Chapter XIV: Part 14

The work done in carrying a unit magnetic pole once round a circuit
conveying a current is called the "line integral of magnetic force"
along that path. If, for instance, we carry a unit pole in a circular
path of radius r once round an infinitely long straight filamentary
current I, the line integral is 4[pi]I. It is easy to prove that this
is a general law, and that if we have any currents flowing in a
conductor the line integral of magnetic force taken once round a path
linked with the current circuit is 4[pi] times the total current
flowing through the circuit. Let us apply this to the case of an
endless solenoid. If a copper wire insulated or covered with cotton or
silk is twisted round a thin rod so as to make a close spiral, this
forms a "solenoid," and if the solenoid is bent round so that its two
ends come together we have an endless solenoid. Consider such a
solenoid of mean length l and N turns of wire. If it is made endless,
the magnetic force H is the same everywhere along the central axis and
the line integral along the axis is Hl. If the current is denoted by
I, then NI is the total current, and accordingly 4[pi]NI = Hl, or H =
4[pi]NI/l. For a thin endless solenoid the axial magnetic force is
therefore 4[pi] times the current-turns per unit of length. This holds
good also for a long straight solenoid provided its length is large
compared with its diameter. It can be shown that if insulated wire is
wound round a sphere, the turns being all parallel to lines of
latitude, the magnetic force in the interior is constant and the lines
of force therefore parallel. The magnetic force at a point outside a
conductor conveying a current can by various means be measured or
compared with some other standard magnetic forces, and it becomes then
a means of measuring the current. Instruments called galvanometers and
ammeters for the most part operate on this principle.

_Thermal Effects of Currents._--J.P. Joule proved that the heat produced by a constant current in a given time in a wire having a constant resistance is proportional to the square of the strength of the current. This is known as Joule's law, and it follows, as already shown, as an immediate consequence of Ohm's law and the fact that the power dissipated electrically in a conductor, when an electromotive force E is applied to its extremities, producing thereby a current I in it, is equal to EI.

If the current is alternating or periodic, the heat produced in any
time T is obtained by taking the sum at equidistant intervals of time
of all the values of the quantities Ri^2dt, where dt represents a small
interval of time and i is the current at that instant. The quantity
_
/ T
T^(-1) | i^2dt is called the mean-square-value of the variable
_/ 0

current, i being the instantaneous value of the current, that is, its
value at a particular instant or during a very small interval of time
dt. The square root of the above quantity, or
_ _ _
| / T | 1/2,
| T^(-1) | i^2dt |
|_ _/ 0 _|

is called the root-mean-square-value, or the effective value of the
current, and is denoted by the letters R.M.S.

Currents have equal heat-producing power in conductors of identical resistance when they have the same R.M.S. values. Hence periodic or alternating currents can be measured as regards their R.M.S. value by ascertaining the continuous current which produces in the same time the same heat in the same conductor as the periodic current considered. Current measuring instruments depending on this fact, called hot-wire ammeters, are in common use, especially for measuring alternating currents. The maximum value of the periodic current can only be determined from the R.M.S. value when we know the wave form of the current. The thermal effects of electric currents in conductors are dependent upon the production of a state of equilibrium between the heat produced electrically in the wire and the causes operative in removing it. If an ordinary round wire is heated by a current it loses heat, (1) by radiation, (2) by air convection or cooling, and (3) by conduction of heat out of the ends of the wire. Generally speaking, the greater part of the heat removal is effected by radiation and convection.

If a round sectioned metallic wire of uniform diameter d and length l
made of a material of resistivity [rho] has a current of A amperes
passed through it, the heat in watts produced in any time t seconds is
represented by the value of 4A^2[rho]lt/10^9[pi]d^2, where d and l
must be measured in centimetres and [rho] in absolute C.G.S.
electromagnetic units. The factor 10^9 enters because one ohm is 10^9
absolute electromagnetic C.G.S. units (see UNITS, PHYSICAL). If the
wire has an emissivity e, by which is meant that e units of heat
reckoned in joules or watt-seconds are radiated per second from unit
of surface, then the power removed by radiation in the time t is
expressed by [pi]dlet. Hence when thermal equilibrium is established
we have 4A^2[rho]lt/10^9[pi]d^2 = [pi]dlet, or A^2 =
10^9[pi]^2ed^3/4[rho]. If the diameter of the wire is reckoned in mils
(1 mil = .001 in.), and if we take e to have a value 0.1, an
emissivity which will generally bring the wire to about 60 deg. C., we
can put the above formula in the following forms for circular
sectioned copper, iron or platinoid wires, viz.

A = [root](d^3/500) for copper wires
A = [root](d^3/4000) for iron wires
A = [root](d^3/5000) for platinoid wires.

These expressions give the ampere value of the current which will
bring bare, straight or loosely coiled wires of d mils in diameter to
about 60 deg. C. when the steady state of temperature is reached. Thus,
for instance, a bare straight copper wire 50 mils in diameter (=0.05
in.) will be brought to a steady temperature of about 60 deg. C. if a
current of [root]50^3/500 = [root]250 = 16 amperes (nearly) is passed
through it, whilst a current of [root]25 = 5 amperes would bring a
platinoid wire to about the same temperature.

A wire has therefore a certain safe current-carrying capacity which is determined by its specific resistance and emissivity, the latter being fixed by its form, surface and surroundings. The emissivity increases with the temperature, else no state of thermal equilibrium could be reached. It has been found experimentally that whilst for fairly thick wires from 8 to 60 mils in diameter the safe current varies approximately as the 1.5th power of the diameter, for fine wires of 1 to 3 mils it varies more nearly as the diameter.

_Action of one Current on Another._--The investigations of Ampere in connexion with electric currents are of fundamental importance in electrokinetics. Starting from the discovery of Oersted, Ampere made known the correlative fact that not only is there a mechanical action between a current and a magnet, but that two conductors conveying electric currents exert mechanical forces on each other. Ampere devised ingenious methods of making one portion of a circuit movable so that he might observe effects of attraction or repulsion between this circuit and some other fixed current. He employed for this purpose an astatic circuit B, consisting of a wire bent into a double rectangle round which a current flowed first in one and then in the opposite direction (fig. 5). In this way the circuit was removed from the action of the earth's magnetic field, and yet one portion of it could be submitted to the action of any other circuit C. The astatic circuit was pivoted by suspending it in mercury cups q, p, one of which was in electrical connexion with the tubular support A, and the other with a strong insulated wire passing up it.

Ampere devised certain crucial experiments, and the theory deduced from them is based upon four facts and one assumption.[2] He showed (1) that wire conveying a current bent back on itself produced no action upon a proximate portion of a movable astatic circuit; (2) that if the return wire was bent zig-zag but close to the outgoing straight wire the circuit produced no action on the movable one, showing that the effect of an element of the circuit was proportional to its projected length; (3) that a closed circuit cannot cause motion in an element of another circuit free to move in the direction of its length; and (4) that the action of two circuits on one and the same movable circuit was null if one of the two fixed circuits was n times greater than the other but n times further removed from the movable circuit. From this last experiment by an ingenious line of reasoning he proved that the action of an element of current on another element of current varies inversely as a square of their distance. These experiments enabled him to construct a mathematical expression of the law of action between two elements of conductors conveying currents. They also enabled him to prove that an element of current may be resolved like a force into components in different directions, also that the force produced by any element of the circuit on an element of any other circuit was perpendicular to the line joining the elements and inversely as the square of their distance. Also he showed that this force was an attraction if the currents in the elements were in the same direction, but a repulsion if they were in opposite directions. From these experiments and deductions from them he built up a complete formula for the action of one element of a current of length dS of one conductor conveying a current I upon another element dS' of another circuit conveying another current I' the elements being at a distance apart equal to r.

If [theta] and [theta]' are the angles the elements make with the line
joining them, and [phi] the angle they make with one another, then
Ampere's expression for the mechanical force f the elements exert on
one another is

f = 2II'r^(-2) {cos [phi] - (3/2)cos [theta] cos [theta]'}dSdS'.

This law, together with that of Laplace already mentioned, viz. that
the magnetic force due to an element of length dS of a current I at a
distance r, the element making an angle [theta] with the radius vector
o is IdS sin [theta]/r^2, constitute the fundamental laws of
electrokinetics.

Ampere applied these with great mathematical skill to elucidate the mechanical actions of currents on each other, and experimentally confirmed the following deductions: (1) Currents in parallel circuits flowing in the same direction attract each other, but if in opposite directions repel each other. (2) Currents in wires meeting at an angle attract each other more into parallelism if both flow either to or from the angle, but repel each other more widely apart if they are in opposite directions. (3) A current in a small circular conductor exerts a magnetic force in its centre perpendicular to its plane and is in all respects equivalent to a magnetic shell or a thin circular disk of steel so magnetized that one face is a north pole and the other a south pole, the product of the area of the circuit and the current flowing in it determining the magnetic moment of the element. (4) A closely wound spiral current is equivalent as regards external magnetic force to a polar magnet, such a circuit being called a finite solenoid. (5) Two finite solenoid circuits act on each other like two polar magnets, exhibiting actions of attraction or repulsion between their ends.

Ampere's theory was wholly built up on the assumption of action at a distance between elements of conductors conveying the electric currents. Faraday's researches and the discovery of the fact that the insulating medium is the real seat of the operations necessitates a change in the point of view from which we regard the facts discovered by Ampere. Maxwell showed that in any field of magnetic force there is a tension along the lines of force and a pressure at right angles to them; in other words, lines of magnetic force are like stretched elastic threads which tend to contract.[3] If, therefore, two conductors lie parallel and have currents in them in the same direction they are impressed by a certain number of lines of magnetic force which pass round the two conductors, and it is the tendency of these to contract which draws the circuits together. If, however, the currents are in opposite directions then the lateral pressure of the similarly contracted lines of force between them pushes the conductors apart. Practical application of Ampere's discoveries was made by W.E. Weber in inventing the electrodynamometer, and later Lord Kelvin devised ampere balances for the measurement of electric currents based on the attraction between coils conveying electric currents.

_Induction of Electric Currents._--Faraday[4] in 1831 made the important discovery of the induction of electric currents (see ELECTRICITY). If two conductors are placed parallel to each other, and a current in one of them, called the primary, started or stopped or changed in strength, every such alteration causes a transitory current to appear in the other circuit, called the secondary. This is due to the fact that as the primary current increases or decreases, its own embracing magnetic field alters, and lines of magnetic force are added to or subtracted from its fields. These lines do not appear instantly in their place at a distance, but are propagated out from the wire with a velocity equal to that of light; hence in their outward progress they cut through the secondary circuit, just as ripples made on the surface of water in a lake by throwing a stone on to it expand and cut through a stick held vertically in the water at a distance from the place of origin of the ripples. Faraday confirmed this view of the phenomena by proving that the mere motion of a wire transversely to the lines of magnetic force of a permanent magnet gave rise to an induced electromotive force in the wire. He embraced all the facts in the single statement that if there be any circuit which by movement in a magnetic field, or by the creation or change in magnetic fields round it, experiences a change in the number of lines of force linked with it, then an electromotive force is set up in that circuit which is proportional at any instant to the rate at which the total magnetic flux linked with it is changing. Hence if Z represents the total number of lines of magnetic force linked with a circuit of N turns, then -N(dZ/dt) represents the electromotive force set up in that circuit. The operation of the induction coil (q.v.) and the transformer (q.v.) are based on this discovery. Faraday also found that if a copper disk A (fig. 6) is rotated between the poles of a magnet NO so that the disk moves with its plane perpendicular to the lines of magnetic force of the field, it has created in it an electromotive force directed from the centre to the edge or vice versa. The action of the dynamo (q.v.) depends on similar processes, viz. the cutting of the lines of magnetic force of a constant field produced by certain magnets by certain moving conductors called armature bars or coils in which an electromotive force is thereby created.

In 1834 H.F.E. Lenz enunciated a law which connects together the
mechanical actions between electric circuits discovered by Ampere and
the induction of electric currents discovered by Faraday. It is as
follows: If a constant current flows in a primary circuit P, and if by
motion of P a secondary current is created in a neighbouring circuit
S, the direction of the secondary current will be such as to oppose
the relative motion of the circuits. Starting from this, F.E. Neumann
founded a mathematical theory of induced currents, discovering a
quantity M, called the "potential of one circuit on another," or
generally their "coefficient of mutual inductance." Mathematically M
is obtained by taking the sum of all such quantities as ff dSdS' cos
[phi]/r, where dS and dS' are the elements of length of the two
circuits, r is their distance, and [phi] is the angle which they make
with one another; the summation or integration must be extended over
every possible pair of elements. If we take pairs of elements in the
same circuit, then Neumann's formula gives us the coefficient of
self-induction of the circuit or the potential of the circuit on
itself. For the results of such calculations on various forms of
circuit the reader must be referred to special treatises.

H. von Helmholtz, and later on Lord Kelvin, showed that the facts of
induction of electric currents discovered by Faraday could have been
predicted from the electrodynamic actions discovered by Ampere
assuming the principle of the conservation of energy. Helmholtz takes
the case of a circuit of resistance R in which acts an electromotive
force due to a battery or thermopile. Let a magnet be in the
neighbourhood, and the potential of the magnet on the circuit be V, so
that if a current I existed in the circuit the work done on the magnet
in the time dt is I(dV/dt)dt. The source of electromotive force
supplies in the time dt work equal to EIdt, and according to Joule's
law energy is dissipated equal to RI^2dt. Hence, by the conservation
of energy,

EIdt = RI^2dt + I(dV/dt)dt.

If then E = 0, we have I = -(dV/dt)/R, or there will be a current due
to an induced electromotive force expressed by -dV/dt. Hence if the
magnet moves, it will create a current in the wire provided that such
motion changes the potential of the magnet with respect to the
circuit. This is the effect discovered by Faraday.[5]

_Oscillatory Currents._--In considering the motion of electricity in conductors we find interesting phenomena connected with the discharge of a condenser or Leyden jar (q.v.). This problem was first mathematically treated by Lord Kelvin in 1853 (_Phil. Mag._, 1853, 5, p. 292).

If a conductor of capacity C has its terminals connected by a wire of
resistance R and inductance L, it becomes important to consider the
subsequent motion of electricity in the wire. If Q is the quantity of
electricity in the condenser initially, and q that at any time t after
completing the circuit, then the energy stored up in the condenser at
that instant is 1/2q^2/C, and the energy associated with the circuit
is 1/2L(dq/dt)^2, and the rate of dissipation of energy by resistance
is R(dq/dt)^2, since dq/dt = i is the discharge current. Hence we can
construct an equation of energy which expresses the fact that at any
instant the power given out by the condenser is partly stored in the
circuit and partly dissipated as heat in it. Mathematically this is
expressed as follows:--

_ _ _ _
d | q^2 | d | /dq\^2 | /dq\^2
- -- | 1/2 --- | = -- | 1/2L ( -- ) | + R ( -- )
dt |_ C _| dt |_ \dt/ _| \dt/

or

d^2q R dq 1
---- + -- -- + -- q = 0.
dt^2 L dt LC

The above equation has two solutions according as R^2/4L^2 is greater
or less than 1/LC. In the first case the current i in the circuit can
be expressed by the equation

[alpha]^2+[beta]^2
i= Q ------------------ e^(-[alpha]t) [e^([beta]t) - e^(-[beta]t)],
2[beta]
________
/R^2 1
where [alpha] = R/2L, [beta] = / --- - --, Q is the value of q when
\/ 4L^2 LC

t = 0, and e is the base of Napierian logarithms; and in the second
case by the equation

[alpha]^2+[beta]^2
i = Q ------------------ e^(-[alpha]t) sin [beta]t
[beta]
_________
/1 R^2
where [alpha] = R/2L, and [beta] = / -- - ----.
\/ LC 4L^2

These expressions show that in the first case the discharge current of
the jar is always in the same direction and is a transient
unidirectional current. In the second case, however, the current is an
oscillatory current gradually decreasing in amplitude, the frequency n
of the oscillation being given by the expression
_________
1 /1 R^2
n = ----- / -- - ----.
2[pi] \/ LC 4L^2

In those cases in which the resistance of the discharge circuit is
very small, the expression for the frequency n and for the time period
of oscillation R take the simple forms n = 1, 2[pi][root]LC, or T =
1/n = 2[pi][root]LC.

The above investigation shows that if we construct a circuit consisting of a condenser and inductance placed in series with one another, such circuit has a natural electrical time period of its own in which the electrical charge in it oscillates if disturbed. It may therefore be compared with a pendulum of any kind which when displaced oscillates with a time period depending on its inertia and on its restoring force.

The study of these electrical oscillations received a great impetus after H.R. Hertz showed that when taking place in electric circuits of a certain kind they create electromagnetic waves (see ELECTRIC WAVES) in the dielectric surrounding the oscillator, and an additional interest was given to them by their application to telegraphy. If a Leyden jar and a circuit of low resistance but some inductance in series with it are connected across the secondary spark gap of an induction coil, then when the coil is set in action we have a series of bright noisy sparks, each of which consists of a train of oscillatory electric discharges from the jar. The condenser becomes charged as the secondary electromotive force of the coil is created at each break of the primary current, and when the potential difference of the condenser coatings reaches a certain value determined by the spark-ball distance a discharge happens. This discharge, however, is not a single movement of electricity in one direction but an oscillatory motion with gradually decreasing amplitude. If the oscillatory spark is photographed on a revolving plate or a rapidly moving film, we have evidence in the photograph that such a spark consists of numerous intermittent sparks gradually becoming feebler. As the coil continues to operate, these trains of electric discharges take place at regular intervals. We can cause a train of electric oscillations in one circuit to induce similar oscillations in a neighbouring circuit, and thus construct an oscillation transformer or high frequency induction coil.

_Alternating Currents._--The study of alternating currents of electricity began to attract great attention towards the end of the 19th century by reason of their application in electrotechnics and especially to the transmission of power. A circuit in which a simple periodic alternating current flows is called a single phase circuit. The important difference between such a form of current flow and steady current flow arises from the fact that if the circuit has inductance then the periodic electric current in it is not in step with the terminal potential difference or electromotive force acting in the circuit, but the current lags behind the electromotive force by a certain fraction of the periodic time called the "phase difference." If two alternating currents having a fixed difference in phase flow in two connected separate but related circuits, the two are called a two-phase current. If three or more single-phase currents preserving a fixed difference of phase flow in various parts of a connected circuit, the whole taken together is called a polyphase current. Since an electric current is a vector quantity, that is, has direction as well as magnitude, it can most conveniently be represented by a line denoting its maximum value, and if the alternating current is a simple periodic current then the root-mean-square or effective value of the current is obtained by dividing the maximum value by [root]2. Accordingly when we have an electric circuit or circuits in which there are simple periodic currents we can draw a vector diagram, the lines of which represent the relative magnitudes and phase differences of these currents.

A vector can most conveniently be represented by a symbol such as a +
ib, where a stands for any length of a units measured horizontally and
b for a length b units measured vertically, and the symbol i is a sign
of perpendicularity, and equivalent analytically[6] to [root]-1.
Accordingly if E represents the periodic electromotive force (maximum
value) acting in a circuit of resistance R and inductance L and
frequency n, and if the current considered as a vector is represented
by I, it is easy to show that a vector equation exists between these
quantities as follows:--

E = RI + [iota]2[pi]nLI.

Since the absolute magnitude of a vector a + [iota]b is [root](a^2 +
b^2), it follows that considering merely magnitudes of current and
electromotive force and denoting them by symbols (E) (I), we have the
following equation connecting (I) and (E):--

(I) = (E)[root](R^2 + p^2L^2),

where p stands for 2[pi]n. If the above equation is compared with the
symbolic expression of Ohm's law, it will be seen that the quantity
[root](R^2 + p^2L^2) takes the place of resistance R in the expression
of Ohm. This quantity [root](R^2 + p^2L^2) is called the "impedance"
of the alternating circuit. The quantity pL is called the "reactance"
of the alternating circuit, and it is therefore obvious that the
current in such a circuit lags behind the electromotive force by an
angle, called the angle of lag, the tangent of which is pL/R.

_Currents in Networks of Conductors._--In dealing with problems
connected with electric currents we have to consider the laws which
govern the flow of currents in linear conductors (wires), in plane
conductors (sheets), and throughout the mass of a material
conductor.[7] In the first case consider the collocation of a number
of linear conductors, such as rods or wires of metal, joined at their
ends to form a network of conductors. The network consists of a number
of conductors joining certain points and forming meshes. In each
conductor a current may exist, and along each conductor there is a
fall of potential, or an active electromotive force may be acting in
it. Each conductor has a certain resistance. To find the current in
each conductor when the individual resistances and electromotive
forces are given, proceed as follows:--Consider any one mesh. The sum
of all the electromotive forces which exist in the branches bounding
that mesh must be equal to the sum of all the products of the
resistances into the currents flowing along them, or [Sigma](E) =
[Sigma](C.R.). Hence if we consider each mesh as traversed by
imaginary currents all circulating in the same direction, the real
currents are the sums or differences of these imaginary cyclic
currents in each branch. Hence we may assign to each mesh a cycle
symbol x, y, z, &c., and form a cycle equation. Write down the cycle
symbol for a mesh and prefix as coefficient the sum of all the
resistances which bound that cycle, then subtract the cycle symbols of
each adjacent cycle, each multiplied by the value of the bounding or
common resistances, and equate this sum to the total electromotive
force acting round the cycle. Thus if x y z are the cycle currents,
and a b c the resistances bounding the mesh x, and b and c those
separating it from the meshes y and z, and E an electromotive force in
the branch a, then we have formed the cycle equation x(a + b + c) -
by - cz = E. For each mesh a similar equation may be formed. Hence we
have as many linear equations as there are meshes, and we can obtain
the solution for each cycle symbol, and therefore for the current in
each branch. The solution giving the current in such branch of the
network is therefore always in the form of the quotient of two
determinants. The solution of the well-known problem of finding the
current in the galvanometer circuit of the arrangement of linear
conductors called Wheatstone's Bridge is thus easily obtained. For if
we call the cycles (see fig. 7) (x + y), y and z, and the resistances
P, Q, R, S, G and B, and if E be the electromotive force in the
battery circuit, we have the cycle equations

(P + G + R)(x + y) - Gy - Rz = 0,
(Q + G + S)y - G(x + y) - Sz = 0,
(R + S + B)z - R(x + y) - Sy = E.

From these we can easily obtain the solution for (x + y) - y = x,
which is the current through the galvanometer circuit in the form

x = E(PS - RQ)[Delta].

where [Delta] is a certain function of P, Q, R, S, B and G.

_Currents in Sheets._--In the case of current flow in plane sheets, we
have to consider certain points called sources at which the current
flows into the sheet, and certain points called sinks at which it
leaves. We may investigate, first, the simple case of one source and
one sink in an infinite plane sheet of thickness [delta] and
conductivity k. Take any point P in the plane at distances R and r
from the source and sink respectively. The potential V at P is
obviously given by

Q r1
V = -------------log_e --,
2[pi]k[delta] r2

where Q is the quantity of electricity supplied by the source per
second. Hence the equation to the equipotential curve is r1r2 = a
constant.

If we take a point half-way between the sink and the source as the
origin of a system of rectangular co-ordinates, and if the distance
between sink and source is equal to p, and the line joining them is
taken as the axis of x, then the equation to the equipotential line is

y^2 + (x + p)^2
--------------- = a constant.
y^2 + (x - p)^2

This is the equation of a family of circles having the axis of y for a
common radical axis, one set of circles surrounding the sink and
another set of circles surrounding the source. In order to discover
the form of the stream of current lines we have to determine the
orthogonal trajectories to this family of coaxial circles. It is easy
to show that the orthogonal trajectory of the system of circles is
another system of circles all passing through the sink and the source,
and as a corollary of this fact, that the electric resistance of a
circular disk of uniform thickness is the same between any two points
taken anywhere on its circumference as sink and source. These
equipotential lines may be delineated experimentally by attaching the
terminals of a battery or batteries to small wires which touch at
various places a sheet of tinfoil. Two wires attached to a
galvanometer may then be placed on the tinfoil, and one may be kept
stationary and the other may be moved about, so that the galvanometer
is not traversed by any current. The moving terminal then traces out
an equipotential curve. If there are n sinks and sources in a plane
conducting sheet, and if r, r', r" be the distances of any point from
the sinks, and t, t', t" the distances of the sources, then

r r' r" ...
----------- = a constant,
t t' t" ...

is the equation to the equipotential lines. The orthogonal
trajectories or stream lines have the equation

[Sigma]([theta] - [theta]') = a constant,

where [theta] and [theta]' are the angles which the lines drawn from
any point in the plane to the sink and corresponding source make with
the line joining that sink and source. Generally it may be shown that
if there are any number of sinks and sources in an infinite
plane-conducting sheet, and if r, [theta] are the polar co-ordinates
of any one, then the equation to the equipotential surfaces is given
by the equation

[Sigma](A log_er) = a constant,

where A is a constant; and the equation to the stream of current lines
is

[Sigma]([theta]) = a constant.

In the case of electric flow in three dimensions the electric
potential must satisfy Laplace's equation, and a solution is therefore
found in the form [Sigma](A/r) = a constant, as the equation to an
equipotential surface, where r is the distance of any point on that
surface from a source or sink.

_Convection Currents._--The subject of convection electric currents has risen to great importance in connexion with modern electrical investigations. The question whether a statically electrified body in motion creates a magnetic field is of fundamental importance. Experiments to settle it were first undertaken in the year 1876 by H.A. Rowland, at a suggestion of H. von Helmholtz.[8] After preliminary experiments, Rowland's first apparatus for testing this hypothesis was constructed, as follows:--An ebonite disk was covered with radial strips of gold-leaf and placed between two other metal plates which acted as screens. The disk was then charged with electricity and set in rapid rotation. It was found to affect a delicately suspended pair of astatic magnetic needles hung in proximity to the disk just as would, by Oersted's rule, a circular electric current coincident with the periphery of the disk. Hence the statically-charged but rotating disk becomes in effect a circular electric current.

The experiments were repeated and confirmed by W.C. Rontgen (_Wied. Ann._, 1888, 35, p. 264; 1890, 40, p. 93) and by F. Himstedt (_Wied. Ann._, 1889, 38, p. 560). Later V. Cremieu again repeated them and obtained negative results (_Com. rend._, 1900, 130, p. 1544, and 131, pp. 578 and 797; 1901, 132, pp. 327 and 1108). They were again very carefully reconducted by H. Pender (_Phil. Mag._, 1901, 2, p. 179) and by E.P. Adams (id. ib., 285). Pender's work showed beyond any doubt that electric convection does produce a magnetic effect. Adams employed charged copper spheres rotating at a high speed in place of a disk, and was able to prove that the rotation of such spheres produced a magnetic field similar to that due to a circular current and agreeing numerically with the theoretical value. It has been shown by J.J. Thomson (_Phil. Mag._, 1881, 2, p. 236) and O. Heaviside (_Electrical Papers_, vol. ii. p. 205) that an electrified sphere, moving with a velocity v and carrying a quantity of electricity q, should produce a magnetic force H, at a point at a distance [rho] from the centre of the sphere, equal to qv sin [theta]/[rho]^2, where [theta] is the angle between the direction of [rho] and the motion of the sphere. Adams found the field produced by a known electric charge rotating at a known speed had a strength not very different from that predetermined by the above formula. An observation recorded by R.W. Wood (_Phil. Mag._, 1902, 2, p. 659) provides a confirmatory fact. He noticed that if carbon-dioxide strongly compressed in a steel bottle is allowed to escape suddenly the cold produced solidifies some part of the gas, and the issuing jet is full of particles of carbon-dioxide snow. These by friction against the nozzle are electrified positively. Wood caused the jet of gas to pass through a glass tube 2.5 mm. in diameter, and found that these particles of electrified snow were blown through it with a velocity of 2000 ft. a second. Moreover, he found that a magnetic needle hung near the tube was deflected as if held near an electric current. Hence the positively electrified particles in motion in the tube create a magnetic field round it.

_Nature of an Electric Current._--The question, What is an electric current? is involved in the larger question of the nature of electricity. Modern investigations have shown that negative electricity is identical with the electrons or corpuscles which are components of the chemical atom (see MATTER and ELECTRICITY). Certain lines of argument lead to the conclusion that a solid conductor is not only composed of chemical atoms, but that there is a certain proportion of free electrons present in it, the electronic density or number per unit of volume being determined by the material, its temperature and other physical conditions. If any cause operates to add or remove electrons at one point there is an immediate diffusion of electrons to re-establish equilibrium, and this electronic movement constitutes an electric current. This hypothesis explains the reason for the identity between the laws of diffusion of matter, of heat and of electricity. Electromotive force is then any cause making or tending to make an inequality of electronic density in conductors, and may arise from differences of temperature, i.e. thermoelectromotive force (see THERMOELECTRICITY), or from chemical action when part of the circuit is an electrolytic conductor, or from the movement of lines of magnetic force across the conductor.

BIBLIOGRAPHY.--For additional information the reader may be referred
to the following books: M. Faraday, _Experimental Researches in
Electricity_ (3 vols., London, 1839, 1844, 1855); J. Clerk Maxwell,
_Electricity and Magnetism_ (2 vols., Oxford, 1892); W. Watson and
S.H. Burbury, _Mathematical Theory of Electricity and Magnetism_, vol.
ii. (Oxford, 1889); E. Mascart and J. Joubert, _A Treatise on
Electricity and Magnetism_ (2 vols., London, 1883); A. Hay,
_Alternating Currents_ (London, 1905); W.G. Rhodes, _An Elementary
Treatise on Alternating Currents_ (London, 1902); D.C. Jackson and
J.P. Jackson, _Alternating Currents and Alternating Current Machinery_
(1896, new ed. 1903); S.P. Thompson, _Polyphase Electric Currents_
(London, 1900); _Dynamo-Electric Machinery_, vol. ii., "Alternating
Currents" (London, 1905); E.E. Fournier d'Albe, _The Electron Theory_
(London, 1906). (J. A. F.)

FOOTNOTES:

[1] See J.A. Fleming, _The Alternate Current Transformer_, vol. i. p.
519.

[2] See Maxwell, _Electricity and Magnetism_, vol. ii. chap. ii.

[3] See Maxwell, _Electricity and Magnetism_, vol. ii. 642.

[4] _Experimental Researches_, vol. i. ser. 1.

[5] See Maxwell, _Electricity and Magnetism_, vol. ii. S 542, p. 178.

[6] See W.G. Rhodes, _An Elementary Treatise on Alternating Currents_
(London, 1902), chap. vii.

[7] See J.A. Fleming, "Problems on the Distribution of Electric
Currents in Networks of Conductors," _Phil. Mag_. (1885), or Proc.
Phys. Soc. Lond. (1885), 7; also Maxwell, _Electricity and Magnetism_
(2nd ed.), vol. i. p. 374, S 280, 282b.

[8] See _Berl. Acad. Ber._, 1876, p. 211; also H.A. Rowland and C.T.
Hutchinson, "On the Electromagnetic Effect of Convection Currents,"
_Phil. Mag._, 1889, 27, p. 445.

ELECTROLIER, a fixture, usually pendent from the ceiling, for holding electric lamps. The word is analogous to chandelier, from which indeed it was formed.

ELECTROLYSIS (formed from Gr. [Greek: lyein], to loosen). When the passage of an electric current through a substance is accompanied by definite chemical changes which are independent of the heating effects of the current, the process is known as _electrolysis_, and the substance is called an _electrolyte_. As an example we may take the case of a solution of a salt such as copper sulphate in water, through which an electric current is passed between copper plates. We shall then observe the following phenomena. (1) The bulk of the solution is unaltered, except that its temperature may be raised owing to the usual heating effect which is proportional to the square of the strength of the current. (2) The copper plate by which the current is said to enter the solution, i.e. the plate attached to the so-called positive terminal of the battery or other source of current, dissolves away, the copper going into solution as copper sulphate. (3) Copper is deposited on the surface of the other plate, being obtained from the solution. (4) Changes in concentration are produced in the neighbourhood of the two plates or electrodes. In the case we have chosen, the solution becomes stronger near the anode, or electrode at which the current enters, and weaker near the cathode, or electrode at which it leaves the solution. If, instead of using copper electrodes, we take plates of platinum, copper is still deposited on the cathode; but, instead of the anode dissolving, free sulphuric acid appears in the neighbouring solution, and oxygen gas is evolved at the surface of the platinum plate.

With other electrolytes similar phenomena appear, though the primary chemical changes may be masked by secondary actions. Thus, with a dilute solution of sulphuric acid and platinum electrodes, hydrogen gas is evolved at the cathode, while, as the result of a secondary action on the anode, sulphuric acid is there re-formed, and oxygen gas evolved. Again, with the solution of a salt such as sodium chloride, the sodium, which is primarily liberated at the cathode, decomposes the water and evolves hydrogen, while the chlorine may be evolved as such, may dissolve the anode, or may liberate oxygen from the water, according to the nature of the plate and the concentration of the solution.

_Early History of Electrolysis._--Alessandro Volta of Pavia discovered the electric battery in the year 1800, and thus placed the means of maintaining a steady electric current in the hands of investigators, who, before that date, had been restricted to the study of the isolated electric charges given by frictional electric machines. Volta's cell consists essentially of two plates of different metals, such as zinc and copper, connected by an electrolyte such as a solution of salt or acid. Immediately on its discovery intense interest was aroused in the new invention, and the chemical effects of electric currents were speedily detected. W. Nicholson and Sir A. Carlisle found that hydrogen and oxygen were evolved at the surfaces of gold and platinum wires connected with the terminals of a battery and dipped in water. The volume of the hydrogen was about double that of the oxygen, and, since this is the ratio in which these elements are combined in water, it was concluded that the process consisted essentially in the decomposition of water. They also noticed that a similar kind of chemical action went on in the battery itself. Soon afterwards, William Cruickshank decomposed the magnesium, sodium and ammonium chlorides, and precipitated silver and copper from their solutions--an observation which led to the process of electroplating. He also found that the liquid round the anode became acid, and that round the cathode alkaline. In 1804 W. Hisinger and J.J. Berzelius stated that neutral salt solutions could be decomposed by electricity, the acid appearing at one pole and the metal at the other. This observation showed that nascent hydrogen was not, as had been supposed, the primary cause of the separation of metals from their solutions, but that the action consisted in a direct decomposition into metal and acid. During the earliest investigation of the subject it was thought that, since hydrogen and oxygen were usually evolved, the electrolysis of solutions of acids and alkalis was to be regarded as a direct decomposition of water. In 1806 Sir Humphry Davy proved that the formation of acid and alkali when water was electrolysed was due to saline impurities in the water. He had shown previously that decomposition of water could be effected although the two poles were placed in separate vessels connected by moistened threads. In 1807 he decomposed potash and soda, previously considered to be elements, by passing the current from a powerful battery through the moistened solids, and thus isolated the metals potassium and sodium.

The electromotive force of Volta's simple cell falls off rapidly when the cell is used, and this phenomenon was shown to be due to the accumulation at the metal plates of the products of chemical changes in the cell itself. This reverse electromotive force of polarization is produced in all electrolytes when the passage of the current changes the nature of the electrodes. In batteries which use acids as the electrolyte, a film of hydrogen tends to be deposited on the copper or platinum electrode; but, to obtain a constant electromotive force, several means were soon devised of preventing the formation of the film. Constant cells may be divided into two groups, according as their action is chemical (as in the bichromate cell, where the hydrogen is converted into water by an oxidizing agent placed in a porous pot round the carbon plate) or electrochemical (as in Daniell's cell, where a copper plate is surrounded by a solution of copper sulphate, and the hydrogen, instead of being liberated, replaces copper, which is deposited on the plate from the solution).

_Faraday's Laws._--The first exact quantitative study of electrolytic phenomena was made about 1830 by Michael Faraday (_Experimental Researches_, 1833). When an electric current flows round a circuit, there is no accumulation of electricity anywhere in the circuit, hence the current strength is everywhere the same, and we may picture the current as analogous to the flow of an incompressible fluid. Acting on this view, Faraday set himself to examine the relation between the flow of electricity round the circuit and the amount of chemical decomposition. He passed the current driven by a voltaic battery ZnPt (fig. 1) through two branches containing the two electrolytic cells A and B. The reunited current was then led through another cell C, in which the strength of the current must be the sum of those in the arms A and B. Faraday found that the mass of substance liberated at the electrodes in the cell C was equal to the sum of the masses liberated in the cells A and B. He also found that, for the same current, the amount of chemical action was independent of the size of the electrodes and proportional to the time that the current flowed. Regarding the current as the passage of a certain amount of electricity per second, it will be seen that the results of all these experiments may be summed up in the statement that the amount of chemical action is proportional to the quantity of electricity which passes through the cell.

Faraday's next step was to pass the same current through different electrolytes in series. He found that the amounts of the substances liberated in each cell were proportional to the chemical equivalent weights of those substances. Thus, if the current be passed through dilute sulphuric acid between hydrogen electrodes, and through a solution of copper sulphate, it will be found that the mass of hydrogen evolved in the first cell is to the mass of copper deposited in the second as 1 is to 31.8. Now this ratio is the same as that which gives the relative chemical equivalents of hydrogen and copper, for 1 gramme of hydrogen and 31.8 grammes of copper unite chemically with the same weight of any acid radicle such as chlorine or the sulphuric group, SO4. Faraday examined also the electrolysis of certain fused salts such as lead chloride and silver chloride. Similar relations were found to hold and the amounts of chemical change to be the same for the same electric transfer as in the case of solutions.

We may sum up the chief results of Faraday's work in the statements known as Faraday's laws: The mass of substance liberated from an electrolyte by the passage of a current is proportional (1) to the total quantity of electricity which passes through the electrolyte, and (2) to the chemical equivalent weight of the substance liberated.

Since Faraday's time his laws have been confirmed by modern research, and in favourable cases have been shown to hold good with an accuracy of at least one part in a thousand. The principal object of this more recent research has been the determination of the quantitative amount of chemical change associated with the passage for a given time of a current of strength known in electromagnetic units. It is found that the most accurate and convenient apparatus to use is a platinum bowl filled with a solution of silver nitrate containing about fifteen parts of the salt to one hundred of water. Into the solution dips a silver plate wrapped in filter paper, and the current is passed from the silver plate as anode to the bowl as cathode. The bowl is weighed before and after the passage of the current, and the increase gives the mass of silver deposited. The mean result of the best determinations shows that when a current of one ampere is passed for one second, a mass of silver is deposited equal to 0.001118 gramme. So accurate and convenient is this determination that it is now used conversely as a practical definition of the ampere, which (defined theoretically in terms of magnetic force) is defined practically as the current which in one second deposits 1.118 milligramme of silver.

Taking the chemical equivalent weight of silver, as determined by chemical experiments, to be 107.92, the result described gives as the electrochemical equivalent of an ion of unit chemical equivalent the value 1.036 X 10^(-5). If, as is now usual, we take the equivalent weight of oxygen as our standard and call it 16, the equivalent weight of hydrogen is 1.008, and its electrochemical equivalent is 1.044 X 10^(-5). The electrochemical equivalent of any other substance, whether element or compound, may be found by multiplying its chemical equivalent by 1.036 X 10^(-5). If, instead of the ampere, we take the C.G.S. electromagnetic unit of current, this number becomes 1.036 X 10^(-4).

_Chemical Nature of the Ions._--A study of the products of decomposition does not necessarily lead directly to a knowledge of the ions actually employed in carrying the current through the electrolyte. Since the electric forces are active throughout the whole solution, all the ions must come under its influence and therefore move, but their separation from the electrodes is determined by the electromotive force needed to liberate them. Thus, as long as every ion of the solution is present in the layer of liquid next the electrode, the one which responds to the least electromotive force will alone be set free. When the amount of this ion in the surface layer becomes too small to carry all the current across the junction, other ions must also be used, and either they or their secondary products will appear also at the electrode. In aqueous solutions, for instance, a few hydrogen (H) and hydroxyl (OH) ions derived from the water are always present, and will be liberated if the other ions require a higher decomposition voltage and the current be kept so small that hydrogen and hydroxyl ions can be formed fast enough to carry all the current across the junction between solution and electrode.

The issue is also obscured in another way. When the ions are set free at the electrodes, they may unite with the substance of the electrode or with some constituent of the solution to form secondary products. Thus the hydroxyl mentioned above decomposes into water and oxygen, and the chlorine produced by the electrolysis of a chloride may attack the metal of the anode. This leads us to examine more closely the part played by water in the electrolysis of aqueous solutions. Distilled water is a very bad conductor, though, even when great care is taken to remove all dissolved bodies, there is evidence to show that some part of the trace of conductivity remaining is due to the water itself. By careful redistillation F. Kohlrausch has prepared water of which the conductivity compared with that of mercury was only 0.40 X 10^(-11) at 18 deg. C. Even here some little impurity was present, and the conductivity of chemically pure water was estimated by thermodynamic reasoning as 0.36 X 10^(-11) at 18 deg. C. As we shall see later, the conductivity of very dilute salt solutions is proportional to the concentration, so that it is probable that, in most cases, practically all the current is carried by the salt. At the electrodes, however, the small quantity of hydrogen and hydroxyl ions from the water are liberated first in cases where the ions of the salt have a higher decomposition voltage. The water being present in excess, the hydrogen and hydroxyl are re-formed at once and therefore are set free continuously. If the current be so strong that new hydrogen and hydroxyl ions cannot be formed in time, other substances are liberated; in a solution of sulphuric acid a strong current will evolve sulphur dioxide, the more readily as the concentration of the solution is increased. Similar phenomena are seen in the case of a solution of hydrochloric acid. When the solution is weak, hydrogen and oxygen are evolved; but, as the concentration is increased, and the current raised, more and more chlorine is liberated.

An interesting example of secondary action is shown by the common
technical process of electroplating with silver from a bath of
potassium silver cyanide. Here the ions are potassium and the group
Ag(CN)2.[1] Each potassium ion as it reaches the cathode precipitates
silver by reacting with the solution in accordance with the chemical
equation

K + KAg(CN)2 = 2KCN + Ag,

while the anion Ag(CN)2 dissolves an atom of silver from the anode,
and re-forms the complex cyanide KAg(CN)2 by combining with the 2KCN
produced in the reaction described in the equation. If the anode
consist of platinum, cyanogen gas is evolved thereat from the anion
Ag(CN)2, and the platinum becomes covered with the insoluble silver
cyanide, AgCN, which soon stops the current. The coating of silver
obtained by this process is coherent and homogeneous, while that
deposited from a solution of silver nitrate, as the result of the
primary action of the current, is crystalline and easily detached.

In the electrolysis of a concentrated solution of sodium acetate,
hydrogen is evolved at the cathode and a mixture of ethane and carbon
dioxide at the anode. According to H. Jahn,[2] the processes at the
anode can be represented by the equations