soft iron cores n and n', around which is
coiled wire in the same circuit but in opposite directions.
When no current passes the cores are of north polarity,
and the oscillating lever aD is attracted to the one near-
est it. If a current is sent through the coils, the cores be-
come electromagnets of opposite polarity, and the lever
then moves towards the north pole. If the current is re-
versed the lever moves in the opposite direction, and if
the circuit is broken the lever moves towards the nearest
ELECTRICITY AND MAGNETISM.
135
core, as both then become north poles from the inductive
action of the steel magnet, The motion of the lever is con-
9 TO LINE
TO LOCAL
BATTERY
D TO LINE
Fig. 37'
trolled by the two studs at a, the upper of which is con-
nected with a local battery in circuit with a sounder, as in
the common relay. The position of these studs is so regu-
lated that the lever, even when touching the upper one and
closing the local circuit, is nearer ri than n. The moment,
therefore, that the line current ceases, the lever is attracted
by two north poles, but moves towards the nearest, break-
ing the local circuit at the stud. No springs are necessary,
nor does the relay require any adjustment for strength of
current. From the foregoing it is seen that when the
direction of the current is such as to make a north pole
and n> a south, the lever moves and closes the local circuit.
When no current passes in the line, or when it passes in
the opposite direction, the local circuit is open.
In sending two messages at the same time in the same
direction, two keys are used, one reversing the current, send-
ing positive or negative currents, the other sending weak
or powerful. The strength of the current is, therefore,
controlled by one key, its sign by the other. The method
i 3 6
NOTES ON
used by Edison for transmitting is shown in the figure. In
the position shown the battery B has its terminals at N
L1NE and P t the current passing
from B through A" 2 to the
spring 6* and thence to P, If
the key K' is worked cur-
rents of either polarity may
be sent into the line, and
passing through a polarized
relay at the receiving station,
a sounder in the local circuit
is \vorked whenever a cur-
rent in a given direction is
transmitted by A". The
strength of the current is
immaterial, the polarized
Fig. 3 3.
relay answering only to cur-
rents in one direction. As shown in the figure, the circuit
of the battery B\ which is much larger than B, is open.
If, however, the key /v" 2 is depressed the spring 6" comes in
contact with the point m and breaks contact with #, and
as it is separated from A" 2 by the insulating material /, the
current of B now has to pass through B' m and 5 to P,
and is, of course, reinforced by the the powerful current
of B' in the same direction. Whenever A" 3 is depressed,
therefore, the points N and P retain their polarity, but the
current is of three or four times its original strength. In
practice all contacts are made by springs, so that the cir-
cuit is never broken at A%, but one current is followed di-
rectly by the other. The message transmitted by K ' 8 is
received by an ordinary relay in the same circuit with the
polarized relay at the receiving station, the lever of which
is controlled by a spring so adjusted that the weak cur-
rent of B will not cause sufficient magnetism in the elec
ELECTRICITY AND MAGNETISM. 137
tromagnets to attract the armature against the action of
the spring, but when A% is worked the current due to
B + B' easily overcomes it, whether the current be posi-
tive or negative, and the relay, therefore, transmits all
signals made by K *.
The quadruplex is merely an extension of the duplex,
using the diplex or double transmission. If in Fig. 163
(Thompson) the transmitting apparatus just described is
used instead of the keys R and R, and if between A and
B two relays are placed in series, one an ordinary relay
and the other a polarized, the figure would represent the
general arrangement of Edison's quadruplex system widely
used in the United States.
86. Blake's Transmitter.
In most telephone circuits, the receiving instrument is a
Bell telephone, but the transmitting is a modification of
Edison's telephone, known as Blake's Transmitter. The
waves of sound impinge on a metallic diaphragm, caus-
ing it to press with more or less force on a carbon button.
A current from a battery passes through the button and
the varying pressure of the diaphragm causes a varying
resistance in the circuit, and produces in the current fluc-
tuations, corresponding in number and time to the waves of
sound. If this current is passed through a Bell telephone,
the message could be heard. As now used, however, the
battery circuit is entirely local. In this local circua is the
primary coil of an induction coil, the secondary being in
circuit with the line to the next station. Every fluctua-
tion, therefore, in the strength of the local circuit, due to
the change of pressure on the carbon button of the trans-
mitter, induces a current in the secondary coil which
works a Bell telephone at the distant station. The induc-
tion coil is small, but it causes the electromotive force of
I3 8 NOTES ON
the line circuit to be much greater than that due to the
battery and extends the use of the telephone to greater
distances.
87. Telephone Exchanges.
The use of the telephone has been greatly extended by
the system of exchanges. A large number of persons
have telephoae circuits to a central office, where any two
circuits may be joined, thus enabling any two to converse.
A great difficulty in all telephone circuits is due to induc-
tion. The instrument is so extremely delicate that any in-
constant current near it induces sufficiently powerful cur-
rents in the telephone circuit to frequently obliterate a
message entirely. Telegrams may be read in telephones
if the telegraph and telephone circuits approach each
other very closely, and telephone messages may also be
heard in other circuits than that in which they are trans-
mitted. Most of the disturbances commonly attributed to
induction are, however, in all probability due to grounded
telegraph circuits.
ELECTRICITY AND MAGNETISM
139
REFERENCES TO PROF. THOMPSON'S ELEMENTARY LESSONS.
191, Note 26.
353,
Note 45, 46, 47.
ig2,
34-
357,
" 48.
199,
I.
358,
49, 50-
200,
i.
36o,
Si-
201,
2.
36i,
" 52.
202,
3-
362,
53-
203,
4-
i 364 '
54-
204,
5-
367,
55-
237,
6, 7, 8, g, 10.
371,
44 56.
238,
ii.
372,
57-
239,
12.
374,
44 58-
24O,
13-
375,
" 76.
241,
14.
376,
77-
245,
377,
78, 79-
2 4 6,
ii'
378,
" 55-
247,
17.
380,
" 80.
252,
18.
39i,
258,
ig.
392,
" 59-
26l,
20, 21.
393,
262,
51, c.
394,
' " 59, 60.
310,
22, 23, 24,25, 26,
395,
" 61.
27.
396,
41 62.
3 11 ,
28.
397,
" 63.
312,
23.
398,
" 66.
313,
2g.
404,
" 64.
3M,
30.
8 405,
65.
315,
3i6,
3i, 32.
33-
407,
i " 67, 68, 69,
317,
33-
| 409,
71.
3i8,
34, 35, 36, 37, 38.
i 4IO>
41 72.
319,
36.
4ii,
73, 74, 75.
320,
39-
415,
44 81.
324,
40.
425,
44 82.
325^,
41.
ts 426,
" 83.
327,
42.
427,
' 4 84.
338,
42.
428,
44 85.
43-
436,
44 86, 87.
352,'
44-
14 DAY USE
RETURN TO DESK FROM WHICH BORROWED
In* \y fl 1 ^1 Ib^ Lw 1 If
This book is due on the last date stamped below, or
on the date to which renewed.
Renewed books are subject to immediate recall.
f iditt'a/Rfisr
'
JAN 2 1957
^.lillSSRK 66 u-uSaSttoSKmi.
Berkeley
Murdock, J.
B.
M8
Notes on
sleotricity
ana magnet 1
sm
THE UNIVERSITY OF CALIFORNIA LIBRARY