PARTICULAR CASES OF ELECTRIFICATION

84.] Definition.—The electric or electromotive force at a point is the force which would be experienced by a small body charged with the unit of positive electricity and placed at that point, the electrification of the system being supposed to remain undisturbed by the presence of this unit of electricity.

We shall use the German letter E as the symbol of electric force. 85.] Let us now return to the case of a sphere whose radius is r, the exter- nal surface of which is uniformly electrified, the surface-density of the electri- fication being σ. As we have already proved, the whole charge of the sphere is e = 4πr2 σ.

At any point outside the sphere such that the distance from the centre of the sphere is r′ the electromotive force, E, is directed from the centre, and its value is e E = ′2 . rVALUE OF THE POTENTIAL. 72 If the point is close to the surface of the sphere, r′ = r, and e E = 2 = 4πσ, r or the electric force close to the surface of an electrified sphere is at right angles to the surface and is equal to the surface-density multiplied by 4π.

We have already seen that in all cases the electric force close to the surface of a conductor is at right angles to that surface, and is proportional to the surface-density. We now, by means of this particular case, find that the constant ratio of the electric force to the surface-density is 4π for a uniformly electrified sphere, and therefore this is the ratio for a conductor of any form.

The equation E = 4πσ is the complete expression of the law discovered by Coulomb and referred to in Arts. 47 and 81. Value of the Potential. 86.] We must next consider the values of the potential at different distances from a small electrified body. Definition. The electric potential at any point is the work which must be expended in order to bring a body charged with unit of electricity from an infinite distance to that point.

If ψ is the potential at A and ψ ′ that at B, then the work which must be spent by the external agency in overcoming electrical force while carrying a unit of electricity from A to B is ψ ′ − ψ. The quantity ψ ′ − ψ would also represent the work which would be done by the electrical forces in assisting the transfer of the unit of electricity from B to A if the motion were reversed. If the force from B to A were constant and equal to E, then ψ ′ − ψ = BA . E.

In general, the electric force varies as the body moves from B to A, so that we cannot at once apply this method of finding the difference of potentials.POTENTIAL AT A POINT. 73 But, by breaking up the path BA into a sufficient number of parts, we may make these parts so small that the electric force may be regarded as uniform during the passage of the body along any one of these parts. We may then ascertain the parts of the work done in each part of the path, and by adding them together, obtain the whole work done during the passage from B to A. Fig. 19.

Let us suppose a unit of electricity placed at O, and let the distances of the 1 points A, B, C, … Z from O be a, b, c, … z. The electric force at A is 2 , a 1 at B 2 , and so on, all in the direction from O to A. b To find the work which must be done in order to bring a unit of electricity from A to B we must multiply the distance AB by the average of the electro- motive force at the various points between A and B. The least value of the 1 1 force is 2 and the greatest value is 2 · Hence the work required is greater a b AB AB than 2 and less than 2 . Now AB is a − b, and the true value of the work a b is the excess of the potential at B over that at A. Hence if we now write A, B, C, … Z for the potentials at the corresponding points, we may express the work required to bring the unit of electricity from A to B by B − A. This quantity therefore is greater than a−b 1 1 b or ( − ) , 2 b a a a but less than a−b 1 1 a or ( − ) . 2 b a b b We may express this by the double inequality 1 1 a 1 1 a (b − a) b < B − A < (b − a) b.POTENTIAL AT A POINT. Similarly 74 1 1 c 1 1 b (c − b) b < C − B < (c − b) c, a b and so on. The ratios , , &c., are all greater than unity. Let us suppose that b c b the greatest of these ratios is equal to p. The ratios , &c., are the reciprocals a 1 of these; they are therefore all less than unity, but none less than . Hence p 1 1 1 1 1 ( b − a ) p <B − A < ( b − a ) p 1 1 1 1 1 ( c − b ) p <C − B < ( c − b ) p ……… 1 1 1 1 1 − <Z − Y < − p. (z y) p (z y) Adding these inequalities we find 1 1 1 1 1 ( z − a ) p < Z − A < ( z − y ) p. By increasing the number of points between A and Z and making the inter- vals between them smaller we may make the greatest ratio, p, as near to unity as we please, and we may therefore assert that, as the line AZ is more and 1 more minutely divided, the quantity p and its reciprocal approach unity as p their common limit. In the limit, therefore, Z −A= 1 1 − . z a

We have thus found the difference between the potentials at A and Z. To determine the actual value of the potential, say at Z, we must refer to the definition of the potential, that it is the work expended in bringing unit ofPOTENTIAL AT A POINT. 75 electricity from an infinite distance to the given point. We have therefore in the above expression to suppose the point A removed to an infinite distance from O, in which case the potential A is zero, and the reciprocal of the dis- 1 tance, or , is also zero. The equation is therefore reduced to the form a 1 Z= , z or in words, the numerical value of the potential at a given point due to unit of electricity at a given distance is the reciprocal of the number expressing that distance. e If the charge is e, then the potential at a distance z is . z The potential due to a number of charges placed at different distances from the given point is found by adding the potentials due to each separate charge, regard being had to the sign of each potential.