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ELECTROKINETICS
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ELECTROKINETICS , that See also:part of See also:electrical See also:science which is concerned with the properties of electric currents.
See also:Classification of Electric Currents.—Electric currents are classified into (a) See also:conduction currents, (b) convection currents, (c) displacement or See also:dielectric currents. In the See also:case of conduction currents See also:electricity flows or moves through a stationary material See also:body called the conductor. In convection currents electricity is carried from See also:place to place with and on moving material bodies or particles. In dielectric currents there is no continued See also:movement of electricity, but merely a limited displacement through or in the See also:mass of an insulator or dielectric. The path in which an electric current exists is called an electric See also: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 See also: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 See also:form of a thin See also:wire or See also:cylinder, a See also:sheet, See also: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 See also:dimension. Self-closed electric currents taking place in a sheet of conductor are called " eddy currents."
Although in See also:ordinary See also:language the current is said to flow in the conductor, yet according to See also:modern views the real pathway of the See also: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 See also:powers: (I) by the See also:production of a magnetic See also: Hence if dq is the quantity of electricity which flows across any section of the conductor in the See also:element of time dt, the current i=dq/dt. Electric currents may be also classified as See also:constant or variable and as unidirectional or " See also:direct," that is flowing always in the same direction, or " alternating," that is See also:reversing their direction at See also:regular intervals. In the last case the variation of current may follow any particular See also:law. It is called a " periodic current " if the See also: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 See also: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 See also:wave See also:diagram, or by a polar diagram.' In the first case we take a straight See also:line to represent the See also:uniform flow of time, and at small equidistant intervals set up perpendiculars above or below the time See also:axis, representing to See also:scale the current at that instant in one direction or the other; the extremities of these ordinates then define a wavy See also:curve which is called the wave form of the current (fig. I). 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 See also:simple See also:harmonic curve or simple sine curve. If 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 ern, and i is the value of the current at any instant t, and I its maximum value, then in this case we have i= I See also:sin pt. Such a current is called a " sine current " or simple periodic current. In a polar diagram (fig. 2) a number of radial lines are See also: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 See also:complete See also: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 See also:Fourier's theorem it follows that any periodic curve having any wave form can be imitated by the super- 1 See J. A. See also:Fleming, The Alternate Current Transformer, vol. i. p. 519.position of simple sine currents differing in maximum value and in phase. See also: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 See also:standard electrolyte. Limiting our See also:consideration to the case of linear currents or currents flowing in thin cylindrical wires, a See also: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 See also:UNITS, PEIYsICAL). H. C. Oersted discovered in r82o 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 See also:ELECTROMAGNETISM). The unit current in the electromagnetic See also:system of measurement is defined as the current which, flowing in a thin wire See also:bent into the form of a circle of one centimetre in See also:radius, creates a magnetic field having a strength of 27 units at the centre of the circle, and therefore would exert a See also:mechanical force of 27 dynes on a unit magnetic See also:pole placed at that point (see See also:MAGNETISM). Since the length of the circumference of the circle of unit radius is 27 units, this is See also: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 See also:rate that one electrostatic unit of quantity (see See also:ELECTROSTATICS) flows past any section of the wire in one unit of time. The electromagnetic unit of current defined as above is 3 X io'° times larger than the electrostatic unit. In the selection of a See also: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 I See also:ampere was selected, intended originally to be 1/to of the See also:absolute electromagnetic C.G.S. unit of current as above defined. The practical unit of current, called the See also:international ampere, is, however, legally defined at the present time as the continuous unidirectional current which when flowing through a neutral See also:solution of See also:silver nitrate deposits in one second on the See also:cathode or negative pole o•ooiit8 of a gramme of silver. There is See also: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/to part of the absolute electromagnetic unit. A periodic or alternating current is said to have a value of I ampere if when passed through a See also:fine wire it produces in the same time the same heat as a unidirectional continuous current of r 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 .12. This equiheating continuous current is called the effective or See also: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.1VI. F. is defined by See also:Ohm's Iaw, 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 See also:function of the geometrical form of the circuit, of its nature, i.e. material, and of its temperature, but is See also:independent of the electromotive force or current. The resistance (R) is measured in units caIIed 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 A=V/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 See also:motion of heat in a conductor. As a See also: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 See also:change of temperature produced by the current. Our belief, however, in its universality and accuracy rests upon the See also: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 See also: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 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 See also:account the electromotive force (E.M.F.) produced by this variation, and the product Ri 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 See also:physical quality of it in virtue of which energy is dissipated as heat in the circuit when a current flows through it. The See also:power communicated to any electric circuit when a current i is created in it by a continuous unidirectional electromotive force E is equal to Ei, and the energy dissipated as heat in that circuit by the conductor in a small See also:interval of time dt is measured by Ei dt. Since by Ohm's law E = Ri, where 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 Rig, 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 See also: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 zLi2, where 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. ZMv2. The rate at which this conserved energy varies with the current is called the electrokinetic momentum " of this circuit (=Li). Physically interpreted this quantity signifies the number of lines of magnetic See also: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 See also:external points which can in some instances be calculated. See also:Laplace proved that the magnetic force due to an element of length dS of a circuit conveying a current I at a point P at a distance r from the element is expressed by IdS sin 0/See also:r2, where B is the See also: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 See also: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 I is zrrI/r, since allelements 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 zar2I/(r2-hd2)*. Another important case is that of an infinitely See also:long straight current. By summing up the magnetic force due to each element at any point P outside the continuous straight current I, and at a distance d from it, we can show that it is equal to zI/d or is inversely proportional to the distance of the point from the wire. In the above See also:formula the current I is measured in absolute electromagnetic units. If we reckon the current in amperes A, then I=A/Io. 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 4lat circular disk is suspended so as to be See also:free to rotate See also:round a straight current which passes through its centre, and two See also: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 See also: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 N times round the circuit instead of once, we have to insert NA/lo in place of I in all the above formulae. The quantity 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 See also: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 See also:eye by See also:iron filings (see MAGNETISM). If a See also: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). In all cases a magnetic pole of strength M, placed in the field of an electric current, is urged along the lines of force with a mechanical. force equal to MH, where H is the magnetic force. If then we carry a unit magnetic pole against the direction in which it would naturally move we do See also:work. The lines of magnetic force em-bracing 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 I, the line integral is 4irI. It is easy to prove that this is a See also: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 47 times the See also: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 See also:cotton or See also:silk is See also:twisted round a thin See also:rod so as to make a close See also: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 47rNI =Hl, or H =49NI/l. For a thin endless solenoid the axial magnetic force is therefore 47r times the current-turns per unit of length. This holds See also:good also for a long straight solenoid provided its length is large compared with its See also:diameter. It can be shown that if insulated wire is See also:wound round a See also:sphere, the turns being all parallel to lines of See also: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. See also:Instruments called galvanometers and ammeters for the most part operate on this principle. Thermal Effects of Currents.—J. P. See also: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 EL 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 Rig dt, where dt represents a small interval of time and i is the current at that instant. The quantity T_1j i2dt 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 J i2dt l 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 See also:common use, especially for measuring alternating currents. The maximum value of the periodic current can only be deter-See also:mined 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 See also:state of See also: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, (I) by See also:radiation, (2) by See also: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 A amperes passed through it, the heat in See also:watts produced in any time t seconds is represented by the value of 4A2plt/to'lyd2, where d and l must be measured in centimetres and p in absolute C.G.S. electromagnetic units. The See also:factor Io' enters because one ohm is Io' 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 See also:watt-seconds are radiated per second from unit of surface, then the power removed by radiation in the time t is expressed by ardlet. Hence when thermal equilibrium is established we have 4A2plt/IO'7rd2=adlet, or See also:A2=Io'a2ed3/4p. If the diameter of the wire is reckoned in mils (1 mil —ow in.), and if we take e to have a value o•1, an emissivity which will generally bring the wire to about 6o° C., we can put the above formula in the following forms for circular sectioned copper, iron or platinoid wires, viz. A=Jd3/5oo for copper wires A = d3/4000 for iron wires A = d3/5000 for platinoid wires. These expressions give the ampere value of the current which will bring See also:bare, straight or loosely coiled wires of d mils in diameter to about 6o° 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 6o° C. if a current of I/ 503/5oo =AI 25o =16 amperes (nearly) is passed through it, whilst a current of d 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 6o mils in diameter the safe current varies approximately as the 1-5th power of the diameter, for fine wires of r 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 See also: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 See also: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 See also: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 See also: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 See also:crucial experiments, and the theory deduced from them is based upon four facts and one See also:assumption.' He showed (I) 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 ' See See also:Maxwell, Electricity and Magnetism, vol. ii. See also:chap. ii. 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 0 and 0' are the angles the elements make with the line joining them, and 4) 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-'{See also:cos ¢—1 cos 0 cos e')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 0 with the radius vector o is IdS sin B/r2, constitute the fundamental laws of electrokinetics. Ampere applied these with See also: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) Cur-rents in wires See also:meeting at an angle attract each other more into See also: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 See also:shell or a thin circular disk of See also:steel so magnetized that one See also:face is a See also:north pole and the other a See also:south pole, the product of the See also: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 See also: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. See also:Faraday's researches and the discovery of the fact .that the insulating See also: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 See also:contract.' If, therefore, two conductors See also: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. See also:Weber in inventing the electrodynamometer, and later See also:Lord See also:Kelvin devised ampere See also:balance: for the measurement of electric currents based on the attraction between coils
conveying electric currents.
See also:Induction of Electric Currents.—Faraday' 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 See also: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 See also: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 See also:light; hence in their outward progress they cut through the secondary circuit, just as ripples made on the surface of See also:water in a See also:lake by throwing a See also: 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 See also:vice versa. The action of the See also: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 See also:armature bars or coils in which an electromotive force is thereby created. In 1834 H. F. E. See also: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. See also: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 if dSdS' cos ¢/r, where dS and dS' are the elements of length of the two circuits, r is their distance, and 4) 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 See also:special See also:treatises. H. von See also: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 See also: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 RI2dt. Hence, by the conservation of energy, Eldt = RI2dt+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.' Oscillatory Currents.—In considering the motion of electricity in conductors we find interesting phenomena connected with the See also:discharge of a See also:condenser or See also:Leyden See also: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 3. See Maxwell, Electricity and Magnetism, vol. ii. § 542, p. 178. 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 See also:con-denser at that instant is 2qz/C, and the energy associated with the circuit is IL(dq/dt)2, and the rate of dissipation of energy by resistance is R(dq/dt)2, since dgidt =i is the discharge current. Hence we can construct an See also: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 : gz -dt 2CCJ =dtL L \d l> z z +R (d J or dt2+L d+LCgO' The above equation has two solutions according as R2/4L2 is greater or -less than 1/LC. In the first case the current i in the circuit can be expressed by the equation i=Qa±s0`e-°'(eel—e A / z where a=R/2L, (3='V 4L2—, Q is the value of q when t=o, and e is the See also:base of Napierian logarithms; and in the second case by the equation i = Qaz +13a-" sin Ot where a = R/2L, and 13 =' V -4. R2 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 See also:amplitude, the frequency n of the oscillation being given by the expression 1 1 R2 n =sr V LC—41f 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= r, 22r vl~, or T=1/n =z2rs/LC• The above investigation shows that if we construct a circuit consisting of a condenser and inductance placed in See also:series with one another, such circuit has a natural electrical time period of its own in which the electrical See also:charge in it oscillates if disturbed. It may therefore be compared with a pendulum of any See also: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. See also: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 See also: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 See also:gap of an induction coil, then when the coil is set in action we have a series of See also:bright noisy See also:sparks, each of which consists of a See also: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-See also:ball distance a discharge happens. This discharge, how-ever, 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 See also:plate or a rapidly moving film, we have See also: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 loth See also:century by reason of their application in electrotechnicsand 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 See also: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 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 See also:differences of these currents. A vector can most conveniently be represented by a See also:symbol such as ad-Lb, where a stands for any length of a units measured See also:horizon-See also:tally and b for a length b units measured vertically, and the smybol L is a sign of perpendicularity, and equivalent analytically' to -1. Accordingly if E represents the periodic electromotive force (maxi-mum 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 +i2lrnLI. Since the absolute magnitude of a vector a+eb is sl (a2+b2), 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)/y R2+p2L2, where p stands for 2zrn. If the above equation is compared with the symbolic expression of Ohm's law, it will be seen that the quantity v (R2+p2L2) takes the place of resistance R in the expression of Ohm. This quantity v (R2+p2L2) 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.2 In the first case consider the collocation of a number of linear conductors, such as rods or wires of See also: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 ~(E)=E(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 See also: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 See also: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 1 See W. G. See also:Rhodes, An Elementary See also:Treatise on Alternating Currents (See also:London, 1902), chap. vii. 2 See J. A. Fleming, " Problems on the Distribution of Electric Currents in Networks of Conductors," Phil. Mag. (1885), or Proc. Phys. See also:Soc. Lond. (1885), 7; also Maxwell, Electricity and Magnetism (2nd ed.), vol. i. p. 374, § z8o, 282b. 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 deter- minants. The solution of the well- known problem of finding the current in the See also:galvanometer circuit of the arrangement of linear conductors called See also:Wheatstone's See also:Bridge is thus easily ob- tained. For if we See also:call the cycles (see fig. 7) (x+y), y and z, and the resist- ances P, Q, R, S, G and B, and if E be the electromotive force in the battery circuit, we have the cycle equations See also:Ill (P+G+R) (x+y) —Gy—Rz=O, 8 (Q+G+S)y—G(x+y) -Sz=O, F1c.7. (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)o. where i 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 See also: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 See also:infinite plane sheet of thickness S 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 V Q to y' 27rNO r2 where Q is the quantity of electricity supplied by the source per second. Hence the equation to the equipotential curve is rlr2=a constant. If we take a point See also: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 3'2+(x+p)2 =a constant. y2+(x—p)2 This is the equation of a See also:family of circles having the axis of y for a common See also:radical axis, one set of circles surrounding the sink and another set of circles surrounding the source. In See also: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 See also: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 t t ~„ =a constant, is the equation to the equipotential lines. The orthogonal trajectories or stream lines have the equation 1(0—0') =a constant, where 0 and 0' 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, 0 are the polar co-ordinates of any one, then the equation to the equipotential surfaces is given by the equation E(A See also:log,r) =a constant, . where A is a constant; and the equation to the stream or current lines is F.(6) =a constant. In the case of electric flow in three dimensions the electric potential must satisfy La See also:lace's equation, and a solution is therefore found in the formE(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 See also:settle it were first under-taken in the See also:year 1876 by H. A. See also:Rowland, at a See also:suggestion of H. von Helmholtz). After preliminary experiments, Rowland's first apparatus for testing this See also:hypothesis was constructed, as follows: An ebonite disk was covered with radial strips of See also:gold-See also: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 See also: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. See also:Rontgen (Wied. See also:Ann., 1888, 35, p. 264; 1890, 40, p. 93) and by F. Himstedt (Wied. Ann., 1889, 38, p. 56o). Later V. Cremieu again repeated them and obtained negative results (Cont. rend., 1900, 130, p. 1544, and 131, pp. 578 and 797; 1901, 132, pp. 327 and I io8). They were again very carefully reconducted by H. See also:Pender (Phil. Mag., 19or, 2, p. 179) and by E. P. See also: W. See also:Wood (Phil. Mag., 1902, 2, p. 659) provides a confirmatory fact. He noticed that if See also:carbon-dioxide strongly compressed in a steel See also:bottle is allowed to See also:escape suddenly the See also:cold produced solidifies some part of the See also:gas, and the issuing See also:jet is full of particles of carbon-dioxide See also:snow. These by See also:friction against the nozzle are electrified positively. Wood caused the jet of gas to pass through a See also:glass See also: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 See also: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 See also:atom (see MATTER and ELECTRICITY). Certain lines of See also:argument See also: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 See also:density or number per unit of See also: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 See also: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 Berl. Acad. Ber., 1876, p. 211; also H. A. Rowland and C. T. See also:Hutchinson, " On the Electromagnetic Effect of Convection Cur-rents," Phil. Mag., 1889, 27, p. 445. (see See also: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. Additional information and CommentsThere are no comments yet for this article.
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