density of the charge, the forces at P due to the electricity on

these two portions of the surface are σ

AB I
and σ
K

A'B' I 2/2 K

quently the forces exerted at P are equal as well as opposite, and so cancel each other. It is obvious that the whole surface of the sphere may be divided up into similar pairs of mutually compensating elements, and therefore that a uniform spherical charge exerts no force at any point within the sphere.

FIG. 15.

To determine the effect of a uniformly électrified sphere at an external point P (Fig. 16),

and also a for the angle OAP = OA'P' = OPA = OPA' (from the similarity, each to each, of the triangles OAP', OA'P', and OPA, OPA' which follows from the common angle at O in each pair of triangles respectively, and from the relation, OP'

R

R

OP

),

we get, for the areas A and A'

PA': then the electric forces at P due to and A' are

Puts

PA and s' the charges on A

=

but, by the similarity of triangles above referred to, we have both

r

and =

S

R
Op'

and therefore the forces exerted by the elementary areas at a and a' are equal, and their resultant bisects the angle between AP and A'P, that is, it acts along Op. Obviously the whole surface may be divided into similar pairs of elements, one lying on one side and the other on the other of the plane, CPC', perpendicular to OP: the resultant effect of each such pair, and therefore of all the pairs, or of the whole spherical surface, must be along OP, as might have been assumed without other proof than the consideration of geometrical symmetry.

We shall get the effective component of the force due to an element of the surface, that is, the component along OP, by multiplying the total force due to the element by cos a. the effective component due to any element may be written

Hence

and, to get the effect of the whole sphere, we have only to multiply

R2 I

the constant factor o OP2 K

This evidently is 4π.

by the sum of all the values of do.

Hence the force at the point P due to the sphere charged to the uniform surface-density σ is

if Q be put for 4πR2σ, the whole charge of the sphere, and D for the distance OP. Hence it appears that the force at an external point due to a charge distributed uniformly over the surface of a sphere is the same as if the whole charge were concentrated in one point at the centre of the sphere. This result is of very great

importance on account of its frequent application.

If the point P is close to the surface of the sphere, we have D = R, and therefore, in this case

and the force acts along the normal to the surface, outwards if o

is positive, inwards if σ is negative,

It may be noted that, for a given value of the charge, the force is independent of the radius of the sphere.

It follows that a homogeneous sphere, or a sphere that might be subdivided into concentric homogeneous layers, acts at any external point as though its whole mass were concentrated at the centre. To find the action of a homogeneous sphere at an internal point distant from the centre, imagine a spherical surface, concentric with the sphere, drawn through the point: then, the force due to the part outside this surface vanishes, and the part inside acts as though it were concentrated at the centre. The action of the whole sphere is therefore represented by

if p is the volume-density of the sphere. The force is consequently proportional to r.

The case here considered could only occur electrically if the sphere were composed of some non-conducting material, for with a conductor we should have p = 0. K in the last formula must be taken as the dielectric coefficient of the material of the sphere.

20. Power of Points.-The observed facts as to the electric density in conductors, as has been already stated (§ 18), are such as might be deduced mathematically from the idea of two electricities, conceived of as capable of moving easily through conducting substances, but not able to pass through non-conductors, and such that each smallest portion of one repels every other portion of the same kind and attracts every portion of the second kind, according to the law of the inverse square of the distance. The same facts, however, might also be deduced from the conception of stresses in the dielectric field of the nature of a tension between the opposite boundaries and a pressure in directions transverse to the lines of tension. When the opposite boundaries of the field are near together, as, for example, in the case of the field between two metal plates facing each other at a short distance, the transverse force tending to widen the field cannot produce much effect, because of the small length of the field, and the resulting form of the field (or, as it is expressed in the other order of ideas, the electric distribution) is mainly determined by the tension. On the other hand, if the boundaries of the field are very far apart, as in the cases considered in § 19, the effect of the

transverse pressure becomes so great that that of the tension is inappreciable in comparison.

If an electric field is bounded on the one hand by a flat conducting surface of considerable size, say the wall of a room, and on the other by a metal point presented towards the flat surface at a moderate distance, then both the tension and the pressure in the electric field tend to cause a concentration of it about the point, so that the electric force in the air just outside the point becomes very great. But, as will be shown more fully later, the air cannot sustain more than a limited electric force. When the limit is exceeded, the air gives way to the electric stresses, and all signs of electrification cease, almost or quite as completely as if the bounding conductors of the field had been brought into contact.

FIG. 17.

This sudden breaking down of a dielectric medium, known as a disruptive discharge, is accompanied by luminous phenomena and sound. If the medium is gaseous or liquid, its continuity is of course immediately restored after the discharge; in the case of a solid, permanent rupture is the result.

When an electric field is formed between a sharp point and an opposing plate, the stress in the field close to the point may be sufficient to cause disruptive discharge although at other parts of the field the stress may be far less. On this account the arrangement mentioned is often used in the construction of electrical apparatus when it is desired to facilitate discharge through a greater or less thickness of air. On the other hand, points or projections of any kind are avoided as far as possible when it is

wished to prevent such discharge. By connecting a point and a plate with the two sides of an electric machine, the electric field

FIG. 18.

between them is constantly reproduced, and discharge takes place continuously. The air close to the point having given way, the stress in the air farther off has nothing to counterbalance it, and the air consequently moves in a continuous stream away from the point, constituting what is known as the electric wind. This effect is easily made evident by means of a candle-flame (Fig. 17). The breaking down of tension in front of the point may also be illustrated by the electric wind-mill (Fig. 18), the points of which move backwards for the same kind of reason that a man pulling hard at a rope tumbles backwards if the rope breaks.

If the air near the point is charged with solid or liquid particles, such as exist in smoke, these are rapidly deposited on the opposing surface and the smoke disappears. This process has been practically employed to hasten the deposition of dust or fumes from the air.

The production of the electric wind is accompanied by a sort of hissing sound, and in the dark the point exhibits a projecting tuft of violet light if it is positive, and a small brilliant star if it is negative.