A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 7 of 28

68.] In order to simplify the mathematical process, it is con- venient to consider the action of an electrified body, not on another body of any form, but on an indefinitely small body, charged with an indefinitely small amount of electricity, and placed at any point of the space to which the electrical action extends. By making the charge of this body indefinitely small we render insensible its disturbing action on the charge of the first body.

72 ELECTROSTATICS. [69.

Let e be the charge of the small body, and let the force acting on it when placed at the point (#,y, z) be Re, and let the direction- cosines of the force be I, m, n, then we may call R the resultant electrical_Intensity at the point (#, ^, z).

If X, Y, Z denote the components of R, then

X-Rl, Y=Rm, Z=Rn.

In speaking of the resultant electrical intensity at a point, we do not necessarily imply that any force is actually exerted there, but only that if an electrified body were placed there it would be acted on by a force Re, where e is the charge of the body *.

Definition. The Resultant electric Intensity at any point is the force which would be exerted on a small body charged with the unit of positive electricity, if it were placed there without disturbing the actual distribution of electricity.

This force not only tends to move a body charged with electricity, but to move the electricity within the body, so that the positive electricity tends to move in the direction of R and the negative electricity in the opposite direction. Hence the quantity R is also called the E lee t r omo t i ve Intensity at the point (a?, y, z).

When we wish to express the fact that the resultant intensity is a vector, we shall denote it by the German letter (£. If the body is a dielectric, then, according to the theory adopted in this treatise, the electricity is displaced within it, so that the quantity of electricity which is forced in the direction of (£ across unit of area fixed perpendicular to (£ is

£> = —#(£;

where $) is the displacement, (£ the resultant intensity, and K the specific inductive capacity of the dielectric.

If the body is a conductor, the state of constraint is continually giving way, so that a current of conduction is produced and main- tained as long as (£ acts on the medium.

Line-Integral of Electric Intensity, or Electromotive Force along an Arc of a Curve.

69.] The Electromotive force along a given arc AP of a curve is numerically measured by the work which would be done by the

• The Electric and Magnetic Intensity correspond, in electricity and mag- netism, to the intensity of gravity, commonly denoted by g, in the theory of heavy bodies.

70.] ELECTKOMOTIVE FORCE. 73

electric force on a unit of positive electricity carried along the curve from J, the beginning, to P, the end of the arc.

If s is the length of the arc, measured from A, and if the re- sultant intensity 72 at any point of the curve makes an angle e with the tangent drawn in the positive direction, then the work done on unit of electricity in moving along the element of the curve /fo will be Rcostds,

and the total electromotive force E will be

E = / ~R cos e ds,

J

the integration being extended from the beginning to the end of the arc.

If we make use of the components of the intensity, the expres- sion becomes

ds

If X, Y, and Z are such that Xd% + Ydy + Zdz is the complete differential of — F, a function of #, y, z, then

E = fP(Xdx + Ydy + Zdz) = - f*dr = VA- VP ;

J A J A

where the integration is performed in any way from the point A to the point P, whether along the given curve or along any other line between A and P.

In this case V is a scalar function of the position of a point in space, that is, when we know the coordinates of the point, the value of V is determinate, and this value is independent of the position and direction of the axes of reference. See Art. 16.

On Functions of the Position of a Point.

In what follows, when we describe a quantity as a function of the position of a point, we mean that for every position of the point the function has a determinate value. We do not imply that this value can always be expressed by the same formula for all points of space, for it may be expressed by one formula on one side of a given surface and by another formula on the other side.

On Potential Functions.

70.] The quantity Xdx+ Ydy + Zdz is an exact differential whenever the force arises from attractions or repulsions whose in- tensity is a function of the distances from any number of points.

74 ELECTEOSTATICS. [71.

For if r-L be the distance of one of the points from the point (#, y, z\ and if Rl be the repulsion, then

with similar expressions for Yl and ^, so that

and since ^ is a function of rx only, R1 dr^ is an exact differential of some function of r^ , say — V^ .

Similarly for any other force R^ acting from a centre at dis- tance r2, X2dx + Y2dy + Zzdz = R2dr2 = — <*JJ.

But X = Xj + Xg-f&c. and T and # are compounded in the same way, therefore

Xdx+ Ydy+Zdz = -d^-d^+be. - -dV.

The integral of this quantity, under the condition that it vanishes at an infinite distance, is called the Potential Function.

The use of this function in the theory of attractions was intro- duced by Laplace in the calculation of the attraction of the earth. Green, in his essay ' On the Application of Mathematical Analysis to Electricity/ gave it the name of the Potential Function. Gauss, working independently of Green, also used the word Potential. Clausius and others have applied the term Potential to the work which would be done if two bodies or systems were removed to an infinite distance from one another. We shall follow the use of the word in recent English works, and avoid ambiguity by adopting the following definition due to Sir W. Thomson.

Definition of Potential. The Potential at a Point is the work which would be done on a unit of positive electricity by the elec- tric forces if it were placed at that point without disturbing the electric distribution, and carried from that point to an infinite distance : or, what comes to the same thing, the work which must be done by an external agent in order to bring the unit of positive electricity from an infinite distance (or from any place where the potential is zero) to the given point.

71.] Expressions for the Resultant Intensity and its components in terms of the Potential.

Since the total electromotive force along any arc AB is

72.] POTENTIAL^ 75

if we put ds for the arc AB we shall have for the force resolved in the direction of ds,

.72 cos e = =- :

ds

whence, by assuming ds parallel to each of the axes in succession, we get

X--— . r-- — Z--—'

dx dy dz '

E

ldV ~~ Ida

dV

dV

T 7

r

We shall denote the intensity itself, whose magnitude, or tensor, is E and whose components are X, Y, Z, by the German letter ($, as in Arts. 17 and 68.

The Potential at all Points within a Conductor is the same. 72.] A conductor is a body which allows the electricity within it to move from one part of the body to any other when acted on by electromotive force. When the electricity is in equilibrium there can be no electromotive force -acting within the conductor. Hence E = 0 throughout the whole space occupied by the con- ductor. From this it follows that

dV dV dV

^=°> *=°' ^=°; and therefore for every point of the conductor

r=c,

where C is a constant quantity.

Since the potential at all points within the substance of the conductor is C, the quantity C is called the Potential of the con- ductor. C may be denned as the work which must be done by external agency in order to bring a unit of electricity from an infinite distance to the conductor, the distribution of electricity being supposed not to be disturbed by the presence of the unit.

It will be shewn at Art. 246 that in general when two bodies of different kinds are in contact, an electromotive force acts from one to the other through the surface of contact, so that when they are in equilibrium the potential of the latter is higher than that of the former. For the present, therefore, we shall suppose all our conductors made of the same metal, and at the same temperature.

If the potentials of the conductors A and B be VA and VB re- spectively, then the electromotive force along a wire joining A and B will be F-F

76 ELECTROSTATICS. [73.

in the direction AS, that is, positive electricity will tend to pass from the conductor of higher potential to the other.

Potential, in electrical science, has the same relation to Elec- tricity that Pressure, in Hydrostatics, has to Fluid, or that Tem- perature, in Thermodynamics, has to Heat. Electricity, Fluids, and Heat all tend to pass from one place to another, if the Poten- tial, Pressure, or Temperature is greater in the first place than in the second. A fluid is certainly a substance, heat is as certainly not a substance, so that though we may find assistance from ana- logies of this kind in forming clear ideas of formal relations of electrical quantities, we must be careful not to let the one or the other analogy suggest to us that electricity is either a substance like water, or a state of agitation like heat.

Potential due to any Electrical System.

73.] Let there be a single electrified point charged with a quantity e of electricity, and let r be the distance of the point #', y', / from it, then A> ra>

7= / Edr = ~dr = -• Jr Jr r2 r

Let there be any number of electrified points whose coordinates are (#15 yl5 ^), (#2) y^ z2), &c. and their charges elt e2, &c., and let their distances from the point (#', y'> z) be rlt r2, &c., then the potential of the system at (#', y\ /) will be

Let the electric density at any point (UK, y> z) within an elec- trified body be p, then the potential due to the body is

7 = (j(p-dxdydz',

where r = {(cc-xj + (y-yj + (z-zj}^

the integration being extended throughout the body.

On the Proof of the Law of the Inverse Square.

74 «.] The fact that the force between electrified bodies is inversely as the square of the distance may be considered to be established by Coulomb's direct experiments with the torsion-balance. The results, however, which we derive from such experiments must be regarded as affected by an error depending on the probable error of each experiment, and unless the skill of the operator be very great,

PROOF OF THE LAW OF FORCE. 77

the probable error of an experiment with the torsion-balance is considerable.

A far more accurate verification of the law of force may be deduced from an experiment similar to that described at Art 32 (Exp. VII).

Cavendish, in his hitherto unpublished work on electricity, makes the evidence of the law of force depend on an experiment of this kind.

He fixed a globe on an insulating1 support, and fastened two hemispheres by glass rods to two wooden frames hinged to an axis so that the hemispheres, when the frames were brought together, formed an insulated spherical shell concentric with the globe.

The globe could then be made to communicate with the hemispheres by means of a short wire, to which a silk string was fastened so that the wire could be removed without discharging the apparatus.

The globe being in communication with the hemispheres, he charged the hemispheres by means of a Leyden jar, the potential of which had been previously measured by an electrometer, and immediately drew out the communicating wire by means of the silk string, removed and discharged the hemispheres, and tested the electrical condition of the globe by means of a pith ball electro- meter.

No indication of any charge of the globe could be detected by the pith ball electrometer, which at that time (1773) was considered the most delicate electroscope.

Cavendish next communicated to the globe a known fraction of the charge formerly communicated to the hemispheres, and tested the globe again with his electrometer.

He thus found that the charge of the globe in the original experiment must have been less than ^ of the charge of the whole apparatus, for if it had been greater it would have been detected by the electrometer.

He then calculated the ratio of the charge of the globe to that of the hemispheres on the hypothesis that the repulsion is inversely as a power of the distance differing slightly from 2, and found that if this difference was -^ there would have been a charge on the globe equal to -^ of that of the whole apparatus, and therefore capable of being detected by the electrometer.

74 b.~\ The experiment has recently been repeated at the Cavendish Laboratory in a somewhat different manner.

The hemispheres were fixed on an insulating stand, and the globe

78 ELECTROSTATICS. [74 6.

fixed in its proper position within them by means of an ebonite ring. By this arrangement the insulating support of the globe was never exposed to the action of any sensible electric force, and therefore never became charged, so that the disturbing effect of electricity creeping along the surface of the insulators was entirely removed.

Instead of removing the hemispheres before testing the potential of the globe, they were left in their position, but discharged to earth. The effect of a given charge of the globe on the electro- meter was not so great as if the hemispheres had been removed, but this disadvantage was more than compensated by the perfect security afforded by the conducting vessel against all external electric disturbances.

The short wire which made the connexion between the shell and the globe was fastened to a small metal disk which acted as a lid to a small hole in the shell, so that when the wire and the lid were lifted up by a silk string, the electrode of the electrometer could be made to dip into the hole and rest on the globe within.

The electrometer was Thomson's Quadrant Electrometer described in Art. 219. The case of the electrometer and one of the electrodes were always connected to earth, and the testing electrode was con- nected to earth till the electricity of the shell had been discharged.

To estimate the original charge of the shell, a small brass ball was placed on an insulating support at a considerable distance from the shell.

The operations were conducted as follows : —

The shell was charged by communication with a Leyden jar.

The small ball was connected to earth so as to give it a negative charge by induction, and was then left insulated.

The communicating wire between the globe and the shell was removed by a silk string.

The shell was then discharged, and kept connected to earth.

The testing electrode was disconnected from earth, and made to touch the globe, passing through the hole in the shell.

Not the slightest effect on the electrometer could be observed.

To test the sensitiveness of the apparatus the shell was discon- nected from earth and the small ball was discharged to earth. The electrometer then showed a positive deflection, D.

The negative charge of the brass ball was about -fa of the ori- ginal charge of the shell, and the positive charge induced by the ball when the shell was put to earth was about . J of that of the balL

74 £.] PEOOF OF THE LAW OF FORCE. 79

Hence when the ball was put to earth the potential of the shell, as indicated by the electrometer, was about T|F of its original potential.

But if the repulsion had been as rq~2, the potential of the globe would have been —0-1478 q of that of the shell by equation 22, p. 81.

Hence if + d be the greatest deflexion of the electrometer which could escape observation, and D the deflexion observed in the second part of the experiment, q cannot exceed

±lil.

~ 72 D

Now even in a rough experiment D was more than 300 d, so that q cannot exceed 1

• 21600*

Theory of the Experiment.

74 c.~\ To find the potential at any point due to a uniform spherical shell, the repulsion between two units of matter being any given function of the distance.

Let 0 (/•) be the repulsion between two units at distance /-, and let/(r) be such that

(r)rfr. (1)

Let the radius of the shell be a, and its surface density o-, then, if a denotes the whole mass of the shell,

a = 477 a2 a-. (2)

Let b denote the distance of the given point from the centre of the shell, and let r denote its distance from any given point of the shell.

If we refer the point on the shell to spherical coordinates, the pole being the centre of the shell, and the axis the line drawn to the given point, then

r2 = 02 + 52-2<z£cos0. (3)

The mass of the element of the shell is

<razsin6d(t>dO, (4)

and the potential due to this element at the given point is

aa2s{nof-&ded<i>; (5)

and this has to be integrated with respect to <j> from <j> = 0 to $ = 2 TT, which gives

(6)

r which has to be integrated from 0 = 0 to 0 = it.

80 ELECTROSTATICS. [74 C.

Differentiating (3) we find

rdr = absmOdO. (?)

Substituting the value of dd in (6) we obtain

2w<r !/»<&•, (8)

the integral of which is

-f(rj}, (9)

when rx is the greatest value of r, which is always a + b, and rL is the least value of r, which is b — a when the given point is out- side the shell and a— b when it is within the shell.

If we write a for the whole charge of the shell, and V for its potential at the given point, then for a point outside the shell

For a point on the shell itself

P=8

and for a point inside the shell

We have next to determine the potentials of two concentric spherical shells, the radii of the outer and inner shells being a and b, and their charges a and ft.

Calling the potential of the outer shell A, and that of the inner J5, we have by what precedes

In the first part of the experiment the shells communicate by the short wire and are both raised to the same potential, say V.

By putting A = £ = F, and solving the equations (13) and (14) for ft, we find the charge of the inner shell

fl -

In the experiment of Cavendish, the hemispheres forming the outer shell were removed to a distance which we may suppose in-

746.] PROOF OF THE LAW OF FORCE. 81

finite, and discharged. The potential of the inner shell (or globe) would then become

In the form of the experiment as repeated at the Cavendish Laboratory the outer shell was left in its place, but connected to earth, so that A = 0. In this case we find for the potential of the inner shell in terms of V

./»/_. I\ -Cl ~ Z\ x

(17)

74 d."\ Let us now assume, with Cavendish, that the law offeree is some inverse power of the distance, not differing much from the inverse square, and let us put

(f)(r) = rq~2; (18)

then f(r) = — L~f*+\ (19)

If we suppose q to be small, we may expand this by the ex- ponential theorem in the form

and if we neglect terms involving q2, equations (16) and (17) be- come

<»«>

from which we may determine q in terms of the results of the experiment.

740.] Laplace gave the first demonstration that no function of the distance except the inverse square satisfies the condition that a uniform spherical shell exerts no force on a particle within it *.

If we suppose that /3 in equation (15) is always zero, we may apply the method of Laplace to determine the form of f(r). We have by (15),

Differentiating twice with respect to £, and dividing by #, we find

f"(a + b] =f"(a — b). If this equation is generally true

/" (r) = <?0, a constant.

• Mec. Cel, I. 2. VOL. I. G

82 ELECTEOSTATTCS. [75.

Hence, /'(r) = <V + <?i5

and by (l) f° <t>(r)dr = £& = CQ+ ^-,

jr r r

We may observe, however, that though the assumption of Cavendish, that the force varies as some power of the distance, may appear less general than that of Laplace, who supposes it to be any function of the distance, it is the only one consistent with the fact that similar figures can be electrified so as to have similar electrical properties.

For if the force were any function of the distance except a power of the distance, the ratio of the force at two different distances would not be a function of the ratio of the distances, but would depend on the absolute value of the distances, and would therefore involve the ratios of these distances to an absolutely fixed length.

Indeed Cavendish himself points out that on his own hypothesis as to the constitution of the electric fluid, it is impossible for the distribution of electricity to be accurately similar in two conductors geometrically similar, unless the charges are proportional to the volumes. For he supposes the particles of the electric fluid to be closely pressed together near the surface of the body, and this is equivalent to supposing that the law of repulsion is no longer the inverse square, but that as soon as the particles come into contact, their repulsion begins to increase at a much greater rate with any further diminution of their distance.

Surface-Integral of Electric Induction, and Electric Displacement through a surface.

75.] Let R be the resultant intensity at any point of the surface, and e the angle which R makes with the normal drawn towards the positive side of the surface, then R cos e is the component of the intensity normal to the surface, and if dS is the element of the surface, the electric displacement through dS will be, by Art. 68,

— KRcoscdS. 4?r

since we do not at present consider any dielectric except air, K— 1.

We may, however, avoid introducing at this stag-e the theory of

electric displacement, by calling R cos e dS the Induction through

the element dS. This quantity is well known in mathematical

76.] ELECTRIC INDUCTION. 83

physics, but the name of induction is borrowed from Faraday. The surface-integral of induction is

R cos e dSt

and it appears by Art. 21, that if X, Y, Z are the components of R, and if these quantities are continuous within a region bounded by a closed surface S, the induction reckoned from within outwards is

f dY dZ

the integration being extended through the whole space within the surface.

Induction through a Closed Surface due to a Single Centre of Force.

76.] Let a quantity e of electricity be supposed to be placed at a point 0, and let r be the distance of any point P from 0, the force at that point is R — er~2 in the direction OP.

Let a line be drawn from 0 in any direction to an infinite dis- tance. If 0 is without the closed surface this line will either not cut the surface at all, or it will issue from the surface as many times as it enters. If 0 is within the surface the line must first issue from the surface, and then it may enter and issue any number of times alternately, ending by issuing from it.

Let e be the angle between OP and the normal to the surface drawn outwards where OP cuts it, then where the line issues from the surface, cos e will be positive, and where it enters, cos € will be negative.

Now let a sphere be described with centre 0 and radius unity, and let the line OP describe a conical surface of small angular aperture about 0 as vertex.

This cone will cut off a small element d<* from the surface of the sphere, and small elements dS19 dS2, &c. from the closed surface at the different places where the line OP intersects it.

Then, since any one of these elements dS intersects the cone at a distance r from the vertex and at an obliquity e,

dS = r2 sec e ^w ; and, since R = er~2, we shall have

R cos c dS = ±e d<* ;

the positive sign being taken when r issues from the surface, and the negative where it enters it.

If the point 0 is without the closed surface, the positive values

G 2

84 ELECTROSTATICS. [77.

are equal in number to the negative ones, so that for any direction of r, ^.R cos e dS = 0,

and therefore / / R cos e dS = 0,

the integration being extended over the whole closed surface.

If the point 0 is within the closed surface the radius vector OP first issues from the closed surface, giving a positive value of e da>, and then has an equal number of entrances and issues, so that in this case 2 R cos e dS = e da.

Extending the integration over the whole closed surface, we shall include the whole of the spherical surface, the area of which is 4 TT,

so that rr rr

I I R cos e dS = e I I d® = kite.

Hence we conclude that the total induction outwards through a closed surface due to a centre of force e placed at a point 0 is zero when 0 is without the surface, and ±ne when 0 is within the surface.

Since in air the displacement is equal to the induction divided by 4 TT, the displacement through a closed surface, reckoned out- wards, is equal to the electricity within the surface.

Corollary. It also follows that if the surface is not closed but is bounded by a given closed curve, the total induction through it is co e, where G> is the solid angle subtended by the closed curve at 0. This quantity, therefore, depends only on the closed curve, and the form of the surface of which it is the boundary may be changed in any way, provided it does not pass from one side to the other of the centre of force.

On the Equations of Laplace and Poisson.

77.] Since the value of the total induction of a single centre of force through a closed surface depends only on whether the centre is within the surface or not, and does not depend on its position in any other way, if there are a number of such centres elt e2, &c. within the surface, and £/, £/, &c. without the surface,

we shall have r r

I I Rcost dS = 47T0;

where e denotes the algebraical sum of the quantities of electricity at all the centres of force within the closed surface, that is, the total electricity within the surface, resinous electricity being reck- oned negative.

j3a.] EQUATIONS OF LAPLACE AND POISSON. 85

If the electricity is so distributed within the surface that the density is nowhere infinite, we shall have by Art. 64,

4 TT e = 4 TT / / / p dx dy dz, and by Art. 75,

If we take as the closed surface that of the element of volume dx dy dz, we shall have, by equating these expressions,

dX dY dZ

-T -- \r -j -- h -7- = 47rp;

dx dy dz

and if a potential V exists, we find by Art. 7 1 , d27

This equation, in the case in which the density is zero, is called Laplace's Equation. In its more general form it was first given by Poisson. It enables us, when we know the potential at every point, to determine the distribution of electricity. We shall denote, as in Art. 26, the quantity

.

T5 + T2 + TT bv - oar dy2 dz2

and we may express Poisson's equation in words by saying that the electric density multiplied by 4w is the concentration of the potential. Where there is no electrification, the potential has no concentration, and this is the interpretation of Laplace's equation.

By Art. 72, V is constant within a conductor. Hence within a conductor the volume-density is zero, and the whole charge must be on the surface.

If we suppose that in the superficial and linear distributions of electricity the volume-density p remains finite, and that the elec- tricity exists in the form of a thin stratum or a narrow fibre, then, by increasing p and diminishing the depth of the stratum or the section of the fibre, we may approach the limit of true superficial or linear distribution, and the equation being true throughout the process will remain true at the limit, if interpreted in accordance with the actual circumstances.

Variation of the Potential at a Charged Surface. 78 a.] The potential function, F, must be physically continuous in the sense defined in Art. 7, except at the bounding surface of

86 ELECTBOSTATICS. [78 a.

two different media, in which case, as we shall see in Art. 246, there may be a difference of potential between the substances, so that when the electricity is in equilibrium, the potential at a point in one substance is higher than the potential at the contiguous point in the other substance by a constant quantity, C, depending on the natures of the two substances and on their temperatures.

But the first derivatives of V with respect to #, y, or z may be discontinuous, and, by Art. 8, the points at which this discontinuity occurs must lie in a surface, the equation of which may be expressed in the form 0 = 0 (#} y^ z) = o. (l)

This surface separates the region in which $ is negative from the region in which 0 is positive.

Let VL denote the potential at any given point in the negative region, and V2 that at any given point in the positive region, then at any point in the surface at which $ = 0, and which may be said to belong to both regions,

r1 + c=%, (2)

where C is the constant excess of potential, if any, in the substance on the positive side of the surface.

Let I, m, n be the direction-cosines of the normal v2 drawn from a given point of the surface into the positive region. Those of the normal vl drawn from the same point into the negative region will be — I, — m, and — n.

The rates of variation of V along the normals are

(3) (4)

dv± ax ay dz

dK 7dK d\ dK

T*-= l-r^ + m-^- +n-jl-. dv2 dx dy dz

Let any line be drawn on the surface, and let its length, measured from a fixed point in it, be s, then at every point of the surface, and therefore at every point of this line, V^— T[ = C. Differentiating this equation with respect to s, we get

_

dx " dx dsdy " dy ds \4& " dz > ds

and since the normal is perpendicular to this line , dx dy dz

786.] POTENTIAL NEAR A CHARGED SURFACE. 87

From (3), (4), (5), (6) we find

AVt AT, djr drt.

_

dy dy

dV, ,dV, dV^

-- r1 = n 13-T + -T^} ' (9)

dz dz \dv± dv^'

If we consider the variation of the electromotive intensity at a point in passing through the surface, that component of the in- tensity which is normal to the surface may change abruptly at the surface, but the other two components parallel to the tangent plane remain continuous in passing through the surface.

783.] To determine the charge of the surface, let us consider a closed surface which is partly in the positive region and partly in the negative region, and which therefore encloses a portion of the surface of discontinuity.

The surface integral, r r

II R cos c ffSt

extended over this surface, is equal to 4 ire, where e is the quantity of electricity within the closed surface. Proceeding as in Art. 2 1 , we find

S, (2)

where the triple integral is extended throughout the closed surface, and the double integral over the surface of discontinuity.

Substituting for the terms of this equation their values from

But by the definition of the volume- density, p, and the surface- density, «,

• ^ ^ (J g)

Hence, comparing the last terms of these two equations,

0. (13)

This equation is called the characteristic equation of V at an elec- trified surface of which the surface-density is o-.

88 ELECTKOSTATICS. [78 C.

78<?.] If V is a function of #,y, z which, throughout a given con- tinuous region of space, satisfies Laplace's equation

and if throughout a finite portion of this region V is constant and equal to C, then 7 must be constant and equal to C throughout the whole region in which Laplace's equation is satisfied.

If 7 is not equal to C throughout the whole region, let 8 be the surface which bounds the finite portion within which 7 = C.

At the surface 8, 7 = C.

Let v be a normal drawn outwards from the surface 8. Since 8 is the boundary of the continuous region for which F= C, the value of Fas we travel from the surface along the normal begins

d7 to differ from C. Hence -=- just outside the surface may be posi-

tive or negative, but cannot be zero except for normals drawn from the boundary line between a positive and a negative area.

But if v is the normal drawn inwards from the surface S, 7f—C

• W and -j-y = 0.

dv

Hence, at every point of the surface except certain boundary lines, d7 d7'

is a finite quantity, positive or negative, and therefore the surface S has a continuous distribution of electricity over all parts of it except certain boundary lines which separate positively from nega- tively charged areas.

Laplace's equation is not satisfied at the surface S except at points lying on certain lines on the surface. The surface S there- fore, within which 7 — C, includes the whole of the continuous region within which Laplace's equation is satisfied.

Force Acting on a Charged Surface.

79.] The general expression for the components of the force acting on a charged body parallel to the three axes are of the form

(14) to y But at a charged surface p is infinite, and X is discontinuous, so

with similar expressions for B and C, the components parallel to and z.

.79-] FORCE ACTING ON A CHARGED SURFACE. 89

that we cannot calculate the force directly from expressions of this form.

We have proved, however, that the discontinuity affects only that component of the intensity which is normal to the charged surface, the other two components being- continuous.

Let us therefore assume the axis of x normal to the surface at the given point, and let us also assume, at least in the first part of our investigation, that X is not really discontinuous, but that it changes continuously from X1 to X2 while x changes from x^ to a?2. If the result of our calculation gives a definite limiting value for the force when x.2— x^ is diminished without limit, we may consider it correct when #2 = x^ , and the charged surface has no thickness.

Substituting for p its value as found in Art. 77,

Integrating this expression with respect to x from x = x± to x = #2 it becomes

This is the value of A for a stratum parallel to yz of which the thickness is #2 — $lt

Since Y and Z are continuous, -r= — f- -=- is finite, and since X

9 dy dz

is also finite,

where C is the greatest value of (-j- + -j-)X between x — «j and

• = *„. y