Two Parallel Planes

89.] Another simple case of electrification is that in which the electrodes are two parallel plane surfaces at a distance c. We shall suppose the dimen- sions of these surfaces to be very great compared with the distance betweenFORCE BETWEEN TWO PARALLEL PLANES.

them, and we shall consider the electrical action only in that part of the space between the planes whose distance from the edges of the plates is many times greater than c.

Let A be the potential of the upper plane in the figure, and B that of the lower plane. Then the electric force at any point Fig. 20.

P between the planes, and not A−B near the edge of either plane, is , acting from A to B. The electric den- c sity on the upper plane is found by Coulomb’s Law by dividing this quantity by 4π . If σ be the surface density A−B . (1) 4πc The surface density on the plane B is equal to this in magnitude but opposite in sign.

Let us now consider the quantity of electricity on an area S, which we may suppose cut out from the upper plane by an imaginary closed curve. Multiplying S into σ, we find σ= A−B S. (2) 4πc The quantity of electricity on an equal area of the plane B taken exactly op- posite to S will be −e. The energy of the electrification of these two portions of electricity is, by Art. 31, e= Q = 12 {Ae + B(−e)} = 12 (A − B)e. (3) Expressing this in terms of e it becomes 2π 2 e c. (4) S If c, the distance between the surfaces, be made to increase to c ′ the charges of the surfaces remaining the same, the energy will become Q= Q′ = 2π 2 ′ e c . S (5)ATTRACTED DISK ELECTROMETERS. 79 The augmentation of the potential energy is Q′ − Q = 2π 2 ′ e (c − c), S (6) and this is the work done by external agency in pulling the planes asunder against the electric attraction. If F is the electric attraction between the two areas S, F (c ′ − c) = 2π 2 ′ e (c − c), S (7) or 2π 2 e . (8) S 90.] This result gives us the best experimental method of measuring the quantity of electricity on the area S, for by this equation F = e=√ FS . 2π (9)

In this expression F is the force of attraction on the area S determined in dynamical measure from observation of its effects. S is the area of the surface and π is the ratio of the circumference of a circle to its diameter. The difference between the potentials, A and B, of the two planes is easily found in terms of e by means of equation (2), thus, A − B = 4πc e 8πF = c√ . S S (10)

91.] In Sir William Thomson’s attracted disk electrometers a disk is so arranged that when in its proper position the surface of the disk forms part of a much larger plane surface extending for a considerable distance on all sides of the disk. The part of the surface outside the moveable disk is called the Guard Ring and the surface of the disk and guard ring together may be considered as the surface of a large disk, part of which, near its centre, is moveable. Opposite this disk is placed another disk having its surface parallel

INVERSE PROBLEM OF ELECTROSTATICS.

to the first disk and much larger than the moveable disk. The electrification of the moveable disk is then the same as that of a small portion of one of the large opposed planes taken at a considerable distance from the edge of the plane, and only very small corrections are needed to make the formulæ already given apply to the case of the moveable disk. The distribution of electrification and of electric force near the edges of the large disks is by no means so simple. It is calculated in Art. 202 of my larger Treatise, and the lines of force and equipotential surfaces are shown in Plate V at the end of this book.

92.] The direct problem of electrostatics—the problem which the circum- stances of every electrostatic experiment present to us—may be stated as fol- lows. A system of insulated conductors is given in form and position, and the electric charge of each conductor is given, required the distribution of elec- tricity on each conductor and the electric potential at any point of the field. The mathematical difficulties of the solution of this problem have been overcome hitherto only in a small number of cases, and it is only by a study of what we may call the inverse problem that the results we possess have been obtained.

In the inverse problem, a possible distribution of potential being given, it is required to find the forms, positions, and charges of a system of conductors which shall be consistent with this distribution of potential. Any number of solutions of this latter problem may be obtained by tak- ing, instead of the electrified bodies of the original distribution, any set of equipotential surfaces surrounding them, and supposing these surfaces to be the surfaces of conductors, the charge of each conductor being equal to the sum of the charges of all the bodies of the original distribution which it en- closes.

Every electrical problem of which we know the solution has been con- structed by an inverse process of this kind. It is therefore of great importance to the electrician that he should know what results have been obtained in this way, since the only method by which he can expect to solve a new problem is by reducing it to one of the cases in which a similar problem has been constructed by the inverse process.