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428 ELEMENTS OF ELECTRICITY.

E. M. F. will be generated in the side A, that is, there will be in-
duced in the coil a current, which, viewed from the point P (a
point on the horizontal axis of the coil perpendicular to the meri-
dian), will be counter-clockwise in direction. As B passes the
position A, and A passes the position B, the direction of the E. M.

F. in B and in A, and consequently the direction of the current in
the coil, is reversed, but at this same instant the opposite face of
the coil is presented to P, so that viewed from P, the current
flowing around the coil is always in the same direction. This cur-
rent is pulsating. It is zero when the plane of the coil is at right
angles to the magnetic meridian, and it is a maximum when this
plane coincides with the meridian, hence it rises and falls with
every half revolution of the coil. At the instant when the plane
of the coil coincides with the magnetic meridian, the instrument
is in principle the same as a tangent galvanometer (Par. 373),
and at all times it may be regarded as a tangent galvanometer
traversed by a current whose value is a mean of the instantane-
ous values of the current. The suspended needle will be deflected
accordingly.

The induced E. M. F. will vary directly with the rate of cutting
of the lines of force embraced by the coil. The number embraced
is Trr 2 H, r being the mean radius of the coil. The rate at which
these are cut varies with , the angular velocity of the coil, and
with n, the number of turns in the coil. If R be the resistance of
the coil, the current through it is proportional to



The field produced at the center of the coil is (Par. 354)
/ _ T v 2wn - 27r 2 r#ra 2 co

J ~~ 1 x ~r ~w~

If the needle be deflected through an angle 6, we have (Par. 146)



whence

,->

ft =

tan 5



ELECTRO-MAGNETICS. 429

But rco is the actual velocity of a point at the extremity of the
horizontal diameter of the coil. Calling this v, we have



D

R = r - -.V

tan 6

whence we see that the resist-

ance of the coil is equal to the product of a velocity by a numerical
factor. In the expression above, n and r are constants of the in-
strument and w and 5 are determined by observation. It will be
noted that it is not necessary to know the strength of the needle or
the intensity of the field H. If v be expressed in centimeters per
second, R will be in absolute units of resistance.

In actually carrying out the above determination, many
delicate refinements were observed. These are described in detail
in the Report of the British Association for the Advancement of
Science for the year 1864.

Resistance has been measured absolutely by several other
methods.

543. The Ohm. As a result of the experiment outlined in the
preceding paragraph, the investigators became possessed of a coil
of wire whose resistance in absolute units was accurately known.
The absolute unit being excessively small, the next step was to
select a practical unit which should be based upon this absolute
unit. It has been shown (Par. 284) that the need for a unit of
resistance had been felt for some time. The resistance coils made
by Ohm could not be standardized. In 1860 Siemens defined as
a unit of resistance a column of pure mercury one meter long and
one square millimeter in cross-section, the mercury being at a
temperature of C. Electricians had become accustomed to
this unit and the German scientists especially were loath to give
it up. The practical unit of resistance, the ohm, was accordingly
chosen so as to agree as nearly as possible with Siemen's unit, and
was defined as 10 9 absolute units of resistance, or (Par. 291) as
the resistance of a column of mercury, one millimeter in cross-
section and 106.3 centimeters in length, at a temperature of C.
It was later found more convenient to retain the length of the
column but to specify the quantity of mercury in terms of weight,
or as 14.4521 grams.

544. The Ampere. We have seen above (Par. 538) that the
absolute unit of current had been determined from the tangent



430 ELEMENTS OF ELECTRICITY.

galvanometer. It remains now to fix the practical unit of current.
The existing practical standard of E. M. F. was that of the Daniell
cell (Par. 206). This applied to the practical unit of resistance,
the ohm, should drive through it the unit current. This current
was found to be very nearly one-tenth of the absolute unit. The
practical unit of current, the ampere, was therefore selected as
exactly one-tenth of the absolute unit. Its definition has already
been given (Par. 228).

545. The Volt. The selection of the primary practical units
of resistance and current also fixed the volt, the practical unit of
E. M. F. From Ohm's law, E = IR. Since / = 1Q- 1 absolute units
of current and # = 10 9 absolute units of resistance, ' = 10- 1 Xl0 9
= 10 8 . The volt was therefore defined as 10 8 absolute units of
E. M. F.

546. Resume. The following resume will show the thread of
connection between the successive steps in the adoption of the
absolute and the practical electro-magnetic units.

(a) The absolute unit of current was determined by means of
the tangent galvanometer.

(b) The absolute resistance of a coil of wire was determined by
rotating the coil.

(c) From this was determined the absolute unit of resistance.

(d) This was found to be about .954 XlO~ 9 of Siemen's mercury
unit, already in use.

(e) To disturb this standard as little as possible, the practical
unit of resistance, the ohm, was taken as 10 9 absolute units of
resistance.

(f) The existing practical standard of E. M. F. was that of the
Daniell cell (1.07 volts) and it was desirable to disturb this as
little as possible.

(g) A Daniell cell applied to a circuit of one ohm drove through
it a current which was very slightly greater than one-tenth of the
absolute unit- of current.

(h) The practical unit of current, the ampere, was taken as
exactly one-tenth of the absolute unit of current.

(i) The selection of the practical units of resistance and current
involved that of E. M. F., the volt, since the three units are
bound together by Ohm's law. The volt is, therefore, 10 8 absolute
units of E. M. F.



ELECTRO-MAGNETICS. 431

547. Comparison of the Dimensional Formulae in the Two
Systems. A comparison of the dimensional formulae of the units
in the two systems will point to the contradictory conclusion that
they do not agree. As an example, let us compare the dimensional
formulae of the units of quantity.

In the electro-static system, we have from Coulomb's laws for
the force exerted between Jtwo equal quantities Q (Par. 56),
F = Q*/L 2 , whence Q = L vF. In mechanics it is shown that
force = mass X acceleration, or F=MxL/T 2 . Substituting this
value of F in the expression above, we have for the electro-static
dimensional formula of quantity

(I)



In the electro-magnetic system, Q = IxT = (E/R)xT. In
Par. 540 it was shown that E=FxL/Q and that R = L/T,
whence the electro-magnetic dimensional formula of quantity is

Q=VALL (ii)

Comparing (I) and (II), we see at once that they are not the
same, and that the ratio of (I) to (II) is L/T, a velocity.

In a similar manner may be determined the dimensional
formulae of the remaining units of current, capacity, potential
resistance, and inductance as given in the following table:

Unit Electro-static Electro-magnetic Ratio

Current LVW2./T* VluTL/T L/T = V

Quantity LVM.L/T v^LL L/T = V

Capacity L T Z /L L*/T* = V*

Potential VWX/T LVM.L/T 2 T/L = l/V

Resistance T/L L/T T*/L* = 1/V*

Inductance T 2 /L L T 2 /L 2 = 1/V 2

The V which enters all of these ratios has been determined in
widely different ways by a number of observers and found to be
3xl0 10 , or thirty billion, centimeters per second. This is the
velocity of light.

548. Explanation of Lack of Agreement. It is on the face of
it absurd that like quantities should have different dimensional
formulae, and also that these formulae should contain such
irrational quantities as the square root of a mass and of a length.
Consideration will show that this state of affairs results from our
failure to take into account in the formulae above the dielectric



432 ELEMENTS OF ELECTRICITY.

coefficient K (Par. 90) in the case of the electro-static units, and
the permeability /* (Par. 392) in the case of the electro-magnetic
units. The medium being air, these factors are both unity and
hence are of no arithmetical effect, but in omitting them we are
not justified in ignoring their dimensions. What these dimensions
are, we do not know, but that- they account for the lack of agree-
ment in the dimensional formulae of the two systems the following
will show.

In the preceding paragraph, in determining the dimensional
formula of the electro-static unit of quantity, our assumed ex-
pression for the force between two equal quantities Q should have
been (Par. 90) F=Q*/K.L 2

whence Q = L^/K.VMJ./T (I)

Likewise, in determining the dimensional formula of the electro-
magnetic unit of quantity, the expression for the force between
two equal magnetic poles should have been (Par. 133)

Whence m

The force exerted by a current /, flowing in a circular coil of
radius L, upon a pole m at the center of the coil is proportional to
ml/L (Par. 355), whence

/ = F.L/m

Substituting the value of m above and multiplying by T, we
have, since Q = IxT _

Q = VM.L/VV (ii)

Equating the second members of (I) and (II) and solving



T

We see then that while the dimensions of the separate factors
K and n are unknown, the reciprocal of the square root of their
product is a velocity, and therefore they can not be disregarded.

This velocity, as stated in the preceding paragraph, is the
velocity of light, and is also the velocity with which electric
waves travel through space. As will be shown later it has an
important bearing on Maxwell's electro-magnetic theory of light,
which is that light is really due to the passage of electric waves
through the ether.



ELECTRO-MECHANICS. 433



PART V.
ELECTRO-MECHANICS.



CHAPTER 40.

DIRECT CURRENT GENERATORS.

549. Electro-Mechanics. Electro-Mechanics, the subject which
we are now to take up, is a more or less artificial division in-
tended to embrace the production of electric currents by machin-
ery, a consideration of these mechanically generated currents, and
finally their employment to operate other machines.

Electricity, no matter how produced, is always the same agent
and the principles which have been developed in the preceding
pages suffice to explain all the facts which we shall now bring out.
The currents produced by machines are, however, more or less
pulsating and are often alternating, that is, they periodically
(usually many times a second), change their direction. These
rapid changes in the current give rise to certain phenomena which
renders it desirable to consider these currents in detail.

550. Classes of Electrical Machines. Electrical machines are
primarily of two classes, generators and motors. The former, also
called dynamos, transform mechanical energy into electrical energy
and therefore deliver electrical energy to a circuit. On the other
hand, motors transform electrical energy into mechanical energy
and therefore receive electrical energy from a circuit.

Machines are further classed according as they are designed to
deal with direct currents or with alternating currents. We shall
now consider generators of the former class.

551. Coil Rotating in a Magnetic Field. Suppose CD, Fig. 262,
to be a coil in the magnetic field NS and free to rotate about the
axis A B. Suppose its initial position to be, as shown in the figure,
with its plane perpendicular to the lines of force of the field. It



434



ELEMENTS OF ELECTRICITY.



now embraces the maximum number of these lines, and the first
effect of rotation about AB, whether clockwise or counter-clock-
wise, will be to decrease this number. This change will develop
in the coil an induced E. M. F. whose direction may be determined
by application of the rule given in Par. 421. It is simpler, however,
to apply the right hand rule given in Par. 422, whence we see at
once that whether the rotation be clockwise or counter-clockwise,
as the side C of the coil rotates 180 from the position C to the




Fig. 262.

position D, there is an E. M. F. induced in C from rear to front,
while as it rotates from D to (7, the E. M. F. induced is from front
to rear. The E. M. F. is reversed in direction whenever the coil
passes through the perpendicular plane, and is zero when the
coil lies in it, for which reason this plane is called the neutral
plane.

552. Calculation of E. M. F. of Rotating Coil. The E. M. F.

induced by the rotation of a coil in a magnetic field is from Par.
426 equal to the rate of decrease of the number of lines of force
embraced by the coil. If the field be uniform, this E. M. F. may
be calculated as follows. Let ab, Fig. 263, be the primary position
of the coil, its plane at right angles to the field. Its E. M. F. at
any point, such as d, is measured by the rate of decrease at that
point of the number of lines embraced.

Let the total field embraced by ab be N. Let the coil make n
revolutions per second, that is, let its angular velocity be 2vn.
If ca, the radius of the circle described by a, be R, the actual
velocity of a is 2irnR per second. At h the coil is moving at right
angles across the field with a velocity which in one second would



ELECTRO-MECHANICS.



435



carry it a distance hk. The total width of the field being 2R, it



would be crossed in



= seconds, and in this time N lines



of force would be cut by each side of the coil, therefore, the E. M. F.
being generated at h is

9 \7

- E =^ = 2 N

a



fer



i lia




\ ' \ .




\ ' \ 1




\ ' 1 '









:g:



b

Fig. 263.

If the coil consists of S turns, the E. M. F. is 2-jrnNS. To con-
vert this to volts, it must be divided by 10 8 (Par. 427), whence
finally

volts.



Should the coil at d continue to move for one second in the
same direction and at the same rate as at d, it would move a dis-
tance de=hk and in doing so would cut across the lines between
/ and e. If the angle dca through which the coil has turned from
its primary position be 0, then fe=de. sin 6. At the same time,
the other side g of the coil is cutting across the field at this same
rate, the total decrease being 2de. sin d. Since de=hk, the E. M. F.
being generated at d is

2mrNS . .



Or placing C for the coefficient of sin 0,

E=C. sin

For the present it is sufficient to bear in mind that the E. M. F.
generated by a coil rotating in a uniform field varies as the sine
of the angle through which the coil has turned from its primary
position at right angles to the field, that is, from the neutral plane.



436



ELEMENTS OF ELECTRICITY.



553. Production of Current by Rotating Coil. In Fig. 264 let
CD represent a coil rotating about the axis XY in the magnetic
field NS, and suppose that instead of being a closed coil, the end
C terminates in a ring A, and the end D in a ring B, these rings
being attached to the axis upon which the coil rotates, but being
insulated from it and from each other. In Par. 551 it was shown
that as the coil rotates 180 from its present position at right




Fig. 264.

angles to the field, an E. M. F. is generated from rear to front in C
and from front to rear in D. No current is produced because the
circuit is broken between the rings. If now a metal strip E be
pressed against the ring A, and a second strip F be pressed against
B, and these strips be connected by a wire, the circuit will be
completed and a current will flow through the coil and wire as
indicated by the arrows. A and B are collector rings, E and F are
brushes, and the wire connecting these brushes is the external
circuit.

554. Alternating Current. The resistance of the arrangement
just described being constant, the current in the external circuit
varies directly with the E. M. F. generated in the coil and this,
we have seen (Par. 552), varies as the sine of the angle through
which the coil has rotated from its position in the neutral plane.



ELECTRO-MECHANICS.



437



Thus, at the instant shown in Fig. 264, the current in C is zero,
but as C moves, a current flows towards A, reaching its maximum
value when the coil has turned through 90 or has become parallel
to the lines of force of the field. From this point, the current
diminishes and is again zero when C has turned through 180 or
has reached the position D. As C passes this point, the current
again starts up, but it is now reversed, that is, it flows away from
instead of towards A, and it consequently is also reversed in the
external circuit. If the original direction be considered positive,
this last must be considered negative. The current therefore
reaches a negative maximum when C has turned through 270,
returns to zero when C reaches its primary position, and again
reverses as C passes through this position.

To an E. M. F. and current which thus pass through these
periodic fluctuations and reversals, the term alternating is applied.

555. Graphic Representation of Alternating E. M. F. and Cur-
rent. In Fig. 265 let B represent the cross-section of a coil rotat-
ing about as a center and in a uniform field whose positive




Fig. 265.



direction, as indicated by the arrows, is downwards. AD is there-
fore the neutral plane. Should the coil start at A, the direction
of the induced E. M. F. is independent of the direction of rotation,
that is, whether the coil rotates in a clockwise or in a counter-
clockwise direction, the E. M. F. will act out from the plane of the
paper. However, to conform to the trigonometric convention as
to the direction in which angles are to be measured, we shall
assume the rotation to be counter-clockwise. From Par. 552, the
induced E. M. F. at any point B is proportional to B N, the sine
of the angle 6 through which the coil has rotated. If, therefore,
we lay off on a horizontal axis, A A, distances proportional to the



438 ELEMENTS OF ELECTRICITY.

angles through which the coil has turned, and at the points so
determined erect ordinates upon which we lay off distances pro-
portional to the sines of the corresponding angles, the sine or
harmonic curve drawn through the extremities of these ordinates
will represent the successive values of the E. M. F. For example,
the point R is determined by laying off AM proportional to AB,
and MR proportional (in this case equal) to NB.

The curve shows what was stated in the preceding paragraph,
that is, that the E. M. F. is zero at A, rises to a maximum when the
coil reaches C, decreases to zero at D where it reverses, reaches a
negative maximum at E and returns to zero at A, and so on.

In the case under consideration, the E. M. F. acting towards
the observer is considered positive, but this is purely a matter of
convention and it is immaterial whether we regard it as positive
or negative provided that the E. M. F. induced as the coil rotates
from A to D be opposite in sign to that induced as it rotates from
D to A. If the direction of the field be reversed, the direction of
the E. M. F. is also reversed.

Since the current varies directly with the E. M. F., we may take
this same sine curve as representing the current also, or we may
represent the current by another sine curve of the same periodicity
but of different amplitude. From Ohm's law, I=E/R, we see
that / and E are numerically equal only when R is unity. If R
be less than unity, / is numerically greater than E and would be
represented by the outer broken curve in Fig. 265. If R be greater
than unity, / would be represented by the inner broken curve.

Reflection will show that the abscissae of these sine curves may
also be laid off on a scale of time, the distance A A corresponding
to the time of one complete revolution of the coil.

556. Rectification of Alternating Current. Fig. 266 represents
the same arrangement of a coil rotating in a magnetic field as
described in Par. 553, only in this case the ends of the coil termi-
nate in the copper semicircles or segments A and B instead of in
two separate rings. These segments are likewise mounted upon
the shaft of the coil, insulated from it and from each other. The
brush E presses against the segment A; the brush F against the
segment B. For simplicity of description, suppose the rotation
to be clockwise. As C moves from its present position to the
position D, the induced E. M. F. acts towards A, and current will
therefore enter the external circuit by the brush E and leave it



ELECTRO-MECHANICS.



439



by the brush F. As C, having reached the position D, passes
through the neutral plane, the induced E. M. F. becomes zero
and immediately thereafter reverses, that is, acts from A and




Fig. 266.

towards B. But also, as the coil passes through the neutral plane
the brushes slip across the gap between the segments and E is
now in contact with B, while F is in contact with A, therefore,




Fig. 267.

current still flows out into the external circuit through the brush
E and the direction of the current in the external circuit remains
unchanged.



440 ELEMENTS OF ELECTRICITY.

This is shown graphically in Fig. 267. The sine curve A repre-
sents the alternating current (and E. M. F.) in the coil. B
represents the current in the external circuit, the negative loops
of the curve A having been reversed and made positive.

An alternating current which has thus been made unidirectional
is said to be rectified. The split ring, or arrangement of copper
segments by which this is brought about, is a commutator, and the
process is called commutation or rectification. We shall see later
that an alternating current may be rectified otherwise than by a
commutator.

557. Increase in Number of Turns of Coil. If the rotating
coil, instead of consisting of a single turn, be composed of several




Fig. 268.

as shown in Fig. 268, an approximately equal E. M. F. will be
induced in each. Examination of the figure will show that these
turns being connected in series, the total E. M. F. is the sum of
the separate E. M. F.s, or increases in proportion to the number
of turns. The resultant E. M. F. of the coil is represented graph-
ically by a sine curve of the same periodicity as the curves in Fig.
267 but of an amplitude greater in proportion to the number of
turns.

Although the E. M. F. is thus increased by increasing the
number of turns, practical considerations place a limit upon the
number that may be added. Thus, the resistance of the coil
increases directly with the number of turns and it is important
that this resistance should be kept very small. The diameter of
the wire, already large, must therefore be increased, and the wire
is further enlarged by an insulating covering. In the actual



ELECTRO-MECHANICS.



441



machines, the space in which these wires are wound is restricted,
being usually a narrow groove or slot in the surface of a cylindrical
body, and therefore the number of turns seldom exceeds six or
eight.

558. Increase in Number of Coils. For a considerable portion
of the time during the rotation of the single coil described in the
preceding paragraphs, the induced E. M. F. is small, and twice
during each complete revolution it is zero. If there be mounted
upon the same axis a second coil whose plane is at right angles to
that of the first (Fig. 269), the induced E. M. F. in this second




Fig, 269.

coil will be a maximum at the instant when it is zero in the first
coil, and also it will be zero in the second coil when it is a maximum
in the first. By a suitably arranged commutator we may always
draw current from that coil whose E. M. F. is the greater, and
thus avoid the periodic dropping to zero. For example, suppose
the commutator to consist of four segments to which the coils are
connected as indicated in the figure. At the instant represented,
the E. M. F. in A A, the vertical coil, is zero, and that in BB, the
horizontal coil, is a maximum, and it is this latter coil which is
sending current out into the external circuit. As the coils rotate,
the E. M. F. in BB decreases, that in AA increases, and these
reach equality when the coils have turned through an angle of 45.
At that moment, the brushes are across the gap between the seg-
ments and in contact with both. At the next instant, the brushes
are in contact with the segments connected to the A A coil, in
which coil the E. M. F. is rising to a maximum. These changes



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