A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 24 of 28

284.] In any system of conductors in which there are no internal electromotive forces the heat generated by currents distributed in accordance with Ohm's Law is less than if the currents had been distributed in any other manner consistent with the actual con- ditions of supply and outflow of the current.

The heat actually generated when Ohm's Law is fulfilled is mechanically equivalent to 2Pp Qq, that is, to the sum of the products of the quantities of electricity supplied at the different external electrodes, each multiplied by the potential at which it is supplied.

CHAPTEE VII.

CONDUCTION IN THREE DIMENSIONS.

Notation of Electric Currents.

285.] AT any point let an element of area dS be taken normal to the axis of #, and let Q units of electricity pass across this area from the negative to the positive side in unit of time, then, if

-^becomes ultimately equal to u when dS is indefinitely diminished, cl/o

u is said to be the Component of the electric current in the direction of x at the given point.

In the same way we may determine v and w, the components of the current in the directions of y and z respectively.

286.] To determine the component of the current in any other direction OR through the given point 0, let I, m, n be the direction- cosines of OR ; then if we cut off from the axes of x, y. z portions

equal to r r , r

T > — > and — I m n

respectively at A, B and <?, the triangle ABC will be normal to OR.

The area of this triangle ABC will be

o

Fig. 23. and by diminishing r this area may be diminished

without limit.

The quantity of electricity which leaves the tetrahedron ABCO by the triangle ABC must be equal to that which enters it through the three triangles OBC, OCA, and OAB.

The area of the triangle OBC is J — , and the component of

2'8;.] COMPONENT AND RESULTANT CURRENTS. 377

the current normal to its plane is u, so that the quantity which enters through this triangle is \ r2 —

inn

The quantities which enter through the triangles OCA and OAB respectively are v w

\r*—;, and \ r2 j- • •I1 Im

If y is the component of the velocity in the direction OR, then the quantity which leaves the tetrahedron through ABC is

Imn

Since this is equal to the quantity which enters through the three other triangles,

Imn

,,. , . , multiplying by — ^— , we get

w\ (l)

If we put u? + v2 + w2 = F2,

and make £', m', n' such that

u = /T, t? = wT, and w = n'Y ;

then y = T(ll'+mm' + nn\ (2)

Hence, if we define the resultant current as a vector whose magnitude is T, and whose direction-cosines are /', m\ n', and if y denotes the current resolved in a direction making an angle Q with that of the resultant current, then

y = T cos 0 ; (3)

shewing that the law of resolution of currents is the same as that of velocities, forces, and all other vectors.

287.] To determine the condition that a given surface may be a surface of flow, let

F(x, y,z) = \ (4)

be the equation of a family of surfaces any one of which is given by making A. constant ; then, if we make

d\

d\

dx -*- dy ' dt "3V5' the direction-cosines of the normal, reckoned in the direction in which A increases, are

l=N — > m = N-r-> n = N^r' (6)

dx dy dz

378

CONDUCTION IN THREE DIMENSIONS.

[288.

..,( dX d\ dX') y = N lu — + v~ + w-rl> I dx dy dz

(?)

Hence, if y is the component of the current normal to the surface, dX dX

~dy ' ~~ dz j

If y = 0 there will be no current through the surface, and the surface may be called a Surface of Flow, because the lines of motion are in the surface.

288.] The equation of a surface of flow is therefore dX dX

w ~^ |" v ~~^

dx dy

If this equation is true for all values of A, all the surfaces of the family will be surfaces of flow.

289.] Let there be another family of surfaces, whose parameter is A', then, if these are also surfaces of flow, we shall have

<M_ dX' dX' dx dy dz

2-o.

dz

(8)

= 0.

(9)

If there is a third family of surfaces of flow, whose parameter is A", then dK,, d,, d,,

U 1_ V— — -L.W—T- = 0. (10)

ax dy dz

If we eliminate between these three equations, u, v, and w dis- appear together, and we find

dX dX dX

dx ' dy dz

7 \ / 7' 7 \ t

aX aX aX dx

dy dz dX" dX" dX"

dx

dz

= 0;

(11)

or

(12)

A" =«, (A, A')!

that is, X" is some function of A and A'.

290.] Now consider the four surfaces whose parameters are A, A + dA, X', and A' + 8 A'. These four surfaces enclose a quadrilateral tube, which we may call the tube 8A.8A'. Since this tube is bounded by surfaces across which there is 110 flow, we may call it a Tube of Flow. If we take any two sections across the tube, the quantity which enters the tube at one section must be equal to the quantity which leaves it at the other, and since this quantity is therefore the same for every section of the tube, let us call it LbX.bX' where L is a function of A and A', the parameters which determine the particular tube.

293-] TUBES OF FLOW. 379

291.] If bS denotes the section of a tube of flow by a plane normal to x, we have by the theory of the change of the inde- pendent variables,

^ dy dz dz dy ' and by the definition of the components of the current

ubS = LSX.b'. (14)

Hence u = L(——-—d-

^dy dz dz dy

,dXdX' d\dX\

(15)

, f w •=. Jj ( )

\dx dn dti dm /

O* "I 1 T- /WA M-A M-A M'A \

Similarly v = L (— — - — )

\dz doo dx dz '

d_^

dy dy dx

292.] It is always possible when one of the functions A or A' is known, to determine the other so that L may be equal to unity. For instance, let us take the plane of yz, and draw upon it a series of equidistant lines parallel to y, to represent the sections of the family ' by this plane. In other words, let the function A' be determined by the condition that when sc = 0 Ax= z. If we then make L = I, and therefore (when x = 0)

: / U

then in the plane (x = 0) the amount of electricity which passes through any portion will be

jjudydz =jjd\dX'. (16)

Having determined the nature of the sections of the surfaces of flow by the plane of yz^ the form of the surfaces elsewhere is determined by the conditions (8) and (9). The two functions A and A" thus determined are sufficient to determine the current at every point by equations (15), unity being substituted for L.

On Lines of Flow.

293.] Let a series of values of A and of A' be chosen, the suc- cessive differences in each series being unity. The two series of surfaces defined by these values will divide space into a system of quadrilateral tubes through each of which there will be a unit current. By assuming the unit sufficiently small, the details of the current may be expressed by these tubes with any desired amount of minuteness. Then if any surface be drawn cutting the

380 CONDUCTION IN THREE DIMENSIONS. [294.

system of tubes, the quantity of the current which passes through this surface will be expressed by the number of tubes which cut it, since each tube carries unity of current.

The actual intersections of the surfaces may be called Lines of Flow. When the unit is taken sufficiently small, the number of lines of flow which cut a surface is approximately equal to the number of tubes of flow which cut it, so that we may consider the lines of flow as expressing not only the direction of the current but its strength, since each line of flow through a given section corresponds to a unit current.

On Current- Sheets and Current-Functions.

294.] A stratum of a conductor contained between two con- secutive surfaces of flow of one system, say that of A', is called a Current- Sheet. The tubes of flow within this sheet are deter- mined by the function A. If A^ and XP denote the values of A at the points A and P respectively, then the current from right to left across any line drawn on the sheet from A to P is Ap — A^. If AP be an element, ds, of a curve drawn on the sheet, the current which crosses this element from right to left is

d\ _ — d*. ds

This function A, from which the distribution of the current in the sheet can be completely determined, is called the Current- Function.

Any thin sheet of metal or conducting matter bounded on both sides by air or some other non-conducting medium may be treated as a current-sheet, in which the distribution of the current may be expressed by means of a current-function. See Art. 647.

Equation of ' Continuity?

295.] If we differentiate the three equations (15) with respect to

#, ^, z respectively, remembering that L is a function of A and A',

we find du dv dw _

^+^+^=

The corresponding equation in Hydrodynamics is called the Equation of 'Continuity/ The continuity which it expresses is the continuity of existence, that is, the fact that a material sub- stance cannot leave one part of space and arrive at another, without going through the space between. It cannot simply vanish in the

295-] EQUATION OF CONTINUITY. 381

one place and appear in the other, but it must travel along- a con- tinuous path, so that if a closed surface be drawn, including the one place and excluding the other, a material substance in passing from the one place to the other must go through the closed surface. The most general form of the equation in hydrodynamics is

where p signifies the ratio of the quantity of the substance to the volume it occupies, that volume being in this case the differential element of volume, and (pu), (pv), and (pw) signify the ratio of the quantity of the substance which crosses an element of area in unit of time to that area, these areas being normal to the axes of x, y, and z respectively. Thus understood, the equation is applicable to any material substance, solid or fluid, whether the motion be continuous or discontinuous, provided the existence of the parts of that sub- stance is continuous. If anything, though not a substance, is subject to the condition of continuous existence in time and space, the equation will express this condition. In other parts of Physical Science, as, for instance, in the theory of electric and magnetic quantities, equations of a similar form occur. We shall call such equations ' equations of continuity ' to indicate their form, though we may not attribute to these quantities the properties of matter, or even continuous existence in time and space.

The equation (17), which we have arrived at in the case of electric currents, is identical with (18) if we make p = 1, that is, if we suppose the substance homogeneous and incompressible. The equation, in the case of fluids, may also be established by either of the modes of proof given in treatises on Hydrodynamics. In one of these we trace the course and the deformation of a certain element of the fluid as it moves along. In the other, we fix our attention on an element of space, and take account of all that enters or leaves it. The former of these methods cannot be applied to electric currents, as we do not know the velocity with which the electricity passes through the body, or even whether it moves in the positive or the negative direction of the current. All that we know is the algebraical value of the quantity which crosses unit of area in unit of time, a quantity corresponding to (pu) in the equation (18). We have no means of ascertaining the value of either of the factors p or u, and therefore we cannot follow a par- ticular portion of electricity in its course through the body. The other method of investigation, in which we consider what passes

382 CONDUCTION IN THREE DIMENSIONS. [296.

through the walls of an element of volume, is applicable to electric currents, and is perhaps preferable in point of form to that which we have given, but as it may be found in any treatise on Hydro- dynamics we need not repeat it here.

Quantity of Electricity which passes through a given Surface.

296.] Let T be the resultant current at any point of the surface. Let dS be an element of the surface, and let € be the angle between T and the normal to the surface, then the total current through

the surface will be r r

I JTcoscdS,

the integration being extended over the surface.

As in Art. 21, we may transform this integral into the form

in the case of any closed surface, the limits of the triple integration being those included by the surface. This is the expression for the total efflux from the closed surface. Since in all cases of steady currents this must be zero whatever the limits of the integration, the quantity under the integral sign must vanish, and we obtain in this way the equation of continuity (17).

CHAPTER VIII.

RESISTANCE AND CONDUCTIVITY IN THREE DIMENSIONS.

On the most General Relations between Current and Electro- motive Force.

297.] LET the components of the current at any point be u, v, w.

Let the components of the electromotive force be X, Yt Z.

The electromotive force at any point is the resultant force on a unit of positive electricity placed at that point. It may arise (1) from electrostatic action, in which case if Fis the potential,

dv dv dr.

X = -fo' r=~di' Z = ~Tz' W

or (2) from electromagnetic induction, the laws of which we shall afterwards examine ; or (3) from thermoelectric or electrochemical action at the point itself, tending to produce a current in a given direction.

We shall in general suppose that X, Y, Z represent the com- ponents of the actual electromotive force at the point, whatever be the origin of the force, but we shall occasionally examine the result of supposing it entirely due to variation of potential.

By Ohm's Law the current is proportional to the electromotive force. Hence X, Y, Z must be linear functions of u, v, w. We may therefore assume as the equations of Resistance,

, A , V

.)

(2) Z=

We may call the coefficients R the coefficients of longitudinal resistance in the directions of the axes of coordinates.

The coefficients P and Q may be called the coefficients of trans- verse resistance. They indicate the electromotive force in one direction required to produce a current in a different direction.

384 RESISTANCE AND CONDUCTIVITY. [298.

If we were at liberty to assume that a solid body may be treated as a system of linear conductors, then, from the reciprocal property (Art. 281) of any two conductors of a linear system, we might shew that the electromotive force along z required to produce a unit current parallel to y must be equal to the electromotive force along y required to produce a unit current parallel to z. This would shew that Px = Q19 and similarly we should find P2 = Q2, and P3 = Q3. When these conditions are satisfied the system of co- efficients is said to be Symmetrical. When they are not satisfied it is called a Skew system.

We have great reason to believe that in every actual case the system is symmetrical, but we shall examine some of the con- sequences of admitting the possibility of a skew system.

298.] The quantities u, v, w may be expressed as linear functions of X, Y, Z by a system of equations, which we may call Equations of Conductivity,

v = qsX+r2Y+j?lZ, > (3)

we may call the coefficients r the coefficients of Longitudinal con- ductivity, and p and q those of Transverse conductivity.

The coefficients of resistance are inverse to those of conductivity. This relation may be defined as follows :

Let [PQR\ be the determinant of the coefficients of resistance, and [pqr] that of the coefficients of conductivity, then

[PQR] [pqr] = 1, (6)

[PQR] A = (P2 P3- ft 50 , [pqr] Pl = (AA-& r^ (7)

&c. &c.

The other equations may be formed by altering the symbols P, Q, R3P) q, r, and the suffixes 1, 2, 3 in cyclical order.

Rate of Generation of Heat.

299.] To find the work done by the current in unit of time in overcoming resistance, and so generating heat, we multiply the components of the current by the corresponding components of the electromotive force. We thus obtain the following expressions for Wt the quantity of work expended in unit of time :

300.] COEFFICIENTS OF CONDUCTIVITY. 385

(8) uv') (9)

By a proper choice of axes, either of the two latter equations may be deprived of the terms involving the products of u, v, w or of Jf, Yt Z. The system of axes, however, which reduces W to the form

is not in general the same as that which reduces it to the form

It is only when the coefficients Plt P2, P3 are equal respectively to Q19 Q2, Q3 that the two systems of axes coincide.

If with Thomson * we write

P=S+T, Q = S-1 and p = s +t, q = s — t

then we have

l 2,s,

and [PQR] rt = R2 R9-S* + T^

(13)

[p<^K=-tf,

If therefore we cause Slt $2 , S3 to disappear, s1 will not also dis- appear unless the coefficients T are zero.

Condition of Stability.

300.] Since the equilibrium of electricity is stable, the work spent in maintaining the current must always be positive. The conditions that W may be positive are that the three coefficients Rlt R2y R3, and the three expressions

must all be positive.

There are similar conditions for the coefficients of conductivity.

• Trans. R. S. Edin., 1853-4, p. 165.

VOL. I. C C

386 EESISTANCE AND CONDUCTIVITY. [3OI.

Equation of Continuity in a Homogeneous Medium.

301.] If we express the components of the electromotive force as the derivatives of the potential Vt the equation of continuity du dv dw

becomes in a homogeneous medium

d27

1 Jx2 2 dy2 3 dz2 1 dy dz

If the medium is not homogeneous there will be terms arising from the variation of the coefficients of conductivity in passing from one point to another.

This equation corresponds to Laplace's equation in an isotropic medium.

302.] If we put

LA* ft I — •>/>«'>« I O O O O 4 O 2 ___ /%• a 2 ^^ M a a f 1 7 I

/ o — / -i Tn I o -p u o-i Oo Oq"™~ » 1 "i — 'O 09 ' Q oo , I 1 / I

J i Z O ' L 6 6 J.I 4<i OO' /

and f AB~\ = A,A9A^+2 B, B« B» — A, B2 — A2 B2 — A3 B2, (18)

L J J.^01 X A w XJL A Ja *>O'\y

where [W]^ = r2r3 — s-f,

(19)

and so on, the system A, B will be inverse to the system r, s, and if we make

A: x2 +A2y* + A3z2+2Blyz+2B2zx+2 B3 xy = [AS] p2, (20) we shall find that

r=T~-

477 p

is a solution of the equation.

In the case in which the coefficients T are zero, the coefficients A and B become identical with R and S. When T exists this is not the case.

In the case therefore of electricity flowing out from a centre in an infinite, homogeneous, but not isotropic, medium, the equipotential surfaces are ellipsoids, for each of which p is constant. The axes of these ellipsoids are in the directions of the principal axes of con- ductivity, and these do not coincide with the principal axes of resistance unless the system is symmetrical.

By a transformation of this equation we may take for the axes of #, y, z the principal axes of conductivity. The coefficients of the forms 8 and B will then be reduced to zero, and each coefficient

3°3«] SKEW SYSTEM. 387

of the form A will be the reciprocal of the corresponding coefficient of the form r. The expression for p will be

303.] The theory of the complete system of equations of resist- ance and of conductivity is that of linear functions of three vari- ables, and it is exemplified in the theory of Strains *, and in other parts of physics. The most appropriate method of treating it is that by which Hamilton and Tait treat a linear and vector function of a vector. We shall not, however, expressly introduce Quaternion notation.

The coefficients T^ T2, Ts may be regarded as the rectangular components of a vector T, the absolute magnitude and direction of which are fixed in the body, and independent of the direction of the axes of reference. The same is true of tlt tz, t3, which are the components of another vector t.

The vectors T and t do not in general coincide in direction.

Let us now take the axis of z so as to coincide with the vector T, and transform the equations of resistance accordingly. They will then have the form

v, j | )

(23)

Z — S2 u + S1

It appears from these equations that we may consider the elec- tromotive force as the resultant of two forces, one of them depending only on the coefficients E and 8, and the other depending on T alone. The part depending on R and 8 is related to the current in the same way that the perpendicular on the tangent plane of an ellipsoid is related to the radius vector. The other part, depending on T, is equal to the product of T into the resolved part of the current perpendicular to the axis of T, and its direction is per- pendicular to T and to the current, being always in the direction in which the resolved part of the current would lie if turned 90° in the positive direction round T.

If we consider the current and T as vectors, the part of the electromotive force due to T is the vector part of the product, Tx current.

The coefficient T may be called the Rotatory coefficient. "We have reason to believe that it does not exist in any known sub-

• See Thomson and Tait's Natural PhHo'orfi/, § 154. C C 2

388

RESISTANCE AND CONDUCTIVITY.

[304.

stance. It should be found, if anywhere, in magnets, which have a polarization in one direction, probably due to a rotational phe- nomenon in the substance.

304.] Assuming then that there is no rotatory coefficient, we shall shew how Thomson's Theorem given in Art. 100 may be extended to prove that the heat generated by the currents in the system in a given time is a unique minimum.

To simplify the algebraical work let the axes of coordinates be chosen so as to reduce expression (9), and therefore also in this case expression (10), to three terms; and let us consider the general characteristic equation (16) which thus reduces to d2F d2F d2F_

rl J™2 f r2 ~7Tz "^ T3 ^2 — * ( /

Also, let

and let

5, c be three functions of #, y, z satisfying the condition da db dc , .

-r + T- + T- — ° » (25)

dx du dz v '

(26)

c =—

Finally, let the triple-integral

tf + R^)dxdydz (27)

be extended over spaces bounded as in the enunciation of Art. 100 ; such viz. that ^is constant over certain portions or else the normal component of the vector «, #, c is given, the latter condition being accompanied by the further restriction that the integral of this component over the whole bounding surface must be zero : then W will be a minimum when

u = 0, v = 0, w = 0. For we have in this case

and therefore, by (26),

W

dF

dF

=///(-.£

-!///("s + 'f+«S'"**'- <28'

305.] EXTENSION OF THOMSON'S THEOREM. 389

du dv dw to + Ty + Tz = °' the third term vanishes by virtue of the conditions at the limits.

The first term of (28) is therefore the unique minimum value of W.

305.] As this proposition is of great importance in the theory of electricity, it may be useful to present the following proof of the most general case in a form free from analytical operations.

Let us consider the propagation of electricity through a conductor of any form, homogeneous or heterogeneous.

Then we know that

(1) If we draw a line along the path and in the direction of the electric current, the line must pass from places of high potential to places of low potential.

(2) If the potential at every point of the system be altered in a given uniform ratio, the currents will be altered in the same ratio, according to Ohm's Law.

(3) If a certain distribution of potential gives rise to a certain distribution of currents, and a second distribution of potential gives rise to a second distribution of currents, then a third distribution in which the potential is the sum or difference of those in the first and second will give rise to a third distribution of currents, such that the total current passing through a given finite surface in the third case is the sum or difference of the currents passing through it in the first and second cases. For, by Ohm's Law, the additional current due to an alteration of potentials is independent of the original current due to the original distribution of potentials.

(4) If the potential is constant over the whole of a closed surface, and if there are no electrodes or intrinsic electromotive forces within it, then there will be no currents within the closed surface, and the potential at any point within it will be equal to that at the surface.

If there are currents within the closed surface they must either be closed curves, or they must begin and end either within the closed surface or at the surface itself.

But since the current must pass from places of high to places of low potential, it cannot flow in a closed curve.

Since there are no electrodes within the surface the current cannot begin or end within the closed surface, and since the potential at all points of the surface is the same, there can be no current along lines passing from one point of the surface to another.

390 RESISTANCE AND CONDUCTIVITY. [306.

Hence there are no currents within the surface, and therefore there can be no difference of potential, as such a difference would produce currents, and therefore the potential within the closed surface is everywhere the same as at the surface.

(5) If there is no electric current through any part of a closed surface, and no electrodes or intrinsic electromotive forces within the surface, there will be no currents within the surface, and the potential will be uniform.

We have seen that the currents cannot form closed curves, or begin or terminate within the surface, and since by the hypothesis they do not pass through the surface, there can be no currents, and therefore the potential is constant.

(6) If the potential is uniform over part of a closed surface, and if there is no current through the remainder of the surface, the potential within the surface will be uniform for the same reasons.

(7) If over part of the surface of a body the potential of every point is known, and if over the rest of the surface of the body the current passing through the surface at each point is known, then only one distribution of potentials at points within the body can exist.

For if there were two different values of the potential at any point within the body, let these be V\ in the first case and V2 in the second case, and let us imagine a third case in which the potential of every point of the body is the excess of potential in the first case over that in the second. Then on that part of the surface for which the potential is known the potential in the third case will be zero, and on that part of the surface through which the currents are known the currents in the third case will be zero, so that by (6) the potential everywhere within the surface will be zero, or there is no excess of V^ over T2i or the reverse. Hence there is only one possible distribution of potentials. This proposition is true whether the solid be bounded by one closed surface or by several.

On the Approximate Calculation of the Resistance of a Conductor of a given Form.

306.] The conductor here considered has its surface divided into three portions. Over one of these portions the potential is main- tained at a constant value. Over a second portion the potential has a constant value different from the first. The whole of th^remainder of the surface is impervious to electricity. We may suppose the

306.] RESISTANCE OF A WIRE OF VARIABLE SECTION. 391

conditions of the first and second portions to be fulfilled by applying- to the conductor two electrodes of perfectly conducting material, and that of the remainder of the surface by coating it with per- fectly non-conducting material.

Under these circumstances the current in every part of the conductor is simply proportional to the difference between the potentials of the electrodes. Calling this difference the electro- motive force, the total current from the one electrode to the other is the product of the electromotive force by the conductivity of the conductor as a whole, and the resistance of the conductor is the reciprocal of the conductivity.

It is only when a conductor is approximately in the circumstances above defined that it can be said to have a definite resistance, or conductivity as a whole. A resistance coil, consisting of a thin wire terminating in large masses of copper, approximately satisfies these conditions, for the potential in the massive electrodes is nearly constant, and any differences of potential in different points of the same electrode may be neglected in comparison with the difference of the potentials of the two electrodes.

A very useful method of calculating the resistance of such con- ductors has been given, so far as I know, for the first time, by Lord Rayleigh, in a paper on the Theory of Resonance *.

It is founded on the following considerations.

If the specific resistance of any portion of the conductor be changed, that of the remainder being unchanged, the resistance of the whole conductor will be increased if that of the portion is increased, and diminished if that of the portion be diminished.

This principle may be regarded as self-evident, but it may easily be shewn that the value of the expression for the resistance of a system of conductors between two points selected as electrodes, increases as the resistance of each member of the system in- creases.

It follows from this that if a surface of any form be described in the substance of the conductor, and if we further suppose this surface to be an infinitely thin sheet of a perfectly conducting substance, the resistance of the conductor as a whole will be diminished unless the surface is one of the equipotential surfaces in the natural state of the conductor, in which case no effect will be produced by making it a perfect conductor, as it is already in electrical equilibrium.

• Phil. Trans., 1871, p. 77. See Art. 102.

392 RESISTANCE AND CONDUCTIVITY. [306.

If therefore we draw within the conductor a series of surfaces, the first of which coincides with the first electrode, and the last with the second, while the intermediate surfaces are bounded by the non-conducting surface and do not intersect each other, and if we suppose each of these surfaces to be an infinitely thin sheet of perfectly conducting matter, we shall have obtained a system the resistance of which is certainly not greater than that of the original conductor, and is equal to it only when the surfaces we have chosen are the natural equipotential surfaces.

To calculate the resistance of the artificial system is an operation of much less difficulty than the original problem. For the resist- ance of the whole is the sum of the resistances of all the strata contained between the consecutive surfaces, and the resistance of each stratum can be found thus :

Let dS be an element of the surface of the stratum, v the thick- ness of the stratum perpendicular to the element, p the specific resistance, E the difference of potential of the perfectly conducting surfaces, and dC the current through dS, then

dC=E — dS. (1)

Pv

and the whole current through the stratum is

the integration being extended over the whole stratum bounded by the non-conducting surface of the conductor. Hence the conductivity of the stratum is

<">

and the resistance of the stratum is the reciprocal of this quantity. If the stratum be that bounded by the two surfaces for which the function F has the values F and F+ dF respectively, then

and the resistance of the stratum is

dF

-VFdS

p

To find the resistance of the whole artificial conductor, we have only to integrate with respect to F, and we find

307.] RESISTANCE OF A WIRE OF VARIABLE SECTION. 393

-VFdS P

The resistance E of the conductor in its natural state is greater than the value thus obtained, unless all the surfaces we have chosen are the natural equipotential surfaces. Also, since the true value of R is the absolute maximum of the values of Rl which can thus be obtained, a small deviation of the chosen surfaces from the true equipotential surfaces will produce an error of R which is com- paratively small.

This method of determining a lower limit of the value of the resistance is evidently perfectly general, and may be applied to conductors of any form, even when p, the specific resistance, varies in any manner within the conductor.

The most familiar example is the ordinary method of determining the resistance of a straight wire of variable section. In this case the surfaces chosen are planes perpendicular to the axis of the wire, the strata have parallel faces, and the resistance of a stratum of section S and thickness ds is

and that of the whole wire of length s is

. <•)

where S is the transverse section and is a function of s.

This method in the case of wires whose section varies slowly with the length gives a result very near the truth, but it is really only a lower limit, for the true resistance is always greater than this, except in the case where the section is perfectly uniform.

307.] To find the higher limit of the resistance, let us suppose a surface drawn in the conductor to be rendered impermeable to electricity. The effect of this must be to increase the resistance of the conductor unless the surface is one of the natural surfaces of flow. By means of two systems of surfaces we can form a set of tubes which will completely regulate the flow, and the effect, if there is any, of this system of impermeable surfaces must be to increase the resistance above its natural value.