1911 Encyclopædia Britannica/Electrokinetics

by John Ambrose Fleming

ELECTROKINETICS, that part of electrical science which is concerned with the properties of electric currents.

Classification of Electric Currents.—Electric currents are classified into (a) conduction currents, (b) convection currents, (c) displacement or dielectric currents. In the case of conduction currents electricity flows or moves through a stationary material body called the conductor. In convection currents electricity is carried from place to place with and on moving material bodies or particles. In dielectric currents there is no continued movement of electricity, but merely a limited displacement through or in the mass of an insulator or dielectric. The path in which an electric current exists is called an electric circuit, and may consist wholly of a conducting body, or partly of a conductor and insulator or dielectric, or wholly of a dielectric. In cases in which the three classes of currents are present together the true current is the sum of each separately. In the case of conduction currents the circuit consists of a conductor immersed in a non-conductor, and may take the form of a thin wire or cylinder, a sheet, surface or solid. Electric conduction currents may take place in space of one, two or three dimensions, but for the most part the circuits we have to consider consist of thin cylindrical wires or tubes of conducting material surrounded with an insulator; hence the case which generally presents itself is that of electric flow in space of one dimension. Self-closed electric currents taking place in a sheet of conductor are called “eddy currents.”

Although in ordinary language the current is said to flow in the conductor, yet according to modern views the real pathway of the energy transmitted is the surrounding dielectric, and the so-called conductor or wire merely guides the transmission of energy in a certain direction. The presence of an electric current is recognized by three qualities or powers: (1) by the production of a magnetic field, (2) in the case of conduction currents, by the production of heat in the conductor, and (3) if the conductor is an electrolyte and the current unidirectional, by the occurrence of chemical decomposition in it. An electric current may also be regarded as the result of a movement of electricity across each section of the circuit, and is then measured by the quantity conveyed per unit of time. Hence if $dq$ is the quantity of electricity which flows across any section of the conductor in the element of time $dt,$ the current $i=dq/dt.$

Electric currents may be also classified as constant or variable and as unidirectional or “direct,” that is flowing always in the same direction, or “alternating,” that is reversing their direction at regular intervals. In the last case the variation of current may follow any particular law. It is called a “periodic current” if the cycle of current values is repeated during a certain time called the periodic time, during which the current reaches a certain maximum value, first in one direction and then in the opposite, and in the intervals between has a zero value at certain instants. The frequency of the periodic current is the number of periods or cycles in one second, and alternating currents are described as low frequency or high frequency, in the latter case having some thousands of periods per second. A periodic current may be represented either by a wave diagram, or by a polar diagram.^[1] In the first case we take a straight line to represent the uniform flow of time, and at small equidistant intervals set up perpendiculars above or below the time axis, representing to scale the current at that instant in one direction or the other; the extremities of these ordinates then define a wavy curve which is called the wave form of the current (fig. 1). It is obvious that this curve can only be a single valued curve. In one particular and important case the form of the current curve is a simple harmonic curve or simple sine curve. If ${\text{T}}$ represents the periodic time in which the cycle of current values takes place, whilst $n$ is the frequency or number of periods per second and $p$ stands for $2\pi n,$ and $i$ is the value of the current at any instant $t,$ and I its maximum value, then in this case we have $i={\text{I }}\sin pt.$ Such a current is called a “sine current” or simple periodic current.

Fig. 1	Fig. 2

In a polar diagram (fig. 2) a number of radial lines are drawn from a point at small equiangular intervals, and on these lines are set off lengths proportional to the current value of a periodic current at corresponding intervals during one complete period represented by four right angles. The extremities of these radii delineate a polar curve. The polar form of a simple sine current is obviously a circle drawn through the origin. As a consequence of Fourier’s theorem it follows that any periodic curve having any wave form can be imitated by the superposition of simple sine currents differing in maximum value and in phase.

Definitions of Unit Electric Current.—In electrokinetic investigations we are most commonly limited to the cases of unidirectional continuous and constant currents (C.C. or D.C.), or of simple periodic currents, or alternating currents of sine form (A.C.). A continuous electric current is measured either by the magnetic effect it produces at some point outside its circuit, or by the amount of electrochemical decomposition it can perform in a given time on a selected standard electrolyte. Limiting our consideration to the case of linear currents or currents flowing in thin cylindrical wires, a definition may be given in the first place of the unit electric current in the centimetre, gramme, second (C.G.S.) of electromagnetic measurement (see Units, Physical). H. C. Oersted discovered in 1820 that a straight wire conveying an electric current is surrounded by a magnetic field the lines of which are self-closed lines embracing the electric circuit (see Electricity and Electromagnetism). The unit current in the electromagnetic system of measurement is defined as the current which, flowing in a thin wire bent into the form of a circle of one centimetre in radius, creates a magnetic field having a strength of $2\pi$ units at the centre of the circle, and therefore would exert a mechanical force of $2\pi$ dynes on a unit magnetic pole placed at that point (see Magnetism). Since the length of the circumference of the circle of unit radius is $2\pi$ units, this is equivalent to stating that the unit current on the electromagnetic C.G.S. system is a current such that unit length acts on unit magnetic pole with a unit force at a unit of distance. Another definition, called the electrostatic unit of current, is as follows: Let any conductor be charged with electricity and discharged through a thin wire at such a rate that one electrostatic unit of quantity (see Electrostatics) flows past any section of the wire in one unit of time. The electromagnetic unit of current defined as above is 3 × 10¹⁰ times larger than the electrostatic unit.

In the selection of a practical unit of current it was considered that the electromagnetic unit was too large for most purposes, whilst the electrostatic unit was too small; hence a practical unit of current called 1 ampere was selected, intended originally to be 1/10 of the absolute electromagnetic C.G.S. unit of current as above defined. The practical unit of current, called the international ampere, is, however, legally defined at the present time as the continuous unidirectional current which when flowing through a neutral solution of silver nitrate deposits in one second on the cathode or negative pole 0.001118 of a gramme of silver. There is reason to believe that the international unit is smaller by about one part in a thousand, or perhaps by one part in 800, than the theoretical ampere defined as 1/10 part of the absolute electromagnetic unit. A periodic or alternating current is said to have a value of 1 ampere if when passed through a fine wire it produces in the same time the same heat as a unidirectional continuous current of 1 ampere as above electrochemically defined. In the case of a simple periodic alternating current having a simple sine wave form, the maximum value is equal to that of the equiheating continuous current multiplied by ${\sqrt {2}}.$ This equiheating continuous current is called the effective or root-mean-square (R.M.S.) value of the alternating one.

Resistance.—A current flows in a circuit in virtue of an electromotive force (E.M.F.), and the numerical relation between the current and E.M.F. is determined by three qualities of the circuit called respectively, its resistance (R), inductance (L), and capacity (C). If we limit our consideration to the case of continuous unidirectional conduction currents, then the relation between current and E.M.F. is defined by Ohm’s law, which states that the numerical value of the current is obtained as the quotient of the electromotive force by a certain constant of the circuit called its resistance, which is a function of the geometrical form of the circuit, of its nature, i.e. material, and of its temperature, but is independent of the electromotive force or current. The resistance (R) is measured in units called ohms and the electromotive force in volts (V); hence for a continuous current the value of the current in amperes (A) is obtained as the quotient of the electromotive force acting in the circuit reckoned in volts by the resistance in ohms, or ${\text{A}}={\text{V}}/{\text{R}}.$ Ohm established his law by a course of reasoning which was similar to that on which J. B. J. Fourier based his investigations on the uniform motion of heat in a conductor. As a matter of fact, however, Ohm’s law merely states the direct proportionality of steady current to steady electromotive force in a circuit, and asserts that this ratio is governed by the numerical value of a quality of the conductor, called its resistance, which is independent of the current, provided that a correction is made for the change of temperature produced by the current. Our belief, however, in its universality and accuracy rests upon the close agreement between deductions made from it and observational results, and although it is not derivable from any more fundamental principle, it is yet one of the most certainly ascertained laws of electrokinetics.

Ohm’s law not only applies to the circuit as a whole but to any part of it, and provided the part selected does not contain a source of electromotive force it may be expressed as follows:—The difference of potential (P.D.) between any two points of a circuit including a resistance ${\text{R}},$ but not including any source of electromotive force, is proportional to the product of the resistance and the current $i$ in the element, provided the conductor remains at the same temperature and the current is constant and unidirectional. If the current is varying we have, however, to take into account the electromotive force (E.M.F.) produced by this variation, and the product ${\text{R}}i$ is then equal to the difference between the observed P.D. and induced E.M.F.

We may otherwise define the resistance of a circuit by saying that it is that physical quality of it in virtue of which energy is dissipated as heat in the circuit when a current flows through it. The power communicated to any electric circuit when a current is created in it by a continuous unidirectional electromotive force ${\text{E}}$ is equal to ${\text{E}}i,$ and the energy dissipated as heat in that circuit by the conductor in a small interval of time $dt$ is measured by ${\text{E}}idt.$ Since by Ohm’s law ${\text{E}}={\text{R}}i,$ where ${\text{R}}$ is the resistance of the circuit, it follows that the energy dissipated as heat per unit of time in any circuit is numerically represented by ${\text{R}}i^{2},$ and therefore the resistance is measured by the heat produced per unit of current, provided the current is unvarying.

Inductance.—As soon as we turn our attention, however, to alternating or periodic currents we find ourselves compelled to take into account another quality of the circuit, called its “inductance.” This may be defined as that quality in virtue of which energy is stored up in connexion with the circuit in a magnetic form. It can be experimentally shown that a current cannot be created instantaneously in a circuit by any finite electromotive force, and that when once created it cannot be annihilated instantaneously. The circuit possesses a quality analogous to the inertia of matter. If a current $i$ is flowing in a circuit at any moment, the energy stored up in connexion with the circuit is measured by ${\tfrac {1}{2}}{\text{L}}i^{2},$ where ${\text{L}},$ the inductance of the circuit, is related to the current in the same manner as the quantity called the mass of a body is related to its velocity in the expression for the ordinary kinetic energy, viz. ${\tfrac {1}{2}}{\text{M}}v^{2}.$ The rate at which this conserved energy varies with the current is called the “electrokinetic momentum” of this circuit ($={\text{L}}i$). Physically interpreted this quantity signifies the number of lines of magnetic flux due to the current itself which are self-linked with its own circuit.

Magnetic Force and Electric Currents.—In the case of every circuit conveying a current there is a certain magnetic force (see Magnetism) at external points which can in some instances be calculated. Laplace proved that the magnetic force due to an element of length $d{\text{S}}$ of a circuit conveying a current ${\text{I}}$ at a point ${\text{P}}$ at a distance ${\text{r}}$ from the element is expressed by ${\text{I}}d{\text{S}}\sin \theta /r^{2},$ where $\theta$ is the angle between the direction of the current element and that drawn between the element and the point. This force is in a direction perpendicular to the radius vector and to the plane containing it and the element of current. Hence the determination of the magnetic force due to any circuit is reduced to a summation of the effects due to all the elements of length. For instance, the magnetic force at the centre of a circular circuit of radius $r$ carrying a steady current ${\text{I}}$ is $2\pi {\text{I}}/r,$ since all elements are at the same distance from the centre. In the same manner, if we take a point in a line at right angles to the plane of the circle through its centre and at a distance $d,$ the magnetic force along this line is expressed by $2\pi r^{2}{\text{I}}/(r^{2}+d^{2})^{\frac {3}{2}}.$ Another important case is that of an infinitely long straight current. By summing up the magnetic force due to each element at any point ${\text{P}}$ outside the continuous straight current ${\text{I}}$ and at a distance $d$ from it, we can show that it is equal to $2{\text{I}}/d$ or is inversely proportional to the distance of the point from the wire. In the above formula the current ${\text{I}}$ is measured in absolute electromagnetic units. If we reckon the current in amperes ${\text{A}},$ then ${\text{I}}={\text{A}}/10.$

It is possible to make use of this last formula, coupled with an experimental fact, to prove that the magnetic force due to an element of current varies inversely as the square of the distance. If a flat circular disk is suspended so as to be free to rotate round a straight current which passes through its centre, and two bar magnets are placed on it with their axes in line with the current, it is found that the disk has no tendency to rotate round the current. This proves that the force on each magnetic pole is inversely as its distance from the current. But it can be shown that this law of action of the whole infinitely long straight current is a mathematical consequence of the fact that each element of the current exerts a magnetic force which varies inversely as the square of the distance. If the current flows ${\text{N}}$ times, round the circuit instead of once, we have to insert ${\text{NA}}/10$ in place of ${\text{I}}$ in all the above formulae. The quantity ${\text{NA}}$ is called the “ampere-turns” on the circuit, and it is seen that the magnetic field at any point outside a circuit is proportional to the ampere-turns on it and to a function of its geometrical form and the distance of the point.

There is therefore a distribution of magnetic force in the field of every current-carrying conductor which can be delineated by lines of magnetic force and rendered visible to the eye by iron filings (see Magnetism). If a copper wire is passed vertically through a hole in a card on which iron filings are sprinkled, and a strong electric current is sent through the circuit, the filings arrange themselves in concentric circular lines making visible the paths of the lines of magnetic force (fig. 3). In the same manner, by passing a circular wire through a card and sending a strong current through the wire we can employ iron filings to delineate for us the form of the lines of magnetic force (fig. 4).

Fig. 3.	Fig. 4.

In all cases a magnetic pole of strength ${\text{M}},$ placed in the field of an electric current, is urged along the lines of force with a mechanical force equal to ${\text{MH}},$ where ${\text{H}}$ is the magnetic force. If then we carry a unit magnetic pole against the direction in which it would naturally move we do work. The lines of magnetic force embracing a current-carrying conductor are always loops or endless lines.

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 ${\text{I}},$ the line integral is $4\pi {\text{I}}.$ 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 ${\text{N}}$ turns of wire. If it is made endless, the magnetic force ${\text{H}}$ is the same everywhere along the central axis and the line integral along the axis is ${\text{H}}l.$ If the current is denoted by ${\text{I}},$ then ${\text{NI}}$ is the total current, and accordingly $4\pi {\text{NI}}={\text{H}}l,$ or ${\text{H}}=4\pi {\text{NI}}/l,$ 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 ${\text{E}}$ is applied to its extremities, producing thereby a current ${\text{I}}$ in it, is equal to ${\text{EI.}}$

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

$\left[{\text{T}}^{-1}\int _{0}^{\text{T}}i^{2}dt\right]^{{\frac {1}{2}},}$

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 $p$ has a current of ${\text{A}}$ amperes passed through it, the heat in watts produced in any time $t$ seconds is represented by the value of $4{\text{A}}^{2}\rho lt/10^{9}\pi d^{2},$ 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 10⁹ 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 $4{\text{A}}^{2}plt/10^{9}\pi d^{2}=\pi dlet,$ or ${\text{A}}^{2}=10^{9}\pi ^{2}ed^{3}/4\rho .$ 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° C., we can put the above formula in the following forms for circular sectioned copper, iron or platinoid wires, viz.

${\text{A}}={\sqrt {d^{2}/500}}$ for copper wires

${\text{A}}={\sqrt {d^{2}/4000}}$ for iron wires

${\text{A}}={\sqrt {d^{2}/5000}}$ 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° 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° C. if a current of ${\sqrt {50^{3}/500}}={\sqrt {250}}=16$ amperes (nearly) is passed through it, whilst a current of ${\sqrt {25}}=5$ 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 Ampère in connexion with electric currents are of fundamental importance in electrokinetics. Starting from the discovery of Oersted, Ampère 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. Ampère 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.

Fig. 5. Ampère 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 $d{\text{S}}$ of one conductor conveying a current ${\text{I}}$ upon another element $d{\text{S}}'$ of, another circuit conveying another current ${\text{I}}'$ 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 Ampère’s expression for the mechanical force $f$ the elements exert on one another is

$f{=}2{\text{II}}'r^{\_2}\{\cos \phi -{\tfrac {3}{2}}\cos \theta \cos \theta '\}d{\text{S}}d{\text{S}}'.$

This law, together with that of Laplace already mentioned, viz. that the magnetic force due to an element of length $d{\text{S}}$ of a current ${\text{I}}$ at a distance $r$, the element making an angle $\theta$ with the radius vector $o$ is ${\text{I}}d{\text{S}}\sin \theta /r^{2}$, constitute the fundamental laws of electrokinetics.

Ampère 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.

Ampère’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 Ampère. 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 Ampère’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 ${\text{Z}}$ represents the total number of lines of magnetic force linked with a circuit of ${\text{N}}$ turns, then $-{\text{N}}(dZ/dt)$ 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
Fig. 6. of a magnet ${\text{NO}}$ 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 Ampère and the induction of electric currents discovered by Faraday. It is as follows: If a constant current flows in a primary circuit ${\text{P}}$, and if by motion of ${\text{P}}$ a secondary current is created in a neighbouring circuit ${\text{S}}$, 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 ${\text{M}}$, called the “potential of one circuit on another,” or generally their “coefficient of mutual inductance.” Mathematically ${\text{M}}$ is obtained by taking the sum of all such quantities as ƒƒ$d{\text{S}}d{\text{S}}'\cos \phi /r$, where $d{\text{S}}$ and $d{\text{S}}'$ 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 Ampère assuming the principle of the conservation of energy. Helmholtz takes the case of a circuit of resistance ${\text{R}}$ 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 ${\text{V}}$, so that if a current ${\text{I}}$ existed in the circuit the work done on the magnet in the time $dt$ is ${\text{I}}(d{\text{V}}/dt)dt.$ The source of electromotive force supplies in the time $dt$ work equal to ${\text{EI}}dt$, and according to Joule’s law energy is dissipated equal to ${\text{RI}}^{2}dt$. Hence, by the conservation of energy,

${\text{EI}}dt{=}{\text{RI}}^{2}dt+{\text{I}}(d{\text{V}}/dt)dt.$

If then ${\text{E}}{=}{\text{O}}$ we have ${\text{I}}{=}-(d{\text{V}}/dt)/{\text{R}}$, or there will be a current due to an induced electromotive force expressed by $-d{\text{V}}/dt$. 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 ${\text{C}}$ has its terminals connected by a wire of resistance ${\text{R}}$ and inductance ${\text{L}}$, it becomes important to consider |the subsequent motion of electricity in the wire. If ${\text{Q}}$ is the quantity of electricity in the condenser initially, and $g$ that at any time $t$ after completing the circuit, then the energy stored up in the condenser at that instant is ${\tfrac {1}{2}}q^{2}/{\text{C,}}$ and the energy associated with the circuit is ${\tfrac {1}{2}}{\text{L}}(dq/dt)^{2},$ and the rate of dissipation of energy by resistance is ${\text{R}}(dq/dt)^{2},$ 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:—

$-{\frac {d}{dt}}\left[{\tfrac {1}{2}}{\frac {q^{2}}{\text{C}}}\right]{=}{\frac {d}{dt}}\left[{\tfrac {1}{2}}{\text{L}}\left({\frac {dq}{dt}}\right)^{2}\right]+{\text{R}}\left({\frac {dq}{dt}}\right)^{2}$

or

${\frac {d^{2}q}{dt^{2}}}+{\frac {\text{R}}{\text{L}}}{\frac {dq}{dt}}+{\frac {1}{\text{LC}}}q{=}0.$

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

$i{=}{\text{Q}}{\frac {\alpha ^{2}+\beta ^{2}}{2\beta }}e^{-\alpha }(e^{\beta t}-e^{-\beta t}),$

where $\alpha {=}{\text{R}}/2{\text{L}},\,\,\beta {=}{\sqrt {\frac {{\text{R}}^{2}}{4{\text{L}}^{2}}}}-{\frac {1}{\text{LC}}}$, ${\text{Q}}$ is the value of $q$ when $t=0,$ and $e$ is the base of Napierian logarithms; and in the second case by the equation

$i{=}{\text{Q}}{\frac {\alpha ^{2}+\beta ^{2}}{\beta }}e^{-\alpha t}\sin \beta t$

where

$\alpha {=}{\text{R}}/2{\text{L}},$⁠and⁠$\beta {=}{\sqrt {{\frac {1}{\text{LC}}}-{\frac {{\text{R}}^{2}}{4{\text{L}}^{2}}}}}\cdot$

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

$n{=}{\frac {1}{2\pi }}{\sqrt {{\frac {1}{\text{LC}}}-{\frac {{\text{R}}^{2}}{4{\text{L}}^{2}}}}}\cdot$

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 ${\text{R}}$ take the simple forms $n{=}1,\,\,2\pi {\sqrt {\text{LC}}}$ or ${\text{T}}{=}1/n{=}2\pi {\sqrt {\text{LC}}}$}}

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 ${\sqrt {2}}.$ 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 $a+\iota b$ where $a$ stands for any length of $a$ units measured horizontally and $b$ for a length $b$ units measured vertically, and the smybol $\iota$ is a sign of perpendicularity, and equivalent analytically^[6] to ${\sqrt {-1}}.$ Accordingly if ${\text{E}}$ represents the periodic electromotive force (maximum value) acting in a circuit of resistance ${\text{R}}$ and inductance ${\text{L}}$ and frequency $n,$ and if the current considered as a vector is represented by ${\text{I,}}$ it is easy to show that a vector equation exists between these quantities as follows:—

${\text{E}}{=}{\text{RI}}+\iota 2\pi n{\text{LI}}.$

Since the absolute magnitude of a vector $a+\iota b$ is ${\sqrt {(a^{2}+b^{2})}},$ it follows that considering merely magnitudes of current and electromotive force and denoting them by symbols ${\text{(E) (I)}}$ we have the following equation connecting ${\text{(E)}}$ and ${\text{(I)}}$:—

$({\text{I}}){=}{\text{(E)}}/{\sqrt {{\text{R}}^{2}+p^{2}{\text{L}}^{2}}},$

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 ${\sqrt {({\text{R}}^{2}+p^{2}{\text{L}}^{2})}}$ takes the place of resistance ${\text{R}}$ in the expression of Ohm. This quantity ${\sqrt {({\text{R}}^{2}+p^{2}{\text{L}}^{2})}}$ is called the “impedance” of the alternating circuit. The quantity $p{\text{L}}$ 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 $p{\text{L}}/{\text{R}}.$

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 $\sum {\text{(E)}}=\sum {\text{(C.R.)}}$ 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 ${xyz}$ are the cycle currents, and ${abc}$ the resistances bounding the mesh $x,$ and $b$ and $c$ those separating it from the meshes $y$ and $z,$ and ${\text{E}}$ an electromotive force in the branch $a,$ then
Fig. 7. we have formed the cycle equation $x(a+b+c)-by-cz{=}{\text{E}}.$ 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 ${\text{P, Q, R, S, G}}$ and ${\text{B}},$ and if ${\text{E}}$ be the electromotive force in the battery circuit, we have the cycle equations

$({\text{P}}+{\text{G}}+{\text{R}})(x+y)-{\text{G}}y-{\text{R}}z{=}0,$ $({\text{Q}}+{\text{G}}+{\text{S}})y-{\text{G}}(x+y)-{\text{S}}z{=}0,$ $({\text{R}}+{\text{S}}+{\text{B}})z-{\text{R}}(x+y)-{\text{S}}y{=}{\text{E}}.$

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

$x{=}{\text{E}}({\text{PS}}-{\text{RQ}})\Delta ,$

where $\Delta$ is a certain function of ${\text{P, Q, R, S, B}}$ and ${\text{G}}.$

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 ${\text{P}}$ in the plane at distances ${\text{R}}$ and $r$ from the source and sink respectively. The potential ${\text{V}}$ at ${\text{P}}$ is obviously given by

${\text{V}}{=}{\frac {\text{Q}}{2\pi k\delta }}\log {\frac {r_{1}}{r_{2},}}$

where ${\text{Q}}$ is the quantity of electricity supplied by the source per second. Hence the equation to the equipotential curve is $r_{1}r_{2}{=}$ 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

${\frac {y^{2}+(x+p)^{2}}{y^{2}+(x-p)^{2}}}{=}$ a constant.

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

${\frac {{rr'r''}{\text{ . . . }}}{{tt't''}{\text{ . . . }}}}{=}$a constant.

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

$\textstyle \sum \displaystyle (\theta -\theta '){=}$ 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

$\textstyle \sum \displaystyle ({\text{A}}\log _{e}r){=}$ a constant,

where ${\text{A}}$ is a constant; and the equation to the stream or current lines is

$\textstyle \sum \displaystyle (\theta ){=}$ a constant.</math>

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 $\textstyle \sum \displaystyle ({\text{A}}/r){=}$ 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. Röntgen (Wied. Ann., 1888, 35, p. 264; 1890, 40, p. 93) and by F. Himstedt (Wied. Ann., 1889, 38, p. 560). Later V. Crémieu 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 ${\text{H}},$ at a point at a distance $p$ 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.) 

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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. § 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, §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.