The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 18 of 39

C f . . AB OA . . (OP AB \

• \am AP+BP+0P8m~\0A • APTBP)>

where 0 is the centre of the bowl, and A, B are the points in which a plane through P and the axis of the bowl cuts the circular rim.

Find the density of electricity at a point on either side of the bowl and shew that the

capacity is

a . . . — (a + sin a),

IT

where a is the radius of the sphere, and 2a is the angle subtended at the centre.

94. Two spheres are charged to potentials VQ and Vx. The ratio of the distances of any point from the two limiting points of the spheres being denoted by ev and the angle between them by £, prove that the potential at the point £, 77 is

sinh(n + |)(/3 + »?) 0

• VX J{2 (cosh , - cos Q) 2 sinh\n + llp + l} P* («» 0 • *".

where 77 = a, ij = - /3 are the equations of the spheres. Hence find the charge on either sphere.

f.^c*«— eijffl&Slfcg'-.cfl'-^

CHAPTER IX

STEADY CURRENTS IN LINEAR CONDUCTORS

Physical Principles.

338. If two conductors charged with electricity to different potentials are connected by a conducting wire, we know that a flow of electricity will take place along the wire. This flow will tend to equalise the potentials of the two conductors, and when these potentials become equal the flow of electricity will cease. If we had some means by which the charges on the conductors could be replenished as quickly as they were carried away by conduction through the wire, then the current would never cease. The con- ductors would remain permanently at different potentials, and there would be a steady flow of electricity from one to the other. Means are known by which two conductors can be kept permanently at different potentials, so that a steady flow of electricity takes place through any conductor or conductors joining them. We accordingly have to discuss the mathematical theory of such currents of electricity.

We shall begin by the consideration of the flow of electricity in linear conductors, by a linear conductor being meant one which has a definite cross-section at every point. The commonest instance of a linear conductor is a wire.

339. Definition. The strength of a current at any point in a wire or other linear conductor, is measured by the number of units of electricity which flow across any cross-section of the conductor per unit time.

If the units of electricity are measured in Electrostatic Units, then the current also will be measured in Electrostatic Units. These, however, as will be explained later, are not the units in which currents are usually measured in practice.

Let P, Q be two cross-sections of a linear conductor in which a steady current is flowing, and let us suppose that no other conductors touch this conductor between P and Q. Then, since the current is, by hypothesis, steady, there must be no accumulation of electricity in the region of the

338-341] Physical Principles 301

conductor between P and Q. Hence the rate of flow into the section of the conductor across P must be exactly equal to the rate of flow out of this section across Q. Or, the currents at P and Q must be equal. Hence we speak of the current in a conductor, rather than of the current at a point in a conductor. For, as we pass along a conductor, the current cannot change except at points at which the conductor is touched by other conductors.

Ohm's Law.

340. In a linear conductor in which a current is flowing, we have electricity in motion at every point, and hence must have a continuous variation in potential as we pass along the conductor. This is not in opposition to the result previously obtained in Electrostatics, for in the previous analysis it had to be assumed that the electricity was at rest. In the present instance, the electricity is not at rest, being in fact kept in motion by the difference of potential under discussion.

The analogy between potential and height of water will perhaps help. A lake in which the water is at rest is analogous to a conductor in which electricity is in equi- librium. The theorem that the potential is constant over a conductor in which electricity is in equilibrium, is analogous to the hydrostatic theorem that the surface of still water must all be at the same level. A conductor through which a current of electricity is flowing finds its analogue in a stream of running water. Here the level is not the same at all points of the river — it is the difference of level which causes the water to flow. The water will flow more rapidly in a river in which the gradient is large than in one in which it is small. The electrical analogy to this is expressed by Ohm's Law.

Ohm's Law. The difference of potential between any two points of a wire or other linear conductor in which a current is flowing, stands to the current flowing through the conductor in a constant ratio, which is called the resistance between the two points.

It is here assumed that there is no junction with other conductors between these two points, so that the current through the conductor is a definite quantity.

341. Thus if C is the current flowing between two points P, Q at which the potentials are VP, VQ, we have

VP-VQ=CR (264),

where R is the resistance between the points P and Q. Very delicate experiments have failed to detect any variation in the ratio

(fall of potential)/(current),

as the current is varied, and this justifies us in speaking of the resistance as a definite quantity associated with the conductor. The resistance depends naturally on the positions of the two points by which the current enters and leaves the conductor, but when once these two points are fixed the resistance

302

Steady Currents in Linear Conductors [ch. ix

is independent of the amount of current. In general, however, the resistance of a conductor varies with the temperature, and for some substances, of which selenium is a notable example, it varies with the amount of light falling on the conductor.

The Voltaic Cell.

342. The simplest arrangement by which a steady flow of electricity can be produced is that known as a Voltaic Cell. This is represented diagram- matically in Fig. 95. A voltaic cell consists essentially of two conductors

Fig. 95.

A, B of different materials, placed in a liquid which acts chemically on at least one of them. On establishing electrical contact between the two ends of the conductors which are out of the liquid, it is found that a continuous current flows round the circuit which is formed by the two conductors and the liquid, the energy which is required to maintain the current being derived from chemical action in the cell.

To explain the action of the cell, it will be necessary to touch on a subject of which a full account would be out of place in the present book. As an experimental fact it is found that two conductors of dissimilar material, when placed in contact, have different potentials when there is no flow of electricity from one to the other*, although of course the potential over the whole of either conductor must be constant. In the light of this experimental fact, let us consider the conditions prevailing in the voltaic cell before the two ends a, b of the conductors are joined.

So long as the two conductors A, B and the liquid C do not form a closed circuit, there can be no flow of electricity. Thus there is electric equilibrium,

• For a long time there has been a divergence of opinion as to whether this difference of potential is not due to the chemical change at the surfaces of the conductors, and therefore dependent on the presence of a layer of air or other thud substance between the conductors. It seems now to be almost certain that this is the case, but the question is not one of vital importance as regards the mathematical theory of electric currents.

341-344] Physical Principles 303

and the three conductors have definite potentials VA,VB,VC. The difference of potential between the two " terminals " a, b is VA — VB, but the peculiarity of the voltaic cell is that this difference of potential is not equal to the difference of potential between the two conductors when they are placed in contact and are in electrical equilibrium without the presence of the liquid G. Thus on electrically joining the points a, b in the voltaic cell electrical equilibrium is an impossibility, and a current is established in the circuit which will continue until the physical conditions become changed or the supply of chemical energy is exhausted.

Electromotive Force.

343. Let A, B, G be any three conductors arranged so as to form a closed circuit. Let VAB be the contact difference of potential between A and B when there is electric equilibrium, and let VBC, VCA have similar meanings.

If the three substances can be placed in a closed circuit without any current flowing, then we can have equilibrium in which the three conductors will have potentials VA, VB, V0, such that

VA-VB=VAB; VB-VC=VBC; VC-VA = VGA.

Thus we must have

vAB+vBO+vCA = o,

a result known as Volta's Law.

If, however, the three conductors form a voltaic cell, the expression on the left-hand of the above equation does not vanish, and its value is called the electromotive force of the cell. Denoting the electromotive force by E, we have

VAB + VB0+VCA = E (265).

We accordingly have the following definition :

Definition. The Electromotive Force of a cell is the algebraic sum of the discontinuities of potential encountered in passing in order through the series of conductors of which the cell is composed.

Clearly an electromotive force has direction as well as magnitude. It is usual to speak of the two conductors which pass into the liquid as the high-potential terminal and the low-potential terminal, or sometimes as the positive and negative terminals. Knowing which is the positive or high- potential terminal, we shall of course know the direction of the electromotive force.

344. If the conductors G, A of a voltaic cell ABG are separated, and then joined by a fourth conductor D, such that there is no chemical action between D and the conductors G or A, it will easily be seen that the sum of the discontinuities in the new circuit is the same as in the old.

304 Steady Currents in Linear Conductors [ch. ix

For by hypothesis CD A can form a closed circuit in which no chemical action can occur, and therefore in which there can be electric equilibrium.

Hence we must have

rcD + VDA + VAC=0 (266).

Moreover the sum of all the discontinuities in the circuit is VAB+VBC + VCD + VDA

= VAB+VBC- VAC, by equation (266)

= Vab+Vbc+Vca

= E, by equation (265),

proving the result. A similar proof shews that we may introduce any series of conductors between the two terminals of a cell, and so long as there is no chemical action in which these new conductors are involved, the sum of all the discontinuities in the circuit will be constant, and equal to the electromotive force of the cell.

Let ABC... MN be any series of conductors, including a voltaic cell, and let the material of N be the same as that of A. UN and A are joined we obtain a closed circuit of electromotive force E, such that

VAB + V£C + ... + VJIN + VNA = E.

Moreover VNA = 0, since the material of N and A is the same. Thus the relation may be rewritten as

VAB + V£C+... + VM„=E (267).

In the open series of conductors ABC ... MN, there can be no current, so that each conductor must be at a definite uniform potential. If we denote the potentials by VA, VB, ... VM> VN, we have

*A~ 'B~ *AB>

*M ~ 'N = Kmy

Hence equation (267) becomes

VA-VN = E.

We now see that the electromotive force of a cell is the difference of potential between the ends of the cell when the cell forms an open circuit, and the materials of the two ends are the same.

A series of cells, joined in series so that the high-potential terminal of one is in electrical contact with the low-potential terminal of the next, and so on, is called a battery of cells, or an " electric battery" arranged in series.

It will be clear from what has just been proved, that the electromotive force of such a battery of cells is equal to the sum of the electromotive forces of the separate cells of the series.

344, 345] Units 305

Units.

345. On the electrostatic system, a unit current has been defined to be a current such that an electrostatic unit of electricity crosses any selected cross-section of a conductor in unit time. For practical purposes, a different unit, known as the ampere, is in use. The ampere is equal very approximately to 3 x 109 electrostatic units of current (see below, § 587).

To form some idea of the actual magnitude of this unit, it may be stated that the amount of current required to ring an electric bell is about half an ampere. About the same amount is required to light a 50 c.P. 100- volt metallic filament incandescent lamp.

As an electromotive force is of the same physical nature as a difference of potential, the electrostatic unit of electromotive force is taken to be the same as that of potential. The practical unit is about 3^ of the electrostatic unit, and is known as the volt (see below, § 587).

It may be mentioned that the electromotive force of a single voltaic cell is generally intermediate between one and two volts ; the electromotive force which produces a perceptible shock in the human body is about 30 volts, while an electromotive force of 500 volts or more is dangerous to life. Both of these latter quantities, however, vary enormously with the condition of the body, and particularly with the state of dryness or moisture of the skin. The electoomotive force used to work an electric bell is commonly 6 or 8 volts, while an electric light installation will generally have a voltage of about 100 or 200 volts.

The unit of resistance, in all systems of units, is taken to be a resistance such that unit difference of potential between its extremities produces unit current through the conductor. We then have, by Ohm's Law,

difference of potential at extremities ,„„ .

current = r . x (268).

resistance v '

In the practical system of units, the unit of resistance is called the ohm. From what has already been said, it follows that when two points having a potential-difference of one volt are connected by a resistance of one ohm, the current flowing through this resistance will be one ampere. In this case the difference of potential is ^^ electrostatic units, and the current is 3 x 109 electrostatic units, so that by relation (268), it follows that one ohm must be

equal to = — =-^m electrostatic units of resistance (see below, § 587).

Some idea of the amount of this unit may be gathered from the statement that the resistance of a mile of ordinary telegraph wire is about 10 ohms. The resistance of a good telegraph insulator may be billions of ohms.

20

306 Steady Currents in Linear Conductors [oh. ix

Physical Theoeies of Conduction.

Electron-theory of conduction.

345 a. As has been already explained (§ 28), the modern view of electricity regards a current of electricity as a material flow of electric charges. In all conductors except a small class known as electrolytic conductors (see below, § 345 b), these charged bodies are believed to be identical with the electrons.

In a solid some of the electrons are supposed to be permanently bound to particular atoms or molecules, whilst others, spoken of as "free" electrons, move about in the interstices of the solid, continually having their courses changed by collisions with the molecules. Both kinds of electrons will be influenced by the presence of an electric field. It is probable that the restricted motions of the " bound " electrons account for the phenomenon of inductive capacity (§151) whilst the unrestricted motion of the free electrons explains the phenomenon of electric conductivity.

Even when no electric forces are applied, the free electrons move about through a solid, but they move at random in all directions, so that as many electrons move from right to left as from left to right and the resultant current is nil. If an electric force is applied to the conductor, each electron has superposed on to its random motion a motion impressed on it by the electric force, and the electrons as a whole are driven through the conductor by the continued action of the electric force. If it were not for their collisions with the molecules of the conductor, the electrons would gain indefinitely in momentum under the action of the impressed electric force, but the effect of collisions is continually to check this growth of momentum.

Let us suppose that there are N electrons per unit length of the conductor, and that at any moment these have an average forward velocity u through the material of the conductor. If m is the mass of each electron, the total momentum of the moving electrons will be Nmu. The rate at which this total momentum is checked by collisions will be proportional to N and to u, and may be taken to be Nyu. The rate at which the momentum is increased by the electric forces acting is NXe, where X is the electric intensity and e is the charge, measured positively, of each electron. Thus we have the equation

■j- (Nmu) = NXe - Ny a (a).

In unit time the number of electrons which pass any fixed point in the conductor is Nu, so that the total flow of electricity per unit time past any point is Neu. This is by definition equal to the current in the conductor, so that if we call this i, we have

Neu = i (b).

345 a, 345 b] Electrolytic Conduction 307

This enables us to reduce equation (a) to the form

• N"x-^i)  (o). 
  

dt m V Ne2

The equation shews that if a steady electric force is applied, such that the intensity at any point is X, the current will not increase indefinitely but will remain stationary after it has reached a value i given by

i = X.

7

If V is the potential at any point of a conducting wire, and if s is a

dV coordinate measured along the wire, we have X = — — , so that

ds ~Ne*1' Integrating between any two points P and Q of the conductor, we have

This is the electron-theory interpretation of equation (264), and explains how the truth of Ohm's Law is involved in the modern conception of the nature of an electric current. It will be noticed that on this view of the matter, Ohm's Law is only true for steady currents.

We notice that the resistance of the conductor, on this theory, is <y/Ne2 per unit length. Thus, generally speaking, bodies in which there are many free electrons ought to be good conductors, and conversely.

The charge on the electron being 4*774 xlO-10 electrostatic units, we may notice that a current of one ampere (3 x 109 electrostatic units of current) is one in which 6%3 x 1018 electrons pass any given point of the conductor every second. Consider a conductor in which the number of electrons per cubic centimetre is 1021 (cf. § 615, below). Then in a wire of 1 square mm. cross-section there are 1019 electrons per unit length, so that the average velocity of these when the wire is conveying a current of 1 ampere is of the order of one cm. per sec. This average velocity is superposed on to a random velocity which is known to be of the order of magnitude of 107 cms. per sec, so that the additional velocity produced by even a strong current is only very slight in comparison with the normal velocity of agitation of the electrons.

Electrolytic conduction.

345 b. Besides the type of electric conduction just explained, there is a second, and entirely different type, known as Electrolytic conduction, the distinguishing characteristic of which is that the passage of a current is accompanied by chemical change in the conductor.

For instance, if a current is passed through a solution of potassium chloride in water, it will be found that some of the salt is divided up by the passage of the current into its chemical constituents, and that the potassium

20—2

308 Steady Currents in Linear Conductors [ch. ix

appears solely at the point at which the current leaves the liquid, while the chlorine similarly appears at the point at which the current enters. It thus appears that during the passage of an electric current, there is an actual transport of matter through the liquid, chlorine moving in one direction and potassium in the other. It is moreover found by experiment that the total amount, whether of potassium or chlorine, which is liberated by any current is exactly proportional to the amount of electricity which has flowed through the electrolyte.

These and other facts suggested to Faraday the explanation, now universally accepted, that the carriers of the current are identical with the matter which is transported through the electrolyte. For instance, in the foregoing illustration, each atom of potassium carries a positive charge to the point where the current leaves the liquid, while each atom of chlorine, moving in the direction opposite to that of the current, carries a negative charge. The process is perhaps explained more clearly by regarding the total current as made up of two parts, first a positive current and second a negative current flowing in the reverse direction. Then the atoms of chlorine are the carriers of the negative current, and the atoms of potassium are the carriers of the positive current.

Electrolytes may be solid, liquid, or gaseous, but in most cases of importance they are liquids, being solutions of salts or acids. The two parts into which the molecule of the electrolyte is divided are called the ions (Icov), that which carries the positive current being called the positive ion, and the other being called the negative ion. The point at which the current enters the electrolyte is called the anode, the point at which it leaves is called the cathode. The two ions are also called the anion or cation according as they give up their charges at the anode or cathode respectively. Thus we have

The anion carries — charge against current, and delivers it at the anode,

The cation carries + charge with current, and delivers it at the cathode.

When potassium chloride is the electrolyte, the potassium atom is the cation, and the chlorine atom is the anion. If experiments are performed with different chlorides (say of potassium, sodium, and lithium), it will be found that the amount of chlorine liberated by a given current is in every case the same, while the amounts of potassium, sodium, or lithium, being exactly those required to combine with this fixed amount of chlorine, are necessarily proportional to their atomic weights. This suggests that each atom of chlorine, no matter what the electrolyte may be in which it occurs, always carries the same negative charge, say — e, while each atom of potassium,

345 b, 345 c] Electrolytic Conduction 309

sodium, or lithium carries the same positive charge, say + E. Moreover E and e must be equal, or else each undissociated molecule of the electrolyte would have to be supposed to carry a charge E — e, whereas its charge is known to be nil.

It is found to be a general rule that every anion which is chemically monovalent carries the same charge —e, while every monovalent cation carries a charge -f e. Moreover divalent ions carry charges + 2e, trivalent ions carry charges + Se, and so on.

As regards the actual charges carried, it is found that one ampere of

current flowing for one second through a salt of silver liberates 0'001118

grammes of silver. Silver is monovalent and its atomic weight is 107-92

(referred to 0 = 16), so that the amount of any other monovalent element of

atomic weight m deposited by the same current will be 0-00001036 x m

grammes. It follows that the passage of one electrostatic unit of electricity

•n u • n ru *■ - 0-00001036 x m _ ,c will result in the liberation of — — , or 345 x 10~15 x m grammes

of the substance.

We can calculate from these data how many ions are deposited by one unit of current, and hence the amount of charge carried by each ion. It is found that, to within the limits of experimental error, the negative charge carried by each monovalent anion is exactly equal to the charge carried by the electron. It follows that each monovalent anion has associated with it one electron in excess of the number required to give it zero charge, while each monovalent cation has a deficiency of one electron ; divalent ions have an excess or deficiency of two electrons, and so on.

345 c. Ohm's Law appears, in general, to be strictly true for the resist- ance of electrolytes. In the light of the explanation of Ohm's Law given in § 345 a, this will be seen to suggest that the ions are free to move as soon as an electric intensity, no matter how small, begins to act on them. They must therefore be already in a state of dissociation ; no part of the electric intensity is required to effect the separation of the molecule into ions.

Other facts confirm this conclusion, such as for instance the fact that various physical properties — electric conductivity, colour, optical rotatory power, etc. — are additive in the sense that the amount possessed by the whole electrolyte is the sum of the amounts known to be possessed by the separate ions.

We may therefore suppose that as soon as an electric force begins to act, all the positive ions begin to move in the direction of the electric force, while all the negative ions begin to move in the opposite direction. Let us suppose the average velocities of the positive and negative ions to be u, v respectively, and let us suppose that there are N of each per unit length of the electrolyte measured along the path of the current. Then across any cross-section of the electrolyte there pass in unit time Nu positive ions each carrying a charge se

310

Steady Currents in Linear Conductors [ch. ix

in the direction in which the current is measured, and Nv negative ions each carrying a charge - se in the reverse direction, s being the valency of each ion. It follows that the total current is given by

i = Nse(u+v) (d).

Each unit of time Nu positive ions cross a cross-section close to the anode, having started from positions between this cross-section and the anode. Thus each unit of time Nu molecules are separated in the neigh- bourhood of the anode, and similarly Nv molecules are separated in the neighbourhood of the cathode. The concentration of the salt is accordingly weakened both at the anode and at the cathode, and the ratio of the amounts of these weakenings is that of u : v. This provides a method of determining the ratio of u : v.

Also equation (d) provides a method of determining u + v, for i can be readily measured, and Nse is the total charge which must be passed through the electrolyte to liberate the ions in unit length, and this can be easily determined.

Knowing u + v and the ratio u : v, it is possible to determine u and v. The following table gives results of the experiments of Kohlrausch on three chlorides of alkali metals, for different concentrations, the current in each case being such as to give a potential fall of 1 volt per centimetre.

Concentration

Potassium

chloride

Sodium chloride

Lithium

chloride

u

V

u V

u

V

0

660

690

450 690

360

690

•0001

654

681

448 681

356

681

•001

643

670

440 670

343

670

•01

619

644

415 644

318

644

•03

597

621

390 623

298

619

•1

564

589

360 592

259

594

[The unit in every case is a velocity of 10 ~6 cms. per second.]

We notice that when the solution is weak, the velocity of the chlorine ion is the same, no matter which electrolyte it has originated in. This gives, perhaps, the best evidence possible that the conductivity of the electrolyte is the sum of the conductivities of the chlorine and of the metal separately.

By arranging for the ions to produce discoloration of the electrolyte as they move through it, Lodge, Whetham and others have been able to observe the velocity of motion of the ions directly, and in all cases the observed velocities have agreed, within the limits of experimental error, with the theoretically determined values.

3-15C-346] Kirchhoff's Laws 311

Conduction through gases.

345 d. In a gas in its normal state, an electric current cannot be carried in either of the ways which are possible in a solid or a liquid, and it is consequently found that a gas under ordinary conditions conducts electricity only in a very feeble degree. If however Rontgen rays are passed through the gas, or ultra-violet light of very short wave-length, or a stream of the rays from radium or one of the radio-active metals, then it is found that the gas acquires considerable conducting powers, for a time at least. For this kind of conduction it is found that Ohm's Law is not obeyed, the relation between the current and the potential-gradient being an extremely complex one.

The complicated phenomena of conduction through gases can all be explained on the hypothesis that the gas is conducting only when " ionised," and the function of the Rontgen rays, ultra-violet light, etc. is supposed to be that of dividing up some of the molecules into their component ions. The subject of conduction through gases is too extensive to be treated here. In what follows it is assumed that the conductors under discussion are not gases, so that Ohm's Law will be assumed to be obeyed throughout.

Kirchhoff's Laws.

346. Problems occur in which the flow of electricity is not through a single continuous series of conductors : there may be junctions of three or more conductors at which the current of electricity is free to distribute itself between different paths, and it may be important to determine how the electricity will pass through a network of conductors containing junctions.

The first principle to be used is that, since the currents are supposed steady, there can be no accumulation of electricity at any point, so that the sum of all the currents which enter any junction must be equal to the sum of all the currents which leave it. Or, if we introduce the convention that currents flowing into a junction are to be counted as positive, while those leaving it are to be reckoned negative, then we may state the principle in the form :

The algebraic sum of the currents at any junction must be zero.

From this law it follows that any network of currents, no matter how complicated, can be regarded as made up of a number of closed currents, each of uniform strength throughout its length. In some conductors, two or more of these currents may of course be superposed.

Let the various junctions be denoted by A, B, C, ..., and let their potentials be VA, VB, V0, .... Let RAB be the resistance of any single con- ductor connecting two junctions A and B, and let GAB be the current flowing

312 Steady Currents in Linear Conductors [ch. ix

through it from A to B. Let us select any path through the network of conductors, such as to start from a junction and bring us back to the starting point, say ABC...NA. Then on applying Ohm's Law to the separate con- ductors of which this path is formed, we obtain (§ 341)

V — V — H 7?

rA rB — ^AB^ABi

*B ~ *C = ^BC-^BOt

By addition we obtain 2(7^=0 (269),

where the summation is taken over all the conductors which form the closed circuit.

In this investigation it has been assumed that there are no discontinuities of potential, and therefore no batteries, in the selected circuit. If dis- continuities occur, a slight modification will have to be made. We shall treat points at which discontinuities occur as junctions, and if A is a junction of this kind, the potentials at A on the two sides of the surface of separation between the two conductors will be denoted by VA and VA. Then, by Ohm's Law, we obtain for the falls of potential in the different conductors of the circuit,

VA' -VB= GABRAB,

VB — Va=z L>BCKBG, etc.,

and by addition of these equations

$(VA'-VA) = tCR.

The left-hand member is simply the sum of all the discontinuities of potential met in passing round the circuit, each being measured with its proper sign. It is therefore equal to the sum of the electromotive forces of all the batteries in the circuit, these also being measured with their proper signs.

Thus we may write %CR = t£ (270),

where the summation in each term is taken round any closed circuit of conductors, and this equation, together with

2C=0 (271),

in which the summation now refers to all the currents entering or leaving a single junction, suffices to determine the current in each conductor of the network.

Equation (271) expresses what is known as Kirchhoff's First Law, while equation (270) expresses the Second Law.

346-348] Kirchhoff's Laws 313

Conductors in Series.

347. When all the conductors form a single closed circuit, the current through each conductor is the same, say C, so that equation (270) becomes

CXR = IE.

The sum 2i2 is spoken of as the "resistance of the circuit," so that the current in the circuit is equal to the total electromotive force divided by the total resistance. Conductors arranged in such a way that the whole current passes through each of them in succession are said to be arranged "in series."

Conductors in Parallel.

348. It is possible to connect any two points A, B by a number of conductors in such a way that the current divides itself between all these

Fig. 96.

conductors on its journey from A to B, no part of it passing through more than one conductor. Conductors placed in this way are said to be arranged "in parallel."

Let us suppose that the two points A, B are connected by a number of conductors arranged in parallel. Let R1} R2, ... be the resistances of the conductors, and Cu (72, ... the currents flowing through them. Then if VA, VB are the potentials at A and B, we have, by Ohm's Law,

VA — VB=C1R1 = C2R2=....

The total current which enters at A is C1 + Ca + ..., say C. Thus we have

K-V -2l-£l= = G