KLECTROSTATICAL THEORY.] ELECTRICITY 25 We may make here the important remark that, so long as p or a- is not infinite, the integrals in (7) and (&) are finite and continuous. This depends on the fact, which we cannot stop to prove, that the part of the potentia.1 at P, contributed by an infinitely small portion of electricity surrounding P, is infinitely small. In practice, therefore, the electric potential is always continuous; for although we may in theory speak of discrete points and electrified lines where finite electri fication is condensed into infinitely small space, yet no such cases ever occur iu nature. It may also be shown for any electrical system of finite extent, that, as the distance of P from 0, any fixed point at a finite dis tance from the system is increased indefinitely, the potential at P approaches more and more nearly the value pr , where M is the algebraical sum of all the electricity in the system, and I) the distance of P from O. Hence at any point infinitely distant f r:im O, V = 0. We next proceed to prove the following proposition, which will form the basis of the subsequent theory : The surface integral of electric induction taken all over the surface inclosing any space is equal to 4?r times the alge braical sum of all the. electricity in that space. P>y the electric induction across any element of the sur face (taken so small that the resultant force at every point of it may be regarded as uniform) is meant the product of the area of the element into the component of the result ant force in the direction of the normal to the element which is drawn outwards with respect to the inclosed space. Thus tiS being an element of surface, e the angle between the positive direction of the resultant force Hand the outward normal v, and E the sxim of all the electricity in the inclosed space, the proposition in symbols is (9). We shall prove it in the manner most naturally suggested by the theory of electrical elements acting at a distance, by first showing that it is true for a single element e either outside or inside the surface. Let us suppose e to be at a point P, fig. 1 1 , within S, which for greater generality we may suppose to be a re-entrant surface. Draw a small cone of vertical solid angle dw at P, and let it cut the surface in the elements QR, Q li , Q"lf ; let the outward normals to these be QM, Q M , Q"M". The ele ments of the surface integral contri buted by QR, Q R , Q"R" are obviously p R!1 QR = - &c . but &c.; I t Q Vr- , , and Fig. 11. hence the three elements cos t cos < of the integral become + cdu,-edu, +edco ; and the sum is cdia. Adding now the contributions from all the little cones which fill up the solid angle of iir about P, we get j] R cos dS = e/fda = lire. Had the point P been outside, the numbers of emergences and entrances would have been equal, the contribution of each cone zero, and on the whole j7RcosrfS = 0. Combining these results, we see that the proposition is true for a single element. Hence, by summation for all the elements, we can at once extend it to any electrical system ; for all the elements external to S give zero, and all the internal elements give 47r2e = 47rE. Let us apply the above proposition to the space enclosed by the infinitely small parallelepiped whose centre is at xyz, and the co-ordinates of whose jyigles are x^^dx, ij^dt/,z.^dz. The con tributions to the surface integral from the two faces perpendicular to the, -axis are - , fd r dx d*V and (s ~ T d^ Adding these and the four parts from the remaining sides, and equat ing to 4ny<7,n/?A/:, which is the 4irE in this case, we have <**V <FV , d"-V . -j-i + j-f + -y^- + 4irp = , dx* dy* dz* or, is it is usually abbreviated, - ....... (10). Equation (10), originally found by Laplace for the casep 0, and extended by Poisson, lias been called the characteristic equation of the potential. It may be applied at any point where p is finite and the electric force continuous. It might be shown by examining the inte- dV grals representing X, Y,Z,and , &c., that the electric force is con- ctx tinuous wherever there is finite volume density. Equation (10) may be looked on either as an equation to determine the potential when p is given, or as an equation to determine p whenV is given. We shall have occasion to use it in both ways. The characteristic equation cannot be applied in the form (10) when the resultant force is discontinuous. This will be found to be the case at a surface on which electricity is distributed with finite surface density. Let us consider the values of the resultant force at two points, P and Q, infinitely near each other, but on opposite sides of such a surface. Resolve the resultant force tangentially and normally to the surface. If we consider the part of the force which arises from an infinitely small circular disc, whose radius, though infinitely small, is yet infinitely great compared with the distance between P and Q, we see that infinitely little is contributed to the tangential component at P or Q by this disc, while it can be readily shown that the part of the nor mal component arising therefrom is 2?ro-, directed from the disc in each case, when o- is the surface density. Hence, since the part of the resultant force arising from all the rest of the electrified system obviously is not discontinuous be tween P and Q, the tangential component is continuous when we pass through an electrified surface, but the normal com ponent is suddenly altered by 47rcr. For a thorough investigation of these points the reader is referred to Gauss or Green. We can arrive very readily at the amount of the discontinuity of the normal force by applying (9) to the cylinder formed by carrying an infinitely short generating line round the element dS, so that one end of the cylinder is on one side of rfS and the other on the other, the lateral dimensions being infinitely small, but still infinitely greater than the longitudinal. The only part of the integral which is of the order of rfS is the part arising from the two ends ; hence if N,N be the value of the normal components on the two sides of S, we clearly get (N - N ) dS = 4*WS , or X - N = 4 . If Vj, V 2 denote the potentials on the two sides of S, and v^ , v t the normals to d$, drawn towards these sides respectively, then we may obviously write our equation Condi- tions at snr f aC e ni). Written in this form the equation has been called the surface char- Surface actcristic equation of the potential. It may be looked upon as a chanv- surface condition, which must be fulfilled by tlie values of V on the teristi^ two sides of an electrified surface on which the surface density <r is equation, given, and where, in consequence, there is discontinuity in the first differential coefficients of V ; or it may be looked on as an equation to determine a when Y : and V 2 are given. We have seen that we can draw through every point of Level the electric field a line of force. A surface constructed so surface, that the potential at ever}- point of it has the same value is called an equipotential or level surface. We can obviously draw such a surface passing through every point of the field. It is clear, too, that the line of force at every point must be perpendicular to the level surface passing through that point. For since no work is done on a unit of + electricity in passing from one point of a level surface to a neighbouring point, there can be no component of the resultant force tan gential to the surface; in other words, the direction of the resultant force is perpendicular to the surface. This is ex pressed otherwise by saying that the lines of force are orthogonal trajectories to the level surface. If we take a small portion of a level suiface, and draw Tui>es through every point of the boundary a line of force, we shall force, thus generate a tubular suiface which will cut orthogonally every level surface which it meets. Such a surface is called a tube of force. Let a tube of force cut two level .surfaces in the elements (AS and cZS . Apply to the space contained by the part of the
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