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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

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46