POTENTIAL AS A SUM OF QUOTIENTS.

603

the change in the value of 2. Since any movement can be resolved into a succession of small movements, we may omit the word small, and the proposition will still be true. As r increases to infinity, 2 will diminish to zero. Hence, denotes the work from distance

r to infinite distance.

2°

As regards sign, is the work done by electrical force when a unit of positive electricity is carried from distance to infinite distance.

1

Next suppose several quantities, 41, 42, &c., to be collected at different points, 01, 02, &c. Let P be any other point, and let O1P="1, O2P=r2, &c. Then in the passage of a unit of electricity from P to infinite distance, the electrical work is, by the preceding section,

which we will denote by 2, the symbol being read "the sum of such terms as."

Σ is therefore the general expression for the potential at a point due to any quantity of electricity distributed in any manner; in other words, the potential is equal to the sum of the quotients obtained by dividing each element of electricity by its distance from the point. The distances are essentially positive. If the electricity is not all of one sign, some of the quotients,, will be positive and others negative, and their algebraical sum is to be taken.

612. Application to Sphere. In the case of a charged conducting sphere, all the elements q are equally distant from the centre of the sphere, and the sum of the quotients, when we are computing the potential at the centre, will be, Q denoting the charge, and R the radius of the sphere. But the potential is the same at all points in a conductor. is therefore the potential of a sphere of radius R, with charge Q, when uninfluenced by any other electricity than its

own.

R

613. Capacity. The electrical capacity of a conductor is the quantity of electricity required to charge it to unit potential, when it is not influenced by any other electricity besides its own charge and the electricity which this induces in neighbouring conductors. Or,

since, in these circumstances, potential varies directly as charge, capacity may be defined as the quotient of charge by potential. Let C denote capacity, V potential, and Q charge, then we have

Q
R'

But we have seen that, for a sphere of radius R, at a distance from other conductors or charged bodies, V. Hence C=R; that is, the capacity of a sphere is numerically equal to its radius.

This is a particular instance of the general proposition that the capacities of similar conductors are as their linear dimensions; which may be proved as follows:

:

Let the linear dimensions of two similar conductors be as 1: n. Divide their surfaces similarly into very small elements, which will of course be equal in number. Then the areas of corresponding elements will be as 1 : n2, and, if the electrical densities at corresponding points be as 1 : x, the charges on corresponding elements. are as 1 n2x. The potential at any selected point of either conductor is the sum of such terms as 2 (§ 611). Selecting the corresponding point in the other conductor, and comparing potentials, the values of q are as 1 : n2x, and the values of r are as 1:n; therefore the values of are as 1 nx. Hence the potentials of the two conductors are as 1 : nx. If they are equal, we have nx=1, and therefore n2x=n; that is, the charges on corresponding elements, and therefore also on the whole surfaces, are as 1 : n.

We shall see, in the next chapter, that the capacity of a conductor may be greatly increased by bringing it near to another conductor connected with the earth.

614. Connection between Potential and Induced Distribution.-In the circumstances represented in Fig. 335 (§ 563), if we suppose the influencing body C to be positively charged, the potential due to this charge will be algebraically greater at the near end A of the influence conductor than at the remote end B. The induced electricity on A B must be so distributed as to balance this difference, in fact the potential due to this induced electricity is negative at A and positive at B. All cases of induced electricity upon conductors fall under the rule that the potential at all parts of a conductor must be the same, and hence, wherever the potential due to the influencing

electricity is algebraically greatest, the potential due to the electricity on the influenced conductor must be algebraically least.

As there can be no force in the interior of a conductor, the force at any point in the interior, due to the influencing electricity, must be equal and opposite to the force due to the electricity on the surface of the conductor. This holds, whether the conductor be solid or hollow. A hollow conductor thus completely screens from external electrical forces all bodies placed in its interior.

615. Electrical Images.—If a very large plane sheet of conducting material be connected with the earth, and an electrified body be placed in front of it near its middle, the plate will completely screen all bodies behind it from the force due to the electrified body. The induced electricity on the plate therefore exerts, at all points behind the plate, a force equal and opposite to that of the electrified body, or, what is the same thing, a force identical with that which the electrified body would exert if its electricity were reversed in sign. But electricity distributed over a plane surface must act symmetrically towards both sides. Hence the force which the induced electricity exerts in front, is identical with that which would be exerted by a body precisely similar to the given electrical body, symmetrically placed behind the plane, and charged with the opposite electricity. The total force at any point in front of the plane is the resultant of the force due to the given electrified body, and the force due to this imaginary image. The name and the idea of electrical images, of which this is one of the simplest examples, are due to Sir W. Thom

son.

CHAPTER XLVII.

ELECTRICAL CONDENSERS.

616. Condensers.-The process called condensation of electricity consists in increasing the capacity of a conductor by bringing near it another conductor connected with the earth. The two conductors are usually thin plates or sheets of metal, placed parallel to one another, with a larger plate of non-conducting material between them.

A

C

Let A and B (Fig. 377) be the two conducting plates, of which A, called the collecting plate, is connected with the conductor of the machine, and B, called the condensing plate, with the earth; and let C be the non-conducting plate (or dielectric) which separates them. Then, if the machine has been turned. until the limit of charge is attained, the surface of B which faces towards A is covered with negative electricity, drawn from the earth, and held by the attraction of the positive electricity of A; and, conversely, the surface of A which faces. towards B, is covered with positive electricity, held there by the attraction of the negative of B, in addition to the charge which would reside upon it if the conductor were at the existing potential, and B and C were absent. In fact, the electrical density on the face of A, as well as the whole charge of A, would, in this latter case, be almost inappreciable, in comparison with those which exist in the actual circumstances. By condensation of electricity, then,

Fig. 377.-Theoretical Condenser.

That inderen which takes the smallest charge!)

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CALCULATION OF CAPACITY.

607

we are to understand increase-usually enormous increase of electrical density on a given surface, attained without increase of potential. If two conducting plates, in other respects alike, but one with, and the other without a condensing plate, be connected by a wire, and the whole system be electrified, the two plates will have the same potential, but nearly the whole of the charge will reside upon the face of that which is accompanied by a condensing plate.

V

617. Calculation of Capacity of Condenser.-The lines of force between the two plates A and B are everywhere sensibly straight and perpendicular to the plates, with the exception of a very small space round the edge, which may be neglected. The tubes of force (§ 607) are therefore cylinders, and the intensity of force is constant at all parts of their length. Also, since the potential of the plate B is zero, if we take V to denote the potential of the plate A, which is the same as the potential of the conductor, and t to denote the thickness of the intervening plate C, the rate at which potential varies along a line of force is, which is therefore (§ 602) the expression for the force at any point between the plates A, B. The whole space between the plates may be regarded as one cylindrical tube of force of cross-section S equal to the area of either plate, the two ends of the tube being the inner faces of the plates. The quantities of electricity + Q residing on these faces are therefore equal, but of opposite sign (§ 610); and as the force changes from nothing toin passing from one side to the other of the electricity which resides on either of these surfaces, we have (§ 607)

Hence the capacity of the plate A, being, by definition, equal to
is equal to

S

4 t

We should, however, explain that, if the intervening plate C is a solid or liquid, we are to understand by t not the simple thickness, but the thickness reduced to an equivalent of air, in a sense which will be explained further on (§ 624). This reduced thickness is, in the case of glass, about half the actual thickness.

If s denote an element of area of A, and q the charge residing