42 ELECTRICITY KLECTRIC CURRENT. drawn from the first to the second side, and a the electric surface density at time t. If we consider the particular case of homogeneous isotropic r . *i * v dV . dV dV media, and suppose further that X= - y-^ Y= - -v- Z~ - -77 > these equations reduce to and -r4 + k, dv V dp dz* dt (8). In the last equation r l and V 2 are the potentials on the two sides of a boundary between media of specific resistance k^ and k%. In the particular case of steadj motion, the right-hand sides of (7) and (8) are zero. The analytical treatment of problems about steady currents is therefore precisely analogous to that of problems about electrostatical equilibrium, steady flow of heat, hydrodyna mics, &c. : to every solution in one such physical subject corre sponds a solution in each of the others. Many valuable details on this subject are to be found in Thomson s papers oil electrostatics and magnetism. Results The consequences of Ohm s law have been followed of Ohm g OU |; mathematically, and verified in a variety of cases. We shall notice a few which are interesting, either from the accuracy of the experimental results, or from the interest or practical importance of some method or prin ciple involved. Applica- In the case of a steady current in a uniform linear con- tion to ductor, say a wire, it is obvious that the potential must uform f a u uniformly in the direction in which the current is conduc- nowr i n g- Hence, if we suppose the wire stretched out tors. straight, and erect at different points lines perpendicular to it, representing the potential at each point, the locus of the extremities of these lines will be a straight line. This may be arrived at by integrating equation (5), which be- d 2 V comes in this case ;/:;" = 0, x being measured along the wire sup- Sosed to be straight ; we thus get for the potential V, at any point istant x from the origin, at which potential is YO, (9). If V be taken as ordinate, this represents a straight line, the /IT. tangent of whose inclination to the x-axis is - , or - i<Jc. u> We cannot apply Ohm s law at the junction of two different substances. The condition of continuity of course applies; in other words, if the flow has become steady, the current is the same at all points of the circuit, whether homogeneous or not. We shall see, when we come to discuss electromotive force, that there is a con stant difference between the potentials at two points in finitely near each other, but on opposite sides of the boundary between two conductors of different material. If we knew this potential difference for each point of heterogeneous contact in the circuit, we could draw the complete potential curve for the circuit by apply in Ohm s law to- each conductor separately. The diagram (fig. 19) represents (on the con- tact theory, as held ^ a by Ohm, see Origin of Electromotive Force) the fall of potentials and the discontinuities in a voltaic circuit, consisting of zinc, water, and copper, in which the current flows from Cu to Zn across the junction of the metals. We assume for the present that Ohm s law applies to the liquid conductor. Let us denote by V Q , Y K , <fec. the potentials at Q and R, &e., or what is the same thing, the ordinates BQ, BR, &c., in our diagram. Then applying Ohm s law to the homo geneous parts of the circuit, we have V v - VQ = CU , V a - V B = CS, V T - V D = CR", where R , S, R", denote the Pi resistances of the zinc, the water, and the copper respec tively. Now, denoting V v - V c , the potential difference, or as it is sometimes called, the " contact force " between Zn and Cu by E 2C , and so on, let us add the above three equations; we thus get E = Ezc + E A2 + EC A = C(R + R" + S). Here E is called the ivhole electromotive force of the circuit, being the sum of all the discontinuities of potential, taken with their proper signs, or, what is equivalent to the same thing, the whole amount of work which would be done by a unit of + electricity, in passing round the whole circuit once, supposing it to get over the discon tinuities without gain or loss of work. Denning E in this way, we may extend Ohm s law to a heterogeneous circuit, the resistance R being now the sum of all the resistances of the different parts, or the whole resistance. In accordance with this definition, if we take two points, p and q (fig. 19) in the Cu and Zn respectively, the whole electromotive force will be V p - V, + E zc and the current will be given by Vp-Vf + Ezc-KC (10), where R is the whole resistance of pq. V p - V 7 is some times called the " external," and E zc the " internal " electro motive force. If p, q include more than one contact of heterogeneous metals, we have only to add on the left- hand side of (10) the corresponding internal electromotive force for each discontinuity. If p and q be connected by wires of the same metal, say copper, to the electrodes of a Thomson s electrometer, then the electrometer will indicate a potential difference, V p -V, + E zc , and notV p -V ? as might at first sight be suspected. 1 No electricity can flow through the electro meter, hence the copper wire attached at p, and the pair of quadrants to which it leads (we may suppose the quadrants made of copper, but iu reality it does not matter, see below, Origin of Electromotive Force), will be at potential V p . But owing to the contact force between the Zn and Cu at q, the wire from q and the quadrant to which it leads will be at potential V, E zc . It appears, therefore, that the electro meter indication corresponds to the ivhole electromotive force between p and q, and is proportional to the whole resistance between p and q, no matter what metals the circuit may include. 2 This conclusion was verified by Kohlrausch. His method rested on the principle of Volta s condensing electroscope. He used an accumulator consisting of a fixed plate B, and an Verifier equal movable plate A, which could be lowered to a very small tion by fixed distance from B, and raised to a considerable distance, so as Kohl- to touch a fixed wire leading to a Dellmann s electrometer. The rauscL plate A was lowered and connected with p, while q and the fixed plate were connected with the ground ; the connection with j> was then removed, and A raised, its potential thereby greatly increasing owing to its greatly diminished capacity. This increased potential was measured by the electrometer, with which A was in connection through the fixed wire. In one of Kohlrausch s experiments, he found for the electromotive force between a fixed point of the metallic circuit and four points, such that the resistance between each adjacent pair was very nearly equal, the values S5, 1 81, 2-09, 370; the values calculated by Ohm s law were 93, 1 86, 2 80, 373. He also examined the fluid part of the circuit, and still found a good agreement between theory and experiment. (See AViedemann, 102.) The laws of current distribution in a network of linear Networl circuits were first studied by Kirchhoff. He laid down of 1 DeaI two general principles which are very convenient in prac- tors< tical calculations. I. The algebraical sum of all the currents flowing from any node of the network is zero. II. If we go round any circuit of the network, then no 1 It is supposed that all the wires are at the same temperature. 2 Tliis more general statement follows at once from the above reasoning in conjunction with Yolta s law (;/ below, Origin of Elec
tromotive Force).