KLECTROSTATICAL THEORY.] ELECTRICITY In other words, an ,th part of the potential energy is lost. When a battery of jars is discharged through a circuit in which there is a fine wire of large resistance, the greater part of the potential energy lost in the dis charge appears as heat in the fine wire. Riess made elaborate experiments on the heating of wires by the dis charge in this way, and the results of his experiments are in agreement with the formulae which we have just given, (See Heating Effects.) ry We may also arrange a battery of jars by first charging ie=: - each separately to potential V in the usual way, and then connecting them in series, so that the outer coating of each jar is in metallic connection with the inner coating of the next. In such an arrangement of jars, it is obvious that in passing from the outer coating of the last at potential zero to the inner coating of the first, the potential will rise to wV. When we come to discharge such a series, the electromotive force to begin with is nV, so that for any purpose in which great initial electromotive force is required this combination has great advantages over n jars abreast. The " striking distance," for instance, i.e., the greatest distance at which the discharge by spark will just take place through air, is much greater. On the other hand, the quantity of electricity which passes is less, bsing only CV instead of nCV ; the whole loss of potential energy in a complete discharge is, however, the same. The case which we have been discussing must be care fully distinguished from that of a series of jars charged by
- ,.j e . "cascade," where n uncharged jars are connected up in
succession as in last case, and the first charged by con nection with the electric machine to potential V, while the outer coating of the last of the series is connected to earth, and the rest of the jars insulated. The whole electro motive force in this case is clearly only V, and, if all the jars be similar, the potential difference between the coatings in each is -- ; the charge on the inner coating of the first is CV CV 2 therefore and the whole potential energy only A . n n The arrangement is, therefore, not so good as a single jar fully charged by the same machine. It was fancied by Franklin, who invented this method of charging, that some advantage was gained by it in the time of charging, the notion being that the overflow was caught by the successive jars and that electricity w r as thereby saved. Charging by cascade was treated by Green. Some of the experiments of Riess bear on the matter (vide Mascart, 190, 191), which, after all, is simple enough. an ,l In the theory of accumulators, or condensers as they i,l are often called, much stress has been laid on the differ ence between "free 1 and " bound" electricity. To illus-
- > trate the meaning of these terms, let us trke a case where
the calculations can be carried out in detail. Suppose we have two concentric spherical shells, an inner, A, and an outer, B. Let the outer radius of A be a, and the inner and outer radii of B be & and c, so that the thickness of the latter is c - b. Vc, shall suppose that we can, when we please, connect the inside sphere with the earth. It is clear that there can never be any electricity on the inner surface of A. Let the charges on the other surfaces in order be E, F, G. Let us suppose in the first in stance that A is at potential V, and B at zero. Then we have to find E, F, G. Draw a surface in the substance of B; no lines of force cross it, therefore the whole amount of electricity within is zero. Hence F = - E . Also, considering the external space, which is inclosed between two surfaces of zero potential, we see tha* G -= 0. Thus, F F since A is at potential V. we have - .= V . a b E - ^L V = P V /"where j>- ; ^ } . . . (51). . b-a . b-a In this case, then, there is no electrification on the outside of B, and an electric pendulum suspended there would give no indication. Let us now connect A. with the arth, so that its potential becomes zero ; we have now to find the charges and potentials, our datum being that the whole charge on B is - E . As before, we have F= - E , but G is no longer zero. We have, however, F + G - - E . Hence G - E = - E . V V f" Also, since A is at zero potential, - - + = , therefore G = The potential of B is . p + c , or -2L- c p + c p+c rp In this process, therefore, a quantity E - F/, or -- V, of electri- P ~T C city has flowed away to earth from A, and a quantity V has passed from the inner to the outer surface of B, while the potential has altered, on A from V to 0, and on B from to V. Suppose now we connect B with the earth, thus reducing it to zero potential. Since the charge on A remains the same, and that on the inner coating of B is equal and opposite to it, it follows that now the charges on A, &e. , are V, " V, 0, where q denotes -*- ; and the potentials of A and B are - V and 0. After another p + c c pair of such operations the charges will be - - V, &c., and the - a ~|3 potential,-? V ; after a third, charges, ? V, &c., and potential, C C C i V . Hence the charges and potentials go on decreasing in geo metrical progression._ Amounts of electricity ilow away from A equal to qV, q 1 V, q ? V, &c., in the successive operations, and equal amounts of opposite signs are discharged from B. The sum of all these discharges is the whole original charge on A, tor j Hence by an infinite number of alternate connections we shall finally discharge the jar completely. The elec tricity which flows out at each contact is called the " free electricity," and that which remains behind the " bound electricity." The quantity which we have denoted by ^Capacity- is clearly the capacity of a spherical Leyden jar ; it in- of spheri- creases indefinitely as the distance between the conduct- cal ^ ing surfaces decreases, and is very nearly proportional to the surface of the inside coating, v/hen the distance is small compared with the radius of either surface. It is very easy to extend our reasoning to any con denser. If, in fact, <7 n , <7 12 , q. 2Z be the coefficients of self and mutual in duction for the armatures, then this potential after operating n (7 2 . n / <7* n 1 I V , the charges, q u ( - I V and 111 <7-22/ ll ?22/ a y ant i tj ie amounts of electricity which leave 1 and 2 in the nth operation are TCu ( gll<7i!2 g
In 22
respectively. We must not omit one more interesting case. If we Coaxial have two infinite coaxial cylinders of radii a and I cylinders. (b > a), then obviously the potential is symmetrical about the common axi?, and Laplace s equation becomes d?V_ l dV ^dr^ + r ~d7 =0> The integral of this is V = C log r + D. Let the inner oylin be at potential Vj, the outer at potential V 2 , then j-,, ,--. . log a - log b log a - log b Hence the surface density on the inner cylinder is given by 4* dr ~ . b
4ira log -