A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 23 of 28

The dissipation will be very small in the case of so feeble a state of polarization, and it will take place by a very slow absorption of the gases and diffusion through the liquid. The rate of this dissipation is indicated by the exceedingly feeble current which still continues to flow without any visible separation of gases.

If we neglect this dissipation for the short time during which the state of polarization is set up, and if we call Q the total quantity of electricity which is transmitted by the current during this time, then if A is the area of one of the electrodes, and <r the density of the deposit, supposed uniform,

Q = A<r.

If we now disconnect the electrodes of the electrolytic apparatus from the Daniell's cell, and connect them with a galvanometer capable of measuring the whole discharge through it, a quantity of electricity nearly equal to Q will be discharged as the polari- zation disappears.

271.] Hence we may compare the action of this apparatus, which is a form of Hitter's Secondary Pile, with that of a Leyden jar.

Both the secondary pile and the Leyden jar are capable of being charged with a certain amount of electricity, and of being after- wards discharged. During the discharge a quantity of electricity nearly equal to the charge passes in the opposite direction. The difference between the charge and the discharge arises partly from dissipation, a process which in the case of small charges is very slow, but which, when the charge exceeds a certain limit, becomes exceedingly rapid. Another part of the difference between the charge and the discharge arises from the fact that after the electrodes have been connected for a time sufficient to produce an apparently complete discharge, so that the current has completely disappeared, if we separate the electrodes for a time, and afterwards connect them, we obtain a second discharge in the same direction as the original discharge. This is called the residual discharge, and is a phenomenon of the Leyden jar as well as of the secondary pile.

The secondary pile may therefore be compared in several respects to a Leyden jar. There are, however, certain important differences. The charge of a Leyden jar is very exactly proportional to the

271. COMPARISON WITH LEYDEN JAR. 301

electromotive force of the charge, that is, to the difference of potentials of the two surfaces, and the charge corresponding to unit of electromotive force is called the capacity of the jar, a constant quantity. The corresponding quantity, which may be called the capacity of the secondary pile, increases when the electromotive force increases.

The capacity of the jar depends on the area of the opposed surfaces, on the distance between them, and on the nature of the substance between them, but not on the nature of the metallic surfaces themselves. The capacity of the secondary pile depends on the area of the surfaces of the electrodes, but not on the distance between them, and it depends on the nature of the surface of the electrodes, as well as on that of the fluid between them. The maximum difference of the potentials of the electrodes in each element of a secondary pile is very small compared with the maxi- mum difference of the potentials of those of a charged Leyden jar, so that in order to obtain much electromotive force a pile of many elements must be used.

On the other hand, the superficial density of the charge in the secondary pile is immensely greater than the utmost superficial density of the charge which can be accumulated on the surfaces of a Leyden jar, insomuch that Mr. C. F. Varley *, in describing the construction of a condenser of great capacity, recommends a series of gold or platinum plates immersed in dilute acid as prefer- able in point of cheapness to induction plates of tinfoil separated by insulating material.

The form in which the energy of a Leyden jar is stored up is the state of constraint of the dielectric between the conducting surfaces, a state which I have already described under the name of electric polarization, pointing out those phenomena attending this state which are at present known, and indicating the im- perfect state of our knowledge of what really takes place. See Arts. 62, 111.

The form in which the energy of the secondary pile is stored up is the chemical condition of the material stratum at the surface of the electrodes, consisting of the ions of the electrolyte and the substance of the electrodes in a relation varying from chemical combination to superficial condensation, mechanical adherence, or simple juxtaposition.

The seat of this energy is close to the surfaces of the electrodes,

• Specification of C. F. Varley, 'Electric Telegraphs, &c.,' Jan. 1860.

362 ELECTROLYTIC POLARIZATION.

and not throughout the substance of the electrolyte, and the form in which it exists may be called electrolytic polarization.

After studying the secondary pile in connexion with the Leyden jar, the student should again compare the voltaic battery with some form of the electrical machine, such as that described in Art. 211.

Mr. Varley has lately * found that the capacity of one square inch is from 175 to 542 microfarads and upwards for platinum plates in dilute sulphuric acid, and that the capacity increases with the electromotive force, being about 175 for 0.02 of a Daniell's cell, and 542 for 1.6 Daniell's <cells.

But the comparison between the Leyden jar and the secondary pile may be carried still farther, as in the following experiment, due to Buff f. It is only when the glass of the jar is cold that it is capable of retaining a charge. At a temperature below 100°C the glass becomes a conductor. If a test-tube containing mercury is placed in a vessel of mercuiy, and if a pair of electrodes are connected, one with the inner and the other with the outer portion of mercury, the arrangement .constitutes a Leyden jar which will hold a charge at ordinary temperatures. If the electrodes are con- nected with those of a voltaic battery, no current will pass as long as the glass is cold, but if the apparatus is gradually heated a current will begin to pass, and will increase rapidly in intensity as the temperature rises, though the glass remains apparently as hard as ever.

This current is manifestly electrolytic, for if the electrodes are disconnected from the battery, and connected with a galvanometer, a considerable reverse current passes, due to polarization of the surfaces of the glass.

If, while the battery is in action the apparatus is cooled, the current is stopped by the cold glass as before, but the polarization of the surfaces remains. The mercury may be removed, the surfaces may be washed with nitric acid and with water, and fresh mercuiy introduced. If the apparatus is then heated, the current of polar- ization appears as soon as the glass is sufficiently warm to conduct it.

We may therefore regard glass at 100°C, though apparently a solid body, as an electrolyte, and there is considerable reason to believe that in most instances in which a dielectric has a slight degree of conductivity the conduction is electrolytic. The

• Proc. E. S. Jan. 12, 1871.

t Annalen der Chemie und Pharmacie, bd. xc. 257 (1854;.

272.] CONSTANT VOLTAIC ELEMENTS. 363

existence of polarization may be regarded as conclusive evidence of electrolysis, and if the conductivity of a substance increases as the temperature rises, we have good grounds for suspecting that it is electrolytic.

On Constant Voltaic Elements.

272.] When a series of experiments is made with a voltaic battery in which polarization occurs, the polarization diminishes during the time the current is not flowing, so that when it begins to flow again the current is stronger than after it has flowed for some time. If, on the other hand, the resistance of the circuit is diminished by allowing the current to flow through a short shunt, then, when the current is again made to flow through the ordinary circuit, it is at first weaker than its normal strength on account of the great polarization* produced by the use of the short circuit.

To get rid of these irregularities in the current, which are exceedingly troublesome in experiments involving exact measure- ments, it is necessary to get rid of the polarization, or at least to reduce it as much as possible.

It does not appear that there is much polarization at the surface of the zinc plate when immersed in a solution of sulphate of zinc or in dilute sulphuric -acid. The principal seat of polarization is at the surface of the negative metal. When the fluid in which the negative metal is immersed is dilute sulphuric acid, it is seen to become covered with bubbles of hydrogen gas, arising from the electrolytic decomposition of the fluid. Of course these bubbles, by preventing the fluid from touching the metal, diminish the surface of contact and increase the resistance of the circuit. But besides the visible bubbles it is certain that there is a thin coating of hydrogen, probably not in a free state, adhering to the metal, and as we have seen that this coating is able to produce an elec- tromotive force in the reverse direction, it must necessarily diminish the electromotive force of the battery.

Various plans have been adopted to get rid of this coating of hydrogen. It may be diminished to some extent by mechanical means, such as stirring the liquid, or rubbing the surface of the negative plate. In Smee's battery the negative plates are vertical, and covered with finely divided platinum from which the bubbles of hydrogen easily escape, and in their ascent produce a current of liquid which helps to brush off other bubbles as they are formed.

A far more efficacious method, however, is to employ chemical

364: ELECTROLYTIC POLARIZATION.

means. These are of two kinds. In the batteries of Grove and Bunsen the negative plate is immersed in a fluid rich in oxygen, and the hydrogen, instead of forming a coating on the plate, combines with this substance. In Grove's battery the plate is of platinum immersed in strong nitric acid. In Bunsen's first battery it is of carbon in the same acid. Chromic acid is also used for the same purpose, and has the advantage of being free from the acid fumes produced by the reduction of nitric acid.

A different mode of getting rid of the hydrogen is by using copper as the negative metal, and covering the surface with a coat of oxide. This, however, rapidly disappears when it is used as the negative electrode. To renew it Joule has proposed to make the copper plates in the form of disks, half immersed in the liquid, and to rotate them slowly, so that the air may act on the parts exposed to it in turn.

The other method is by using as the liquid an electrolyte, the cation of which is a metal highly negative to zinc.

In Daniel Fs battery a copper plate is immersed in a saturated solution of sulphate of copper. When the current flows through the solution from the zinc to the copper no hydrogen appears on the copper plate, but copper is deposited on it. When the solution is saturated, and the current is not too strong, the copper appears to act as a true cation, the anion S O| travelling towards the zinc.

When these conditions are not fulfilled hydrogen is evolved at the cathode, but immediately acts on the solution, throwing down copper, and uniting with SO4 to form oil of vitriol. When this is the case, the sulphate of copper next the copper plate is replaced by oil of vitriol, the liquid becomes colourless, and polarization by hydrogen gas again takes place. The copper deposited in this way is of a looser and more friable structure than that deposited by true electrolysis.

To ensure that the liquid in contact with the copper shall be saturated with sulphate of copper, crystals of this substance must be placed in the liquid close to the copper, so that when the solution is made weak by the deposition of the copper, more of the crystals may be dissolved.

We have seen that it is necessary that the liquid next the copper should be saturated with sulphate of copper. It is still more necessary that the liquid in which the zinc is immersed should be free from sulphate of copper. If any of this salt makes its way to the surface of the zinc it is reduced, and copper is deposited

272.]

THOMSON'S FORM OF DANIELL'S CELL.

365

on the zinc. The zinc, copper, and fluid then form a little circuit in which rapid electrolytic action goes on, and the zinc is eaten away by an action which contributes nothing to the useful effect of the battery.

To prevent this, the zinc is immersed either in dilute sulphuric acid or in a solution of sulphate of zinc, and to prevent the solution of sulphate of copper from mixing- with this liquid, the two liquids are separated by a division consisting of bladder or porous earthen- ware, which allows electrolysis to take place through it, but effectually prevents mixture of the fluids by visible currents.

In some batteries sawdust is used to prevent currents. The experiments of Graham, however, shew that the process of diffusion goes on nearly as rapidly when two liquids are separated by a division of this kind as when they are in direct contact, provided there are no visible currents, and it is probable that if a septum is employed which diminishes the diffusion, it will increase in exactly the same ratio the resistance of the element, because elec- trolytic conduction is a process the mathematical laws of which have the same form as those of diffusion, and whatever interferes with one must interfere equally with the other. The only differ- ence is that diffusion is always going on, whereas the current flows only when the battery is in action.

In all forms of Daniell's battery the final result is that the sulphate of copper finds its way to the zinc and spoils the battery. To retard this result indefinitely, Sir W. Thomson* has constructed Daniell's battery in the following form.

SIPHON

ELECTRODES

LEVEL or SIPHON

Fig. 22.

In each cell the copper plate is placed horizontally at the bottom

• Proc. B. S., Jau. 19, 1871.

366 ELECTROLYTIC POLARIZATION. [272.

and a saturated solution of sulphate of zinc is poured over it. The zinc is in the form of a grating and is .placed horizontally near the surface of the solution. A glass tube is placed vertically in the solution with its lower end just above the surface of the copper plate. Crystals of sulphate of copper are dropped down this tube, and, dissolving in the liquid, form a solution of greater density than that of sulphate of zinc alone, so that it cannot get to the zinc except by diffusion. To retard this process of diffusion, a siphon, consisting of a glass tube stuffed with cotton wick, is placed with one extremity midway between the zinc and copper, and the other in a vessel outside the cell, so that the liquid is very slowly drawn off near the middle of its depth. To supply its place, water, or a weak solution of sulphate of zinc, is added above when required. In this way the greater part of the sulphate of copper rising through the liquid by diffusion is drawn off by the siphon before it reaches the zinc, and the zinc is surrounded by liquid nearly free from sulphate of copper, and having a very slow downward motion in the cell, which still further retards the upward motion of the sulphate of copper. During the action of the battery copper is deposited on the copper plate, and SO4 travels slowly through the liquid to the zinc with which it combines, forming sulphate of zinc. Thus the liquid at the bottom becomes less dense by the deposition of the copper, and the liquid at the top becomes more dense by the addition of the zinc. To prevent this action from changing the order of density of the strata, and so producing instability and visible currents in the vessel, care must be taken to keep the tube well supplied with crystals of sulphate of copper, and to feed the cell above with a solution of sulphate of zinc suffi- ciently dilute to be lighter than any other stratum of the liquid in the cell.

Daniell's battery is by no means the most powerful in common use. The electromotive force of Grove's cell is 192,000,000, of Daniell's 107,900,000 and that of Bunsen's 188,000,000.

The resistance of Daniell's cell is in general greater than that of Grove's or Bunsen's of the same size.

These defects, however, are more than counterbalanced in all cases where exact measurements are required, by the fact that Daniell's cell exceeds every other known arrangement in constancy of electromotive force. It has also the advantage of continuing in working order for a long time, and of emitting no gas.

CHAPTER VI.

LINEAR ELECTRIC CURRENTS,

On Systems of Linear Conductors.

273.] ANY conductor may be treated aa a linear conductor if it is arranged so that the current must always pass in the same manner between two portions of its surface which are called its electrodes. For instance, a mass of metal of any form the surface of which is entirely covered with insulating material except at two places, at which the exposed surface of the conductor is in metallic contact with electrodes formed of a perfectly conducting material, may be treated as a linear conductor. For if the current be made to enter at one of these electrodes and escape at the other the lines of flow will be determinate, and the relation between electromotive force, current and resistance will be expressed by Ohm's Law, for the current in every part of the mass will be a linear function of E. But if there be more possible electrodes than two, the conductor may have more than one independent current through it, and these may not be conjugate to each other. See Art. 282.

Ohm's Law.

274.] Let E be the electromotive force in a linear conductor from the electrode A± to the electrode A2. (See. Art. 69.) Let C be the strength of the electric current along the conductor, that is to say, let C units of electricity pass across every section in the direction A1 A2 in unit of time, and let R be the resistance of the conductor, then the expression of Ohm's Law is

E=CR. (1)

Linear Conductors arranged in Series.

275.] Let Alt A2 be the electrodes of the first conductor and let the second conductor be placed with one of its electrodes in contact

368 LINEAR ELECTRIC CURRENTS, [276.

with A2, so that the second conductor has for its electrodes A2, A%. The electrodes of the third conductor may be denoted by AB and A4.

Let the electromotive forces along these conductors be denoted by 2£12, E23, E3i, and so on for the other conductors.

Let the resistances of the conductors be

Then, since the conductors are arranged in series so that the same current C flows through each, we have by Ohm's Law,

Ei2 = CR12, E^^CR^ EM = CR3,. (2)

If E is the resultant electromotive force, and R the resultant resistance of the system, we must have by Ohm's Law,

E = CR. (3)

Now E=E12 + E23 + E^ (4)

the sum of the separate electromotive forces, = C(R12 + R23 + R3±) by equations (2). Comparing this result with (3), we find

R = Rl2 + R^ + Ru. (5)

Or, the resistance of a series of conductors is the sum of the resistances of the conductors taken separately.

Potential at any Point of the Series.

Let A and C be the electrodes of the series, B a point between them, a, c, and 6 the potentials of these points respectively. Let R! be the resistance of the part from A to B> R2 that of the part from B to C, and R that of the whole from A to (7, then, since

a—b = RlC, b—c = R2C, and a—c = RC, the potential at B is

which determines the potential at B when the potentials at A and C are given.

Resistance of a Multiple Conductor.

276.] Let a number of conductors ABZt ACZ, ADZ be arranged side by side with their extremities in contact with the same two points A and Z. They are then said to be arranged in multiple arc.

Let the resistances of these conductors be Rl, R2J R3 respect*

277-] SPECIFIC RESISTANCE AND CONDUCTIVITY. 369

ively, and tlie currents Clt C2, C3, and let the resistance of the multiple conductor be R, and the total current C. Then, since the potentials at A and Z are the same for all the conductors, they have the same difference, which we may call E. We then have E=C1R1 = C2R2 = C3R3 = CR,

whence =i + i+i- (7)

Or, the reciprocal of the resistance of a multiple conductor is the sum of the reciprocals of the component conductors.

If we call the reciprocal of the resistance of a conductor the conductivity of the conductor, then we may say that the con- ductivity of a multiple conductor is the sum of the conductivities of the component conductors.

Current in any Branch of a Multiple Conductor. From the equations of the preceding article, it appears that if (?! is the current in any branch of the multiple conductor, and R1 the resistance of that branch,

4=CJr, (8)

where C is the total current, and R is the resistance of the multiple conductor as previously determined.

Longitudinal Resistance of Conductors of Uniform Section.

277.] Let the resistance of a cube of a given material to a current parallel to one of its edges be />, the side of the cube being unit of length, p is called the ' specific resistance of that material for unit of volume.'

Consider next a prismatic conductor of the same material whose length is I, and whose section is unity. This is equivalent to I cubes arranged in series. The resistance of the conductor is there- fore I p.

Finally, consider a conductor of length I and uniform section s. This is equivalent to s conductors similar to the last arranged in multiple arc. The resistance of this conductor is therefore

ju#.

s

When we know the resistance of a uniform wire we can determine VOL. i. B b

370 LINEAR ELECTEIC CURRENTS. [27^.

the specific resistance of the material of which it is made if we can measure its length and its section.

The sectional area of small wires is most accurately determined by calculation from the length, weight, and specific gravity of the specimen. The determination of the specific gravity is sometimes inconvenient, and in such cases the resistance of a wire of unit length and unit mass is used as the e specific resistance per unit of weight/

If r is this resistance, I the length, and m the mass of a wire, then

On the Dimensions of the Quantities involved in these Equations.

278.] The resistance of a conductor is the ratio of the electro- motive force acting on it to the current produced. The conduct- ivity of the conductor is the reciprocal of this quantity, or in other words, the ratio of the current to the electromotive force producing it.

Now we know that in the electrostatic system of measurement the ratio of a quantity of electricity to the potential of the con- ductor on which it is spread is the capacity of the conductor, and is measured by a line. If the conductor is a sphere placed in an unlimited field, this line is the radius of the sphere. The ratio of a quantity of electricity to an electromotive force is therefore a line, but the ratio of a quantity of electricity to a current is the time during which the current flows to transmit that quantity. Hence the ratio of a current to an electromotive force is that of a line to a time, or in other words, it is a velocity.

The fact that the conductivity of a conductor is expressed in the electrostatic system of measurement by a velocity may be verified by supposing a sphere of radius r charged to potential F", and then connected with the earth by the given conductor. Let the sphere contract, so that as the electricity escapes through the conductor the potential of the sphere is always kept equal to T. Then the charge on the sphere is rV at any instant, and the current is

-j- (rF), but, since V is constant, the current is jr V, and the

electromotive force through the conductor is V.

The conductivity of the conductor is the ratio of the current to

the electromotive force, or — , that is, the velocity with which the

Cvt

radius of the sphere must diminish in order to maintain the potential

28o.] SYSTEM OF LINEAR CONDUCTORS. 371

constant when the charge is allowed to pass to earth through the conductor.

In the electrostatic system, therefore, the conductivity of a con- ductor is a velocity, and of the dimensions [LT~1].

The resistance of the conductor is therefore of the dimensions

IL-*T.

The specific resistance per unit of volume is of the dimension of [T], and the specific conductivity per unit of volume is of the dimension of [27"1].

The numerical magnitude of these coefficients depends only on the unit of time, which is the same in different countries.

The specific resistance per unit of weight is of the dimensions

279.] We shall afterwards find that in the electromagnetic system of measurement the resistance of a conductor is expressed by a velocity, so that in this system the dimensions of the resist- ance of a conductor are [XT7"1].

The conductivity of the conductor is of course the reciprocal of this.

The specific resistance per unit of volume in this system is of the dimensions [X2!7"1], and the specific resistance per unit of weight is of the dimensions [L1T1M~.

On Linear Systems of Conductors in general.

280.] The most general case of a linear system is that of n points, Al} A2,...An} connected together in pairs by \n(nr-) linear conductors. Let the conductivity (or reciprocal of the re- sistance) of that conductor which connects any pair of points, say Ap and Aq, be called Kpq, and let the current from Ap to Aq be Cpq. Let Pp and Pq be the electric potentials at the points Ap and Aq respectively, and let the internal electromotive force, if there be any, along the conductor from Ap to Aq be Epq.

The current from Ap to Aq is, by Ohm's Law,

Ctq = KM(Pt-Pq + Kpq). . (1)

Among these quantities we have the following sets of relations :

The conductivity of a conductor is the same in either direction,

*•„ = *„. (a)

The electromotive force and the current are directed quantities, so that ^U = -^p> and Cpq = -Cqp. (3)

Let Plt P2, ...Pn be the potentials at A19 A2, ... An respectively, and let Qlt Q2,... Qn be the quantities of electricity which enter

B b 2

372 LINEAR ELECTRIC CURRENTS. [280.

the system in unit of time at each of these points respectively. These are necessarily subject to the condition of ' continuity '

&+&... + &=<), (4)

since electricity can neither be indefinitely accumulated nor pro- duced within the system.

The condition of ' continuity ' at any point Ap is

Qp = Cpl + Cp2 + &c. + Opn. (5)

Substituting the values of the currents in terms of equation (l), this becomes

Epn). (6)

The symbol Kpp does not occur in this equation. Let us therefore give it the value

Kpp = - (Kpl + Kp2 + &c. + Kpn) ; (7)

that is, let Kpp be a quantity equal and opposite to the sum of all the conductivities of the conductors which meet in Ap. We may then write the condition of continuity for the point Ap,

Epn-Qp. (8)

By substituting 1 , 2, &c. n for p in this equation we shall obtain n equations of the same kind from which to determine the n potentials Plt P2, &c., Pn.

Since, however, if we add the system of equations (8) the result is identically zero by (3), (4) and (7), there will be only n — 1 in- dependent equations. These will be sufficient to determine the differences of the potentials of the points, but not to determine the absolute potential of any. This, however, is not required to calcu- late the currents in the system.

If we denote by D the determinant

(9)

and by Dpq, the minor of Kpq, we find for the value of Ip— Pn,

KqnEqn-Qq)Dpq + &c. (10) In the same way the excess of the potential of any other point, say Aq, over that of An may be determined. We may then de- termine the current between Ap and Aq from equation (l), and so solve the problem completely.

282 a.] SYSTEM OF LINEAR CONDUCTORS. 373

281.] We shall now demonstrate a reciprocal property of any two conductors of the system, answering to the reciprocal property we have already demonstrated for statical electricity in Art. 88.

The coefficient of Qq in the expression for Pp is -^. That of Qp in the expression for Pq is -~ •

Now Dpq differs from Dqp only by the substitution of the symbols such as Kqp for Kpq. But, by equation (2), these two symbols are equal, since the conductivity of a conductor is the same both ways. Hence Dpq=-.Dqp. (ll)

It follows from this that the part of the potential at Ap arising from the introduction of a unit current at Aq is equal to the part of the potential at Aq arising from the introduction of a unit current at Ap.

We may deduce from this a proposition of a more practical form.

Let A, B, C, D be any four points of the system, and let the effect of a current Q, made to enter the system at A and leave it at B, be to make the potential at C exceed that at D by P. Then, if an equal current Q be made to enter the system at C and leave it at D, the potential at A will exceed that at B by the same quantity P.

If an electromotive force E be introduced, acting in the conductor from A to B, and if this causes a current C from X to Y, then the same electromotive force E introduced into the conductor from X to Y will cause an equal current C from A to B.

The electromotive force E may be that of a voltaic battery intro- duced between the points named, care being taken that the resist- ance of the conductor is the same before and after the introduction of the battery.

282 a.] If an electromotive force E^ act along the conductor Ap Aq, the current produced along another conductor of the system A A is easily found to be

Kn K,t Epq (Dre+D,t-Drq-D.r) * D.

There will be no current if

j)rp + Z)sq—£rq—J)sp = 0. (12)

But, by (11), the same equation holds if, when the electromotive force acts along ArAt, there is no current in ApAq. On account of this reciprocal relation the two conductors referred to are said to be conjugate.

The theory of conjugate conductors has been investigated by

374 LINEAR ELECTRIC CURRENTS. [282 I.

Kirchhoff, who has stated the conditions of a linear system in the following manner, in which the consideration of the potential is avoided.

(1) (Condition of 'continuity.') At any point of the system the sum of all the currents which flow towards that point is zero.

(2) In any complete circuit formed by the conductors the sum of the electromotive forces taken round the circuit is equal to the sum of the products of the current in each conductor multiplied by the resistance of that conductor.

We obtain this result by adding equations of the form (l) for the complete circuit, when the potentials necessarily disappear.

*282 £.] If the conducting wires form a simple network and if we suppose that a current circulates round each mesh, then the actual current in the wire which forms a thread of each of two neighbouring meshes will be the difference between the two currents circulating in the two meshes, the currents being reckoned positive when they circulate in a direction opposite to the motion of the hands of a watch. It is easy to establish in this case the following proposition : — Let x be the current, E the electromotive force, and R the total resistance in any mesh ; let also y, z, ... be currents circulating in neighbouring meshes which have threads in common with that in which x circulates, the resistances of those parts being ,?, ^, . . . ; then

Rx—sy—tz—&c. = E.

To illustrate the use of this rule we will take the arrangement known as Wheatstone's Bridge, adopting the figure and notation of Art. 347. We have then the three following equations repre- senting the application of the rule in the case of the three circuits OBC, OCA, OAB in which the currents #, y> z respectively circulate, viz. (a + p+y),,. _yy -/3* = JS;

— y <g + (b + y + a)y — az = 0,

— J3 x — ay + (c + a + (3)z = 0.

From these equations we may now determine the value of x— y the galvanometer current in the branch OA, but the reader is referred to Art. 347 et seq. where this and other questions connected with Wheatstone's Bridge are discussed.

Heat Generated in the System. 283.] The mechanical equivalent of the quantity of heat generated

• [Extracted from notes of Professor Maxwell's lectures by Mr. J. A. Fleming, B.A., St. John's College.]

284.] GENERATION OF HEAT. 375

in a conductor whose resistance is R by a current C in unit of time is, by Art. 242, Jff = RC*. (13)

We have therefore to determine the sum of such quantities as RC2 for all the conductors of the system.

For the conductor from Ap to Aq the conductivity is Kpq, and the resistance R^, where K R -\ (IA\

pQ ' pq V /

The current in this conductor is, according to Ohm's Law,

We shall suppose, however, that the value of the current is not that given by Ohm's Law, but Xpq, where

xp, = cp,+ rf,. (16)

To determine the heat generated in the system we have to find the sum of all the quantities of the form

or JH=2{RpqCpq + 2RpqCpqYpq + RpqYpq}. (17) Giving Cpq its value, and remembering the relation between Kt,,{ and Rpq, this becomes

-P,) (Cpq + 2Tfg) + Epq Y\q}. (18)

Now since both C and X must satisfy the condition of continuity

at Ap, we have Qp = Cpl + CP2 + &c. + Cpn, (19)

Qp = Xpl+XP2 + bc. + Xpn, (20)

therefore 0 = Ypl + YP2 + &c. + Tpn . (21)

Adding together therefore all the terms of (18), we find

2(JZM*»M) = SP,Q, + S5M7V (22)

Now since R is always positive and Y2 is essentially positive, the last term of this equation must be essentially positive. Hence the first term is a minimum when Y is zero in every conductor, that is, when the current in every conductor is that given by Ohm's Law.

Hence the following theorem :