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to the other requires the expenditure of one erg of work.

Unit li . A conductor offers unit resistance when

unit potential difference between its ends causes unit
current to flow through it.

I 'nit Capacity. A conductor has unit capacity when unit
quantity charges it to unit potential.

295. Practical Electrical Units Several of the ab-
solute electromagnetic units are inconveniently small and
others inconveniently large for practical use. Hence the
following multiples and sub-multiples of them have been
generally adopted as practical units :

Current. The ampere, equal to 10" 1 C.G.S. units ; it is
practically represented by the current which will deposit
silver from silver nitrate at the rate of 0.001118 gm. per
second (210).

Quantity. The coulomb, equal to 10" 1 C.G.S. units of
quantity ; it is the quantity conveyed by a current of one
ampere in one second.

Electromotive Force. The volt, equal to 10 s C.G.S. units ;
it is 1000/1434 of the E.M.F. of a standard Clark cell at
. (200).

Resistance. The ohm, equal to 10 9 C.G.S. unite ; a volt
produces an ampere through a resistance of an ohm ; prac-
tically represented by the resistance of a uniform column of
mercury 106.3 cms. in length and 14.4521 gms. mass at
C. (220).

Capacity. The farad, equal to 10~ 9 C.G.S. unite ; it is
the capacity of a condenser which is charged to a potential
of one volt by one coulomb. The microfarad, chiefly used
in practice, is one-millionth of a farad, or 10~ 15 C.G.S. unite.



342 ELECTRICITY AND MAGNETISM.

Work. The joule, equal to 10 7 ergs ; it is represented by
the energy expended per second by one ampere in one
ohm.

Power. The watt, equal to 10 7 ergs per second ; it is
equivalent to the power of a current of one ampere flowing
under an electric pressure of one volt, or one joule per
second ; approximately T J of a horse power.

Induction. The henry, equal to 10 9 C.G.S. units ; it is
the induction in a circuit when the electromotive force
induced in this circuit is one volt, while the inducing
current varies at the rate of one ampere per second (338).

The prefixes kilo- and milli- combined with any of the
preceding units signify a thousand and a thousandth respec-
tively. Thus a kilowatt is a thousand watts, and a millivolt
is a thousandth of a volt. The prefixes mega- and micro-
signify a million and a millionth respectively. Thus, a
megohm is a million ohms, and a microfarad is a millionth
of a farad.



ELECT ROD YNAMICS. 343



CHAPTER XXII.

ELECTRODYNAMICS.

296. Electrodynamics. The term electrodynamics is
applied to that part of the science of electricity which
is concerned with the force exerted by one current on
another. The reciprocal action between conductors con-
veying currents was discovered by Ampere in 1821, shortly
after Oersted's discovery of the reciprocal action between
a current and a magnet. So far as demonstrated, the forces
are between the conductors conveying the currents rather
than between the currents themselves. Every conductor
through which a current is flowing is surrounded by a
magnetic field, and the magnetic fields of two such con-
ductors react on each other.

297- Magnetic Fields about Parallel Currents (Th.,
385). The reciprocal action between conductors carry-
ing currents is purely magnetic, and may be accounted for
by the stresses set up in the surrounding medium. The
magnetic field about a single conductor is composed of
concentric circles (284) ; but when the fields of two con-
ductors are in part superposed, the composite magnetic
figures will l)e those due to the resultant of the two sets
of forces in every part of the field. Moreover, these figures
will exhibit attraction or repulsion between the conductors
according to the relative directions of the currents through
them.



344



ELECTRICITY AND MAGNETISM.



Fig. 159 is the field developed by iron filings about two
parallel wires passing through the two holes and with the
currents flowing in the same direction. In addition to
the distortion of the small circles immediately about the
conductors, showing that they are crowded together on
the outward sides and elongated between the wires, there
are continuous curves enclosing both circuits. These are
due to the coalescence of a number of circles belonging
to the two currents. The conductors are drawn together
by the tension along these lines of force. Midway between
the two is a region where the magnetic forces represented
by the circles are in opposite directions, and here the field
is neutral.





Fig. 159.



Fig. 160.



Fig. 160 is the field about two parallel conductors with
the currents flowing in opposite directions. It is the same
as the field through the centre of a circular conductor and
perpendicular to its plane. Midway between the two wires
the lines of force have the same direction in space, and
produce over a small area a uniform field. This is the
field utilized in the tangent galvanometer. The circles
about the wires are all excentric, but there are no lines
common to the two conductors ; the resiliency of these
lines, or their tendency to recover from the distortion,
forces the conductors apart.



ELECTRODYNAMICS.



345



298. Laws of Parallel and Oblique Currents.

I. Parallel conductors conveying currents in the same
'lir>"-tion attract each other ; if the currents are in opposite
7//v - //i;//x tln'ij repel each other.

This law is true for t\vo portions of the same circuit or
for two independent circuits. It depends on the relation
of the two magnetic fields and not on their independent
origin.

II. Two conductors' crossing obliquely attract each other
if the currents in them both flow toward the point of cr

or away from it ; but they repel if one flows toward and the
oth'.'r away from this point.

The motion always tends to make the
conductors not only parallel, but coinci-
dent. If two flat spirals, like the one in
Fig. 161, be suspended by long wires so
that their planes are parallel, or make a
small angle with each other, they will ex-
hibit mutual attraction and repulsion in a




marked manner.



Flg * I6K



III. The force between two parallel conductors is propor-
tional to the product of the current strengths, to the length of
the portions taken, and inversely as the
distance between them.

P
299. Ampere's Stand. For tfie

purpose of demonstrating the fore-
going laws, Ampere designed a stand
to hold a movable frame cany ing
a current (Fig. 162). At a and b
are mercury cups into which dip the
terminals of the balanced frame.
Another conductor placed parallel to either side of this




162.



346



ELECTRICITY AND MAGNETISM.



r4
ft*



Fig. 163.



rectangle, or obliquely to it, will show attraction or repul-
sion ; the same apparatus will serve to show the reaction
of a magnet on a current.

Such a circuit as the one shown in the
figure tends to set its plane^at right angles
to the magnetic meridian, with the current
flowing down on the east and up on the
west side of it. The direction of its own
field will then coincide with that of the
horizontal component of the earth's field.
Fig. 163 is an example of an astatic cir-
cuit that is not affected by terrestrial mag-
netism. The left side constitutes a south
pole and the right side a north pole ; that
is, the lines of force on the right of the
figure are directed toward an observer looking
at the figure, and away from him on the left.
Therefore the right side is repelled by the N-
seeking pole of a magnet and the left side is
attracted.



30O. Electromagnetic Rotations. - - A
large number of different devices have been
designed for the purpose of showing that
continuous rotations may be produced by the^
action between a magnet and a circuit, or in-
between two parts of the same circuit. In the
earlier apparatus one part of the circuit was
made movable, and the circuit was kept closed
by making connection with a liquid conductor
like mercury. Fig. 164 is one of the forms
designed by Faraday ; a copper wire is hung by a hook at
the top, and the lower end dips into a cup of mercury M



\ \AM1CS. '



347




Fig 165. .



surrounding the pole of a magnet. If the current flows
down through the wire, the lower end will rotate around the
pole clockwise.

Barlow's wheel (Fig.
IHo) is another device to
secure continuous rotation
by the action between a
magnet and a current. Con-
tact is made by mercury in
the trough (7, and the ac-
tion of the magnetic field
is on the radial current from the mercury to the axis A of
r r the copper wheel.

3O1. Electrodynamometfers. The
electrodynamometer is an instrument
designed originally by Weber to meas-
ure the strength of a current by the
electrodynamic action between two
coils of wire, one fixed and the other
movable about a vertical axis through
its own plane. The coils are set with
their magnetic axes at right angles
(Fig. 166), and the free coil moves in
a direction to make their axes coincide.
Let AB be a single convolution of
the fixed coil, and CD one of the sus-
pended coil. The ends a and b of the
latter dip into mercury cups and the
two coils are in series, as shown by
the arrows. The movable coil is sus-
pended by silk threads, or on a point resting in a jewel, and
a helix is rigidly connected with it and with the torsion





Fig. 166.



348



ELECTRICITY AND MAGNETISM.



head T above. The movable conductor is then subjected
to a system of forces tending to turn it in the direction
indicated.

When the coil CD is deflected by sending a current
through the instrument, the torsion head is turned by hand
so as to bring the coil back to its zero or initial position.

The couple due to the action be-
tween the coils is then offset by
the couple of torsion of the helix.
Now the couple of torsion is pro-
portional to the angle of torsion by
Hooke's law, the forces of resti-
tution being, within elastic limits,
proportional to the distortion itself.
The electrodynamic action between
the coils is proportional to the
square of the current,
since doubling the current
doubles it through both
coils, and therefore quad-
ruples the force. The
square of the current is
Flg I67> then proportional to the

angle through which the counterbalancing helix is twisted, or

P = A 2 D,

i= A*SD.

A is a constant depending on the windings and the helix.
Since this expression is the common equation of a parabola,
if the currents and twists of the helix are plotted as coor-
dinates, the resulting curve will be a parabola. The twist
D may be expressed in any convenient divisions of a circle
into equal parts.




ELECTRODYNAMICS.



349



Fig. 167 is one form of the complete instrument, showing
the coils, the helix, and the scale at the top with the
pointers. The movable coil is raised so that the suspend-
ing point is lifted out of the jewel bearing.

The fixed coil may be considered as furnishing a mag-
netic field corresponding to that of the permanent magnet
in the d'Arsniival galvanometer; but in this instrument
the field reverses with the current, and therefore the deflec-
tion is in the same direction whether the current goes in
one direction through the instrument or the other. It may
therefore be used with alternating or reversing currents as
well as with direct ones.




Fig. 168.



302. Kelvin Balances. The justly celebrated' instru-
ments of Lord Kelvin for measuring currents operate by
means of the eleetrodyiiainie action between parallel fixed
and movable coils. This action is counterbalanced by
adjustable weights or sliders instead of the torsion of a
helix. They are therefore dependent on the force of
gravity.

The coils are ring-shaped and horizontal. The movable
rings E and F (Fig. 168) are attached to the ends of
a horizontal balance beam, which is supported by two



350 ELECTRICITY AND MAGNETISM.

trunnions a and >, each hung by an elastic ligament of
fine copper wires. These are utilized to pass the current
into and out of the movable coils. A, B and (7, D are two
pairs of fixed coils, connected as shown, so that the mov-
able ring on either side is attracted by one fixed ring and
repelled by the other. When a current passes through the
six coils in series, the beam tends to rise at F and sink at
E. It is brought back to zero by sliding a weight to the
right along a graduated horizontal arm attached to the
beam of the balance. The weights are so adjusted that
the readings on this arm give the current either directly
or else by means of a table of double square roots. The
current is proportional to the square root of the reading
on a scale of equal parts. These balances, like the elec-
trodynamometer, may be used for alternating as well as for
direct currents.






303. Convection Currents. Two concurring parallel
currents attract and two like electric charges repel each
other. According to Maxwell, the electrodynamic attraction
should exactly equal the electrostatic repulsion when the
electrical charges move with the velocity of light. Ac-
cording to Faraday, a stream of particles carrying electric
charges has a magnetic effect like a current of electricity.
This was demonstrated to be true by Rowland in 1876, who
found that a charged disk, when rapidly rotated, had a
feeble magnetic effect equivalent to a circular current.
Conversely convection currents are acted on by magnets.
The electric arc behaves like a flexible conductor. It may
even be ruptured by the deflecting influence of a powerful
magnet. Elihu Thomson has utilized this effect to extin-
guish an arc started by lightning on an electric lighting
circuit.



FL ECTR OD TNA MICS. 351

^L*-*

304. The Hall Effect. In 1880 Hall discovered that
when a current flows through a very thin strip of metal
in a powerful magnetic field, with its plane perpendicular
to the lines of force, an E.M.F. appears to be developed in
the strip at right angles both to the field and to the direc-
tion of the current. The result is that the lines connecting
equipotential points are no longer at right angles to the
lines of flow, or the equipotential lines and the current
lines are both displaced. The displacement is in one direc-
tion in gold and bismuth, and in the other in iron and tel-
lurium. S. P. Thompson has shown that bismuth, which
exhibits the Hall effect in a marked degree, undergoes a
change of resistance in a magnetic field. The increase of
resistance shown by bismuth is so marked that this prop-
erty is utilized to measure the strength of the magnetic
field in which it is placed.



352 ELECTRICITY AND MAGNETISM.



CHAPTER XXIII.

ELBCTROMAGNETISM.

305. Solenoids. Since a circular current is equiva-
lent to a plane magnetic shell, if we build up a cylinder
of such equal circular currents, all parallel to one another
and with their similar faces all turned in the same direc-
tion, we shall have the equivalent of a cylindrical magnet.
Such a system of circular currents
constitutes a solenoid (Fig. 169).
The practical solenoid is simply a
helix of a large number of flat turns
close together. Each turn of the helix may be resolved
into a plane circular current, ABC (Fig. 170), and a linear
current AC perpendicular to the plane of the
circle. The entire helix of n turns is there-
fore equivalent to n circular currents and a
linear current along the axis of the helix. If
the current returns along the axis, as in the
figure, the external field is due to the circular
elements only.

If such a solenoid be suspended on an Ampere's stand
it will set its axis in the magnetic meridian when a cur-
rent is passed through it. It is therefore equivalent to a
magnet, and its poles can be determined by Maxwell's rule
(283). Its poles will be attracted and repelled by a magnet
like a magnetic needle. The direction of the current is
with or against watch hands according as its S-seeking or
N-seeking pole is presented to the observer.




FL ECTEOMAGNETISM.



353



306. Effect of introducing Iron. When iron is placed
in a magnetic field it becomes magnetized by induction
(^ ")'). If, therefore, a bar of iron be introduced into a
solenoid conveying a current, it will be magnetized by the
magnetic force along the axis of the helix. The presence
of the iron not only confines the lines of induction more
closely to the helix, but it greatly increases the number of




Fig. 171



them, as represented in the solenoids of Fig. 171. These
solenoids are left-handed, but their poles may be deter-
mined in the usual way by the application of the "rule of
thumb " iS2.



3O7. Electromagnets. Directly after Oersted's dis-
covery Arago and Davy independently discovered that iron
and steel may be magnetized by inserting bars or strips of
them into a coil of insulated wire through which an electric
current circulates. If the bar be of soft iron it will exhibit
notable magnetic effects only so long as the current flows
through the magnetizing coil. The loss of magnetization
is nut complete when the current is interrupted; the small
amount remaining is called residual magnetism.



354



ELECTRICITY AND MAGNETISM.




Fig. 172.



Such temporary magnets produced by the magnetic in-
duction within a solenoid or magnetizing helix are called
electromagnets. When properly proportioned they are much
more powerful than permanent magnets. The polarity and
the relation of the poles to the direction of the current
may be determined by one of the usual rules.

3O8. Horseshoe Magnet. - - The most common form
of electromagnet is the horseshoe type (Fig. 172). The
windings on the two iron cylinders or
cores must be in a direction to make
the two poles of opposite signs. It
is the same as if the two cores were
straightened out and the bar wound
\ continuously from end to end.

The armature, not shown in the
figure, consists of a flat bar like the yoke at the other
end, and extending across from pole to pole. Its cross-
section should be equal to that of
the cores. As a rule, the cores, the
yoke, and the armature should form
a nearly closed magnetic circuit
(261). If a ring be wound contin-
uously with a right-handed helix
so as to form a closed circuit, and
if connection be made with it at two
points diametrically opposite (Fig.
173), and a divided current be sent
through, there will be a consequent
south pole where the current enters
and a north pole where it leaves the
ring. The lines of force about it are those of Fig. 138,
The poles are consequent because they belong to two




ELECTRON A GNETLW. 355

magnetic circuits, or to a divided circuit through the
iron.

3O9. Magnetic Permeability. - The effect of placing
iron in a magnetic field is to increase greatly the number
of lines of induction running through the space occupied
by the iron. When these lines of magnetic induction
traverse the iron it is magnetized. The increase in the
number of lines due to the iron may amount to several
thousand per square centimetre.

Let < i r> stand for the induction, or the number of lines
per square centimetre, through the iron. Then the ratio
between , l o and t\J is called the permeability, or



where fi stands for the ' -mieability. It expresses the fact
that iron transmits t^ inductive effect better than air, or
is more permeable. .Magnetic induction is /JL times the
magnetic force.

310. Magnetic Susceptibility. The intensity of mag-
netization is the pole strength per unit area of the polar
surface (:2(jf>). Magnetic susceptibility is the ratio between
the intensity of magnetization and the strength of the
field, or in symbols,

K = <?/<%'.

The conception involved in permeability rather than the
one in susceptibility is the modern one derived from

Faraday.

311. Relation between /JL and K. Let 8?6 be the num-
ber of lines of magnetic force existing in the air before the
introduction of the iron. Then the iron adds to these the



356 ELECTRICITY AND MAGNETISM.

lines due to a magnet of pole strength m. Hence, if * is
the sectional area of the uniformly magnetized bar,



or 66 = ag + 47r- = a? + 47T c y.



Wherefore, -1 4-4,r ,

c%^~ ^

and fji = 1 + 47T/c.

Susceptibility may be negative ; but while permeability may

be less than unity, it is never negative.

312. Paramagnetic and Diamagnetic Substances
compared (J. J. T., 257;. - The concept involved in
permeability permits a clear extinction, to be drawn be-
tween paramagnetic and diamagv >tic substances. Para-
magnetic substances are those whose. * ermeability is greater
than unity ; and since the permeability of air is practically
unity, paramagnetic substances are those more permeable
__ to lines of magnetic in-
duction than air. On the
other hand, diamagnetic
substances have a permea-
bility less than unity, or
are less permeable than
air. Permeability ex-
Fig. > 74 - presses* the number of

magnetic lines in the medium for every line of magnet-
izing force applied to produce them.

Paramagnetic substances concentrate the magnetic lines
and diamagnetic substances diffuse them. If iron be placed
in a magnetic field, it will cause more lines of induction to
pass through than through air; but if bismuth be placed





ELECTRON A GXETISM.



357




Fig. 175.



there instead of iron, fewer lines will pass through it than
through the air previous to its introduction.

If an iron sphere be placed in a uniform magnetic field
(Fig. 174) the effort of the lines will* be to run as much as
possible through the sphere.
This action proceeds on the
principle that the potential
energy of a system always
tends to as small a value as
possible; for when the same
number of lines pass through
iron as through air they have
less riieiLrv in unit volume of iron.

If the sphere in Fig. 175 be bismuth the effort of the
magnetic lines will be to avoid it. There are fewer lines
of induction in it than in air. For the same number of
lines the energy per unit volume is greater in bismuth than
in air.

Wlien the lines of force pass from air to a paramagnetic
substance they are bent away from the normal in the sul>-
stance; but when they pass from air to a diamagnetic
substance they are bent toward the normal.

313. Movement of Paramagnetic and Diamagnetic
Bodies in a Magnetic Field. Faraday examined the
magnetic behavior of a large num-
ber of bodies in the intense field
between the pointed poles of a
powerful electromagnet. A small
bar of iron suspended between the
poles (Fig. 176) turns in the axial
direction ah, while a bar of bismuth
sets its longer axis in the equatorial direction, cd. If the




Fig. 176.



358 ELECTRICITY AND MAGNETISM.

bismuth is in the form of a cube or lump it is repelled
toward one side. Iron moves into the stronger parts of the
field and bismuth into the weaker. They are examples of
the two classes into which bodies are divided with respect
to the action of magnetism on them.

These movements may be satisfactorily explained by the
relative permeability of the body and the medium in which
it is suspended. Feebly magnetic bodies behave as if they
were diamagnetic when surrounded by a more highly mag-
netic medium. A small glass tube containing a weak solu-
tion of ferric chloride is paramagnetic in air ; but when
suspended in a stronger solution of ferric chloride, it takes
a cross-position like a diamagnetic body. When, therefore,
any substance assumes the equatorial position, the only
inference which can justly be drawn from this behavior is
that its permeability is less than that of the air or other
medium surrounding it.

In general, liquids are diamagnetic ; liquid oxygen and
solutions of salts of the magnetic metals are exceptions.

314. Magne-crystallic Action. In crystalline bodies
the permeability may vary with the direction. Such a
substance is said to be ceolotropic. Tyndall found that the
magnetic axis or line of greatest permeability in a crystal
is in general an axis of greatest density, and it is this axis
that tends to place itself either along the magnetic field
or across it according as the crystal is paramagnetic or
diamagnetic.

Directions of unequal induction or permeability can
be produced artificially by pressure. Thus, a small roll
of powdered bismuth, made adhesive by mixing with
gum-water, will set itself across the field between the
poles of the excited magnet ; but if it be squeezed flat by



359

mechanical pressure, it \vill then turn in the axial position.
The lines of pressure transverse to the thickness are then
the lines of closest proximity of the particles and the lines
of most powerful induction.


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