A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 25 of 28

The resistance of each of the tubes may be calculated by the method already given for a fine wire, and the resistance of the whole conductor is the reciprocal of the sum of the reciprocals of the resistances of all the tubes. The resistance thus found is greater

394 RESISTANCE AND CONDUCTIVITY. [307.

than the natural resistance, except when the tubes follow the natural lines of flow.

In the case already considered, where the conductor is in the form of an elongated solid of revolution, let us measure x along the axis, and let the radius of the section at any point be b. Let one set of impermeable surfaces be the planes through the axis for each of which <p is constant, and let the other set be surfaces of revolution for which ^2 _ ^^ (9)

where ^ is a numerical quantity between 0 and 1 .

Let us consider a portion of one of the tubes bounded by the surfaces $ and <f) + d(f), \fr and \j/--cl\ls, x and x + dx.

The section of the tube taken perpendicular to the axis is

ydyd$ — \Wd^d$. (10)

If 9 be the angle which the tube makes with the axis

The true length of the element of the tube is dx sec 0, and its true section is

so that its resistance is

(1,)

e tube is dx sec 0, an cos *,

Let A = /£ dx, and £ = f £ (g)' Ax, (13)

the integration being extended over the whole length, #, of the conductor, then the resistance of the tube d^r d(p is

s

and its conductivity is

To find the conductivity of the whole conductor, which is the sum of the conductivities of the separate tubes, we must integrate this expression between <£ = 0 and 0 = 2 TT, and between ^ = 0 and {j = I . The result is

which may be less, but cannot be greater, than the true con- ductivity of the conductor.

HIGHER AND LOWER LIMITS. 395

When -=- is always a small quantity — will also be small, and we cix /±

may expand the expression for the conductivity, thus

The first term of this expression, — r-, is that which we should

A.

have found by the former method as the superior limit of the con- ductivity. Hence the true conductivity is less than the first term but greater than the whole series. The superior value of the resistance is the reciprocal of this, or

If, besides supposing the flow to be guided by the surfaces <$> and \l/, we had assumed that the flow through each tube is proportional to d\l/ d$, we should have obtained as the value of the resistance under this additional constraint

(17)

which is evidently greater than the former value, as it ought to be, on account of the additional constraint. In Lord Rayleigh's paper this is the supposition made, and the superior limit of the resistance there given has the value (17), which is a little greater than that which we have obtained in (16).

308.] We shall now apply the same method to find the correction which must be applied to the length of a cylindrical conductor of radius a when its extremity is placed in metallic contact with a massive electrode, which we may suppose of a different metal.

For the lower limit of the resistance we shall suppose that an infinitely thin disk of perfectly conducting matter is placed between the end of the cylinder and the massive electrode, so as to bring the end of the cylinder to one and the same potential throughout. The potential within the cylinder will then be a function of its length only, and if we suppose the surface of the electrode where the cylinder meets it to be approximately plane, and all its dimen- sions to be large compared with the diameter of the cylinder, the distribution of potential will be that due to a conductor in the form of a disk placed in an infinite medium. See Arts. 151, 177.

If E is the difference of the potential of the disk from that of the distant parts of the electrode, C the current issuing from the

396 RESISTANCE AND CONDUCTIVITY. [309.

surface of the disk into the electrode, and p' the specific resistance of the electrode ; then if Q is the amount of electricity on the disk, which we assume distributed as in Art. 151, we have

p'C= 1.47r<2 = 2v~, by Art. 151,

7T

2

= 4«J0. (18)

Hence, if the length of the wire from a given point to the electrode is Z-, and its specific resistance p, the resistance from that point to any point of the electrode not near the junction is

and this may be written

~ ira2 v p 4 ' '

where the second term within brackets is a quantity which must be added to the length of the cylinder or wire in calculating its resistance, and this is certainly too small a correction.

To understand the nature of the outstanding error we may observe, that whereas we have supposed the flow in the wire up to the disk to be uniform throughout the section, the flow from the disk to the electrode is not uniform, but is at any point in- versely proportional to the minimum chord through that point. In the actual case the flow through the disk will not be uniform, but it will not vary so much from point to point as in this supposed case. The potential of the disk in the actual case will not be uniform, but will diminish from the middle to the edge.

309.] We shall next determine a quantity greater than the true resistance by constraining the flow through the disk to be uniform at every point. We may suppose electromotive forces introduced for this purpose acting perpendicular to the surface of the disk.

The resistance within the wire will be the same as before, but in the electrode the rate of generation of heat will be the surface- integral of the product of the flow into the potential. The rate of

flow at any point is — :r, and the potential is the same as that of ira2

an electrified surface whose surface-density is or, where

C p f ^

27ro-=-^, (20)

TI a*

p' being the specific resistance.

309.] CORRECTION FOR THE ENDS OP THE WIRE. 397

We have therefore to determine the potential energy of the electrification of the disk with the uniform surface-density a-.

• The potential at the edge of a disk of uniform density o- is easily found to be 4<zo-. The work done in adding a strip of breadth da at the circumference of the disk is Zitavda . 4#cr, and the whole potential energy of the disk is the integral of this,

or P=^a?o*. (21)

In the case of electrical conduction the rate at which work is done in the electrode whose resistance is R' is C2R'. But from the general equation of conduction the current across the disk per unit area is of the form 1 fry

~~7 fa

47T

or -7-0-.

P

Hence the rate at which work is done is

7''

We have therefore

<?2.72'=i?P, (22) whence, by (20) and (21), and the correction to be added to the length of the cylinder is f' 8 a 73^' this correction being greater than the true value. The true cor- rection to be added to the length is therefore — an, where n is a TT 8 number lying between - and — , or between 0.785 and 0.849- 4 3 7T fLord Rayleigh, by a second approximation, has reduced the superior limit of n to 0.8282. * See a Paper by Professor Cayley, London Math. Soc. Proc. vi. p. 47. f Phil. Mag., Nov. 1872. Lord Rayleigh subsequ superior limit. See London Math. Soc. Proc. viii. p. 74. CHAPTEE IX. CONDUCTION THROUGH HETEROGENEOUS MEDIA. On the Conditions to be Fulfilled at the Surface of Separation between Two Conducting Media. 310.] THERE are two conditions which the distribution of currents must fulfil in general, the condition that the potential must be continuous, and the condition of ' continuity' of the electric currents. At the surface of separation between two media the first of these conditions requires that the potentials at two points on opposite sides of the surface, but infinitely near each other, shall be equal. The potentials are here understood to be measured by an elec- trometer put in connexion with the given point by means of an electrode of a given metal. If the potentials are measured by the method described in Arts. 222, 246, where the electrode terminates in a cavity of the conductor filled with air, then the potentials at contiguous points of different metals measured in this way will differ by a quantity depending on the temperature and on the nature of the two metals. The other condition at the surface is that the current through any element of the surface is the same when measured in either medium. Thus, if T[ and 7J are the potentials in the two media, then at any point in the surface of separation ^ = rn (i) and if %, vlf w^ and ^2, v2, w2 are the components of currents in the two media, and I, m, n the direction-cosines of the normal to the surface of separation, u^+v^m+w^ = U2l + v2m + w2n. (2) In the most general case the components M, v, w are linear 3 1 0.] SURFACE-CONDITIONS. 399 functions of the derivatives of F, the forms of which are given in the equations v ^ (3) where X, Y, Z are the derivatives of V with respect to x9 y> z respectively. Let us take the case of the surface which separates a medium having these coefficients of conduction from an isotropic medium having a coefficient of conduction equal to r. Let X', Y', Z' be the values of X, Y} Z in the isotropic medium, then we have at the surface r=r, (4) or Xdx + Tdy + Zdz = X'dx + Tdy + Z'dz, (5) when Idx + mdy + ndz = 0. (6) This condition gives J'=JT+47r<T^ T— Y+4iT(rm, Z'=Z+4ir<Tnf (?) where <r is the surface-density. We have also in the isotropic medium u' = r X', v' = rY', vf = rZ', (8) and at the boundary the condition of flow is ul + v'm + w'n = ul-\-vm + wn, (9) or r(lX+m Y+nZ+kita) = /(r1X+j*37+ q2Z) + m(q3X+ rJ+PlZ] + n(p2X+ g, Y+ r^Z\ (10) whence 47TO-/ = (J(r1-r) + «08 + »Jp2)J + (^3 + mirz-rf + nqJY + (lq2 + mj)l-^n(r3^r))Z. (11) The quantity o- represents the surface-density of the charge on the surface of separation. In crystallized and organized sub- stances it depends on the direction of the surface as well as on the force perpendicular to it. In isotropic substances the coeffi- cients p and q are zero, and the coefficients / are all equal, so that 477(7 = (i - i) (IX+mY+nZ), (12) where rx is the conductivity of the substance, r that of the external medium, and /, m, n the direction-cosines of the normal drawn towards the medium whose conductivity is r. When both media are isotropic the conditions may be greatly 400 CONDUCTION IN HETEROGENEOUS MEDIA. [SH- simplified, for if Jc is the specific resistance per unit of volume, then 1 d7 I dV 1 dV u = —-j-r-> # = — 7T-> w = — j-j-> (13) k dx k ay k dz and if v is the normal drawn at any point of the surface of separa- tion from the first medium towards the second, the condition of continuity is 1 d7^ J_ d72 k± dv ~~ $2 dv If 01 and 02, are the angles which the lines of flow in the first and second media respectively make with the normal to the surface of separation, then the tangents to these lines of flow are in the same plane with the normal and on opposite sides of it, and /&! tan 01 = £2 tan 02 . (15) This may be called the law of refraction of lines of flow. 311.] As an example of the conditions which must be fulfilled when electricity crosses the surface of separation of two media, let us suppose the surface spherical and of radius a, the specific resistance being ^ within and Jc2 without the surface. Let the potential, both within and without the surface, be ex- panded in solid harmonics, and let the part which depends on the surface harmonic St be (1) (2) within and without the sphere respectively. At the surface of separation where r = a we must have Fi=F2 and i£ iS. (3) &! dr k% dr From these conditions we get the equations These equations are sufficient, when we know two of the four quantities A19 A.2t J$lt -Z?2> ^° deduce the other two. Let us suppose Al and J3L known, then we find the following expressions for A2 and B2, 4 _^1(f _l- ^(2 Ml) SPHERICAL SHELL. 401 In this way we can find the conditions which each term of the harmonic expansion of the potential must satisfy for any number of strata bounded by concentric spherical surfaces. 312.] Let us suppose the radius of the first spherical surface to be al3 and let there be a second spherical surface of radius a2 greater than al9 beyond which the specific resistance is £3. If there are no sources or sinks of electricity within these spheres there will be no infinite values of F", and we shall have Bl = 0. We then find for A3 and j53, the coefficients for the outer medium, - (6) The value of the potential in the outer medium depends partly on the external sources of electricity, which produce currents in- dependently of the existence of the sphere of heterogeneous matter within, and partly on the disturbance caused by the introduction of the heterogeneous sphere. The first part must depend on solid harmonics of positive degrees only, because it cannot have infinite values within the sphere. The second part must depend on harmonics of negative degrees, because it must vanish at an infinite distance from the centre of the sphere. Hence the potential due to the external electromotive forces must be expanded in a series of solid harmonics of positive degree. Let A» be the coefficient of one of these, of the form Then we can find A19 the corresponding coefficient for the inner sphere by equation (6), and from this deduce A2, -#2> an^ -^3- Of these jB3 represents the effect on the potential in the outer medium due to the introduction of the heterogeneous spheres. Let us now suppose £3 = &lt so that the case is that of a hollow shell for which k = k% , separating an inner from an outer portion of the same medium for which k = k±. If we put VOL. I. D d 402* CONDUCTION IN HETEROGENEOUS MEDIA. then A = & The difference between AB the undisturbed coefficient, and Al its value in the hollow within the spherical shell, is A3. (8) Since this quantity is always positive whatever be the values ©f k-L and /fc2, it follows that, whether the spherical shell conducts better or worse than the rest of the medium, the electrical action in the space occupied by the shell is less than it would otherwise be. If the shell is a better conductor than the rest of the medium it tends to equalize the potential all round the inner sphere. If it is a worse conductor, it tends to prevent the electrical currents from reaching1 the inner sphere at all. The case of a solid sphere may be deduced from this by making 0j_ = 0, or it may be worked out independently. 313.] The most important term in the harmonic expansion is that in which i = 1, for which (9) The case of a solid sphere of resistance £2 may be deduced from this by making av = 0. We then have (10) a,3 A.. It is easy to shew from the general expressions that the value of .Z?3 in the case of a hollow sphere having a nucleus of resistance #1, surrounded by a shell of resistance ^2, is the same as that of a uniform solid sphere of the radius of the outer surface, and of resistance K, where , ., = 8 "' 3 14-] MEDIUM CONTAINING SMALL SPHERES. 403 314.] If there are n spheres of radius «x and resistance &lt placed in a medium whose resistance is £2> at such distances from each other that their effects in disturbing- the course of the current may be taken as independent of each other, then if these spheres are all contained within a sphere of radius «2, the potential at a great distance from the centre of this sphere will be of the form. 7= (Ar + n3-^)coB6t (12) where the value of £ is The ratio of the volume of the n small spheres to that of the sphere which contains them is The value of the potential at a great distance from the sphere may therefore be written (15) Now if the whole sphere of radius a2 had been made of a material of specific resistance K, we should have had That the one expression should be equivalent to the other, 2^ + flg + X*!-^) lf /17\ = 2> This, therefore, is the specific resistance of a compound medium consisting of a substance of specific resistance k.2> in which are disseminated small spheres of specific resistance klt the ratio of the volume of all the small spheres to that of the whole being p. In order that the action of these spheres may not produce effects depending on their interference, their radii must be small compared with their distances, and therefore^? must be a small fraction. This result may be obtained in other ways, but that here given involves only the repetition of the result already obtained for a single sphere. When the distance between the spheres is not great compared ^ _ ^ with their radii, and when —j - ~ is considerable, then other terms enter into the result, which we shall not now consider. In consequence of these terms certain systems of arrangement of D d 2, 404 CONDUCTION IN HETEROGENEOUS MEDIA. [315. the spheres cause the resistance of the compound medium to be different in different directions. Application of the Principle of Images. 315.] Let us take as an example the case of two media separated by a plane surface, and let us suppose that there is a source 8 of electricity at a distance a from the plane surface in the first medium, the quantity of electricity flowing from the source in unit of time being 8. If the first medium had been infinitely extended the current at any point P would have been in the direction SP, and the TTf O 7 potential at P would have been — where E = — - and rL = SP. In the actual case the conditions may be satisfied by taking a point 7, the image of 8 in the second medium, such that IS is normal to the plane of separation and is bisected by it. Let r2 be the distance of any point from 7, then at the surface of separation '!-*, (1) dv dv Let the potential V± at any point in the first medium be that due to a quantity of electricity E placed at S9 together with an imaginary quantity E2 at 7, and let the potential F2 at any point of the second medium be that due to an imaginary quantity El at 8, then if E K K FT = f- — and Fo = — » (3) r T r the superficial condition V^ = F2 gives TlJ - 777 777 / A \ and the condition A1 dv k% dv gives j- (E—Ez) =-^Elt (6) whence E± = 7-^5 E2 = -^ — T^' (^) The potential in the first medium is therefore the same as would be produced in air by a charge E placed at 8, and a charge U2 at 7 on the electrostatic theory, and the potential in the second medium is the same as that which would be produced in air by a charge El at S. 3 1 7-] STRATUM WITH PARALLEL SIDES. 405 The current at any point of the first medium is the same as would have been produced by the source S together with a source 2~ * S ftj -f- A^ placed at / if the first medium had been infinite, and the current at any point of the second medium is the same as would have been 2& 8 produced by a source 2 placed at/S if the second medium had (#1 -t Kt ) been infinite. We have thus a complete theory of electrical images in the case of two media separated by a plane boundary. Whatever be the nature of the electromotive forces in the first medium, the potential they produce in the. first medium may be found by combining their direct effect with the effect of their image. If we suppose the second medium a perfect conductor, then k.2 = 0, and the image at / is equal and opposite to the source at 8. This is the case of electric images, as in Thomson's theory in electrostatics. If we suppose the second medium a perfect insulator, then £2 = oo, and the image at /is equal to the source at S and of the same sign. This is the case of images in hydrokinetics when the fluid is bounded by a rigid plane surface. 316.] The method of inversion, which is of so much use in electrostatics when the. bounding surface is supposed to be that of a perfect conductor, is not applicable to the more general case of the surface separating two conductors of unequal electric resist- ance. The method of inversion in two dimensions is, however, applicable, as well as the more general method of transformation in two dimensions given in Art. 190*. Conduction through a Plate separating Two Media. 317.] Let us next consider the effect of a plate of thickness AB of a medium whose resist- ance is /£2, and separating \. two media whose resist- ances are k^ and /£3, in ~T~ ~f~ J~ altering the potential due to a source S in the first medium. The potential will be FiS- 24- * See Kirchhoff, Pogg. Ann. Ixiv. 497, and Ixvii. 344 ; Quincke, Pogg. xcvii. 382; and Smith, Proc. R. S. £din., 1869-70, p. 79. 406 CONDUCTION IN HETEROGENEOUS MEDIA. equal to that due to a system of charges placed in air at certain points along the normal to the plate through S. Make AI= SA, 5/i = SB, A^ = TL A, BI2 = ^ B, AJ2 = I2 A, &c. ; then we have two series of points at distances from each other equal to twice the thickness of the plate. 318.] The potential in the first medium at any point P is equal to HILL W + P7 + ?t + Pt +&c" ^ that at a point P' in the second w r // // 4 and that at a point P" in the third where /, I', &c. represent the imaginary charges placed at the points /, &c., and the accents denote that the potential is to be taken within the plate. Then, by the last Article, for the surface through A we have, (11) t #2 + ATj A?2 r K\ For the surface through B we find ^3 "l~ ^2 2 Similarly for the surface through ^4 again, and for the surface through 5, we find for the potential in the first medium, ' "-1-. (15) 3 1 9-] STRATIFIED CONDUCTORS. 407 For the potential in the third medium we find (16) If the first medium is the same as the third, then ^ = /£3 and p = p', and the potential on the other side of the plate will be If the plate is a very much better conductor than the rest of the medium, p is very nearly equal to 1. If the plate is a nearly perfect insulator, p is nearly equal to — 1, and if the plate differs little in conducting power from the rest of the medium, p is a small quantity positive or negative. The theory of this case was first stated by Green in his ' Theory of Magnetic Induction' (Essay, p. 65). His result, however, is correct only when p is nearly equal to 1 *. The quantity g which he uses is connected with p by the equations 2p _ \-kt *g _ *!— *8 = 3— p *! + 2V p' 2+ff *!+V If we put p = — - , we shall have a solution of the problem of 1 + 27TK the magnetic induction excited by a magnetic pole in an infinite plate whose coefficient of magnetization is K. On Stratified Conductors. 319.] Let a conductor be composed of alternate strata of thick- ness c and c' of two substances whose coefficients of conductivity are different. Required the coefficients of resistance and" conduc- tivity of the compound conductor. Let the plane of the strata be normal to Z. Let every symbol relating to the strata of the second kind be accented, and let every symbol relating to the compound conductor be marked with a bar thus, T. Then T= X = X', (c + c')u = cu 4- cV, T= Y = 7', We must first determine u, u , v, v'9 Z and Z' in terms of J, Tand w from the equations of resistance, Art. 297, or those * See Sir W. Thomson's 'Note on Induced Magnetism in a Plate,' Camb. and Dub. Math. Journ., Nov. 1845, or Reprint, art. ix. § 156. 408 CONDUCTION. IN: HETEROGENEOUS MEDIA. [320-. of conductivity, Art. 298. If we put D for the determinant of the coefficients of resistance, we find Similar equations with, the symbols accented give the values of u , if and Zf . Having1 found u, v and w in terms of X, Y and Z, we may write down the equations of conductivity of the stratified c c' If we make k = — and h'= —-,, we find h + h> h+h' cps + c'pj kJi(q,—q, #3 = _ eg* + c'q.3' M'(p2-pz') fe- c + c' (h+h')(c + cf) c + c' 320.] If neither of the two substances of which the strata are formed has the rotatory property of Art. 303, the value of any P or p will be equal to that of its corresponding Q or q. From this it follows that in the stratified conductor also Pi = ?i» Pz = £2* Ps = ?3> or there is no rotatory property developed by stratification, unless it exists in the materials. 321.] If we now suppose that there is no rotatory property, and also that the axes of x, y and z- are the principal axes, then the p and q coefficients vanish, and _ c I o — — If we begin with both substances isotropic, but of different 322.] STRATIFIED CONDUCTORS. 409 conductivities, then the result of stratification will be to make the resistance greatest in the direction of a normal to the strata, and the resistance in all directions in the plane of the strata will be equal. 322.] Take an isotropic substance of conductivity r, cut it into exceedingly thin slices of thickness «, and place them alternately with slices of a substance whose conductivity is <?, and thickness k^a. Let these slices be normal to x. Then cut this compound con- ductor into thicker slices, of thickness b, normal to y, and alternate these with slices whose conductivity is s and thickness kzb. Lastly, cut the new conductor into still thicker slices, of thick- ness c, normal to z, and alternate them with slices whose con- ductivity is s and thickness k%c. The result of the three operations will be to cut the substance whose conductivity is r into rectangular parallelepipeds whose dimensions are #, 6 and c, where b is exceedingly small compared with e, and a is exceedingly small compared with b, and to embed these parallelepipeds in the substance whose conductivity is s, so that they are separated from each other k-^a in the direction of x, k2b in that of y> and k^c in that of z. The conductivities of the conductor so formed in the directions of x, yy and z are to be found by three applications in order of the results of Art. 321. We thereby obtain _ (L+ (1 -f- The accuracy of this investigation depends upon the three dimen- sions of the parallelepipeds being of different orders of magnitude, so that we may neglect the conditions to be fulfilled at their edges and angles. If we make £1} £2 an^ ^3 eac^ 3r+5s : If r = 0, that is, if the medium of which the parallelepipeds are made is a perfect insulator, then 410 CONDUCTION IN HETEROGENEOUS MEDIA. [323. If r — oo, that is, if the parallelepipeds are perfect conductors, ri = T s> r2 == f *> r3 = 2s. In every case, provided ^ — £2 — /£3, it may be shewn that rlt r2 and r3 are in ascending order of magnitude, so that the greatest conductivity is in the direction of the longest dimensions of the parallelepipeds, and the greatest resistance in the direction of their shortest dimensions. 323.] In a rectangular parallelepiped of a conducting solid, let there be a conducting channel made from one angle to the opposite, the channel being a wire covered with insulating material, and let the lateral dimensions of the channel be so small that the conductivity of the solid is not affected except on account of the current conveyed along the wire. Let the dimensions of the parallelepiped in the directions of the coordinate axes be a, b} c, and let the conductivity of the channel, extending from the origin to the point (a be), be abcK. The electromotive force acting between the extremities of the channel is aX+bY+ cZ, and if C' be the current along the channel C'=Kabc(aX+bY+cZ). The current across the face be of the parallelepiped is 6 en, and this is made up of that due to the conductivity of the solid and of that due to the conductivity of the channel, or leu = bc or u = (/-! + Ka2) X+ ( j»3 + Ka b) Y+ (q.2 + Kca) Z. In the same way we may find the values of v and w. The coefficients of conductivity as altered by the effect of the channel will be Pt + Kbc, jp^ + Kca, ql + Kb c, £2 -f Kca, In these expressions, the additions to the values of pl9 &c., due to the effect of the channel, are equal to the additions to the values of q1, &c. Hence the values of p^ and qt cannot be rendered unequal by the introduction of linear channels into every element of volume of the solid, and therefore the rotatory property of Art. 303, if it does not exist previously in a solid, cannot be introduced by such means. 324.] COMPOSITE CONDUCTOR. 411 3.24.] To construct a framework of linear conductors which shall have any given coefficients of conductivity forming a symmetrical system. Let the space be divided into equal small cubes, of which let the figure represent one. Let the coordinates of the points 0, L, M, Nt and their potentials be as follows : — x y z Potential Lt 0 000 X+Y+Z L 0 1 1 X 1/101 Y N 1 I 0 Z Let these four points be connected by six conductors, OL, OM, ON, MN, NL, LM, of which the conductivities are respectively A, 3, C, P, Q, X, The electromotive forces along these conductors will be Y+Z, Z+X, J+7, 7-Z, Z-X, X-7, and the currents A(7+Z), 3(Z + X)9 C(X+7), P(Y-Z], Q(Z-X), R(X-7). Of these currents, those which convey electricity in the positive direction of x are those along LH, LN, OM and ON, and the quantity conveyed is / 71 , fv \ f) i f}\ ~y i [ /~i T?\V j_ / 7? _ d\ P Similarly v = (C—R)X +(C+A w=(3-Q)X +(A-P)Y whence we find by comparison with the equations of conduction, Art. 298, 4 A = T., + n— /i + 2»,, 4P = ro + r,— r, — 20,, CHAPTER X. CONDUCTION IN DIELECTRICS. 325.] WE have seen that when electromotive force acts on a dielectric medium it produces in it a state which we have called electric polarization, and which we have described as consisting of electric displacement within the medium in a direction which, in isotropic media, coincides with that, of the electromotive force, combined with a superficial charge on every element of volume into which we may suppose the dielectric divided, which is negative on the side towards which the force acts, and positive on the side from which it acts. When electromotive force acts on a conducting, medium it also produces what is called an electric current. Now dielectric media, with very few, if any, exceptions, are also more or less imperfect conductors, and many media which are not good insulators exhibit phenomena of dielectric induction. Hence we are led to study the state of a medium in which induction and conduction are going on at the same time. For simplicity we shall suppose the medium isotropic at every point, but not necessarily homogeneous at different points. In this case, the equation of Poisson becomes,- by Art. 83, d /rrdY\ d ,^dV\ d / ^dV\ -r (K-r) + -7-(A.T-J + -T-(K-7-) + 4'np=0, (l) ax ^ ax' dy ^ dy/ dz v &&' where K is the ( specific inductive capacity.' The f equation of continuity' of electric currents becomes dx^r dos dy\r dy* dz.^r dz' dt where r is the specific resistance referred to unit of volume. When K or r is discontinuous, these equations must be trans- formed into those appropriate to surfaces of discontinuity. 326.] THEORY OF A CONDENSER. 413 In a strictly homogeneous medium r and K are both constant, so that we find P dp whence p = Ce *r'; (4) Kr -- or, if we put T = — , p = Ce T . (5) This result shews that under the action of any external electric forces on a homogeneous medium, the interior of which is originally charged in any manner with electricity, the internal charges will die away at a rate which does not depend on the external forces, so that at length there will be no charge of electricity within the medium, after which no external forces can either produce or maintain a charge in any internal portion of the medium, pro- vided the relation between electromotive force, electric polarization and conduction remains the same. When disruptive discharge occurs these relations cease to be true, and internal charge may be produced. On Conduction through a Condenser. 326.] Let C be the capacity of a condenser, R its resistance, and E the electromotive force which acts on it, that is, the difference of potentials of the surfaces of the metallic electrodes. Then the quantity of electricity on the side from which the electromotive force acts will be CU, and the current through the substance of the condenser in the direction of the electromotive force will be •— If the electrification is supposed to be produced by an electro- motive force E acting in a circuit of which the condenser forms part, and if -~ represents the current in that circuit, then do W=*L+C — - (6) Let a battery of electromotive force EQ and resistance rt be introduced into this circuit, then Hence, at any time tl9 ' where Z\=-£^-. (8) 414 CONDUCTION IN DIELECTRICS. [327» Next, let the circuit rt be broken for a time t2, -h. E( = E2) = E^ TZ where T2 = CR. (9) Finally, let the surfaces of the condenser be connected by means of a wire whose resistance is rs for a time tz) E( = Zt) = Et*~* where T3 = g^ . (10) If §3 is the total discharge through this wire in the time t3 , In this way we may find the discharge through a wire which is made to connect the surfaces of a condenser after being charged for a time tfl5 and then insulated for a time t2. If the time of charging is sufficient, as it generally is, to develope the whole charge, and if the time of discharge is sufficient for a complete discharge, the discharge is