The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 8 of 39

12. Two insulated fixed condensers are at given potentials when alone in the electric field and charged with quantities Elt E2 of electricity. Their coefficients of potential are pn, pn, P22- But if they are surrounded by a spherical conductor of very large radius R at potential zero with its centre near them, the two conductors require charges E{, EJ to

produce the given potentials. Prove, neglecting -^ , that

E2'-E2 Pn-pu'

13. Shew that the locus of the positions, in which a unit charge will induce a given charge on a given uninsulated conductor, is an equipotential surface of that conductor supposed freely electrified.

14. Prove (i) that if a conductor, insulated in free space and raised to unit potential, produce at any external point P a potential denoted by (P), then a unit charge placed at P in the presence of this conductor uninsulated will induce on it a charge — (P) ;

(ii) that if the potential at a point Q due to the induced charge be denoted by (PQ), then (PQ) is a symmetrical function of the positions of P and Q.

15. Two small uninsulated spheres are placed near together between two large parallel planes, one of which is charged, and the other connected to earth. Shew by figures the nature of the disturbance so produced in the uniform field, when the line of centres is (i) perpendicular, (ii) parallel to the planes.

16. A hollow conductor A is at zero potential, and contains in its cavity two other insulated conductors, B and C, which are mutually external : B has a positive charge, and C is uncharged. Analyse the different types of lines of force within the cavity which are possible, classifying with respect to the conductor from which the line starts, and the conductor at which it ends, and proving the impossibility of the geometrically possible types which are rejected.

Hence prove that B and C are at positive potentials, the potential of G being less than that of B.

17. A portion P of a conductor, the capacity of which is C, can be separated from the conductor. The capacity of this portion, when at a long distance from other bodies, is c. The conductor is insulated, and the part P when at a considerable distance from the remainder is charged with a quantity e and allowed to move under the mutual attraction up to it ; describe and explain the changes which take place in the electrical energy of the system.

Examples 113

18. A conductor having a charge $1 is surrounded by a second conductor with charge Q2. The inner is connected by a wire to a very distant uncharged conductor. It is then disconnected, and the outer conductor connected. Shew that the charges Qi, Q2, are now

m+n + mn'' m + n '

where G, C(l+m) are the coefficients of capacity of the near conductors, and Cn is the capacity of the distant one.

19. If one conductor contains all the others, and there are n+l in all, shew that there are n + 1 relations between either the coefficients of potential or the coefficients of induction, and if the potential of the largest be V0, and that of the others Vu V2, ... V„, then the most general expression for the energy is ^GV02 increased by a quadratic function of Vt- V0, V2— V0, ... Vn— V0 ; where C is a definite constant for all positions of the inner conductors.

20. The inner sphere of a spherical condenser (radii a, b) has a constant charge E, and the outer conductor is at potential zero. Under the internal forces the outer conductor contracts from radius b to radius by. Prove that the work done by the electric forces is

21. If, in the last question, the inner conductor has a constant potential V, its charge being variable, shew that the work done is

*(bi-a)(b-a)'

and investigate the quantity of energy supplied by the battery.

22. With the usual notation, prove that

Pn+P23>Pl2+Pl3 PllP23>Pl2Pl3-

23. Shew that if p„., pr3, pss be three coefficients before the introduction of a new conductor, and p„\ pra', pa8' the same coefficients afterwards, then

(PrrPsa-Prr'pss) <t (Pn~Pn?-

24. A system consists of p + q + 2 conductors, Alt A2, ... Ap, Bu B2, ... Bq, C, D. Prove that when the charges on the A's and on C, and the potentials of the B's and of C are known, there cannot be more than one possible distribution in equilibrium, unless C is electrically screened from D.

25. A, B, C, D are four conductors, of which B surrounds A and D surrounds C. Given the coefficients of capacity and induction

(i) of A and B when C and D are removed, (ii) of C and D when A and B are removed, (iii) of B and D when A and G are removed, determine those for the complete system of four conductors.

26. Two equal and similar conductors A and B are charged and placed symmetrically with regard to each other ; a third moveable conductor C is earned so as to occupy

j. • 8

114 Systems of Conductors [oh. iv

successively two positions, one practically wholly within A, the other within B, the positions being similar and such that the coefficients of potential of C in either position are p, q, r in ascending order of magnitude. In each position C is in turn connected with the conductor surrounding it, put to earth, and then insulated. Determine the charges on the conductors after any number of cycles of such operations, and shew that they ultimately lead to the ratios

l:-0:/9»-l, where /3 is the positive root of

rx2 — qx + p - r = 0.

27. Two conductors are of capacities Gx and C2, when each is alone in the field. They are both in the field at potentials Vx and F2 respectively, at a great distance r apart. Prove that the repulsion between the conductors is

Ci^(rr1-c2r2)(rr2-c1r1)

As far as what power of - is this result accurate ?

28. Two equal and similar insulated conductors are placed symmetrically with regard to each other, one of them being uncharged. Another insulated conductor is made to touch them alternately in a symmetrical manner, beginning with the one which has a charge. If elt e2 be their charges when it has touched each once, shew that their charges, when it has touched each r times, are respectively

ex2 r to _«.\2r-l-J 0.2

2ej - e2

m^tt- «&{'-<?■?;}■

29. Three conductors Ai} A2 and As are such that A3 is practically inside A%. Ax is alternately connected with A2 and A3 by means of a fine wire, the first contact being with A3. Ai has a charge E initially, A2 and A3 being uncharged. Prove that the charge on Ax after it has been connected n times with A2 is

M. fi , °(y-/3)/"-H3Yt-1)

a+/S\ ^I3(a + y)\a + yj J » where a, ft y stand for pn -pi2, P22—P12 and JO33 -p\2 respectively.

30. Two spheres, radii a, b, have their centres at a distance c apart. Shew that neglecting (a/c)6 and (6/c)6,

lb3 1 la3

*u=£?; fn=V P22=b

1

CHAPTER Y

DIELECTRICS AND INDUCTIVE CAPACITY

125. Mention has already been made (§ 84) of the fact, discovered originally by Cavendish, and afterwards rediscovered by Faraday, that the capacity of a conductor depends on the nature of the dielectric substance between its plates.

Let us imagine that we have two parallel plate condensers, similar in all respects except that one has nothing but air between its plates while in the other this space is filled with a dielectric of inductive capacity K. Let us suppose that the two high-potential plates are connected by a wire, and also the two low-potential plates. Let the condensers be charged, the potential of the high-potential plates being Tf, and that of the low-potential plates being V0.

Then it is found that the charges possessed by the two condensers are not equal. The capacity per unit area of the air-condenser is lj&ird ; that of the other condenser is found to be Kj^ird. Hence the charges per unit area of the two condensers are respectively

^and KVl~

4nrd

4,-rrd

The work done in taking unit charge from the low-potential plate to the high-potential plate is the same in either condenser, namely T^— To, so that the intensity between the plates in either condenser is the same, namely

d '

In the air-condenser this intensity may be regarded as the resultant of the attraction of the negatively charged plate and the repulsion of the positively

charged plate, the law of attraction or repulsion being Coulomb's law — .

Fig. 42.

8—2

116 Dielectrics and Inductive Capacity [ch. v

It is, however, obvious that if we were to calculate the intensity in the second condenser from this law, then the value obtained would be K times

V -V

that in the first condenser, and would therefore be K 1 , °. In point of

V—V

fact, the actual value of the intensity is known to be * , ° .

Thus Faraday's discovery shews that Coulomb's law of force is not of universal validity : the law has only been proved experimentally for air, and it is now found not to be true for dielectrics of which the inductive capacity is different from unity.

This discovery has far-reaching effects on the development of the mathe- matical theory of electricity. In the present book, Coulomb's law was introduced in § 88, and formed the basis of all subsequent investigations. Thus every theorem which has been proved in the present book from § 38 onwards requires reconsideration.

126. We shall follow Faraday in treating the whole subject from the point of view of lines of force. The conceptions of potential, of intensity, and of lines of force are entirely independent of Coulomb's law, and in the present book have been discussed (§§ 30 — 37) before the law was introduced. The conception of a tube of force follows at once from that of a line of force, on imagining lines of force drawn through the different points on a small closed curve. Let us extend to dielectrics one form of the definition of the strength of a tube of force which has already been used for a tube in air, and agree that the strength of a tube is to be measured by the charge enclosed by its positive end, whether in air or dielectric.

In the dielectric condenser, the surface density on the positive plate is

V- V

K ■— — -^ , and this, by definition, is also the aggregate strength of the

tubes per unit area of cross-section. The intensity in the dielectric is

V—V0

— j — - , so that in the dielectric the intensity is no longer, as in air, equal

to 4-7T times the aggregate strength of tubes per unit area, but is equal to 4nrJK times this amount.

Thus if P is the aggregate strength of the tubes per unit area of cross- section, the intensity R is related to P by the equation

£=XP (59)

in the dielectric, instead of by the equation

P = 4ttP (60)

which was found to hold in air.

125-128J Experimental Basis 117

127. Equation (59) has been proved to be the appropriate generalisation of equation (60) only in a very special case. Faraday, however, believed the relation expressed by equation (59) to be universally true, and the results obtained on this supposition are found to be in complete agreement with experiment. Hence equation (59), or some equation of the same significance, is universally taken as the basis of the mathematical theory of dielectrics. We accordingly proceed by assuming the universal truth of equation (59), an assumption for which a justification will be found when we come to study the molecular constitution of dielectrics.

It is convenient to have a single word to express the aggregate strength of tubes per unit area of cross-section, the quantity which has been denoted by P. We shall speak of this quantity as the " polarisation," a term due to ; Faraday. Maxwell's explanation of the meaning of the term " polarisation " is that " an elementary portion of a body may be said to be polarised when it acquires equal and opposite properties on two opposite sides." Faraday explained the properties of dielectrics by means of his conception that the molecules of the dielectric were in a polarised state, and the quantity P is found to measure the amount of the polarisation at any point in the dielectric. We shall come to this physical interpretation of the quantity P at a later stage : for the present we simply use the term " polarisation " as a name for the mathematical quantity P.

This same quantity is called the " displacement " by Maxwell, and under- lying the use of this term also, there is a physical interpretation which we shall come upon later.

128. We now have as the basis of our mathematical theory the following :

Definition. The strength of a tube of force is defined to be the charge enclosed by the positive end of the tube.

Definition. The polarisation at any point is defined to be the aggregate strength of tubes of force per unit area of cross- section.

Experimental Law. The intensity at any point is 4nr/K times the polarisation, where K is the inductive capacity of the dielectric at the point.

In this last relation, we measure the intensity along a line of force, while the polarisation is measured by considering the flux of tubes of force across a small area perpendicular to the lines of force. Suppose, however, that we take some direction 00' making an angle 0 with that of the lines of force. The aggregate strength of the tubes of force which cross an area dS perpendicular to 00' will be P cos OdS, for these tubes are exactly those which cross an area dS cos 6 perpendicular to the lines of force. Thus, consistently with the definition of polarisation, we may say that the polari- sation in the direction 00' is equal to Pcosd. Since the polarisation in

118 Dielectrics and Inductive Capacity [ch. v

any direction is equal to P multiplied by the cosine of the angle between this direction and that of the lines of force, it is clear that the polarisation may be regarded as a vector, of which the direction is that of the lines of force, and of which the magnitude is P.

The polarisation having been seen to be a vector, we may speak of its components /, g, h. Clearly / is the number of tubes per unit area which cross a plane perpendicular to the axis of x, and so on.

The result just obtained may be expressed analytically by the equations

J 4nr * 4?r 4tt

129. The polarisation P being measured by the aggregate strength of tubes per unit area of cross-section, it follows that if a> is the cross-section at any point of a tube of strength e, we have e = &>P. Now we have defined the strength of a tube of force as being equal to the charge at its positive end, so that by definition the strength e of a tube does not vary from point to point of the tube. Thus the product &>P is constant along a tube, or <oKR is constant along a tube, replacing the result that coR is constant in air (§ 56).

The value of the product coP at any point 0 of a tube, being equal to

— — , depends only on the physical conditions prevailing at the point 0.

It is, however, known to be equal to the charge at the positive end of the tube. Hence it must also, from symmetry, be equal to minus the charge at the negative end of the tube. Thus the charges at the two ends of a tube, whether in the same or in different dielectrics, will be equal and opposite, and the numerical value of either is the strength of the tube.

Gauss' Theorem.

130. Let S be any closed surface, and let e be the angle between the direction of the outward normal to any element of surface dS and the direction of the lines of force at the element. The aggregate strength of the tubes of force which cross the element of area dS is P cos e dS, and the integral

fjPcosedS,

which may be called the surface integral of normal polarisation, will measure the aggregate strength of all the tubes which cross the surface S, the strength of a tube being estimated as positive when it crosses the surface from inside to outside, and as negative when it crosses in the reverse direction.

A tube which enters the surface from outside, and which, after crossing

128-1 31 J Gauss' Theorem 119

the space enclosed by the surface, leaves it again, will add no contribution to

1 1 P cos edS, its strength being counted negatively where it enters the

surface, and positively where it emerges. A tube which starts from or ends on a charge e inside the surface S will, however, supply a contribution to

II P cos edS on crossing the surface. If e is positive, the strength of the

tube is e ; and, as it crosses from inside to outside, it is counted positively, and the contribution to the integral is e. Again, if e is negative, the strength of the tube is — e, and this is counted negatively, so that the contribution is again e.

Thus on summing for all tubes,

P cos €dS = E,

//■

where E is the total charge inside the surface. The left-hand member is simply the algebraical sum of the strengths of the tubes which begin or end inside the surface ; the right-hand member is the algebraical sum of the charges on which these tubes begin or end. Putting

the equation becomes 1 1 KB, cos edS= 4nrE.

The quantity R cos e is, however, the component of intensity along the outward normal, the quantity which has been previously denoted by N, so that we arrive at the equation

f(KNdS = 4>7rE (61).

When the dielectric was air, Gauss' theorem was obtained in the form

//

NdS = 4tt#.

Equation (61) is therefore the generalised form of Gauss' Theorem which

must be used when the inductive capacity is different from unity. Since

dV N= — -x— , the equation may be written in the form

dV K^dS = -4>7rE.

on

131. The form of this equation shews at once that a great many results which have been shewn to be true for air are true also for dielectrics other than air.

It is obvious, for instance, that V cannot be a maximum or a minimum at a point in a dielectric which is not occupied by an electric charge : as

120 Dielectrics and Inductive Capacity [ch. v

a consequence all lines of force must begin and end on charged bodies, a result which was tacitly assumed in defining the strength of a tube of force.

A number of theorems were obtained in the discussion of the electrostatic field in air, by taking a Gauss' Surface, partly in air and partly in a con- ductor. Gauss' Theorem was used in the form

ffNdS = 4*7rE,

but we now see that if the inductive capacity of the conductor were not equal to unity, this equation ought to be replaced by equation (61). It is, however, clear that the difference cannot affect the final result ; N is zero inside a conductor, so that it does not matter whether N is multiplied by K or not.

Thus results obtained for systems of conductors in air upon the assumption that Coulomb's law of force holds throughout the field are seen to be true whether the inductive capacity inside the conductors is equal to unity or not.

The Equations of Poisson and Laplace.

132. In § 49, we applied Gauss' theorem to a surface which was formed by a small rectangular parallelepiped, of edges dx, dy, dz, parallel to the axes of coordinates. If we apply the theorem expressed by equation (61) to the same element of volume, we obtain

dy \ dy

where p is the volume density of electrification. This, then, is the generalised form of Poisson's equation: the generalised form of Laplace's equation is obtained at once on putting p = 0.

In terms of the components of polarisation, equation (62) may be written

df dg dh_

dx+dy + dz~p Cbd)'

while if the dielectric is uncharged,

4(£K(8K(£)--*-> m.

1+1+1=° <«*)■

Electric Charges in an infinite homogeneous Dielectric.

133. Consider a charge e placed by itself in an infinite dielectric. If the dielectric is homogeneous, it follows from considerations of symmetry that the lines of force must be radial, as they would be in air. By application

131-135] Gauss' Theorem 121

of equation (61) to a sphere of radius r, having the point charge as centre, it is found that the intensity at a distance r from the charge is

Kr*'

The force between two point charges e, e, at distance r apart in a homo- geneous unbounded dielectric is therefore

ee'

■•(65),

Kr*

and the potential of any number of charges, obtained by integration of this expression, is

v=T^l (66)'

Coulomb's Equation.

134. The strength of a tube being measured by the charge at its end, it follows that at a point just outside a conductor, P, the aggregate strength of the tubes per unit of cross-section, becomes numerically equal to cr, the surface density. We have also the general relation

R- — P

and on replacing P by a, we arrive at the generalised form of Coulomb's equation,

R = ~ (67),

in which K is the inductive capacity at the point under consideration.

Conditions to be satisfied at the Boundary of a Dielectric.

135. Let us examine the conditions which will obtain at a boundary at which the inductive capacity changes abruptly from K1 to K2.

The potential must be continuous in crossing the boundary, for if P, Q, are two infinitely near points on opposite sides of the boundary, the Avork done in bringing a small charge to P must be the same as that done in bringing it to Q. As a consequence of the potential being continuous, it follows that the tangential components of the intensity must also be continuous. For if P, Q are two very near points on different sides of the boundary, and P', Q' a similar pair of points at a small distance away, we have Ve=VQ, and VP' = Vq, so that

PP' QQ' '

The expressions on the two sides of this equation are, however, the two intensities in the direction PP', on the two sides of the boundary, which establishes the result.

122 Dielectrics and Inductive Capacity [ch. v

Also, if there is no charge on the boundary, the aggregate strength of the tubes which meet the boundary in any small area on this boundary is the same whether estimated in the one dielectric or the other, for the tubes do not alter their strength in crossing the boundary, and none can begin or end in the boundary. Thus the normal component of the polarisation is continuous.

136. If Ri is the intensity in the first medium of inductive capacity Kx t measured at a point close to the boundary, and if ex is the angle which the lines of force make with the normal to the boundary at this point, then the normal polarisation in the first medium is

-j— Kx COS 6X. 47T

Similarly, that in the second medium is

-7- Mo cos e2,

so that KiRx cos €i = KoR2 cos e2 (G8).

Since, in the notation already used,

is^cos e1 = N1=—-^,

on

the equation just obtained may be put in either of the forms

K&^KJST, (69),

*s-*s <70>-

In these equations, it is a matter of indifference whether the normal is drawn from the first medium to the second or in the reverse direction ; it is only necessary that the same normal should be taken on both sides of the equation. Relation (70) is obtained at once on applying the generalised form of Gauss' theorem to a small cylinder having parallel ends at infinitesimal distance apart, one in each medium.

137. To sum up, we have found that in passing from one dielectric to another, the surface of separation being uncharged :

(i) the tangential components of intensity have the same values on the two sides of the boundary,

(ii) the normal components of polarisation have the same values.

Or, in terms of the potential, (i) V is continuous,

(ii) Kd-I is continuou, dn

135-138J

Boundary Conditions

123

Refraction of the lines of force.

133. From  the  continuity  of  the  tangential  components  of  intensity,  it 
     

follows :

(i) that the directions of R^ and R2, the intensities on the two sides of the boundary, must lie in a plane containing the normal, and

(ii) that Rx sin ex = R2 sin e2 .

Combining the last relation with equation (68), we obtain

Kx cot ex = K2 cot e2 (71).

From this relation, it appears that if Kx is greater than K2, then €j is greater than e2, and vice versa. Thus in passing from a smaller value of K to a greater value of K, the lines are bent away from the normal. In illustration of this, fig. 43 shews the arrangement of lines of force when a point charge is placed in front of an infinite slab of dielectric (K = 7).

Fig. 43.

124

Dielectrics and Inductive Capacity

[ch. v

A small charged particle placed at any point of this field will experience a force of which the direction is along the tangent to the line of force through the point. The force is produced by the point charge, but its direction will not in general pass through the point charge. Thus we conclude that in a field in which the inductive capacity is not uniform the force between two point charges does not in general act along the line joining them.

139. As an example of the action of a dielectric let us imagine a parallel plate condenser in which a slab of dielectric of thickness t is placed between the plates, its two faces being parallel to the plates and at distances a, b from them, so that a + b + 1 = d, where d is the distance between the plates.

It is obvious from symmetry that the lines of force are straight throughout their path, equation (71) being satisfied by e1 = e2 = 0.

Let <t be the charge per unit area, so that the polari- sation is equal to a everywhere. The intensity, by equation (67), is

R = 47rcr in air,

and

4>7T

R = -~ a in dielectric.

Fig. 44.

Hence the difference of potential between the plates, or the work done in taking unit charge from one plate to the other in opposition to the electric intensity,

4"7r

= 47TO- . a + -j= a . t + 47TCT . 6

= 47ro-je*-(l-i)*j, and the capacity per unit area is

Thus the introduction of the slab of dielectric has the same effect as moving the plates a distance ( 1 — -^ ) t nearer together.

Suppose now that the slab is partly outside the condenser and partly between the plates. Of the total area A of the condenser, let an area B be occupied by the slab of dielectric, an area A — B having only air between the plates.

138-141] Boundary Conditions 125

The lines of force will be straight, except for those which pass near to the edge of the dielectric slab. Neglecting a small correction required by the curvature of these lines, the capacity G of the condenser is given by

c = B A-B

M 4rf{d-(l-£)«

a quantity which increases as B increases. If V is the potential difference and E the charge, the electrical energy

E2

= 401/2 = 1 —

If we keep the charge constant, the electrical energy increases as the slab is withdrawn. There must therefore be a mechanical force tending to resist withdrawal : the slab of dielectric will be sucked in between the plates of the condenser. This, as will be seen later, is a particular case of a general theorem that any piece of dielectric is acted on by forces which tend to drag it from the weaker to the stronger parts of an electric field of force.

Charge on the Surface of a Dielectric.

140. Let dS be any small area of a surface which separates two media of inductive capacities K1} K2, and let this bounding surface have a charge of electricity, the surface density over dS being o\ If we apply

Gauss' Theorem to a small cylinder circumscribing dS we obtain

£+£—*** <72>-

where — in either medium denotes differentiation with respect to the normal drawn away from dS into the dielectric.

141. As we have seen, the surface of a dielectric may be charged by friction. A more interesting way is by utilising the conducting powers of a flame. PlQ' 45-

Let us place a charge e in front of a slab of dielectric as in fig. 43. A flame issuing from a metal lamp held in the hand may be regarded as a conductor at potential zero. On allowing the flame to play over the surface of the dielectric, this surface is reduced to potential zero, and the distribution of the lines of force is now exactly the same as if the face of the dielectric were replaced by a conducting plane at potential zero. The

126 Dielectrics and Inductive Capacity [ch. v

lines of force from the point charge terminate on this plane, so that there must be a total charge — e spread over it. If the plane were actually a conductor this would be simply an induced charge. If, however, the plane is the boundary of a dielectric, the charge differs from an induced charge on a conductor in that it cannot disappear if the original charge e is removed. For this reason, Faraday described it as a " bound " charge. The charge has of course come to the dielectric through the conducting flame.

Molecular Action in a Dielectric.

142. From the observed influence of the structure of a dielectric upon the electric phenomena occurring in a field in which it was J3laced, Faraday was led to suppose that the particles of the dielectric themselves took part in this electric action. After describing his researches on the electric action — " induction " to use his own term — in a space occupied by dielectric he says*:

" Thus induction appears to be essentially an action of contiguous parti- cles, through the intermediation of which the electric force, originating or appearing at a certain place, is propagated to or sustained at a distance...."

" Induction appears to consist in a certain polarised state of the particles, into which they are thrown by the electrified body sustaining the action, the particles assuming positive and negative points or parts...."

"With respect to the term polarity..., I mean at present... a disposition of force by which the same molecule acquires opposite powers on different parts."

And again, later f,

"I do not consider the powers when developed by the polarisation as limited to two distinct points or spots on the surface of each particle to be considered as the poles of an axis, but as resident on large portions of that surface, as they are upon the surface of a conductor of sensible size when it is thrown into a polar state."

" In such solid bodies as glass, lac, sulphur, etc., the particles appear to be able to become polarised in all directions, for a mass when experimented upon so as to ascertain its inductive capacity in three or more directions, gives no indication of a difference. Now, as the particles are fixed in the mass, and as the direction of the induction through them must change with its charge relative to the mass, the constant effect indicates that they can be polarised electrically in any direction."

• Experimental Researches, 1295, 1298, 1304. (Nov. 1837.) t Experimental Researches, 1686, 1688, 1679. (June, 1838.)

141-143]

Molecular Theory

127

" The particles of an insulating dielectric whilst under induction may be compared... to a series of small insulated conductors. If the space round a charged globe were filled with a mixture of an insulating dielectric and small globular conductors, the latter being at a little distance from each other, so as to be insulated, then these would in their condition and action exactly resemble what I consider to be the condition and action of the particles of the insulating dielectric itself. If the globe were charged, these little conductors would all be polar ; if the globe were discharged, they would all return to their normal state, to be polarised again upon the recharging of the globe...."

As regards the question of what actually the particles are which undergo this polarisation, Faraday says* :

" An important inquiry regarding the electric polarity of the particles of an insulating dielectric, is, whether it be the molecules of the particular substance acted on, or the component or ultimate particles, which thus act the part of insulated conducting polarising portions."

"The conclusion I have arrived at is, that it is the molecules of the substance which polarise as wholes; and that however complicated the composition of a body may be, all those particles or atoms which are held together by chemical affinity to form one molecule of the resulting body act as one conducting mass or particle when inductive phenomena and polarisation are produced in the substance of which it is a part."

143. A mathematical discussion of the action of a dielectric constructed as imagined by Faraday, has been given by Mossotti, who utilised a mathe- matical method which had been developed by Poisson for the examination of a similar question in magnetism. For this discussion the molecules are represented provisionally as conductors of electricity.

To obtain a first idea of the effect of an electric field on a dielectric of the kind pictured by Faraday, let us consider a parallel plate condenser,

• + 
  

Fig. 46.

• Experimental Researches, 1699, 1700.

128 Dielectrics and Inductive Capacity [ch. v

having a number of insulated uncharged conducting molecules in the space between the plates. Imagine a tube of strength e meeting a molecule. At the point where this occurs, the tube terminates by meeting a conductor, so that there must be a charge — e on the surface of the molecule. Since the total charge on the molecule is nil there must be a corresponding charge on the opposite surface, and this charge may be regarded as a point of restarting of the tube. The tube then may be supposed to be continually stopped and restarted by molecules as it crosses from one plate of the condenser to the other. At each encounter with a molecule there are induced charges — e, + e on the surface of the molecule. Any such pair of charges, being at only a small distance apart, may be regarded as forming a small doublet, of the kind of which the field of force was investigated in § 64.

144. We have now replaced the dielectric by a series of conductors, the medium between which may be supposed to be air or ether. In the space between these conductors the law of force will be that of the inverse square. In calculating the intensity at any point from this law we have to reckon the forces from the doublets as well as the forces from the original charges on the condenser-plates. A glance at fig. 46 will shew that the forces from the doublets act in opposition to the original forces. Thus for given charges on the condenser-plates the intensity at any point between the plates is lessened by the presence of conducting molecules.