The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 20 of 39

III. (Equation 295.) If the conductors 1, 2 and P, Q are not conjugate, a battery of electromotive force E placed in the arm PQ produces in 1, 2 a fall of potential given by

d2D V-V—K i?°KltpoK^q

K, y2 — JlpqU — ,

dK

and a current from 1 to 2 given by

li 12 li-pQ

All these results and formulae obtain illustration in the results already- obtained for the Wheatstone's Bridge in §§ 351 and 352.

Slowly-varying Currents.

365. All the analysis of the present chapter has proceeded upon the assumption that the currents are absolutely steady, shewing no variation with the time. Changes in the strength of electric currents are in general accompanied by a series of phenomena, which may be spoken of as " induction phenomena," of which the discussion is beyond the scope of the present chapter. If, however, the rate of change of the strength of the currents is very small, the importance of the induction phenomena also becomes very small, so that if the variation of the currents is slow, the analysis of the present chapter will give a close approximation to the truth. This method of dealing with slowly-varying currents will be illustrated by two examples.

I. Discharge of a Condenser through a high Resistance.

366. Let the two plates A, B of a condenser of capacity C be connected by a conductor of high resistance R, and let the condenser be discharged by leakage through this conductor. At any instant let the potentials of the two plates be VA, VB, so that the charges on these plates will be ± G(VA — VB). Let i be the current in the conductor, measured in the direction from A to B.

332 Steady Currents in Linear Conductors [ch. ix

Then, by Ohm's Law,

VA-VB = Ri,

whence we find that the charges on plates A and B are respectively + CRi and — CRi. Since i units leave plate A per unit time, we must have

a differential equation of which the solution is

t i = i0e CR ,

where i0 is the current at time t = 0. The condition that the strength of the current shall only vary slowly is now seen a posteriori to be that OR shall be large.

At time t the charge on the plate A is CRi or

t CRi0e~CR.

This may be written as

t

where Q0 is the charge at time t = 0. Thus both the charge and the current are seen to fall off exponentially with the time, both having the same modulus of decay CR.

Later (§ 516) we shall examine the same problem but without the limita- tion that the current only varies slowly.

II. Transmission of Signals along a Cable.

367. It has already been mentioned that a cable acts as an electrostatic condenser of considerable capacity. This fact retards the transmission of signals, and in a cable of high-capacity, the rate of transmission may be so slow that the analysis of the present chapter can be used without serious error.

Let x be a coordinate which measures distances along the cable, let V, i be the potential at x and the current in the direction of ^-increasing, and let K and R be the capacity and resistance of the cable per unit length, these latter quantities being supposed independent of x.

The section of the cable between points A and B at distances x and x + dx is a condenser of capacity Kdx, and is at the same time a conductor

366-368] Transmission of Signals 333

of resistance R dx. The potential of the condenser is V, so that its charge is VKdx. The fall of potential in the conductor is

so that by Ohm's Law,

dV — ■=- dx = iRdx (301).

The current enters the section i£ at a rate i units per unit time, and

dx

di leaves at a rate of i + ~- dx units per unit time. Hence the charge in this

di section decreases at a rate =- dx per unit time, so that we must have

I (VKdx) = -^dx (302).

Eliminating i from equations (301) and (302), we obtain

d2V dV

368. This equation, being a partial differential equation of the second order, must have two arbitrary functions in its complete solution. We shall shew, however, that there is a particular solution in which V is a function of the single variable x\Jt, and this solution will be found to give us all the information we require.

Let us introduce the new variable u, given by u = xf\ft, and let us assume provisionally that there is a solution V of equation (303) which is a function of u only. For this solution we must have

dV= 1 dV dx2 t du2* dV^dVdu_ Lx_dV dt~ du dt~ 2 V*3 du ' so that equation (303) becomes

du? \ 2 ^/ts du)

= -lKRu~ (304).

du

The fact that this equation involves V and u only, shews that there is an integral of the original equation for which V is a function of u only. This integral is easily obtained, for equation (304) can be put in the form

eO*S~w

du

whence ^=Ce^KRn\

du

in which C is a constant of integration.

334 Steady Currents in Linear Conductors [ch. ix

Integrating this, we find that the solution for V is

V = cfUe-±KRu*du,

in which the lower limit to the integral is a second constant of integration. Introducing a new variable y such that y2 = \KRu2, and changing the constants of integration, we may write the solution in the form

V=V0 + C e-y2dy (305).

J OO

369. We must remember that this is not the general solution of equa- tion (303), but is simply one particular solution. Thus the solution cannot be adjusted to satisfy any initial and boundary conditions we please, but will represent only the solution corresponding to one definite set of initial and boundary conditions. We now proceed to examine what these conditions are.

At time t = 0, the value of x*]t is infinite except at the point x = 0. Thus except at this point, we have V=X when t = 0. At this point the value of xjfjt is indeterminate at the actual instant t = 0, but immediately after this instant assumes the value zero, which it retains through all time. Thus at x = 0, the potential has the constant value

V^Vo + G'Te-y'dy,

J 00

or, say, V = Vu where 0' = 2^~^ .

At x =oo, the value of V is V=V0 through all time.

Thus equation (305) expresses the solution for a line of infinite length which is initially at potential V — T£, and of which the end x — oo remains at this potential all the time, while the end x = 0 is raised to potential Tf by being suddenly connected to a battery-terminal at the instant t = 0.

The current at any instant is given by

1 dV i = — -= -~- , from equation (301),

C"l /KR _^! , . ,onKN

= — d 9 V T e 4t » irorn equation (305),

/fF KRx*

= (^"T0V&rfe"" *" (306)-

We see that the current vanishes only when t — 0 and when t = oo . Thus even within an infinitesimal time of making contact, there will, according to equation (306), be a current at all points along the wire. It must, however, be remembered that equation (306) is only an approxima- tion, holding solely for slowly-varying currents, so that we must not apply

368, 369] Transmission of Signals 335

the solution at the instant t = 0 at which the currents, as given by equation (306), vary with infinite rapidity. For larger values of t, however, we may suppose the current given by equation (306).

The maximum current at any point is found, on differentiating equation (306), to occur at the instant given by

t = \KRx> (307),

so that the further along the wire we go, the longer it takes for the current to attain its maximum value. The maximum value of this current, when it occurs, is

<F->»/s-/"! <308>'

and so is proportional to - . Thus the further we go from the end x = 0, the

CC

smaller the maximum current will be.

We notice that K occurs in expression (307) but not in (308). Thus the electrostatic capacity of a cable will not interfere with the strength of signals sent along a cable, but will interfere with the rapidity of their transmission.

Equation (307) expresses what is commonly called the "KR law" — the retarding effect is proportional to the product of K and R. The theory just developed is commonly spoken of as the Electrostatic Theory of propagation of signals. It was first given by Lord Kelvin in 1855 in a paper* which is notable as having established the theoretical feasibility of an Atlantic cable.

We shall discuss in a later chapter the more general problem of the trans- mission of signals along a wire of any kind. It will then be possible to estimate the degree of error involved in the simple assumptions of the Electrostatic Theory.

EXAMPLES.

1. A length 4a of uniform wire is bent into the form of a square, and the opposite angular points are joined with straight pieces of the same wire, which are in contact at their intersection. A given current enters at the intersection of the diagonals and leaves at an angular point : find the current strength in the various parts of the network, and shew that its whole resistance is equal to that of a length

a\l\

2^2 + 1 of the wire.

2. A network is formed of uniform wire in the shape of a rectangle of sides 2a, 3a, with parallel wires arranged so as to divide the internal space into six squares of sides a, the contact at points of intersection being perfect. Shew that if a current enter the framework by one corner and leave it by the opposite, the resistance is equivalent to that of a length 121a/69 of the wire.

• "On the Theory of the Electrio Telegraph," Proc. Roy. Soc. 1855.

336 Steady Currents in Linear Conductors [ch. ix

3. A fault of given earth-resistance develops in a telegraph line. Prove that the current at the receiving end, generated by an assigned battery at the signalling end, is least when the fault is at the middle of the line.

4. The resistances of three wires BC, CA, AB, of the same uniform section and material, are a, b, c respectively. Another wire from A of constant resistance d can make a sliding contact with BC. If a current enter at A and leave at the point of contact with BC, shew that the maximum resistance of the network is

(a + b + c)d a+b+c+4d'

and determine the least resistance.

5. A certain kind of cell has a resistance of 10 ohms and an electromotive force of ■85 of a volt. Shew that the greatest current which can be produced in a wire whose resistance is 22*5 ohms, by a battery of five such cells arranged in a single series, of which any element is either one cell or a set of cells in parallel, is exactly "06 of an ampere.

6. Six points A, A', B, B', C, C are connected to one another by copper wire whose lengths in yards are as follows: A A' = 16, BC=B'C=l, BC' = B'C' = 2, AB = A'B' = G, AC'=A'C' = 8. Also B and B' are joined by wires, each a yard in length, to the terminals of a battery whose internal resistance is equal to that of r yards of the wire, and all the wires are of the same thickness. Shew that the current in the wire AA' is equal to that which the battery would maintain in a simple circuit consisting of 31r + 104 yards of the wire.

7. Two places A, B are connected by a telegraph line of which the end at A is connected to one terminal of a battery, and the end at B to one terminal of a receiver, the other terminals of the battery and receiver being connected to earth. At a point C of the line a fault is developed, of which the resistance is r. If the resistances of AC, CB be p, q respectively, shew that the current in the receiver is diminished in the ratio

r(p + q) : qr + rp+pq,

the resistances of the battery, receiver and earth circuit being neglected.

8. Two cells of electromotive forces ex, e2 and resistances rx, r2 are connected in parallel to the ends of a wire of resistance B. Shew that the current in the wire is

ei^ + yi rxR + r2R--r1r2'

and find the rates at which the cells are working.

9. A network of conductors is in the form of a tetrahedron PQRS ; there is a battery of electromotive force E in PQ, and the resistance of PQ, including the battery, is R. If the resistances in QR, RP are each equal to r, and the resistances in PS, RS are each equal to Jr, and that in QS=§r, find the current in each branch.

10. A, B, C, D are the four junction points of a Wheatstone's Bridge, and the resistances c, /3, b, y in AB, BD, AC, CD respectively are such that the battery sends no current through the galvanometer in BC. If now a new battery of electromotive force E be introduced into the galvanometer circuit, and so raise the total resistance in that circuit to a, find the current that will flow through the galvanometer.

11. A cable AB, 50 miles in length, is known to have one fault, and it is necessary to localise it. If the end A is attached to a battery, and has its potential maintained at 200 volts, while the other end B is insulated, it is found that the potential of B when

Examples 337

steady is 40 volts. Similarly when A is insulated the potential to which B must be raised to give A a steady potential of 40 volts is 300 volts. Shew that the distance of the fault from A is 19-05 miles.

12. A wire is interpolated in a circuit of given resistance and electromotive force. Find the resistance of the interpolated wire in order that the rate of generation of heat may he a maximum.

13. The resistances of the opposite sides of a TVheatstone's Bridge are a, a' and b, b' respectively. Shew that when the two diagonals which contain the battery and galvano- meter are interchanged,

E E _(a-a')(b-b')(G-R)

G C aa'-bb' '

where G and C are the currents through the galvanometer in the two cases, G and R are the resistances of the galvanometer and battery conductors, and E the electromotive force of the battery.

14. A current G is introduced into a network of linear conductors at A, and taken out at B, the heat generated being IT1. If the network be closed by joining A, B by a resistance r in which an electromotive force E is inserted, the heat generated is H2. Prove that

C2r E*

15. A number N of incandescent lamps, each of resistance r, are fed by a machine of resistance R (including the leads). If the light emitted by any lamp is proportional to the square of the heat produced, prove that the most economical way of arranging the lamps is to place them in parallel arc, each arc containing n lamps, where n is the integer nearest to V 'JVR/r.

16. A battery of electromotive force E and of resistance B is connected with the two terminals of two wires arranged in parallel. The first wire includes a voltameter which contains discontinuities of potential such that a unit current passing through it for a unit time does p units of work The resistance of the first wire, including the voltameter, is R: that of the second is r. Shew that if E is greater than p(B + r)jr, the current through the battery is

E(R + r)-pr Rr + B(R+r)'

17. A system of 30 conductors of equal resistance are connected in the same way as the edges of a dodecahedron. Shew that the resistance of the network between a pair of opposite corners is £ of the resistance of a single conductor.

18. In a network PA, PB, PG, PD, AB, BC, CD, DA, the resistances are a, ft y, 8,

y + 8, d + a, a + /3, /3+y respectively. Shew that, if AD contains a battery of electromotive

force E, the current in BC is

P(ap + y8).E

2/>2<2 + (/3S-ay)2' where P=a + p + y + d, § = /3y + ya + a/3 + aS + /3S + yS.

19. A wire forms a regular hexagon and the angular points are joined to the centre by wires each of which has a resistance - of the resistance of a side of the hexagon.

Shew that the resistance to a current entering at one angular point of the hexagon and

leaving it by the opposite point is

2(ra + 3)

(n + 1)0 + 4)

times the resistance of a side of the hexagon.

J. 22

338 Steady Currents in Linear Conductors [ch. ix

20. Two long equal parallel wires AB, A'B', of length I, have their ends B, B' joined by a wire of negligible resistance, while A, A' are joined to the poles of a cell whose resistance is equal to that of a length r of the wire. A similar cell is placed as a bridge across the wires at a distance x from A, A'. Shew that the effect of the second cell is to increase the current in BB' in the ratio

2 (21 +r) (x+r)l{r(4l+r) + 2x(2l-r)-4:X2}.

21. There are n points 1, 2, ... n, joined in pairs by linear conductors. On introducing a current 0 at electrode 1 and taking it out at 2, the potentials of these are V, V2, ... Pn. If x12 is the actual current in the direction 12, and xx2' any other that merely satisfies the conditions of introduction at 1 and abstraction at 2, shew that

2 (r12a?12a?i2')=( Pi- P2) C=2 (r12x12-),

and interpret the result physically.

If x typify the actual current when the current enters at 1 and leaves at 2, and y typify the actual current when the current enters at 3 and leaves at 4, shew that

2 (r12x12y12) = (Zs - X4) C= ( l\ - Y2) C,

where the X's are potentials corresponding to currents x, and the Y's are potentials corresponding to currents y.

22. A, B, C are three stations on the same telegraph wire. An operator at A knows that there is a fault between A and B, and observes that the current at A when he uses a given battery is i, i' or i", according as B is insulated and C to earth, B to earth, or B and C both insulated. Shew that the distance of the fault from A is

{ka - k'b + {b - aft (ka - k'bft}j(k - k'),

i" i"

where AB = a, BC=b-a, k—-. — -n k' = -l

i -i

23. Six conductors join four points A, B, C, D in pairs, and have resistances a, a, b, /3, c, y, where a, a refer to BC, AD respectively, and so on. If this network be used as a resistance coil, with A, B as electrodes, shew that the resistance caunot lie outside the limits

[^^r-[MGnr+(H)T]

-i

24. Two equal straight pieces of wire A0An, B0Bn are each divided into n equal parts at the points At ... An_i and Bi...Bn_i respectively, the resistance of each part and that of AnBn being R. The corresponding points of each wire from 1 to n inclusive are joined by cross wires, and a battery is placed in A0B0. Shew that, if the current through each cross wire is the same, the resistance of the cross wire AaBa is

{(n-sY + (n-s) + }R

25. If n points are joined two and two by wires of equal resistance r, and two of them are connected to the electrodes of a battery of electromotive force E and resistance R, shew that the current in the wire joining the two points is

2E 2r + nR'

26. Six points A, B, C, D, P, Q are joined by nine conductors AB, AP, BC, BQ, PQ, QC, PD, DC, AD. An electromotive force is inserted in the conductor AD, and a galvanometer in PQ. Denoting the resistance of any conductor XY by rXY, shew that if no current passes through the galvanometer,

(i'bo + rBQ + rCQ) (rAB rDP - rAP rD0) + rBC (rBQ rDP - rAP rCQ) = 0.

Examples 339

27. A network is made by joining the five points 1, 2, 3, 4, 5 by conductors in every possible way. Shew that the condition that conductors 23 and 14 are conjugate is

(A"15 + ^ + E3b + E&) (E^E^ - E13 E2i)

= Eb2 (EbiEls - EuE,b) + Eb3 (E2iEbl - E^E^),

where Er) is conductivity of conductor rs.

28. Two endless wires are each divided into mn equal parts by the successive terminals of mn connecting wires, the resistance of each part being R. There is an identically similar battery in every mth connecting wire, the total resistance of each being the same, and the resistance of each of the other mn — n connecting wires is h. Prove that the current through a connecting wire which is the rth from the nearest battery is

£C(l-tana)(tanra + tanm-ra)/(tana-tanma),

where C is the current through each battery, and sin 2a = hj(k+R).

29. A long line of telegraph wire AAXA2 ... AnAn + 1 is supported by n equidistant insulators at Alt A2, ... An. The end A is connected to one pole of a battery of electro- motive force E and resistance B, and the other pole of this battery is put to earth, as also the other end An + X of the wire. The resistance of each portion AAX, AtA2, ... AnAn+i is the same, R. In wet weather there is a leakage to earth at each insulator, whose resistance may be taken equal to r. Shew that the current strength in APAP + x is

Ecos\x(2n-2p + l)a B cosh (2/i + 1) a + \l~Rr sinh (2?i + 2) a '

where 2 sinh a = ] R\r.

30. A regular polygon A1A2...An is formed of n pieces of uniform wire, each of resistance <r, and the centre 0 is joined to each angular point by a straight piece of the same wire. Shew that, if the point 0 is maintained at zero potential, and the point Ax at potential V, the current that flows in the conductor ArAr+1 is

2 Fsinh a sinh (n - 2r + 1) a a cosh na '

where a is given by the equation

7r

cosh 2a=l+ sin — n

31. A resistance network is constructed of 2n rectangular meshes forming a truncated cylinder of 2n faces, with two ends each in the form of a regular polygon of 2n sides. Each of these sides is of resistance r, and the other edges of resistance R. If the electrodes be two opposite corners, then the resistance is

, , ,. tanh 6

where sinh2 6 = ^.

la

32. A network is formed by a system of conductors joining every pair of a set of n points, the resistances of the conductors being all equal, and there is an electromotive force in the conductor joining the points A1} A2. Shew that there is no current in any conductor except those which pass through Ax or A2, and find the current in these conductors.

22—2

340 Steady Currents in Linear Conductors [ch. ix

33. Each member of the series of n points Ax, A<i,...An is united to its successor by a wire of resistance p, and similarly for the series of n points Bx, B2,...Bn. Each pair of points corresponding in the two series, such as Ar and Br, is united by a wire of resistance R. A steady current i enters the network at Ai and leaves it at Bn. Shew that the current at A\ divides itself between AxAi and A-J5X in the ratio

sinh a + sinh (n — 1) a + sinh {n - 2) a : sinh a + sinh {n — 1) a - sinh (n — 2) a, where cosh a = „ ■ .

34. An underground cable of length a is badly insulated so that it has faults throughout its length indefinitely near to one another and uniformly distributed. The conductivity of the faults is 1/p' per unit length of cable, and the resistance of the cable is p per unit length. One pole of a battery is connected to one end of a cable and the other pole is earthed. Prove that the current at the farther end is the same as if the cable were free from faults and of total resistance

/pp' smhTay'sj

35. Two parallel conducting wires at unit distance are connected by 7i4 1 cross pieces of the same wire, so as to form n squares. A current enters by an outer corner of the first square, and leaves by the diagonally opposite corner of the last. Shew that, if the resistance is that of a length £?i + o„ of the wire,

<*n + l:

a„ + 2

36. A, B are the ends of a long telegraph wire with a number of faults, and C is an intermediate point on the wire. The resistance to a current sent from A is R when C is earth connected, but if C is not earth connected the resistance is S or T according as the end B is to earth or insulated. If R\ S', T' denote the resistances under similar circumstances when a current is sent from B towards A, shew that

T'(R-S) = R'(R-T).

37. The inner plates of two condensers of capacities C, C are joined by wires of resistances R, R' to a point P, and their outer plates by wires of negligible resistance to a point Q. If the inner plates be also connected through a galvanometer, shew that the needle will suffer no sudden deflection on joining P, Q to the poles of a battery, if CR=C'R'.

38. An infinite cable of capacity and resistance K and R per unit length is at zero potential. At the instant t—0 one end is suddenly connected to a battery for an infinitesimal interval and then insulated. Shew that, except for very small values of t, the potential at any instant at a distance x from this end of the cable will be pro- portional to

1 _^R^

7te~ « •

CHAPTEE X

STEADY CURRENTS IN CONTINUOUS MEDIA

Components of Current.

370. In the present chapter we shall consider steady currents of elec- tricity flowing through continuous two- and three-dimensional conductors instead of through systems of linear conductors.

We can find the direction of flow at any point P in a conductor by imagining that we take a small plane of area dS and turn it about at the point P until we find the position in which the amount of electricity crossing it per unit time is a maximum. The normal to the plane when in this position will give the direction of the current at P, and if the total amount of electricity crossing this plane per unit time when in this position is CdS, then C may be defined to be the strength of the current at P.

If I, m, n are the direction-cosines of the direction of the current at P, then the current C may be treated as the superposition of three currents IC, mC, nC parallel to the axes. To prove this we need only notice that the flow across an area dS of which the normal makes an angle 6 with the direc- tion of the current, and has direction-cosines I', m', n', must be CdS cos 0, or

CdS {IV + mm! + nn').

The first term of this expression may be regarded as the contribution from a current IC parallel to the axis Ox, and so on. The quantities IC, mC, nC are called the components of the current at the point P.

Lines and Tubes of Flow.

371. Definition. A line of flow is a line drawn in a conductor such that at every point its tangent is in the direction of the current at the point.

Definition. A tube of flow is a tubular region of infinitesimal cross- section, bounded by lines of flow.

342

Steady Currents in continuous Media [oh. x

It is clear that at every point on the surface of a tube of flow, the current is tangential to the surface. Thus no current crosses the boundary of a tube of flow, from which it follows that the aggregate current flowing across all cross-sections of a tube of flow will be the same.

The amount of this current will be called the strength of the tube.

Thus if G is the current at any point of a tube of flow, and if &> is the cross-section of the tube at that point, then Ceo is constant throughout the length of the tube, and is equal to the strength of the tube.

There is an obvious analogy between tubes of flow in current electricity and tubes of force in statical electricity, the current C corresponding to the polarisation P. In current electricity, Ca> is constant and equal to the strength of the tube of flow, while in statical electricity Pa is constant and equal to the strength of the tube of force (§ 129).

Specific Resistance.

372. The specific resistance of a substance is defined to be the resistance of a cube of unit edge of the substance, the current entering by a perfectly conducting electrode which extends over the whole of one face, and leaving by a similar electrode on the opposite face.

The specific resistances of some substances of which conductors and insulators are frequently made are given in the following table. The units are the centimetre and the ohm.

Dilute sulphuric acid (^ acid at 22° C.) 33. „ „ „ (| acid at 22° C.) 1-6.

Glass (at 200° C.) 2-27xl07.

„ (at 400° C.) 7-35x10*.

Guttapercha, about 3xl014.

If t is the specific resistance of any substance, the resistance of a wire

It of length I and cross-section s will clearly be — .

s

Silver ...

1-61 x 10-s.

Copper

1-64 x 10-6.

Iron (soft)

9-83 x lO-6.

„ (hard)

9-06 x 10~6.

Mercury

... 96-15 xlO"6.

Ohm's Law.

373. In a conductor in which a current is flowing, different points will, in general, be at different potentials. Thus there will be a system of equipotentials and of lines of force inside a conductor similar to those in an electrostatic field. It is found, as an experimental fact, that in a homogeneous conductor, the lines of flow coincide with the lines of force — or, in other words, the electricity at every point moves in the direction of the forces acting on it.

In considering the motion of material particles in general it is not usually true that the motion of the particles is in the direction of the forces acting upon them. The velocity

371-374] Ohm's Law 343

of a particle at the end of any small interval of time is compounded of the velocity at the beginning of the interval together with the velocity generated during the interval. The latter velocity is in the direction of the forces acting on the particle, but is generally insignificant in comparison with the original velocity of the particle. In the particular case in which the original velocity of the particle was very small, the direction of motion at the end of a small interval will be that of the force acting on the particle. If the particle moves in a resisting medium, it may be that the velocity of the particle is kept permanently very small by the resistance of the medium : in this case the direction of motion of the particle at every instant, relatively to the medium, may be that of the forces acting on it.

On the modern view of electricity, a current of electricity is composed of electrons which are driven through a conductor by the electric forces acting on them, and in their motion experience frequent collisions with the molecules of the conductor. The effect of these collisions is continually to check the forward velocity of the electrons, so that this forward velocity is kept small just as if they were moving through a resisting medium of the ordinary kind, and so it comes about that the direction of flow of current is in the direction of the electric intensity (cf. § 345 a).

374. Let us select any tube of force of small cross-section inside a conductor, and let P, Q be any two points on this tube of force, at which the potentials are VP and VQ, the former being the greater. Let these points be so near together that throughout the range PQ the cross-section of the tube of force may be supposed to have a constant value co, while the specific resistance of the material of the conductor may be supposed to have a constant value t.

From what has been said in § 373, it follows that the tube of force under consideration is also a tube of flow. If G denotes the current, then the current flowing through this tube of flow in the direction from P to Q will be Ceo. This current may, within the range PQ, be regarded as flowing through a conductor of cross-section co and of specific resistance t. The

PQ.T

resistance of this conductor from P to Q is accordingly — , while the fall

of potential is VF- Vq. Thus by Ohm's Law

CO

so that PpQ Q — Gr.

If ;r- denotes differentiation along the tube of force, the fraction on the ds

left of the foregoing equation reduces, when P and Q are made to coincide,

dV to — — , so that the equation assumes the form

-d-^=Gr (309).

OS

344 Steady Currents in continuous Media [ch. x

Let I, m, 7i be the direction-cosines of the line of flow at P, and let u, v, w be the components of the current at P, so that u = IG, etc. Then

—- = I ^— = — LUt = — ut, etc.,

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and we see that equation (309) is equivalent to the three equations

u =

ldV\

t dx

v =

ldV

t dy

w =

ldV

t dz ,

.(310).

These equations express Ohm's Law in a form appropriate to flow through a solid conductor.

Equation of Continuity,

375. Since we are supposing the currents to be steady, the amount of current which flows into any closed region must be exactly equal to the amount which flows out. This can be expressed by saying that the integral algebraic flow into any closed region must be nil.

Let any closed surface S be taken entirely inside a conductor. Let I, m, n be the direction-cosines of the inward normal to any element dS of this surface, and let u, v, w be the components of current at this point. Then the normal component of flow across the element dS is lu + mv + nw, and the condition that the integral algebraic flow across the surface S shall be nil is expressed by the equation

//

(lu + mv + mv) dS = 0.

By Green's Theorem (§ 176), this equation may be transformed into

du dv dw\ _

dx dy dz J '

and since this integral has to vanish, whatever the region through which it is taken, each integrand must vanish separately. Hence at every point inside the conductor, we must have

du dv diu _ .

dx dy dz ''

This is the so-called " equation of continuity," expressing that no elec- tricity is created or destroyed or allowed to accumulate during the passage of a steady current through a conductor.

374-377] Equation of Continuity 345

The same equation can be obtained at once on considering the current- flow across the different faces of a small rectangular parallelepiped of edges dx, dy, dz (cf. § 49).

Equation (310) of course expresses that the vector C of which the components are u, v, w, must be solenoidal. The equation of continuity- can accordingly be expressed in the form

div C = 0.

Equation satisfied by the Potential.

376. On substituting in equation (311) the values for u, v, w given by equations (310), we obtain

dx\r dx) dy\rdy) dz\r dz )

The potential must accordingly be a solution of this differential equation. The equation is the same as would be satisfied by the potential in an uncharged dielectric in an electrostatic field, provided the inductive capacity

at every point is proportional to -. If the specific resistance of the con- ductor is the same throughout, the differential equation to be satisfied by the potential reduces to

377. We may for convenience suppose that the current enters and leaves

by perfectly conducting electrodes, and that the conductor through which the

current flows is bounded, except at the electrodes, by perfect insulators. Then,

over the surface of contact between the conductor and the electrodes, the

potential will be constant. Over the remaining boundaries of the conductor,