| (P + G + R) (x + y) − Gy − Rz = 0, (Q + G + S)y − G (x + y) − Sz = 0, (R + S + B)z − R (x + y) − Sy = E. |
From these we can easily obtain the solution for (x + y) − y = x, which is the current through the galvanometer circuit in the form
x = E (PS − RQ) Δ.
where Δ is a certain function of P, Q, R, S, B and G.
Currents in Sheets.—In the case of current flow in plane sheets, we have to consider certain points called sources at which the current flows into the sheet, and certain points called sinks at which it leaves. We may investigate, first, the simple case of one source and one sink in an infinite plane sheet of thickness δ and conductivity k. Take any point P in the plane at distances R and r from the source and sink respectively. The potential V at P is obviously given by
| V = | Q | log e | r1 | , |
| 2πkδ | r2 |
where Q is the quantity of electricity supplied by the source per second. Hence the equation to the equipotential curve is r1r2 = a constant.
If we take a point half-way between the sink and the source as the origin of a system of rectangular co-ordinates, and if the distance between sink and source is equal to p, and the line joining them is taken as the axis of x, then the equation to the equipotential line is
| y² + (x + p)² | = a constant. |
| y² + (x − p)² |
This is the equation of a family of circles having the axis of y for a common radical axis, one set of circles surrounding the sink and another set of circles surrounding the source. In order to discover the form of the stream of current lines we have to determine the orthogonal trajectories to this family of coaxial circles. It is easy to show that the orthogonal trajectory of the system of circles is another system of circles all passing through the sink and the source, and as a corollary of this fact, that the electric resistance of a circular disk of uniform thickness is the same between any two points taken anywhere on its circumference as sink and source. These equipotential lines may be delineated experimentally by attaching the terminals of a battery or batteries to small wires which touch at various places a sheet of tinfoil. Two wires attached to a galvanometer may then be placed on the tinfoil, and one may be kept stationary and the other may be moved about, so that the galvanometer is not traversed by any current. The moving terminal then traces out an equipotential curve. If there are n sinks and sources in a plane conducting sheet, and if r, r′, r″ be the distances of any point from the sinks, and t, t′, t″ the distances of the sources, then