The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 9 of 39

This general result can be seen at once from the theorem of § 121. The introduction of new conductors (the molecules) lessens the energy cor- responding to given charges on the plates, i.e. increases the capacity of the condenser, and so lessens the intensity between the plates.

145. In calculating that part of the intensity which arises from the doublets, it will be convenient to divide the dielectric into concentric spherical shells having as centre the point at which the intensity is required. The volume of the shell of radii r and r + dr is 47rr2 dr, so that the number of doublets included in it will contain r2dr as a factor. The potential produced

Hi COS 0

by any doublet at a point distant r from it is - — - — , so that the intensity will contain a factor — . Thus the intensity arising from all the doublets in

the shell of radii r, r + dr will depend on r through the factor -.r-dr

dr or — r

The importance of the different shells is accordingly the same, as regards

comparative orders of magnitude, as that of the corresponding contributions

fdv to the integral / — . The value of this integral is log r + a constant, and this

143-146] Molecular Theory 129

is infinite when r = 0 and when r = oo . Thus the important contributions come from very small and very large values of r. It can however be seen that the contributions from large values of r neutralise one another, for the term cos 6 in the potentials of the different doublets will be just as often positive as negative.

Hence it is necessary only to consider the contributions from shells for which r is very small, so that the whole field at any point may be regarded as arising entirely from the doublets in the immediate neighbourhood of the point. The force will obviously vary as we move in and out amongst the molecules, depending largely on the nearness and position of the nearest molecules. If, however, we average this force throughout a small volume, we shall obtain an average intensity of the field produced by the doublets, and this will depend only on the strength and number of the doublets in and near to this element of volume. Obviously this average intensity near any point will be exactly proportional to the average strength of the doublets near the point, and this again will be exactly proportional to the strength of the inducing field by which the doublets are produced, so that at any point we may say that the average field of the doublets stands to the total field in a ratio which depends only on the structure of the medium at the point.

146. Now suppose that our measurements are not sufficiently refined to enable us to take account of the rapid changes of intensity of the electric field which must occur within small distances of molecular order of magnitude. Let us suppose, as we legitimately may, that the forces which we measure are forces averaged through a distance which contains a great number of molecules. Then the force which we measure will consist of the sum of the average force produced by the doublets, and of the force produced by the external field. The field which we observe may accordingly be regarded as the superposition of two fields, or what amounts to the same thing, the observed intensity R may be regarded as the resultant of two intensities Rlt M2, where

JR.! is the average intensity arising from the neighbouring doublets,

R2 is the intensity due to the charges outside the dielectric, and to the distant doublets in the dielectric.

These forces, as we have seen, must be proportional to one another, so that each must be proportional to the polarisation P. It follows that P is proportional to R, the ratio depending only on the structure of the medium at the point. If we take the relation to be

R = ^P • (73),

then K is the inductive capacity at the point, and the relation between R and P is exactly the relation upon which our whole theory has been based, j. 9

130 Dielectrics mid Inductive Capacity [ch. v

147. The theory could accordingly be based on Mossotti's theory, instead of on Faraday's assumption, and from the hypothesis of molecular polarisa- tion we should be able to deduce all the results of the theory, by first deducing equation (73) from Mossotti's hypothesis, and then the required results from equation (73) in the way in which they have been deduced in the present chapter.

Thus the influence of the conducting molecules produces physically the same result as if the properties of the medium were altered in the way suggested by Faraday, and mathematically the properties of the medium are in either case represented by the presence of the factor K in equation (73).

Relation between Inductive Capacity and Structure of Medium.

148. The electrostatic unit of force was defined in such a way that the inductive capacity of air was taken as unity. It is now obvious that it would have been more scientific to have taken ether as standard medium, so that the inductive capacity of every medium would have been greater than unity. Unfortunately, the practice of referring all inductive capacities to air as standard has become too firmly established for this to be possible. The difference between the two standards is very slight, the inductive capacity of normal air in terms of ether being 1*000590. Thus the inductive capacity of a vacuum may be taken to be "99941 referred to air.

So long as the molecules are at distances apart which are great compared with their linear dimensions, we may neglect the interaction of the charges induced on the different molecules, and treat their effects as additive. It follows that in a gas K — K0, where K0 is the inductive capacity of free ether, ought to be proportional to the density of the gas. This law is found to be in exact agreement with experiment*.

149. It is, however, possible to go further and calculate the actual value of the ratio of K — K0 to the density. We have seen that this will be a constant for a given substance, so that we shall determine its value in the simplest case : we shall consider a thin slab of the dielectric placed in a parallel plate condenser, as described in § 139. Let this slab be of thickness e, and let it coincide with the plane of yz. Let the dielectric contain n mole- cules per unit volume.

The element dydz will contain nedydz molecules. If each of these is a doublet of strength fju, the element dydz will have a field which will be equivalent at all distant points to that of a single doublet of strength nyuedydz. This is exactly the field which would be produced if the two faces of the slab were charged with electricity of surface density ± ?i/t.

• Boltzmann, Wiener Sitzungsber. 69, p. 812.

147-149] Molecular Theory 131

We can accordingly at once find the field produced by these doublets — it is the same as that of a parallel plate condenser, in which the plates are at distance e apart and are charged to surface density + n/t. There is no intensity except between the plates, and here the intensity of the field is

Thus if R is the total intensity outside the slab, that inside will be R — 4<7rn/jb. If K is the inductive capacity of the material of the slab, and K0 that of the free ether outside the slab, we have

K0R = K(R-4i7rnfi),

so that — T^ = ~R ^ )"

It remains to determine the ratio /j,/R. The potential of a doublet is — while that of the field R may be tak,en to be — Rx + G. Thus the total

potential of a single doublet and the external field is

f^-Rx + C,

and this makes the surface r = a an equipotential if —3 = R. Thus the

surfaces of the molecules will be equipotentials if we imagine the molecules to be spheres of radius a, and the centres of the doublets to coincide with the centres of the spheres, the strength of each doublet being Ra?.

Putting (j, = Ra3, equation (74) becomes*

K-K0

K

= 47rna8.

Now in unit volume of dielectric, the space occupied by the n molecules

4-7T . K — K

is -5- na?. Calling this quantity 6, we have — ^r- - = SO, or, since our calcu-

lations only hold on the hypothesis that 6 is small,

§r = 1 + 30 (75).

If the lines of force went straight across from one plate of the condenser

• Clausius (Mech. Wdrmetheorie, 2, p. 94) has obtained the relation

K-Kp _4w 3 K+2KQ~ 3 Ua' by considering the field inside a sphere of dielectric. The value of K must of course be inde- pendent of the shape of the piece of the dielectric considered. The apparent discrepancy in the two values of K obtained, is removed as soon as we reflect that both proceed on the assumption that K- K0 is small, for the results agree as far as first powers of K- Kq. Pagliani (Accad. dei Lined, 2, p. 48) finds that in point of fact the equation

— 7r-^ = 47rna3 A

agrees better with experiment than the formula of Clausius.

9—2

132

Dielectrics and Inductive Capacity

[CH. V

to the other, the proportion of the length of each which would be inside a conductor would, on the average, be 6. Since there is no fall of a potential inside a conductor, the total fall of potential from one plate to the other would be only 1 — 6 times what it would be if the molecules were absent, and the ratio K/K0 would be 1/(1 — 0) or, if 0 is small, 1 + 6. Since, however, the lines of force tend to run through conductors wherever possible, there is more shortening of lines of force than is shewn by this simple calculation. Equation (75) shews that when the molecules are spherical the effect is three times that given by this simple calculation. For other shapes of molecules the multiplying factor might of course be different.

Equation (75) gives at once a method of determining 0 for substances for which 6 is small, namely gases, but, owing to the unwarranted assumption that the molecules are spherical, the results will be true as regards order of magnitude only. If the dielectric is a gas at atmospheric pressure, the value of n is known, being about 2-705 x 1019, and this enables us to calculate the value of a.

150. The  following  table  gives  series  of  values  of  -~  for  gases  at  atmo- 
     

-fio spheric pressure:

Gas

=7- observed

Autho- rity*

Mean —

■&0

a calculated

(Mossotti's

Theory)

a calculated

(Theory of

Gases) f

Helium

He

1 -0000724

3

1-0000724

•596xl0-8

1-10 x lO-8

Hydrogen

H2

1 -000264 1 -000264

1 2

1-000264

•916xl0-3

1-36x10-8

Oxygen ...

02

1-000543

3

1 -000543

MTxlO"8

1-81 xl0-8

Argon

Ar

1-000566

3

1*000566

1-18X10-8

1-82x10-8

Air ... ...

—

1 -000590 1 -000586

1

2

1-000588

1*19x10-8

l-87xlO-3

Nitrogen

N2

1-000594

3

1-000594

1-20x10-8

1-90x10-8

Carbon Monoxide

CO

1-000690 1 -000694

1

2

1 -000692

1-26x10-8

1-89x10-8

Carbon Dioxide

co2

1 -000946 1 -000984

1 2

1-000965

1-40X10-8

2-31xl0-8

Nitrous Oxide ...

N20

1 -000994 1-001158

1

2

1-001082

1-46x10-8

2-32X10-8

Ethylene ...

C2H4

1-001312 1-001458

1

2

1-001385

1-60x10-8

2-77x10-8

• Authorities :— 1. Boltzmann, Wiener Sitzungsber. 09, p. 795.

2. J. Klemencic, Wiener Sitzungsber. 91, p. 712.

3. These values are calculated from the refractive indices for Sodium Light. | Jeans, Dynamical Theory of Gases, 4th Edition, p. 327.

149-151] Molecular Theory 133

The last two columns give respectively the values of a calculated from equation (75), and the value of a given by the Theory of Gases. The two sets of values do not agree exactly — this could not be expected when we remember the magnitude of the errors introduced in treating the molecules as spherical. But what agreement there is supplies very significant evidence as to the truth of the theory of molecular polarisation.

151. It still remains to explain what physical property of the molecule justifies us in treating its surface as a perfect conductor. It has already been explained that all matter contains a number of negatively charged par- ticles or electrons. These form the outer layers of the atoms and molecules and it is by their motion that the conduction of electricity is effected. In a dielectric there is no conduction, so that each electron must remain permanently associated with the same molecule. There is, however, plenty of evidence that the electrons are not rigidly fixed to the molecules but are free to move within certain limits. The molecule may be regarded as consisting partially or wholly of a cluster of electrons, normally at rest in positions of equilibrium under the various attractions and repulsions present, but capable of vibrating about these positions. Under the influence of an external field of force, the electrons will move slightly from their equilibrium positions — we may imagine that a kind of tidal motion of electrons takes place in the molecule. Obviously, by the time that equilibrium is attained, the outer surface of the molecule must be an equipotential. This, however, is exactly what is required for Mossotti's hypothesis. We may accordingly abandon the conception of conducting spheres, which was only required to make the surface of the molecule an equipotential, and may, without impairing the power of Mossotti's explanation, replace these conducting spheres by shells of electrons. If in some way we can further replace these shells by rings of electrons in rapid orbital motion, the modified hypothesis will be in very close agreement with modern beliefs as to the structure of matter.

On this view, the quantity a tabulated in the sixth column of the table on p. 132, will measure the radius of the outermost shell of electrons. Even outside this outermost shell, however, there will be an appreciable field of force, so that when two molecules of a gas collide there will in general be a considerable distance between their outermost layers of electrons. Thus if the collisions of molecules in a gas are to be regarded as the collisions of elastic spheres, the radius of these spheres must be supposed to be con- siderably greater than a. Now it is the radius of these imaginary elastic spheres which we calculate in the Kinetic Theory of Gases : there is therefore no difficulty in understanding the differences between the two sets of values for a given in the table of p. 132.

It is known that molecules are not in general spherical in shape, but, as we shall see below, there is no difficulty in extending Mossotti's theory to cover the case of non-spherical molecules.

134:

Dielectrics and Inductive Capacity

[CH. V

Anisotropic Media.

152. There are some dielectrics, generally of crystalline structure, in which Faraday's relation between polarisation and intensity is found not to be true. The polarisation in such dielectrics is not, in general, in the same direction as the intensity, and the angle between the polarisation and intensity and also the ratio of these quantities are found to depend on the direction of the field relatively to the axes of the crystal. We shall find that the conception of molecular action accounts for these peculiarities of crystalline dielectrics.

Let us consider an extreme case in which the spherical molecules of fig. 46 are replaced by a number of very elongated or needle-shaped bodies. The lines of force will have their effective lengths shortened by an amount which depends on whether much or little of them falls within the material of the needle-shaped molecules, and, as in § 149, there will be an equation of the form

where 6 is the aggregate volume of the number of molecules which occur in a unit volume of the gas, and s is a numerical multiplier. But it is at once clear that the value of s will depend not only on the shape but also on the orientation of the molecules. Clearly the value of s will be greatest when the needles are placed so that their greatest length lies in the direction of

4

Fig. 46 a.

i

-f+

t

-1 +

i

-1+

Fio. 46 b.

Flo. 46 c,

the lines of force, as in fig. 46 a, and will be least when the needles lie at right angles to this position, as in fig. 46 b. Or to put the matter in another way, a piece of dielectric in which the molecules are needle-shaped and parallel will exhibit different values of K according as the field of force is parallel or at right angles to the lengths of the needles.

152] Anisotropic Media 135

This extreme case illustrates the fundamental property of crystalline dielectrics, but it ought to be understood that in actual substances the values of K do not differ so much for different directions as this extreme case might be supposed to suggest. For instance for quartz, one of the substances in which the difference is most marked, Curie finds the extreme values of K to be 4-55 and 4 -49.

Before attempting to construct a mathematical theory of the behaviour of a crystalline dielectric we may examine the case of a dielectric having needle-shaped molecules placed parallel to one another, but so as to make any angle 0 with the direction of the lines of force, as in fig. 46 c.

It is at once clear that not only are the effective lengths of the lines of force shortened by the presence of the molecules, but also the directions of the lines of force are twisted. It follows that the polarisation, regarded as a vector as in § 128, must in general have a direction different from that of the average intensity R of the field.

To analyse such a case we shall, as in § 146, regard the field near any point as the superposition of two fields :

(i) the field which arises from the doublets on the neighbouring molecules, say a field of components of intensity X1} Yly Z1;

(ii) the field caused by the doublets arising from the distant molecules and from the charges outside the dielectric, say a field of components of intensity Xit Y2, Z2.

Clearly in the case we are now considering, the intensities R1} R2 of these fields will not be in the same direction.

The components of intensity of the whole field are given by

X = Xa + X2, etc.

To discuss the first part of the field, let us regard the whole field as the superposition of three fields, having respectively components (X, 0, 0), (0, Y, 0) and (0, 0, Z). If the molecules are spherical, or if, not being spherical, their orientations in space are distributed at random, then clearly the field of components (X, 0, 0) will induce doublets which will produce simply a field of components (K'X, 0, 0) where K' is a constant. But if the molecules are neither spherical in shape nor arranged at random as regards •their orientations in space, it will be necessary to assume that the induced doublets give rise to a field of components

K'nX, K,X, K\3X.

136 Dielectrics and Inductive Capacity [ch. v

On superposing the doublets induced by the three fields (X, 0, 0), (0, Y, 0) and (0, 0, Z), we obtain

X, = K'nX + K'a Y+ K'S1Z \

Yl = K'uX + K'mY+K'nZ - (76).

Zx = K'13X + K'w Y + K'33Z J

Thus we have relations of the form

4,Trf=KuX + K21Y+KslZ\

I

expressing the relations between polarisation and intensity.

4tt# = Kl2X + KUY + K32Z | (77)

4?r/i = Kl3X + Kw Y + K33Z

These are the general equations for crystalline media. "We shall shortly prove (§ 176) that

K12 = K21, K23 = K32, KS1 = K13 (78),

so that there are not nine, but only six, independent constants.

Non-spherical Molecules.

152 a. A medium in which the molecules are not spherical but are oriented at random can be discussed in a similar way. The whole field (X, Y, Z) may be regarded as the superposition of three fields (X, 0, 0), (0, Y, 0) and (0, 0, Z). The induced doublets produced by the first field will produce a field of com- ponents

(K'X, 0, 0),

the components along Oy and Oz necessarily vanishing on account of the

random orientation of the molecules. The other fields similarly produce

induced fields

(0, K'Y, 0) and (0, 0, K'Z),

whence we readily obtain equations of the form

4>Trf=KX> 4>7ig = KY, ^h = KZ.

Thus Mossotti's theory can readily be extended to non-spherical molecules, but the difficulty remains that according to modern views, a molecule does not consist of layers of electrons at rest, but of systems of electrons in orbital motion. It will not be possible to make the appropriate modification in the theory until the exact nature of this orbital motion is known.

152] Examples 137

EXAMPLES.

1. A spherical condenser, radii a, b, has air in the space between the spheres. The inner sphere receives a coat of paint of uniform thickness t and of a material of which the inductive capacity is K. Find the change produced in the capacity of the condenser.

2. A conductor has a charge e, and V1} V2 are the potentials of two equipotential surfaces completely surrounding it ( Vx > V2). The space between these two surfaces is now filled with a dielectric of inductive capacity K. Shew that the change in the energy of the system is

\e{Jx-n){K-)lK.

3. The surfaces of an air-condenser are concentric spheres. If half the space between the spheres be filled with solid dielectric of specific inductive capacity K, the dividing surface between the solid and the air being a plane through the centre of the spheres, shew that the capacity will be the same as though the whole dielectric were of uniform specific inductive capacity £ (1 +K).

4. The radii of the inner and outer shells of two equal spherical condensers, remote from each other and immersed in an infinite dielectric of inductive capacity K, are respectively a and b, and the inductive capacities of the dielectric inside the condensers are Kit K2. Both surfaces of the first condenser are insulated and charged, the second being uncharged. The inner surface of the second condenser is now connected to earth, and the outer surface is connected to the outer surface of the first condenser by a wire of negligible capacity. Shew that the loss of energy is

Q*{2(b-a)K+aK2} 2Kb{{b-a)K+aK2y

where Q is the quantity of electricity which flows along the wire.

5. The outer coating of a long cylindrical condenser is a thin shell of radius a, and the dielectric between the cylinders has inductive capacity K on one side of a plane through the axis, and K' on the other side. Shew that when the inner cylinder is connected to earth, and the outer has a charge q per unit length, the resultant force on the outer cylinder is

4f(K-K') 7ra(A'+A") per unit length.

6. A heterogeneous dielectric is formed of n concentric spherical layers of specific inductive capacities A'j, A"2, ... A'n, starting from the innermost dielectric, which forms a solid sphere ; also the outermost dielectric extends to infinity. The radii of the spherical boundary surfaces are au a2, ... a„_! respectively. Prove that the potential clue to a quantity Q of electricity at the centre of the spheres at a point distant r from the centre in the dielectric K, is

A,\r aj A8 + 1\a, atJrJ Ana

h

138 Dielectrics and Inductive Capacity [ch. v

7. A condenser is formed by two rectangular parallel conducting plates of breadth b and area A at distance d from each other. Also a parallel slab of a dielectric of thickness t and of the same area is between the plates. This slab is pulled along its length from between the plates, so that only a length x is between the plates. Prove that the electric force sucking the slab back to its original position is

27r£2dbt'(d-t') {A{d-t')+xbt'}*'

where t' = t(K- 1)1 K, K is the specific inductive capacity of the slab, E is the charge, and the disturbances produced by the edges are neglected.

8. Three closed surfaces 1, 2, 3 are equipotentials in an electric field. If the space between 1 and 2 is filled with a dielectric K, and that between 2 and 3 is filled with a dielectric K\ shew that the capacity of a condenser having 1 and 3 for faces is C, given by

C~AK+ BE"

where A, B are the capacities of air-condensers having as faces the surfaces 1, 2 and 2, 3 respectively.

9. The surface separating two dielectrics (Ki, E2) has an actual charge <r per unit area. The electric forces on the two sides of the boundary are F1} F2 at angles c^, c2 with the common normal. Shew how to determine F2, and prove that

E2 cot c2 = E1 cot Ci ( 1

FiFi cos CiJ '

10. The space between two concentric spheres radii a, b which are kept at potentials A, B, is filled with a heterogeneous dielectric of which the inductive capacity varies as the ?ith power of the distance from their common centre. Shew that the potential at any point between the surfaces is

Aan + 1-Bbn + 1 an + 1bn + 1 A-B

an + 1 _ ftn + 1 rn + 1 an + 1 _ £n + 1 '

11. A condenser is formed of two parallel plates, distant h apart, one of which is at zero potential. The space between the plates is filled with a dielectric whose inductive capacity increases uniformly from one plate to the other. Shew that the capacity per unit area is

K2 — K1 Airh log Z2/Z1'

where Kx and E2 are the values of the inductive capacity at the surfaces of the plate. The inequalities of distribution at the edges of the plates are neglected.

12. A spherical conductor of radius a is surrounded by a concentric spherical conducting shell whose internal radius is b, and the intervening space is occupied by a

dielectric whose specific inductive capacity at a distance r from the centre is . If the

inner sphere is insulated and has a charge E, the shell being connected with the earth,

E b(c+r) prove that the potential in the dielectric at a distance r from the centre is — log —} ^ .

Examples 139

13. A spherical conductor of radius a is surrounded by a concentric spherical shell of

radius b, and the space between them is filled with a dielectric of which the inductive

capacity at distance r from the centre is fiep2 p3 where p = rja. Prove that the capacity

of the condenser so formed is

hl 2/^a(ea2-e)_1.

_r

14. If the specific inductive capacity varies as e <*, where r is the distance from a fixed point in the medium, verify that a solution of the differential equation satisfied by the potential is

/a\2r l r ri -]

and hence determine the potential at any point of a sphere, whose inductive capacity is the above function of the distance from the centre, when placed in a uniform field of force.

15. Shew  that  the  capacity  of  a  condenser  consisting  of  the  conducting  spheres  r=a, 
    

r=6, and a heterogeneous dielectric of inductive capacity K=f(d, (p), is

i^^ f f f(&, & sin 0d8 dtp.

16. In an imaginary crystalline medium the molecules are discs placed so as to be all parallel to the plane of ocy. Shew that the components of intensity and polarisation are connected by equations of the form

AttJ = KnX + K2iY ; 4ng = KnX+K22'ir) 4rrh = K33Z.

CHAPTER VI

THE STATE OF THE MEDIUM IN THE ELECTROSTATIC FIELD

153. The whole electrostatic theory has so far been based simply upon Coulomb's Law of the inverse square of the distance. We have supposed that one charge of electricity exerts certain forces upon a second distant charge, but nothing has been said as to the mechanism by which this action takes place. In handling this question there are two possibilities open. We may either assume " action at a distance " as an ultimate explanation — i.e. simply assert that two bodies act on one another across the intervening space, without attempting to go any further towards an explanation of how such action is brought about — or we may tentatively assume that some medium connects the one body with the other, and examine whether it is possible to ascribe properties to this medium, such that the observed action will be transmitted by the medium. Faraday and Maxwell followed the latter course. They refused to admit " action at a distance " as an ultimate explana- tion of electric phenomena, finding such action unthinkable unless transmitted by an intervening medium.

154. It is worth enquiring whether there is any valid a priori argument which compels us to resort to action through a medium. Some writers have attempted to use the phenomenon of Inductive Capacity to prove that the energy of a condenser must reside in the space between the charged plates, rather than on the plates themselves — for, they say, change the medium between the plates, keeping the plates in the same condition, and the energy is changed. A study of Faraday's molecular explanation of the action in a dielectric will shew that this argument proves nothing as to the real question at issue. It goes so far as to prove that when there are molecules placed between electric charges, these molecules themselves acquire charges, and so may be said to be new stores of energy, but it leaves untouched the question of whether the energy resides in the charges on the molecules or in the ether between them.

Again, the phenomenon of induction is sometimes quoted against action at a distance — a small conductor placed at a point P in an electrostatic field shews phenomena which depend on the electric intensity at P. This is taken to shew that the state of the ether at the point P before the introduction of the conductor was in some way different from what it would have been if there had not been electric charges in the neighbourhood. But all that is proved is that the state of the point P after the introduction of the conductor

153, 1 54] The State of the Medium in the Electrostatic Field 141

will be different from what it would have been if there had not been electric charges in the neighbourhood, and this can be explained equally well either by action at a distance or action through a medium. The new conductor is a collection of positive and negative charges : the phenomena under question are produced by these charges being acted upon by the other charges in the field, but whether this action is action at a distance or action through a medium cannot be told.

Indeed, it will be seen that, viewed in the light of the electron-theory and of Faraday's theory of dielectric polarisation, electrical action stands on just the same level as gravitational action. In each case the system of forces to be explained may be regarded as a system of forces between indestructible centres, whether of electricity or of matter, and the law of force is the law of the inverse square, independently of the state of the space between the centres. Now no scientist would claim that there is any A priori proof that gravitation is transmitted through a medium — indeed the trend of opinion at present is quite in the opposite direction — and this fact in itself suffices to shew that there is no a priori means of establishing that electrical action is transmitted through a medium.

Failing an a priori argument, an attempt may be made to disprove action at a distance, or rather to make it improbable, by an appeal to experience. It may be argued that as all the forces of which we have experience in every-day life are forces between substances in contact, therefore it follows by analogy that forces of gravitation, electricity and magnetism, must ultimately reduce to forces between substances in contact — i.e. must be transmitted through a medium. Upon analysis, however, it will be seen that this argument divides all forces into two classes :

(a) Forces of gravitation, electricity and magnetism, which appear to act at a distance.

(/3) Forces of pressure and impact between solid bodies, hydrostatic pressure, etc. which appear to act through a medium.

The argument is now seen to be that because class (/3) appear to act through a medium, therefore class (a) must in reality act through a medium. The argument could, with equal logical force, be used in the exactly opposite direction : indeed it has been so used by the followers of Boscovitch. The Newtonian discovery of gravitation, and of apparent action at a distance, so occupied the attention of scientists at the time of Boscovitch that it seemed natural to regard action at a distance as the ultimate basis of force, and to try to interpret action through a medium in terms of action at a distance. The reversion from this view came, as has been said, with Faraday.

Hertz's subsequent discovery of the finite velocity of propagation of electric action, which had previously been predicted by Maxwell's theory, came to the support of Faraday's view. To see exactly what is meant by this finite velocity of propagation, let us imagine that we place two uncharged conductors A, B at a distance r from one another. By charging A, and so performing work at A, we can induce charges on conductor B, and when this has been done, there will be an attraction between conductors A and B. We can suppose that conductor A is held fast, and that conductor B is allowed to move towards A, work being performed by the attraction from conductor A. We are now recovering from B work which was originally performed at A. The experiments of Hertz shew that a finite time is required before any of the work spent at A becomes available at B. A natural explanation is to suppose that work spent on A assumes the form of energy which spreads itself out through the whole of space, and that the finite time observed before energy becomes available at B is the time required for the first part of the advancing energy to travel from A to B. This explanation involves regarding energy

142 The State of the Medium in the Electrostatic Field [on. vi

as a definite physical entity, capable of being localised in space. It ought to be noticed that our senses give us no knowledge of energy as a physical entity : we experience force, not energy. And the fact that energy appears to be propagated through space with finite velocity does not justify us in concluding that it has a real physical existence, for, as we shall see, the potential appears to be propagated in the same way, and the potential can only be regarded as a convenient mathematical fiction.